Regulation and Stability of Host-Parasite Population Interactions

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Journalof Animal Ecology (1978), 47, 219-247ournalof Animal Ecology (1978), 47, 219-247

REGULATION AND STABILITYOF HOST-PARASITEPOPULATION INTERACTIONS

I. REGULATORY PROCESSES

BY ROY M. ANDERSON AND ROBERT M. MAY

Zoology Department, King's College, London University, London WC2R 2LS, and

Biology Department,Princeton University,Princeton, N.J. 08540, U.S.A.

SUMMARY

(1) Several models describingthe dynamics of host-parasiteassociations are discussed.(2) The models contain the central assumption that the parasite increases the rate of

host mortalities. The parasite induced changes in this rate are formulated as functions

of the parasite numbers per host and hence of the statistical distribution of the parasiteswithin the host population.

(3) The parameters influencingthe ability of the parasite to regulate the growth of its

host's population, and the stability of parasite induced equilibria, are examined for each

model.

(4) Three specific categories of population processes are shown to be of particular

significance in stabilizing the dynamical behaviour of host-parasite interactions and

enhancing the regulatoryrole of the parasite.These categories are overdispersion of parasite numbersper host, nonlinear functional

relationshipsbetween parasite burden per host and host death rate, and density depen-dent constraints on parasite population growth within individual hosts.

INTRODUCTION

Eucaryotic parasites of one kind or another play a part in the natural history of many,if not most, animals. Man is not exempt: considering helminth parasitesalone, roughly200 million people are affectedby the trematode specieswhich cause schistosomiasis and

300 million by the filarial nematode parasites. A multitude of other internal and external

parasitesof man produce effects rangingfrom minor irritationsto major diseases.The relation betweenpopulations of such parasites and their hosts can be regardedas a

particularmanifestation of the general predator-prey interaction. Predator-preytheoryhas received much attention since the work of Lotka and Volterra in the 1920s, and all

contemporary ecology texts give a lot of coverage to the subject. This work, however,tends to drawits inspiration, and its empiricalbasis, from the world of moose and wolves

(with differential equations for continuously overlapping generations) or of arthropod

predatorsand their prey (with difference equations for discrete, non-overlapping gener-ations); see, e.g. the reviewsby May (1976, ch. 4.) and Hassell (1976). That is, the exten-sive predator-prayliterature pertains mainly to situations where (at least in effect) the

predatorskill and eat their prey.Analogous studies of the special kinds of predator-preyrelations that are of parasito-0021-8710/78/0200-0219 $02.00 ©1978 Blackwell Scientific Publications

219

REGULATION AND STABILITYOF HOST-PARASITEPOPULATION INTERACTIONS

I. REGULATORY PROCESSES

BY ROY M. ANDERSON AND ROBERT M. MAY

Zoology Department, King's College, London University, London WC2R 2LS, and

Biology Department,Princeton University,Princeton, N.J. 08540, U.S.A.

SUMMARY

(1) Several models describingthe dynamics of host-parasiteassociations are discussed.(2) The models contain the central assumption that the parasite increases the rate of

host mortalities. The parasite induced changes in this rate are formulated as functions

of the parasite numbers per host and hence of the statistical distribution of the parasiteswithin the host population.

(3) The parameters influencingthe ability of the parasite to regulate the growth of its

host's population, and the stability of parasite induced equilibria, are examined for each

model.

(4) Three specific categories of population processes are shown to be of particular

significance in stabilizing the dynamical behaviour of host-parasite interactions and

enhancing the regulatoryrole of the parasite.These categories are overdispersion of parasite numbersper host, nonlinear functional

relationshipsbetween parasite burden per host and host death rate, and density depen-dent constraints on parasite population growth within individual hosts.

INTRODUCTION

Eucaryotic parasites of one kind or another play a part in the natural history of many,if not most, animals. Man is not exempt: considering helminth parasitesalone, roughly200 million people are affectedby the trematode specieswhich cause schistosomiasis and

300 million by the filarial nematode parasites. A multitude of other internal and external

parasitesof man produce effects rangingfrom minor irritationsto major diseases.The relation betweenpopulations of such parasites and their hosts can be regardedas a

particularmanifestation of the general predator-prey interaction. Predator-preytheoryhas received much attention since the work of Lotka and Volterra in the 1920s, and all

contemporary ecology texts give a lot of coverage to the subject. This work, however,tends to drawits inspiration, and its empiricalbasis, from the world of moose and wolves

(with differential equations for continuously overlapping generations) or of arthropod

predatorsand their prey (with difference equations for discrete, non-overlapping gener-ations); see, e.g. the reviewsby May (1976, ch. 4.) and Hassell (1976). That is, the exten-sive predator-prayliterature pertains mainly to situations where (at least in effect) the

predatorskill and eat their prey.Analogous studies of the special kinds of predator-preyrelations that are of parasito-0021-8710/78/0200-0219 $02.00 ©1978 Blackwell Scientific Publications

219



Regulationand stability of host-parasite nteractions.Iegulationand stability of host-parasite nteractions.I

logical interest are comparatively few. Thus relatively little is known about the effectsthat parasitesmay have upon the population dynamics of their hosts.

In this paperwe explore the dynamicalproperties of some simple models which aim to

capture the essential biological featuresof

host-parasite associations.In

analogy withearlierwork on arthropod systems (Hassell & May 1973; Hassell, Lawton & Beddington1976; Beddington, Hassell & Lawton 1976), we have tried to build the models on bio-

logical assumptions that are rooted in empirical evidence.

The paper is organized as follows. After giving a more precise definition of what we

mean by the term 'parasite',we outline the biological assumptions upon which the 'Basic

model' rests, and expound the dynamical properties of this model. The Basic model

assumes parasites to be distributed independently randomly among hosts; various

patterns of non-random parasite distribution are then discussed, and introduced into

'model A', which is then investigated.The effects of varyingthe way the host's mortality

depends on the density of parasites ('model B'), and of introducing density dependenceinto the parasites' intrinsic death rate ('model C') are explored. The following paper

(May & Anderson 1978) goes on to discuss various destabilizing influences and encom-

passesthe interaction between host and parasite reproductionrates, transmission factors,and time delays.

Throughout, the text aims to be descriptive with results displayed graphically and the

emphasis on the biology of host-parasiteassociations. The mathematical analysis of the

stability properties of the various models is relegatedto appendices.

THE TERM 'PARASITISM'

Parasitism may be regarded as an ecological association between species in which onethe parasite, lives on or in the body of the other, the host. The parasite may spend the

majority of its life in association with one or more host species, or alternatively it may

spend only short periods, adopting a free-livingmode for the major part of its develop-mental cycle. During the parasiticphase of its life cycle, the organism depends upon its

host for the synthesis of one or more nutrients essential for its own metabolism. The

relationshipis usually regardedas obligatory for the parasite and harmful or damagingfor the host. To classify an animal species as parasitic we therefore require that three

conditions be satisfied: utilization of the host as a habitat; nutritionaldependence; and

causing 'harm' to its host.

When one considers such interactions at the population level, the terminology nowused for labelling animal associations appearsratherconfusing and imprecise (see Starr

1975;Askew 1971; Dogiel 1964).This is particularlyapparentwhen one tries to formalize

the nature of the harmful effect of a parasitic species on the population growth of itshost. These difficulties do not always arise. For example, insect parasitoidsas a develop-mental necessity invariablykill their host, the parasite survivingthe death it induced bythe adoption of a free-living mode of life in the adult phase. Askew (1971) has termed

such species 'protean parasites' since they are parasitic as larval forms and free-livingwhen adult.

Other eucaryotic parasitic organisms such as lice, fleas, ticks, mites, protozoa and

helminths exhibit the nutritional and habitat requirementsof a parasite but appearto do

very little harm to their hosts, unless present in very large numbers. Such species do notkill their hosts as a prerequisite for successful development; indeed, in contrast to

parasitoid insects, these organisms are often themselves killed if they cause the death of

logical interest are comparatively few. Thus relatively little is known about the effectsthat parasitesmay have upon the population dynamics of their hosts.

In this paperwe explore the dynamicalproperties of some simple models which aim to

capture the essential biological featuresof

host-parasite associations.In

analogy withearlierwork on arthropod systems (Hassell & May 1973; Hassell, Lawton & Beddington1976; Beddington, Hassell & Lawton 1976), we have tried to build the models on bio-

logical assumptions that are rooted in empirical evidence.

The paper is organized as follows. After giving a more precise definition of what we

mean by the term 'parasite',we outline the biological assumptions upon which the 'Basic

model' rests, and expound the dynamical properties of this model. The Basic model

assumes parasites to be distributed independently randomly among hosts; various

patterns of non-random parasite distribution are then discussed, and introduced into

'model A', which is then investigated.The effects of varyingthe way the host's mortality

depends on the density of parasites ('model B'), and of introducing density dependenceinto the parasites' intrinsic death rate ('model C') are explored. The following paper

(May & Anderson 1978) goes on to discuss various destabilizing influences and encom-

passesthe interaction between host and parasite reproductionrates, transmission factors,and time delays.

Throughout, the text aims to be descriptive with results displayed graphically and the

emphasis on the biology of host-parasiteassociations. The mathematical analysis of the

stability properties of the various models is relegatedto appendices.

THE TERM 'PARASITISM'

Parasitism may be regarded as an ecological association between species in which onethe parasite, lives on or in the body of the other, the host. The parasite may spend the

majority of its life in association with one or more host species, or alternatively it may

spend only short periods, adopting a free-livingmode for the major part of its develop-mental cycle. During the parasiticphase of its life cycle, the organism depends upon its

host for the synthesis of one or more nutrients essential for its own metabolism. The

relationshipis usually regardedas obligatory for the parasite and harmful or damagingfor the host. To classify an animal species as parasitic we therefore require that three

conditions be satisfied: utilization of the host as a habitat; nutritionaldependence; and

causing 'harm' to its host.

When one considers such interactions at the population level, the terminology nowused for labelling animal associations appearsratherconfusing and imprecise (see Starr

1975;Askew 1971; Dogiel 1964).This is particularlyapparentwhen one tries to formalize

the nature of the harmful effect of a parasitic species on the population growth of itshost. These difficulties do not always arise. For example, insect parasitoidsas a develop-mental necessity invariablykill their host, the parasite survivingthe death it induced bythe adoption of a free-living mode of life in the adult phase. Askew (1971) has termed

such species 'protean parasites' since they are parasitic as larval forms and free-livingwhen adult.

Other eucaryotic parasitic organisms such as lice, fleas, ticks, mites, protozoa and

helminths exhibit the nutritional and habitat requirementsof a parasite but appearto do

very little harm to their hosts, unless present in very large numbers. Such species do notkill their hosts as a prerequisite for successful development; indeed, in contrast to

parasitoid insects, these organisms are often themselves killed if they cause the death of

22020



ROY M. ANDERSONAND ROBERTM. MAYOY M. ANDERSONAND ROBERTM. MAY

their host. Although parasite induced host deaths usually result when heavy infections

occur, the precise meaning of the word heavy will very much depend on the size of the

parasite in relation to its host, and the niche and mode of life adopted by the parasitewithin or on the host.

In this paper, we use the term parasite to refer to species which do not kill their hostas a prerequisiteor successful development.This distinguishesparasitesfrom parasitoids.

Parasitesexhibit a wide degree of variabilitybetween species in the degree of harm or

damage they cause to their hosts. At one extreme of the spectrum, parasites merge into

the parasitoid type of association with their close relationship (in population terms) with

predator-preyinteractions. At this extreme, death will invariably result from parasiteinfection, but in contrast to parasitoids such host deaths will also kill the parasitescontained within. At the other end of the spectrumlie the symbiotic forms of association

in which the symbiont lives on or in the host with a degree of nutritional dependenceakin to a permissive gastronomic hospitality. Species at this end of the spectrum cause

negligible, if any, harm to the host even when present in very large numbers.In terms of their population dynamics, there will be differencesbetween parasites at

the two ends of this spectrum; between the parasitoid like parasites and the symbionts.Crofton (1971a, b) has stressed the importance of quantifying these notions, and has

suggested that a useful firststep lies in the definition of a 'lethallevel', which measures the

typical number of parasites required to kill a host. As Crofton notes, quantitativeinformation about such lethal levels is hard to come by.

We shall return to these quantitative questions below: for the presentwe propose that

an organism only be classified as a parasite if it has a detrimental effect on the intrinsic

growth rate of its host population.

BASIC MODEL: BIOLOGICAL ASSUMPTIONS

We define H(t) and P(t) to be the magnitudes of the host and parasite populations,

respectivelyat time t; the averagenumberof parasitesperhost is thenP(t)/H(t). Weassume

that the vast majorityof protozoan and helminth parasitesexhibit continuous populationgrowth, where generations overlap completely. The equations describing the way the

host andparasitepopulations change in time arethus formulated as differentialequations.The basic model is for parasite species which do not reproduce directly within their

definitive or final host, but which produce transmission stages such as eggs, spores or

cysts which, as a developmental necessity, pass out of the host. This type of parasite life

cycle is shown by many protozoan, helminth and arthropod species. In the followingpaper (May & Anderson (1978) in model E) this basic frameworkis modified to encom-

pass species, such as some parasitic protozoa, which have a reproductive phase which

directly contributes to the size of the parasite population within the host.We assume that all parasitic species are capable of multiply infecting a proportion of

the host population and that the birth and death rates of infected hosts are alteredby thenumber of parasites they harbour. The precisefunctional relationship between the num-ber of parasites harboured and the host's chances of surviving or reproducing varies

greatly among different host-parasite associations. The rate of parasite induced hostmortalities may increaselinearlywith parasite burdenor as an exponential or power law

function. Some examples of these functional relationships for protozoan, helminth andarthropod parasites of both vertebrate and invertebratehost species are shown in Fig. 1.Sometimes the relationship may be of a more complex form than suggested by these

their host. Although parasite induced host deaths usually result when heavy infections

occur, the precise meaning of the word heavy will very much depend on the size of the

parasite in relation to its host, and the niche and mode of life adopted by the parasitewithin or on the host.

In this paper, we use the term parasite to refer to species which do not kill their hostas a prerequisiteor successful development.This distinguishesparasitesfrom parasitoids.

Parasitesexhibit a wide degree of variabilitybetween species in the degree of harm or

damage they cause to their hosts. At one extreme of the spectrum, parasites merge into

the parasitoid type of association with their close relationship (in population terms) with

predator-preyinteractions. At this extreme, death will invariably result from parasiteinfection, but in contrast to parasitoids such host deaths will also kill the parasitescontained within. At the other end of the spectrumlie the symbiotic forms of association

in which the symbiont lives on or in the host with a degree of nutritional dependenceakin to a permissive gastronomic hospitality. Species at this end of the spectrum cause

negligible, if any, harm to the host even when present in very large numbers.In terms of their population dynamics, there will be differencesbetween parasites at

the two ends of this spectrum; between the parasitoid like parasites and the symbionts.Crofton (1971a, b) has stressed the importance of quantifying these notions, and has

suggested that a useful firststep lies in the definition of a 'lethallevel', which measures the

typical number of parasites required to kill a host. As Crofton notes, quantitativeinformation about such lethal levels is hard to come by.

We shall return to these quantitative questions below: for the presentwe propose that

an organism only be classified as a parasite if it has a detrimental effect on the intrinsic

growth rate of its host population.

BASIC MODEL: BIOLOGICAL ASSUMPTIONS

We define H(t) and P(t) to be the magnitudes of the host and parasite populations,

respectivelyat time t; the averagenumberof parasitesperhost is thenP(t)/H(t). Weassume

that the vast majorityof protozoan and helminth parasitesexhibit continuous populationgrowth, where generations overlap completely. The equations describing the way the

host andparasitepopulations change in time arethus formulated as differentialequations.The basic model is for parasite species which do not reproduce directly within their

definitive or final host, but which produce transmission stages such as eggs, spores or

cysts which, as a developmental necessity, pass out of the host. This type of parasite life

cycle is shown by many protozoan, helminth and arthropod species. In the followingpaper (May & Anderson (1978) in model E) this basic frameworkis modified to encom-

pass species, such as some parasitic protozoa, which have a reproductive phase which

directly contributes to the size of the parasite population within the host.We assume that all parasitic species are capable of multiply infecting a proportion of

the host population and that the birth and death rates of infected hosts are alteredby thenumber of parasites they harbour. The precisefunctional relationship between the num-ber of parasites harboured and the host's chances of surviving or reproducing varies

greatly among different host-parasite associations. The rate of parasite induced hostmortalities may increaselinearlywith parasite burdenor as an exponential or power law

function. Some examples of these functional relationships for protozoan, helminth andarthropod parasites of both vertebrate and invertebratehost species are shown in Fig. 1.Sometimes the relationship may be of a more complex form than suggested by these

22121




examples. The parasite, for instance, may act in an all-or-nothing manner where lowburdens do not influencethe hosts survival chances but at a given threshold burden thedeath rate rises very rapidlyresultingin certain death for the hosts. In general, however,

where quantitative rate estimates are available the 'harmful' effect of the parasite isusually of the more gradualforms indicated in Fig. 1.

In the majority of host-parasiteassociations it appearsto be the death rate ratherthanthe reproductive rate of the host which is influencedby parasiticinfection. Exceptions tothis general patternareparticularlynoticeable in the associations between larvaldigeneanparasites and their molluscan intermediate hosts. Many parasitic arthropods alsodecrease the reproductive power of their hosts, and in certain cases complete parasiticcastration occurs.

Accordingly, the majority of our models assume that the parasite increases the hostdeath rate. Attention is given to the population consequences of parasite induced

reduction of host reproductive potential (in Model D) in the following paper (May &Anderson 1978).

TABLE1. Descriptionof the principalpopulationparameters sed in the models

Parameter Descriptiona Instantaneoushost birthrate(/host/unitof time).b Instantaneoushost deathrate, wheremortalitiesare due to 'naturalcauses'(/host/unitof

time).a Instantaneousostdeathrate,wheremortalities redueto theinfluence f theparasite

(/host/unitof time).A Instantaneous irthrate of parasite ransmissiontageswherebirthresults ntheproduction

of stages,whichpass out of the host, and areresponsible or transmissionof the parasitewithin the host population(i.e. eggs, cysts, sporesor larvae) (/parasite/unitof time).

# Instantaneousdeath rate of parasiteswithin the host, due to eithernaturalor host induced(immunological)causes (/parasite/unitof time).

Ho Transmission fficiencyconstant,varying nverselywith the proportionof parasite rans-missionstageswhich infectmembersof the host population.

r Instantaneousbirth rate of parasite,whose birth results in the productionof parasiticstageswhichremainwithin the host in whichthey wereproduced /parasite/unit f time).

The two basic equations, for dH/dt and dP/dt, are constructed from several compon-ents, each of which represents specific biological assumptions. A summary of our nota-tion is given in Table 1. These components will now be discussed, one by one.

Thegrowth of the hostpopulationWe assume that the rate of growth of the host population is simply determined by the

natural intrinsic rate of increase in the absence of parasitic infection minus the rate of

parasite induced host mortalities. Both the host reproductive rate a, and the rate of'natural' mortalities b, are represented as constants unaffected by density dependentconstraints on population growth. We use the term 'natural' mortalities to encompass alldeaths due to causes other than parasiticinfection, e.g. predation and senescence.

Ouromission of densitydependentconstraints on host population growthisdeliberate.We recognize that in the real world host population growth will be limited by, amongother factors, intraspecificcompetition for finite resources. Since our aim, however, is to

provide qualitative insights into the mechanisms by which parasites regulate hostpopulation growth, we have excluded the concept of a carrying capacity of the hosts'environmentto simplify algebraicmanipulations. Such simplificationclarifiespredictionsof biological interest.

examples. The parasite, for instance, may act in an all-or-nothing manner where lowburdens do not influencethe hosts survival chances but at a given threshold burden thedeath rate rises very rapidlyresultingin certain death for the hosts. In general, however,

where quantitative rate estimates are available the 'harmful' effect of the parasite isusually of the more gradualforms indicated in Fig. 1.

In the majority of host-parasiteassociations it appearsto be the death rate ratherthanthe reproductive rate of the host which is influencedby parasiticinfection. Exceptions tothis general patternareparticularlynoticeable in the associations between larvaldigeneanparasites and their molluscan intermediate hosts. Many parasitic arthropods alsodecrease the reproductive power of their hosts, and in certain cases complete parasiticcastration occurs.

Accordingly, the majority of our models assume that the parasite increases the hostdeath rate. Attention is given to the population consequences of parasite induced

reduction of host reproductive potential (in Model D) in the following paper (May &Anderson 1978).

TABLE1. Descriptionof the principalpopulationparameters sed in the models

Parameter Descriptiona Instantaneoushost birthrate(/host/unitof time).b Instantaneoushost deathrate, wheremortalitiesare due to 'naturalcauses'(/host/unitof

time).a Instantaneousostdeathrate,wheremortalities redueto theinfluence f theparasite

(/host/unitof time).A Instantaneous irthrate of parasite ransmissiontageswherebirthresults ntheproduction

of stages,whichpass out of the host, and areresponsible or transmissionof the parasitewithin the host population(i.e. eggs, cysts, sporesor larvae) (/parasite/unitof time).

# Instantaneousdeath rate of parasiteswithin the host, due to eithernaturalor host induced(immunological)causes (/parasite/unitof time).

Ho Transmission fficiencyconstant,varying nverselywith the proportionof parasite rans-missionstageswhich infectmembersof the host population.

r Instantaneousbirth rate of parasite,whose birth results in the productionof parasiticstageswhichremainwithin the host in whichthey wereproduced /parasite/unit f time).

The two basic equations, for dH/dt and dP/dt, are constructed from several compon-ents, each of which represents specific biological assumptions. A summary of our nota-tion is given in Table 1. These components will now be discussed, one by one.

Thegrowth of the hostpopulationWe assume that the rate of growth of the host population is simply determined by the

natural intrinsic rate of increase in the absence of parasitic infection minus the rate of

parasite induced host mortalities. Both the host reproductive rate a, and the rate of'natural' mortalities b, are represented as constants unaffected by density dependentconstraints on population growth. We use the term 'natural' mortalities to encompass alldeaths due to causes other than parasiticinfection, e.g. predation and senescence.

Ouromission of densitydependentconstraints on host population growthisdeliberate.We recognize that in the real world host population growth will be limited by, amongother factors, intraspecificcompetition for finite resources. Since our aim, however, is to

provide qualitative insights into the mechanisms by which parasites regulate hostpopulation growth, we have excluded the concept of a carrying capacity of the hosts'environmentto simplify algebraicmanipulations. Such simplificationclarifiespredictionsof biological interest.

22222



ROY M. ANDERSONAND ROBERTM. MAYOY M. ANDERSONAND ROBERTM. MAY 22323

We assume that if the parasite fails to control host population growth, exponentialincrease of the host population occurs until resource limitation results in the gradualapproach to a carrying capacity.

We assume that if the parasite fails to control host population growth, exponentialincrease of the host population occurs until resource limitation results in the gradualapproach to a carrying capacity.

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Size of parasite infection

FIG.1. Someexamplesof the functionalrelationshipbetween he rateof parasite nducedhost mortalities a)andthe parasiteburden i) perhost. (Thestraight inesin graphs a), (b)and

(c)are theleast

squaresbest fit linearmodelsand thecurves nthe

graphs d), (e)and(f)arethe leastsquaresbest fitexponentialmodels).(a) SnailhostLymnea edrosianaAnnan-dale and Prashed)parasitizedby the larvalstagesof the digeneanOrnithobilharziaurkes-tanicumSkrjabin) datafrom Massoud1974); (b) aquaticHemipteranHydrometramyrae(Bueno)parasitizedby the mite HydryphantesenuabilisMarshall)(data from Lanciani1975); (c) laboratorymouse parasitizedby the digeneanFasciolahepatica L.) (datafromHayes,Bailer&Mitovic1973); d)laboratorymouseparasitized ythenematodeHeligmos-omoidespolygyrusDujardin) datafrom Forrester1971);(e) laboratoryrat parasitizedbythe nematodeNippostrongylusrasiliensisYokogawa) (datafrom Hunter& Leigh 1961);(f) laboratorymouse parasitizedby the blood protozoanPlasmodium inckei(Vinkeand

Lips)(datafrom Cox 1966).

Parasite inducedhost mortalitiesIn the Basic model we assume that the rate

of parasite induced host mortalitiesis

linearly proportional to the number of parasites a host harbours(see Fig. l(a)-(c)). Thenumber of host deaths in a small interval of time 6t, among those with i parasites, may


FIG.1. Someexamplesof the functionalrelationshipbetween he rateof parasite nducedhost mortalities a)andthe parasiteburden i) perhost. (Thestraight inesin graphs a), (b)and

(c)are theleast

squaresbest fit linearmodelsand thecurves nthe

graphs d), (e)and(f)arethe leastsquaresbest fitexponentialmodels).(a) SnailhostLymnea edrosianaAnnan-dale and Prashed)parasitizedby the larvalstagesof the digeneanOrnithobilharziaurkes-tanicumSkrjabin) datafrom Massoud1974); (b) aquaticHemipteranHydrometramyrae(Bueno)parasitizedby the mite HydryphantesenuabilisMarshall)(data from Lanciani1975); (c) laboratorymouse parasitizedby the digeneanFasciolahepatica L.) (datafromHayes,Bailer&Mitovic1973); d)laboratorymouseparasitized ythenematodeHeligmos-omoidespolygyrusDujardin) datafrom Forrester1971);(e) laboratoryrat parasitizedbythe nematodeNippostrongylusrasiliensisYokogawa) (datafrom Hunter& Leigh 1961);(f) laboratorymouse parasitizedby the blood protozoanPlasmodium inckei(Vinkeand

Lips)(datafrom Cox 1966).

Parasite inducedhost mortalitiesIn the Basic model we assume that the rate

of parasite induced host mortalitiesis

linearly proportional to the number of parasites a host harbours(see Fig. l(a)-(c)). Thenumber of host deaths in a small interval of time 6t, among those with i parasites, may

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then be representedas ci bt, where o is a constant determiningthe pathogenicity of the

parasite to the host; the correspondingdeath rate among hosts with i parasites is ai. Thetotal rate of loss of hosts in a population of size H(t) is therefore

aH(t) E i. p(i).i =

Herep(i) is the probabilitythat a given host contains i parasites. Clearlyp(i) will dependon i and on various parameters characterizingthe parasite distribution within the host

population. The above sum is, by definition, the averagenumber of parasitesper host attime t:

E i. p(i) Et(i) = P(t)/H(t).i= 0

In short, under the above assumptionsthe net rate of parasite induced host mortalityis

P(t). (1)

Parasitefecundity and transmissionThe rate of production of transmission stages (such as eggs, spores or cysts) per

parasite is defined as A,leading to a net rate for the toal parasite population of

IH(t) E i.p(i) = P(t). (2)i = 0

In the case of direct lifecycle parasites

whereonly

asingle species

of host is utilized,the transmission stages will pass out of the host into the external environment and will

survive in this habitat as resistant stages or free-living larvae awaiting contact with or

ingestion by a member of the host population. While in the external habitat, they will be

subjectto naturalmortalities due to senescence or predation and thus only a proportionof those releasedwill be successful in gaining entry to a new host. The magnitude of this

proportion will depend on the density of the host population in relation to other 'absor-

bers' of the transmission stages, and the proportion may be characterized by the trans-mission factor (cf. MacDonald 1961)

H(t)/(Ho + H(t)).

Here Ho is a constant which, when varied, inversely determines the efficiency oftransmission. When H(t) is large and Ho small, the efficiency approaches unity, where all

the transmission stages produced gain entry to the host population. Conversely when

H(t) is small and Ho large only a small proportion are successful.The net rate at which new parasitesare aquiredwithin the host population is thus

. P(t) . H(t)/(Ho + H(t)). (3)

This term contains the assumption that transmission is virtually instantaneous, no

time delay occurringdue to developmental processes betweenthe birth of a transmission

stage and successful contact with a new host. In some parasite life cycles, transmission

stages are immediately infective to a new host, but in the majority certaindevelopmentalprocesses have to occur before the stage becomes fully infective. The influence of time

delays in the transmission term will be examined as a modification to the basic model in

the following paper (Model F in May & Anderson 1978).

then be representedas ci bt, where o is a constant determiningthe pathogenicity of the

parasite to the host; the correspondingdeath rate among hosts with i parasites is ai. Thetotal rate of loss of hosts in a population of size H(t) is therefore

aH(t) E i. p(i).i =

Herep(i) is the probabilitythat a given host contains i parasites. Clearlyp(i) will dependon i and on various parameters characterizingthe parasite distribution within the host

population. The above sum is, by definition, the averagenumber of parasitesper host attime t:

E i. p(i) Et(i) = P(t)/H(t).i= 0

In short, under the above assumptionsthe net rate of parasite induced host mortalityis

P(t). (1)

Parasitefecundity and transmissionThe rate of production of transmission stages (such as eggs, spores or cysts) per

parasite is defined as A,leading to a net rate for the toal parasite population of

IH(t) E i.p(i) = P(t). (2)i = 0

In the case of direct lifecycle parasites

whereonly

asingle species

of host is utilized,the transmission stages will pass out of the host into the external environment and will

survive in this habitat as resistant stages or free-living larvae awaiting contact with or

ingestion by a member of the host population. While in the external habitat, they will be

subjectto naturalmortalities due to senescence or predation and thus only a proportionof those releasedwill be successful in gaining entry to a new host. The magnitude of this

proportion will depend on the density of the host population in relation to other 'absor-

bers' of the transmission stages, and the proportion may be characterized by the trans-mission factor (cf. MacDonald 1961)

H(t)/(Ho + H(t)).

Here Ho is a constant which, when varied, inversely determines the efficiency oftransmission. When H(t) is large and Ho small, the efficiency approaches unity, where all

the transmission stages produced gain entry to the host population. Conversely when

H(t) is small and Ho large only a small proportion are successful.The net rate at which new parasitesare aquiredwithin the host population is thus

. P(t) . H(t)/(Ho + H(t)). (3)

This term contains the assumption that transmission is virtually instantaneous, no

time delay occurringdue to developmental processes betweenthe birth of a transmission

stage and successful contact with a new host. In some parasite life cycles, transmission

stages are immediately infective to a new host, but in the majority certaindevelopmentalprocesses have to occur before the stage becomes fully infective. The influence of time

delays in the transmission term will be examined as a modification to the basic model in

the following paper (Model F in May & Anderson 1978).

22424




The assumptions ncorporatedn the two equation host-parasitemodel are most

closelylinkedto direct ife cycleparasitic pecies.The populationdynamicsof indirectlife cyclespeciescan also be interpretedn lightof the model'spredictionsf the popu-lationprocessesactingon the intermediate ost or hosts andthe parasitic arvalstages

are subsumed nto the transmission erm(i.e. into the factor Ho in eqn (3)). This isobviouslya major implifying ssumption, articularlynrespecto thetimedelayswhichwill occurduringa parasite'sdevelopmentn its intermediate ost (for a reviewof this

subject,see May 1977).The dynamicalpropertiesof the basicmodelwith time delayswill thus be moreakin to thepopulationdynamicsof indirectifecycle parasites.

Parasite mortalitiesThe deathratefor parasiteswithinthe host populationhas threecomponents.First, hereare ossesdue to naturalhostmortalities.With he intrinsicper capitahost

mortality ateb, suchparasiteosses are at the net rate

00

b. H(t) E i. p(i) = b. P(t). (4)i-= 0

Second, therearelossesfromparasitenducedhost deaths,where heper capitahostloss rate(discussed bove) s takento be ai. The consequentnet mortality ateof para-sitesfrom this cause is

00

acH(t) E i2p(i)_- H(t)Et(i2). (5)i = o

HereE(i2)is the mean-square umberof parasitesperhost, the precisevalueof which

dependson the form of the probabilitydistribution f parasitenumbersperhost,p(i).

Thatis, E (i) dependson themeanparasiteoad,P(t)/H(t),andalso ontheparameter(s)that specifythis distribution.Appendix1 lists the values of this 'secondmoment'forsomecommonlyuseddiscreteprobability istributions,ivingE(i2) n termsof the mean

parasite oad and measuresof over- and under-dispersion.Third,there is a componentof the parasitedeath rategeneratedby naturalparasite

mortalitywithinthe host. Assuminga constantper capita parasite ntrinsicmortalityrate i, these losses make a net contribution f

P(t) (6)

to the overallparasitemortality ate. These 'natural'parasitemortalities ncludedeaths

due to host immunologicalesponses, s well as moreconventionalossesfromparasitesenescence.

BASIC MODEL: DYNAMICS

The biological ingredients discussed above can now be drawn together to give twodifferentialequations, one describingthe rate of change of the host population,

dH/dt =(a-b)H-acP (7)

and the other describing the parasite population dynamics,

dP/dt = (APH/(Ho+ H)) - (b + )P - HEt(i2). (8)

If the parasites are distributed independently randomly among hosts, eqn (8) thenbecomes (see Appendix 1):

The assumptions ncorporatedn the two equation host-parasitemodel are most

closelylinkedto direct ife cycleparasitic pecies.The populationdynamicsof indirectlife cyclespeciescan also be interpretedn lightof the model'spredictionsf the popu-lationprocessesactingon the intermediate ost or hosts andthe parasitic arvalstages

are subsumed nto the transmission erm(i.e. into the factor Ho in eqn (3)). This isobviouslya major implifying ssumption, articularlynrespecto thetimedelayswhichwill occurduringa parasite'sdevelopmentn its intermediate ost (for a reviewof this

subject,see May 1977).The dynamicalpropertiesof the basicmodelwith time delayswill thus be moreakin to thepopulationdynamicsof indirectifecycle parasites.

Parasite mortalitiesThe deathratefor parasiteswithinthe host populationhas threecomponents.First, hereare ossesdue to naturalhostmortalities.With he intrinsicper capitahost

mortality ateb, suchparasiteosses are at the net rate

00

b. H(t) E i. p(i) = b. P(t). (4)i-= 0

Second, therearelossesfromparasitenducedhost deaths,where heper capitahostloss rate(discussed bove) s takento be ai. The consequentnet mortality ateof para-sitesfrom this cause is

00

acH(t) E i2p(i)_- H(t)Et(i2). (5)i = o

HereE(i2)is the mean-square umberof parasitesperhost, the precisevalueof which

dependson the form of the probabilitydistribution f parasitenumbersperhost,p(i).

Thatis, E (i) dependson themeanparasiteoad,P(t)/H(t),andalso ontheparameter(s)that specifythis distribution.Appendix1 lists the values of this 'secondmoment'forsomecommonlyuseddiscreteprobability istributions,ivingE(i2) n termsof the mean

parasite oad and measuresof over- and under-dispersion.Third,there is a componentof the parasitedeath rategeneratedby naturalparasite

mortalitywithinthe host. Assuminga constantper capita parasite ntrinsicmortalityrate i, these losses make a net contribution f

P(t) (6)

to the overallparasitemortality ate. These 'natural'parasitemortalities ncludedeaths

due to host immunologicalesponses, s well as moreconventionalossesfromparasitesenescence.

BASIC MODEL: DYNAMICS

The biological ingredients discussed above can now be drawn together to give twodifferentialequations, one describingthe rate of change of the host population,

dH/dt =(a-b)H-acP (7)

and the other describing the parasite population dynamics,

dP/dt = (APH/(Ho+ H)) - (b + )P - HEt(i2). (8)

If the parasites are distributed independently randomly among hosts, eqn (8) thenbecomes (see Appendix 1):

22525



Regulation and stability of host-parasite nteractions. Iegulation and stability of host-parasite nteractions. I

dP/dt = P(AH/(Ho+ H) -(b + + a) --aP/H). (9)

Eqns (7) and (9) readily yield the equilibrium(dH/dt = dP/dt = 0) host and parasitepopulation values, H* and P*. From eqn (7) the

equilibriummean

parasiteburden is

P*H* = (a-b)la, (10)

whence from eqn (9) H* is

- Ho-(+a+a)Hou + x+ a)'

Provided the host population's intrinsic growth rate is positive (a-b>0), eqn (11)reveals that the parasitesarecapable of regulatingthe growth of the host population onlyif

A> #++a. (12)

l i: It fII

1Itr I

'I600-1

ii I i I I00- I I I I I I

1I II I

" I I I II I

a" I I IiiI I

= 0-1, Ho = 100, a = 0-5, A= 6-0).

This corresponds to the sensible requirement that the parasite 'birth rate' be in excessof the host birth rate (a) plus parasite death rates (both intrinsic, ,u, and due to

parasiteinduced host deaths,o). If this inequality (12) is not satisfied, the host populationgrows exponentially and will eventually be constrained by other regulatory processessuch as finite resources. The parasite will also grow exponentially but at a slower ratethan the host

populationand thus the mean number of

parasites perhost will tend to

zero.

Equations (7) and (9) may be viewed as a generalpair of predator-preyequations, andthe following comments made. (i) In the 'prey' equation, (7), the 'predators' have a

dP/dt = P(AH/(Ho+ H) -(b + + a) --aP/H). (9)

Eqns (7) and (9) readily yield the equilibrium(dH/dt = dP/dt = 0) host and parasitepopulation values, H* and P*. From eqn (7) the

equilibriummean

parasiteburden is

P*H* = (a-b)la, (10)

whence from eqn (9) H* is

- Ho-(+a+a)Hou + x+ a)'

Provided the host population's intrinsic growth rate is positive (a-b>0), eqn (11)reveals that the parasitesarecapable of regulatingthe growth of the host population onlyif

A> #++a. (12)

l i: It fII

1Itr I

'I600-1

ii I i I I00- I I I I I I

1I II I

" I I I II I

a" I I IiiI I

= 0-1, Ho = 100, a = 0-5, A= 6-0).

This corresponds to the sensible requirement that the parasite 'birth rate' be in excessof the host birth rate (a) plus parasite death rates (both intrinsic, ,u, and due to

parasiteinduced host deaths,o). If this inequality (12) is not satisfied, the host populationgrows exponentially and will eventually be constrained by other regulatory processessuch as finite resources. The parasite will also grow exponentially but at a slower ratethan the host

populationand thus the mean number of

parasites perhost will tend to

zero.

Equations (7) and (9) may be viewed as a generalpair of predator-preyequations, andthe following comments made. (i) In the 'prey' equation, (7), the 'predators' have a

22626




constant 'attack rate'; compared with the classic, but unrealistic, Lotka-Volterramodel,this is a destabilizing feature, for it prevents the predators being differentially more

effective at high prey densities. (ii) In the 'predator' equation, (9), we have a kind of

logistic equation, with a predator carrying capacity' proportional to the prey density;this has a stabilizing effect (for a more full discussion and review along these generallines, see May (1976), ch. 4). In order to see how these countervailing tendencies are

resolved,we need to make a stability analysis of the system of eqns (7) and (9).

?(a) (b)

90 -I

';i ~ ~ 240

1 120-t30- \

a) IV \

i

°0 i .,.,. I I I--.zfo 0 2 4 6 8 10 0 4 8 12 16 20

* 50-(c) (d)

25-?

\8-

\ I \

\ /

00 2 4 0 3 62

No. of parasites/host

FIG. 3. Some examplesof observedfrequencydistributionsof parasitenumbersper hostwhichareempiricallydescribed. y the negativebinomial((a) and (b)) or the Poisson((c)and (d)) models. (a) The distributionof the tick IxodestriangulicepsBirula)on a popu-lation of the field mouse Apodemus ylvaticus(L) (data from Randolph 1975); (b) the

distributionof the tapewormCaryophyllaeusaticeps (Pallas)withina populationof thebreamAbramis rama L.)(data romAnderson1974a); c)the distribution f larval tagesofthe nematode Cammallanus xycephalus Wardand Magath)in a;populationof younggizzardshadfish DorosomacepedianumDay) sampled n mid summer datafrom Strom-bergand Crites1974); (d) the distributionof the nematodeAscaridia alli in a populationof chickens whichhadbeenartificiallynfected n the laboratory(datafromNorthamand

Rocha 1958).

For this biologically derivedpair of equations, a rigorous and fully nonlinear stabilityanalysis may be given elegantly. This is done in Appendix 2. The outcome is surprising:the equilibrium point defined by eqns (10) and (II1)is neutrally stable. That is, once

perturbedfrom its equilibriumpoint the system oscillates with a period determinedby the

parameters of the model but with an amplitude dictated for ever after by the initialconditions of the displacement.The result is rigorously 'global', which is to say it holds

constant 'attack rate'; compared with the classic, but unrealistic, Lotka-Volterramodel,this is a destabilizing feature, for it prevents the predators being differentially more

effective at high prey densities. (ii) In the 'predator' equation, (9), we have a kind of

logistic equation, with a predator carrying capacity' proportional to the prey density;this has a stabilizing effect (for a more full discussion and review along these generallines, see May (1976), ch. 4). In order to see how these countervailing tendencies are

resolved,we need to make a stability analysis of the system of eqns (7) and (9).

?(a) (b)

90 -I

';i ~ ~ 240

1 120-t30- \

a) IV \

i

°0 i .,.,. I I I--.zfo 0 2 4 6 8 10 0 4 8 12 16 20

* 50-(c) (d)

25-?

\8-

\ I \

\ /

00 2 4 0 3 62

No. of parasites/host

FIG. 3. Some examplesof observedfrequencydistributionsof parasitenumbersper hostwhichareempiricallydescribed. y the negativebinomial((a) and (b)) or the Poisson((c)and (d)) models. (a) The distributionof the tick IxodestriangulicepsBirula)on a popu-lation of the field mouse Apodemus ylvaticus(L) (data from Randolph 1975); (b) the

distributionof the tapewormCaryophyllaeusaticeps (Pallas)withina populationof thebreamAbramis rama L.)(data romAnderson1974a); c)the distribution f larval tagesofthe nematode Cammallanus xycephalus Wardand Magath)in a;populationof younggizzardshadfish DorosomacepedianumDay) sampled n mid summer datafrom Strom-bergand Crites1974); (d) the distributionof the nematodeAscaridia alli in a populationof chickens whichhadbeenartificiallynfected n the laboratory(datafromNorthamand

Rocha 1958).

For this biologically derivedpair of equations, a rigorous and fully nonlinear stabilityanalysis may be given elegantly. This is done in Appendix 2. The outcome is surprising:the equilibrium point defined by eqns (10) and (II1)is neutrally stable. That is, once

perturbedfrom its equilibriumpoint the system oscillates with a period determinedby the

parameters of the model but with an amplitude dictated for ever after by the initialconditions of the displacement.The result is rigorously 'global', which is to say it holds

22727




true for displacements of arbitrary magnitude (provided the initial H(o) and P(o) are

positive!). Figure 2 illustrates the dynamicalbehaviour of such a host-parasitesystem.This pathological neutral stability property means that the model defined by eqns

(7) and (9) is structurally unstable: the slightest alteration in the form of the various

underlying biological assumptions will precipitate the system either to stability (distur-bances damping back to the equilibrium point) or to instability (oscillations growing

true for displacements of arbitrary magnitude (provided the initial H(o) and P(o) are

positive!). Figure 2 illustrates the dynamicalbehaviour of such a host-parasitesystem.This pathological neutral stability property means that the model defined by eqns

(7) and (9) is structurally unstable: the slightest alteration in the form of the various

underlying biological assumptions will precipitate the system either to stability (distur-bances damping back to the equilibrium point) or to instability (oscillations growing

-I,

\

I

i II

I II I

I

II

\

\I

I II

I II

I

i\ ,

-I,

\

I

i II

I II I

I

II

\

\I

I II

I II

I

i\ ,

N

U)

00

aL

N

U)

00

aL

200-00-

p

Htt

0 5 100

Time

FIG. . Overdispersionf parasiteumberserhost.The imedependentehaviourf thehost(H(t)-solid line)andparasiteP(t)-stippledine)populationsredictedymodelA(eqns7)and 13)).Thepopulationsxhibit ampedscillationsogloballytable quilibria

(a=

3'0,b =

1-0,=

0-1,Ho=

10'0,a =

0-5,A=

6-0,k =

2-0).

Time

FIG. . Overdispersionf parasiteumberserhost.The imedependentehaviourf thehost(H(t)-solid line)andparasiteP(t)-stippledine)populationsredictedymodelA(eqns7)and 13)).Thepopulationsxhibit ampedscillationsogloballytable quilibria

(a=

3'0,b =

1-0,=

0-1,Ho=

10'0,a =

0-5,A=

6-0,k =

2-0).

600-00-

40000

until the host population begins to encounter resource limitation effects as discussed

above, whereupon stable limit cycles typically ensue). Such structural instability makes

the model biologically unrealistic. The Lotka-Volterramodel (althoughhaving a different

biological and mathematical form from eqns (7) and (9)) also has this pathological and

structurally unstable property of neutral stability, for which it has been trenchantlycriticized (e.g. May 1975).

The Basic model defined above is neverthelessvery useful as a point of departure.We

now proceed to modify it by introducing various kinds of density dependences, and of

nonrandom parasite distribution among hosts. In each case, the relation between theunderlying biology and the overall population dynamics is made clear by comparisonwith the razor's edge of the Basic model.

until the host population begins to encounter resource limitation effects as discussed

above, whereupon stable limit cycles typically ensue). Such structural instability makes

the model biologically unrealistic. The Lotka-Volterramodel (althoughhaving a different

biological and mathematical form from eqns (7) and (9)) also has this pathological and

structurally unstable property of neutral stability, for which it has been trenchantlycriticized (e.g. May 1975).

The Basic model defined above is neverthelessvery useful as a point of departure.We

now proceed to modify it by introducing various kinds of density dependences, and of

nonrandom parasite distribution among hosts. In each case, the relation between theunderlying biology and the overall population dynamics is made clear by comparisonwith the razor's edge of the Basic model.

22828

155




DISTRIBUTION OF PARASITES AMONG HOSTS

The parasitologicalliteraturecontains a great deal of information concerning the distri-

bution of parasite numbers within natural populations of hosts. Almost without excep-

tion these observed patterns are over dispersed, where a relatively few members of thehost population harbour the majority of the total parasite population. It has become

customary for parasitologists to fit the negative binomial model to such data, since this

probabilitydistribution has proved to be a good empiricalmodel for a large number of

observed parasite distributions. Figure 3 demonstrates the adequacy of this model in

describing patterns of dispersionwithin host populations.Crofton (1971a)used this observed constancy in the form of parasite distributions as a

basis for a quantitative definition of parasitism at the population level. The author based

this definition on the large number of parasitological processes which could in theory

generate the negative binomial pattern. It is difficult, if not impossible, however, to tryto reach conclusions about the

biologicalmechanisms

generatinga

particulardistribu-

tion patternby simply examiningthe resultant observeddistribution of parasite numbers

per host. A very large numberof both biological and physical processes can generatethe

negative binomial model. Many of these processes have little relevance to the biologiesof parasite life cycles while others are important to both parasiticand free-living organ-isms (Boswell & Patil 1971).

The precise mechanisms giving rise to the patternsshown in Fig. 3(a) and (b) are manyand varied. Two majorfactors are most probably heterogeneity between members of the

host population in exposure to infection (Anderson 1976a) and the influence of past

experiences of infection on the immunological status of a particular host (Anderson

1976b).Suchcomments, however,arepurelyspeculative since detailedexperimentalwork

is requiredto suggest generativeprocesseswhich give rise to observed field patterns.Considerationof the precisemechanisms which generatea particular pattern is not of

overriding importance when examining the qualitative influence of overdispersion on

the dynamical properties of a particular host-parasite interaction. We assume that

heterogeneityin the distribution of parasite numbersper host is the rule rather than the

exception and then proceed to analyse the consequences of such patterns on the popu-lation biology of the system.

From the purelyphenomenological standpoint,the negative binomial has the virtue of

providing a 1-parameter description of the degree of overdispersion, in terms of the

parameterk: the smaller k, the greaterthe degree of parasite clumping. The distribution

is discussed more fully in this light by Bradley & May (1977).It is worth noting, however, that a few reports of random and undispersed distribu-

tions of parasites exist in the literature (Fig. 3(c) and (d)). These patterns are often

observed either within laboratory populations (Northam & Rocha 1958; Anderson,Whitfield & Mills 1977), or within specific strata of a wild host population such as a

particular age class (Anderson 1974b). In other cases random patterns may be observedat a particularpoint in time where the initial invasion of a host population has been

capturedby a sampling programme(Stromberg & Crites 1974).

MODEL A: NONRANDOM PARASITE DISTRIBUTIONS

Overdispersed istributionsIf the parasites are distributedamong the hosts according to a negative binomial, we

can use Appendix 1 to get an explicit expression for E(i2) in eqn (8), which then reads

DISTRIBUTION OF PARASITES AMONG HOSTS

The parasitologicalliteraturecontains a great deal of information concerning the distri-

bution of parasite numbers within natural populations of hosts. Almost without excep-

tion these observed patterns are over dispersed, where a relatively few members of thehost population harbour the majority of the total parasite population. It has become

customary for parasitologists to fit the negative binomial model to such data, since this

probabilitydistribution has proved to be a good empiricalmodel for a large number of

observed parasite distributions. Figure 3 demonstrates the adequacy of this model in

describing patterns of dispersionwithin host populations.Crofton (1971a)used this observed constancy in the form of parasite distributions as a

basis for a quantitative definition of parasitism at the population level. The author based

this definition on the large number of parasitological processes which could in theory

generate the negative binomial pattern. It is difficult, if not impossible, however, to tryto reach conclusions about the

biologicalmechanisms

generatinga

particulardistribu-

tion patternby simply examiningthe resultant observeddistribution of parasite numbers

per host. A very large numberof both biological and physical processes can generatethe

negative binomial model. Many of these processes have little relevance to the biologiesof parasite life cycles while others are important to both parasiticand free-living organ-isms (Boswell & Patil 1971).

The precise mechanisms giving rise to the patternsshown in Fig. 3(a) and (b) are manyand varied. Two majorfactors are most probably heterogeneity between members of the

host population in exposure to infection (Anderson 1976a) and the influence of past

experiences of infection on the immunological status of a particular host (Anderson

1976b).Suchcomments, however,arepurelyspeculative since detailedexperimentalwork

is requiredto suggest generativeprocesseswhich give rise to observed field patterns.Considerationof the precisemechanisms which generatea particular pattern is not of

overriding importance when examining the qualitative influence of overdispersion on

the dynamical properties of a particular host-parasite interaction. We assume that

heterogeneityin the distribution of parasite numbersper host is the rule rather than the

exception and then proceed to analyse the consequences of such patterns on the popu-lation biology of the system.

From the purelyphenomenological standpoint,the negative binomial has the virtue of

providing a 1-parameter description of the degree of overdispersion, in terms of the

parameterk: the smaller k, the greaterthe degree of parasite clumping. The distribution

is discussed more fully in this light by Bradley & May (1977).It is worth noting, however, that a few reports of random and undispersed distribu-

tions of parasites exist in the literature (Fig. 3(c) and (d)). These patterns are often

observed either within laboratory populations (Northam & Rocha 1958; Anderson,Whitfield & Mills 1977), or within specific strata of a wild host population such as a

particular age class (Anderson 1974b). In other cases random patterns may be observedat a particularpoint in time where the initial invasion of a host population has been

capturedby a sampling programme(Stromberg & Crites 1974).

MODEL A: NONRANDOM PARASITE DISTRIBUTIONS

Overdispersed istributionsIf the parasites are distributedamong the hosts according to a negative binomial, we

can use Appendix 1 to get an explicit expression for E(i2) in eqn (8), which then reads

22929



23030 Regulationandstability of host-parasiteinteractions.I

dP/dt = P{(H/(Ho + H)-( u + b + ot)-oa(k+ 1)P/(kH)}

Regulationandstability of host-parasiteinteractions.I

dP/dt = P{(H/(Ho + H)-( u + b + ot)-oa(k+ 1)P/(kH)} (13)13)

As mentioned above, k is the parameterof the negative binomial distribution which givesan inverse measure of the degree of aggregationor contagion of the parasiteswithin the

hosts.The equilibrium host and parasite population values then follow from eqns (7) and

(13):

As mentioned above, k is the parameterof the negative binomial distribution which givesan inverse measure of the degree of aggregationor contagion of the parasiteswithin the

hosts.The equilibrium host and parasite population values then follow from eqns (7) and

(13):

H*IP* = (a- b)a

Ho {#+a+ + (a-b)(k+ l)Ik}l- { + a +a + (a - b)(k+ 1)/k}'

H*IP* = (a- b)a

Ho {#+a+ + (a-b)(k+ l)Ik}l- { + a +a + (a - b)(k+ 1)/k}'

(a)a)

(14)14)

(15)15)

(b)b)

-kk

Zrzrz

0 2 4 0 1 2 0 1 2

k o

FIG.5. ModelA-overdispersed parasitedistributions.Graphs(a) and(b)-the solid linesdenote the boundaries n the k and a parameter pace which separateregionsin whichparametervaluesgive rise to globallystable parasiteregulatedhost equilibria unshadedareas), and unregulatedhost populationgrowth (shaded areas). These boundariesareshownin termsof k anda for two values of A.In graph a) A = 6-0,andin (b) A = 20-0.Inboth (a) and (b), a = 3-0,b = 1 0, # = 0-1, Ho = 10-0.In graphs c), (d) and (e) the solidlines denote the equilibrium ize of the host population,H* (c) and (d)), and the meanparasiteburden,P*/H* (e) for variousvalues of k anda. The shadedregionsdenoteareasinwhichtheparameter aluesof the model leadto unregulated ostpopulationgrowth.Theparameter aluesare as indicated or graphs a) and(b) exceptthat in (c), a = 0-5,A = 6-0,

and in (d), and (e), k = 2-0, A= 6.0.

This equilibriumpoint can exist only if

0 2 4 0 1 2 0 1 2

k o

FIG.5. ModelA-overdispersed parasitedistributions.Graphs(a) and(b)-the solid linesdenote the boundaries n the k and a parameter pace which separateregionsin whichparametervaluesgive rise to globallystable parasiteregulatedhost equilibria unshadedareas), and unregulatedhost populationgrowth (shaded areas). These boundariesareshownin termsof k anda for two values of A.In graph a) A = 6-0,andin (b) A = 20-0.Inboth (a) and (b), a = 3-0,b = 1 0, # = 0-1, Ho = 10-0.In graphs c), (d) and (e) the solidlines denote the equilibrium ize of the host population,H* (c) and (d)), and the meanparasiteburden,P*/H* (e) for variousvalues of k anda. The shadedregionsdenoteareasinwhichtheparameter aluesof the model leadto unregulated ostpopulationgrowth.Theparameter aluesare as indicated or graphs a) and(b) exceptthat in (c), a = 0-5,A = 6-0,

and in (d), and (e), k = 2-0, A= 6.0.

This equilibriumpoint can exist only if

A> .u+a+oe+(a-b)(k+ 1)/k.> .u+a+oe+(a-b)(k+ 1)/k. (16)16)



RoY M. ANDERSONAND ROBERTM. MAYoY M. ANDERSONAND ROBERTM. MAY

The stability of the equilibriumis studied in Appendix 3, where it is shown that distur-

bances undergo damped oscillations back to the equilibrium population values, providedonly the aggregation parameter k is finite and positive. (In the limit k-+o, which

correspondsto the Poisson

limit,the

dampingtimes becomes

infinitely large,and we

recover the neutral stability of the Basic model.)Thus if eqn (16) is satisfied, the host population is effectively regulated by the parasites.

Figure4 illustrates the dynamicalbehaviourof such a system.If the inequality (16) is not fulfilled, then (as discussedpreviously)the host population

grows exponentiallyuntil it encounters environmental carrying capacity limitations. The

parasite population will also grow exponentially but at a slower rate than the hosts, and

thus the mean parasiteburden decreases in size.

The criterion (16) which determines the boundary between host populations that can

be regulatedby their parasites, and those that cannot, is illustratedin Fig. 5.

Figure 5(a) and (b) are two slices through the k- a parameter space, for two different

values of A (and other quantities held constant); they show that as i, the production rateof transmission stages, increases a larger range of k and A values lead to a parasiteregulated host equilibrium. When the parasite's distribution is highly overdispersed(k-+O),regulation is difficultto achieve for any value of the rate of parasiteinduced host

mortalities (a). As a increases, the degree of overdispersionmust decrease, otherwisetoo

many parasitesare lost due to parasite induced host mortalities.

An indication of the influence of o and k on the size of the equilibriumhost and para-site populations, H* and P*, is given in Figure 5 (c), (d) and (e). Figure 5 (d) appearsodd at first glance: an increase in the rate of parasite induced host mortalitymakes for an increase in H*. The reason is that those hosts which die from parasite

infection harbour above average parasite burdens; such deaths thus have relativelymore effect on the parasite population than on the host population. As x increases,this effect becomes increasingly pronounced, leading to a decrease in the mean

parasite burden per host (Fig. 5(e)) and an increase in H* (Fig. 5(d)). Eventually,for a sufficiently large, the parasites can no longer regulatethe host population. Notice

also that as k increases, the parasites are spread more evenly, and hence the net rate of

loss of hosts due to parasite induced mortality rises; i.e. H* decreases as k increases

(Fig. 5(c)).Quantitative estimates of the parameters in eqn (16) are not easy to come by for

natural populations.Table 2 lists

reportedvalues of k for a

varietyof

parasite species.The numbers show

that many parasites, particularly helminths, are highly overdispersedwithin their host

populations, the majority of values tending to be less than 1.0. This typically high degreeof overdispersion tends to confer stability; it also suggests that net losses from host

populations due to parasite infections may be low, since only a few hosts harbour heavyinfections. This is a result of the first importance, to which we will returnin the conclud-

ing discussion.The rate of production of transmission stages by the parasite, i, must exceed the sum

of the hosts reproductiverate, a, plus the host intrinsicgrowthratemultipliedby(k + 1)/k,

plus It and c.Particularlyif k is small, this requires that the parasite must have a much

higher reproductive rate than the host, which accords with traditional beliefs. Repro-

ductive rates of parasitic species, particularly protozoa and helminths, are invariablyhigh and almost without exception very much greater than the host's potential for

reproduction. For example, at the top end of the spectrumlie certain namatode species

The stability of the equilibriumis studied in Appendix 3, where it is shown that distur-

bances undergo damped oscillations back to the equilibrium population values, providedonly the aggregation parameter k is finite and positive. (In the limit k-+o, which

correspondsto the Poisson

limit,the

dampingtimes becomes

infinitely large,and we

recover the neutral stability of the Basic model.)Thus if eqn (16) is satisfied, the host population is effectively regulated by the parasites.

Figure4 illustrates the dynamicalbehaviourof such a system.If the inequality (16) is not fulfilled, then (as discussedpreviously)the host population

grows exponentiallyuntil it encounters environmental carrying capacity limitations. The

parasite population will also grow exponentially but at a slower rate than the hosts, and

thus the mean parasiteburden decreases in size.

The criterion (16) which determines the boundary between host populations that can

be regulatedby their parasites, and those that cannot, is illustratedin Fig. 5.

Figure 5(a) and (b) are two slices through the k- a parameter space, for two different

values of A (and other quantities held constant); they show that as i, the production rateof transmission stages, increases a larger range of k and A values lead to a parasiteregulated host equilibrium. When the parasite's distribution is highly overdispersed(k-+O),regulation is difficultto achieve for any value of the rate of parasiteinduced host

mortalities (a). As a increases, the degree of overdispersionmust decrease, otherwisetoo

many parasitesare lost due to parasite induced host mortalities.

An indication of the influence of o and k on the size of the equilibriumhost and para-site populations, H* and P*, is given in Figure 5 (c), (d) and (e). Figure 5 (d) appearsodd at first glance: an increase in the rate of parasite induced host mortalitymakes for an increase in H*. The reason is that those hosts which die from parasite

infection harbour above average parasite burdens; such deaths thus have relativelymore effect on the parasite population than on the host population. As x increases,this effect becomes increasingly pronounced, leading to a decrease in the mean

parasite burden per host (Fig. 5(e)) and an increase in H* (Fig. 5(d)). Eventually,for a sufficiently large, the parasites can no longer regulatethe host population. Notice

also that as k increases, the parasites are spread more evenly, and hence the net rate of

loss of hosts due to parasite induced mortality rises; i.e. H* decreases as k increases

(Fig. 5(c)).Quantitative estimates of the parameters in eqn (16) are not easy to come by for

natural populations.Table 2 lists

reportedvalues of k for a

varietyof

parasite species.The numbers show

that many parasites, particularly helminths, are highly overdispersedwithin their host

populations, the majority of values tending to be less than 1.0. This typically high degreeof overdispersion tends to confer stability; it also suggests that net losses from host

populations due to parasite infections may be low, since only a few hosts harbour heavyinfections. This is a result of the first importance, to which we will returnin the conclud-

ing discussion.The rate of production of transmission stages by the parasite, i, must exceed the sum

of the hosts reproductiverate, a, plus the host intrinsicgrowthratemultipliedby(k + 1)/k,

plus It and c.Particularlyif k is small, this requires that the parasite must have a much

higher reproductive rate than the host, which accords with traditional beliefs. Repro-

ductive rates of parasitic species, particularly protozoa and helminths, are invariablyhigh and almost without exception very much greater than the host's potential for

reproduction. For example, at the top end of the spectrumlie certain namatode species

23131



TABLE. Values of the negative binomal parameter k observed in natural populationsABLE. Values of the negative binomal parameter k observed in natural populations

Taxonomicgroupof parasite

Platyhelminthes,MonogeneaDigeneaCestoda

Nematoda

Taxonomicgroupof parasite

Platyhelminthes,MonogeneaDigeneaCestoda

Nematoda

Parasitespeciesarasitespecies

Diclidophora enticulataOlsson)Diplostomum asterostei Williams)Caryophyllaeusaticeps Pallas)Schistocephalusolidus Muller)Chandlerellauiscoli von Linstow)

Toxocaracanis(Werner)

Ascaridia alli (Schrank)

Diclidophora enticulataOlsson)Diplostomum asterostei Williams)Caryophyllaeusaticeps Pallas)Schistocephalusolidus Muller)Chandlerellauiscoli von Linstow)

Toxocaracanis(Werner)

Ascaridia alli (Schrank)

Acanthocephala Polymorphus minutus (Groeze)Echinorhynchuslavula Dujardin)

Arthropoda Copeopoda Lepeophtheiruspectoralis (Muller)Chondracanthopsisodosus Muller)Chondracanthopsisodosus

Arachnida Ixodes trianguliceps (Birula)Liponysue acoti(Hirst)

Insecta Pediculushumanus apitis(L)

Acanthocephala Polymorphus minutus (Groeze)Echinorhynchuslavula Dujardin)

Arthropoda Copeopoda Lepeophtheiruspectoralis (Muller)Chondracanthopsisodosus Muller)Chondracanthopsisodosus

Arachnida Ixodes trianguliceps (Birula)Liponysue acoti(Hirst)

Insecta Pediculushumanus apitis(L)

Host species

Gadusvirens L.)Gasterosteus culeatus L.)Abramisbrama L.)Gasterosteus culeatusCulicoides repuscularis

(Malloch)Vulpesvulpes L.)

Gallusgallus (L.)

Gammarusulex (L.)Gasterosteus culeatus L.)Pleuronectes latessa (L.)SebastesmarinusL.)Sebastesmentella L.)Apodemusylvaticus L.)Rattusrattus(L.)Homosapiens

Host species

Gadusvirens L.)Gasterosteus culeatus L.)Abramisbrama L.)Gasterosteus culeatusCulicoides repuscularis

(Malloch)Vulpesvulpes L.)

Gallusgallus (L.)

Gammarusulex (L.)Gasterosteus culeatus L.)Pleuronectes latessa (L.)SebastesmarinusL.)Sebastesmentella L.)Apodemusylvaticus L.)Rattusrattus(L.)Homosapiens

Raa

000

00-

0

0-

000

00-

0

0-



ROY M. ANDERSONAND ROBERTM. MAY 233

such as Haemonchus contortus (Rudolphi), a parasite of sheep, which is capable of

producing 10 000 eggs per day (Crofton 1966).Estimates of a, the rate of parasite induced host mortalities, are more difficult to

extract from the literature. As a rather broad generalization is appears to be widelyaccepted in the parasitological literature that the majority of protozoan and helminth

parasitesdo not cause the death of many hosts in natural populations. However, where

quantitativerate estimates are availablefrom laboratoryexperiments(Fig. 1), it appearsas though many parasite species have the capability of causing high host death rates.

This is particularly apparent in parasite life cycles which utilise invertebrate hosts.

Infected individualswithin a host population have extremely poor survival characteristics

when compared with uninfected hosts (Fig. 6). In natural populations, counter to popu-lar opinion, parasites may play a crucial role in regulating the size of their host popula-tions due to high parasite induced host mortality rates, coupled with small k values.

ROY M. ANDERSONAND ROBERTM. MAY 233

such as Haemonchus contortus (Rudolphi), a parasite of sheep, which is capable of

producing 10 000 eggs per day (Crofton 1966).Estimates of a, the rate of parasite induced host mortalities, are more difficult to

extract from the literature. As a rather broad generalization is appears to be widelyaccepted in the parasitological literature that the majority of protozoan and helminth

parasitesdo not cause the death of many hosts in natural populations. However, where

quantitativerate estimates are availablefrom laboratoryexperiments(Fig. 1), it appearsas though many parasite species have the capability of causing high host death rates.

This is particularly apparent in parasite life cycles which utilise invertebrate hosts.

Infected individualswithin a host population have extremely poor survival characteristics

when compared with uninfected hosts (Fig. 6). In natural populations, counter to popu-lar opinion, parasites may play a crucial role in regulating the size of their host popula-tions due to high parasite induced host mortality rates, coupled with small k values.

05- \C.

c "', I " "t

> 0 15 30

(Week)

g I0o (c)

0 \

0 60 120

(Day)

05- \C.

c "', I " "t

> 0 15 30

(Week)

g I0o (c)

0 \

0 60 120

(Day)Timeime

(b) \\

_ 0

\

0 30 60

(Day)

(d)*

-*

-

I \I's0 20 40

(Day)

(b) \\

_ 0

\

0 30 60

(Day)

(d)*

-*

-

I \I's0 20 40

(Day)

FIG.6. Some examplesof the survivalcharacteristics f infectedhosts as comparedwithuninfectedhosts. (a) SnailhostAustralorbislabratus Say) nfectedwithlarvalstagesof thedigeneanSchistosomamansoni Sambon)(datafrom Pan 1965); (b) snail host Lymnaeagedrosina nfectedwith larvalstagesof the digeneanOrnithobilharziaurkestanicumdatafrom Massoud 1974); (c) mosquito Aedesaegypti (L.) infected with microfilariae f thenematode

Dirofilariammitis

Leidy) datafrom

Kershaw,Lavioipierre&Chalmers

1953);(d) snailhost Australorbislabratusnfectedwith the nematodeDaubaylia otimaca Chit-wood andChitwood) datafrom Chernin1962) solidlines-uninfected hosts,stippled ines

-infected hosts).

A present, however, it is difficult to substantiate this conjecture, because of the diffi-

culty in assessing o in natural populations. Our model suggests that a rough, indirectestimate of c may be obtained from eqn (14):

a = (a-b)(H*/P*). (17)

That is, the rate c is equal to the natural intrinsicgrowth rate of the host populationdivided by the mean parasiteburdenP*/H*. This is a useful result in itself. It suggeststhat when overdispersedburdensof parasites per host, with small mean, are observed inthe field, then x is likely to be relatively high.

FIG.6. Some examplesof the survivalcharacteristics f infectedhosts as comparedwithuninfectedhosts. (a) SnailhostAustralorbislabratus Say) nfectedwithlarvalstagesof thedigeneanSchistosomamansoni Sambon)(datafrom Pan 1965); (b) snail host Lymnaeagedrosina nfectedwith larvalstagesof the digeneanOrnithobilharziaurkestanicumdatafrom Massoud 1974); (c) mosquito Aedesaegypti (L.) infected with microfilariae f thenematode

Dirofilariammitis

Leidy) datafrom

Kershaw,Lavioipierre&Chalmers

1953);(d) snailhost Australorbislabratusnfectedwith the nematodeDaubaylia otimaca Chit-wood andChitwood) datafrom Chernin1962) solidlines-uninfected hosts,stippled ines

-infected hosts).

A present, however, it is difficult to substantiate this conjecture, because of the diffi-

culty in assessing o in natural populations. Our model suggests that a rough, indirectestimate of c may be obtained from eqn (14):

a = (a-b)(H*/P*). (17)

That is, the rate c is equal to the natural intrinsicgrowth rate of the host populationdivided by the mean parasiteburdenP*/H*. This is a useful result in itself. It suggeststhat when overdispersedburdensof parasites per host, with small mean, are observed inthe field, then x is likely to be relatively high.



Regulationandstability of host-parasite nteractions.Iegulationandstability of host-parasite nteractions.I

Underdisperseddistributions

As mentioned earlier, parasites may occasionally be under dispersed, i.e. more evenlydistributed than if they were independently randomly distributed among hosts. Such

underdisperseddistributions may be described by a positive binomial, with a parameterk' that is inversely proportional to the degree of under-dispersion:in the limit k' -0, allhosts carry identical parasite burdens.

As shown in Appendix 1, the appropriate version of eqn (8) is obtained by simplyreplacingk with -k' in eqn (14). It is easily shown (see Appendix 3) that such a system,which amounts to model A with a negativevalue of k, can have no stable equilibrium.

In short, an overdispersedpattern of parasite numbers per host can enable the para-sites stably to regulate the host population. The exact form of the pattern of overdis-

persion is immaterial; the above analysis based on the negative binomial can be repeatedfor other qualitatively similar distributions such as the Neyman type A, with the sameconclusion.

MODEL B: NONLINEAR PARASITE INDUCED MORTALITIES

So far, the parasite inducedhost mortality rate has been taken to be linearlyproportionalto the parasiteburden (cf. Figs l(a)-(c)). Often, however, the relationship between hostdeath rate and parasite burdenis of a more severeform, as indicated in Fig. l(d)-(f).

In model B, we examinethe dynamicalconsequences of a more steeply density depen-dentparasiteinduced deathrate.For specificity,we typifysuch steeper densitydependenceby assumingthat the parasiteinducedmortality rate varies as the square of the numberof

parasites in a host, i.e. as ci2. By recapitulating the arguments that led to eqns (7) and(8), we have now

dH/dt = (a- b)H- HE(i2) (18)

dP/dt = APH/(Ho+ H) - (1 + b)P- HE(i3). (19)

The exact form of the averagevalues of i2 and i3 depends on the parasite distribution

among hosts. In Appendix 3, we indicate the equilibrium population values, and their

stability character, under the various assumptions that the parasites are overdispersed

(negative binomial), independently randomly distributed(Poisson), and underdispersed(positive binomial).

The nonlinearlysevere parasite induced host mortality has two consequences, both ofwhich are illustrated by Fig. 7.

On the one hand, the effect enhances the dynamic stability of an equilibrium point,providedone exists. Thus if the parametersare such that equilibrium populations H* andP* exist, then this point is stable for overdispersed or Poisson parasite distributions

(the latter in contrast to the Basic model), and also for underdisperseddistributions so

long as k' is not too small (in contrast to model A, where all such underdispersedcasesare unstable).

On the other hand, the domain of parameter space for which the equilibriumexists

(i.e. satisfyingthe analogue of eqns (12) and (16)) is always smaller for model B than for

the correspondingmodel A. This can be seen by comparing Figs 5 and 7.All this can be explained intuitively. Comparedwith model A, the more steeply severe

parasite induced host mortality makes it relatively harder for the parasite to check the

Underdisperseddistributions

As mentioned earlier, parasites may occasionally be under dispersed, i.e. more evenlydistributed than if they were independently randomly distributed among hosts. Such

underdisperseddistributions may be described by a positive binomial, with a parameterk' that is inversely proportional to the degree of under-dispersion:in the limit k' -0, allhosts carry identical parasite burdens.

As shown in Appendix 1, the appropriate version of eqn (8) is obtained by simplyreplacingk with -k' in eqn (14). It is easily shown (see Appendix 3) that such a system,which amounts to model A with a negativevalue of k, can have no stable equilibrium.

In short, an overdispersedpattern of parasite numbers per host can enable the para-sites stably to regulate the host population. The exact form of the pattern of overdis-

persion is immaterial; the above analysis based on the negative binomial can be repeatedfor other qualitatively similar distributions such as the Neyman type A, with the sameconclusion.

MODEL B: NONLINEAR PARASITE INDUCED MORTALITIES

So far, the parasite inducedhost mortality rate has been taken to be linearlyproportionalto the parasiteburden (cf. Figs l(a)-(c)). Often, however, the relationship between hostdeath rate and parasite burdenis of a more severeform, as indicated in Fig. l(d)-(f).

In model B, we examinethe dynamicalconsequences of a more steeply density depen-dentparasiteinduced deathrate.For specificity,we typifysuch steeper densitydependenceby assumingthat the parasiteinducedmortality rate varies as the square of the numberof

parasites in a host, i.e. as ci2. By recapitulating the arguments that led to eqns (7) and(8), we have now

dH/dt = (a- b)H- HE(i2) (18)

dP/dt = APH/(Ho+ H) - (1 + b)P- HE(i3). (19)

The exact form of the averagevalues of i2 and i3 depends on the parasite distribution

among hosts. In Appendix 3, we indicate the equilibrium population values, and their

stability character, under the various assumptions that the parasites are overdispersed

(negative binomial), independently randomly distributed(Poisson), and underdispersed(positive binomial).

The nonlinearlysevere parasite induced host mortality has two consequences, both ofwhich are illustrated by Fig. 7.

On the one hand, the effect enhances the dynamic stability of an equilibrium point,providedone exists. Thus if the parametersare such that equilibrium populations H* andP* exist, then this point is stable for overdispersed or Poisson parasite distributions

(the latter in contrast to the Basic model), and also for underdisperseddistributions so

long as k' is not too small (in contrast to model A, where all such underdispersedcasesare unstable).

On the other hand, the domain of parameter space for which the equilibriumexists

(i.e. satisfyingthe analogue of eqns (12) and (16)) is always smaller for model B than for

the correspondingmodel A. This can be seen by comparing Figs 5 and 7.All this can be explained intuitively. Comparedwith model A, the more steeply severe

parasite induced host mortality makes it relatively harder for the parasite to check the

23434




host population's intrinsic propensity to growth. But if it can do so, this density depen-dence results in relatively faster damping of perturbations.

MODEL C: DENSITY DEPENDENCE IN PARASITE POPULATION GROWTH

The growth of a parasite population within a single host is often constrained by theinfluence of density dependent death or reproductive processes. (see Anderson 1976c).Such regulatory mechanisms may be due to either parasite induced host immunological

responses or intraspecific competition for finite resources such as space or nutrients

...... .... ..... .................

0 l«5 3'0 1.530

G.. )l n r p e.............Blin r p site i d m iy............... T e i e e o te di i

butionpattern f the numbersf parasiteserhost,andthe rateof parasitenduced ost

..generatedre unstable. In (a) and (b), the parasitesare underdispersedollowinghe.............

:::..............:.:....:.:.================ == = =.... .. ....................................... ......

.,.,.,.,.,/ ..................................

O ~30,b5 3 0 0 H15 = 0,

Fwithin..ModelB-non-linearparasite inducedortality ates. heinfluencef thedistri-butionpattern of the umbers fparasiteserhost,andthe rate ofarasitenduced ostmortalitiesa) on he numericalizef the host quilibriumopulation nd its globalstability properties. The solid line denotes the values of H*, while the unshaded regionsdenote reasof parameteralues n which hehostopulation growths regulated y theparasitendglobally tablequilibria rise.The ightlyhadependereasefine egionswheretheparasiteails to regulatehe growth f theost population; the darkly haded reasdefine regionswhereocesses onhearasite egulates ost population rowthbut the equilibria

generated are unstable. In (a) and (b), the parasites are underdispersed following thepositive inomial model.hevalueof k' is6e00n (a)and20q0n(b); ngraphsc)the ara-sites arerandomly istributedPoissonmodel); nd)band(e)he parasitesre overdis-persed following the negative binomial modeYl here in (d) k = 8.0 and in (e) k = 2.0 (a =

3'0, b = 10, ,u = 0.1, Ho = 10-0, A = 6 .0).

within or on the host. The severity of an immunological response by the host is usuallyfunctionally related to the degree of antigenic stimulation received (number of parasites).Such responses, whether humoral or cell mediated, tend to increase the death rate of aparasite population and/or reduce its reproductive potential (Anderson & Michel1977; Bradley 1971). Some examples of density dependent parasite death rates forhelminth species in vertebrate hosts are illustrated in Fig. 8.

The influence of such processes on the dynamical properties of host-parasite interac-tions may be examined by appropriate modification of the basic eqns (7) and(8). Forspecificity, we assume the density dependence to be such that natural parasite mortality

host population's intrinsic propensity to growth. But if it can do so, this density depen-dence results in relatively faster damping of perturbations.

MODEL C: DENSITY DEPENDENCE IN PARASITE POPULATION GROWTH

The growth of a parasite population within a single host is often constrained by theinfluence of density dependent death or reproductive processes. (see Anderson 1976c).Such regulatory mechanisms may be due to either parasite induced host immunological

responses or intraspecific competition for finite resources such as space or nutrients

...... .... ..... .................

0 l«5 3'0 1.530

G.. )l n r p e.............Blin r p site i d m iy............... T e i e e o te di i

butionpattern f the numbersf parasiteserhost,andthe rateof parasitenduced ost

..generatedre unstable. In (a) and (b), the parasitesare underdispersedollowinghe.............

:::..............:.:....:.:.================ == = =.... .. ....................................... ......

.,.,.,.,.,/ ..................................

O ~30,b5 3 0 0 H15 = 0,

Fwithin..ModelB-non-linearparasite inducedortality ates. heinfluencef thedistri-butionpattern of the umbers fparasiteserhost,andthe rate ofarasitenduced ostmortalitiesa) on he numericalizef the host quilibriumopulation nd its globalstability properties. The solid line denotes the values of H*, while the unshaded regionsdenote reasof parameteralues n which hehostopulation growths regulated y theparasitendglobally tablequilibria rise.The ightlyhadependereasefine egionswheretheparasiteails to regulatehe growth f theost population; the darkly haded reasdefine regionswhereocesses onhearasite egulates ost population rowthbut the equilibria

generated are unstable. In (a) and (b), the parasites are underdispersed following thepositive inomial model.hevalueof k' is6e00n (a)and20q0n(b); ngraphsc)the ara-sites arerandomly istributedPoissonmodel); nd)band(e)he parasitesre overdis-persed following the negative binomial modeYl here in (d) k = 8.0 and in (e) k = 2.0 (a =

3'0, b = 10, ,u = 0.1, Ho = 10-0, A = 6 .0).

within or on the host. The severity of an immunological response by the host is usuallyfunctionally related to the degree of antigenic stimulation received (number of parasites).Such responses, whether humoral or cell mediated, tend to increase the death rate of aparasite population and/or reduce its reproductive potential (Anderson & Michel1977; Bradley 1971). Some examples of density dependent parasite death rates forhelminth species in vertebrate hosts are illustrated in Fig. 8.

The influence of such processes on the dynamical properties of host-parasite interac-tions may be examined by appropriate modification of the basic eqns (7) and(8). Forspecificity, we assume the density dependence to be such that natural parasite mortality

23535



Regulation andstability of host-parasite nteractions.Iegulationandstability of host-parasite nteractions.I

occurs at a rate ui2,in hosts harbouringi parasites.The model is then eqn (7) for dH/dt,

together with

dP/dt =APH/(Ho + H)- bP- (t + a)HE(i2). (20)

Apart from this modification, the model is exactly as for model A: again the detailedform of eqn (20) will depend on the patternof parasite distribution.

The dynamical behaviour of model C is elucidated in Appendix 3. For a negativebinomial distribution (with parameter k), equilibrium host and parasite populationsexist only if

A > b+ +a+(a + u)(a- b)(1+ k)/(ak). (21)

(a) / (b)0.12- 0*2- /

006- / 0-1 '>. / X

t I i i.0 150000 300000 0 300 600

o)-o

o (c) (d)

' 0-06- * 0-2-

0 / /

0'03- 0 1-

.0 I I rl0 100 200 0 300 600


FIG. 8. Some examplesof the relationshipbetweenthe instantaneousparasite'natural'death rate(u) and parasitedensity (i) withinindividualhosts. The solid linesare the leastsquaresbest fit linearmodels of the form , (i) = ad+bi(a) Calves infected with the gutnematode Ostertagia ostertagi (Stiles); a = 0-0171, b = 0.0308x 10-5 (rate/day) (datafrom Anderson & Michel 1977). (b) Chickens infected with the fowl nematode

Ascaridia lineata (Schneid); d = 0-0856, b = 0-00019 (rate/day) (data from Ackert,Graham, Xolf & Porter1931).(c) Rats infectedwith the tapewormHymenolepis iminuta(Rud); a = 0.00636, b = 0-00032(rate/day)(data from Hesselberg& Andreassen1975).(d) Rats infected with the nematode Heterakisspumosa(Schneider);a = 0-0265, ^ =

0-000037(rate/day)(datafrom Winfield1932).

For independently randomly distributed parasites, the Poisson result follows as thelimit k-- c in eqn (21); for underdispersion(positive binomial with parameterk') one

simply replaces k with -k' in eqn (21). If the equilibriumexists, it is stable for overdis-

persed or Poisson distributions, and also for underdispersionprovided the parameterk' is not too small (explicitly, providedk'> (o+u)/u).

The general pattern of relationship between model C and model A is similar to thatbetween model B and model A. As is made clear by the comparison between eqns (16)and (21), i must be relatively largerif an equilibrium s to exist (i.e. if parasitesare to beable to check host population growth) in model C; this is particularly marked if over-

occurs at a rate ui2,in hosts harbouringi parasites.The model is then eqn (7) for dH/dt,

together with

dP/dt =APH/(Ho + H)- bP- (t + a)HE(i2). (20)

Apart from this modification, the model is exactly as for model A: again the detailedform of eqn (20) will depend on the patternof parasite distribution.

The dynamical behaviour of model C is elucidated in Appendix 3. For a negativebinomial distribution (with parameter k), equilibrium host and parasite populationsexist only if

A > b+ +a+(a + u)(a- b)(1+ k)/(ak). (21)

(a) / (b)0.12- 0*2- /

006- / 0-1 '>. / X

t I i i.0 150000 300000 0 300 600

o)-o

o (c) (d)

' 0-06- * 0-2-

0 / /

0'03- 0 1-

.0 I I rl0 100 200 0 300 600


FIG. 8. Some examplesof the relationshipbetweenthe instantaneousparasite'natural'death rate(u) and parasitedensity (i) withinindividualhosts. The solid linesare the leastsquaresbest fit linearmodels of the form , (i) = ad+bi(a) Calves infected with the gutnematode Ostertagia ostertagi (Stiles); a = 0-0171, b = 0.0308x 10-5 (rate/day) (datafrom Anderson & Michel 1977). (b) Chickens infected with the fowl nematode

Ascaridia lineata (Schneid); d = 0-0856, b = 0-00019 (rate/day) (data from Ackert,Graham, Xolf & Porter1931).(c) Rats infectedwith the tapewormHymenolepis iminuta(Rud); a = 0.00636, b = 0-00032(rate/day)(data from Hesselberg& Andreassen1975).(d) Rats infected with the nematode Heterakisspumosa(Schneider);a = 0-0265, ^ =

0-000037(rate/day)(datafrom Winfield1932).

For independently randomly distributed parasites, the Poisson result follows as thelimit k-- c in eqn (21); for underdispersion(positive binomial with parameterk') one

simply replaces k with -k' in eqn (21). If the equilibriumexists, it is stable for overdis-

persed or Poisson distributions, and also for underdispersionprovided the parameterk' is not too small (explicitly, providedk'> (o+u)/u).

The general pattern of relationship between model C and model A is similar to thatbetween model B and model A. As is made clear by the comparison between eqns (16)and (21), i must be relatively largerif an equilibrium s to exist (i.e. if parasitesare to beable to check host population growth) in model C; this is particularly marked if over-

23636



280()

140,

I

280()

140,

I


(b

I !/ msmI ¹¹¹¹¹

I {{{(fff{{{'.

/§

/ III

/l«l

/III

y iii^ iiill:l:l;!jl,llljjljljj

(b

I !/ msmI ¹¹¹¹¹

I {{{(fff{{{'.

/§

/ III

/l«l

/III

y iii^ iiill:l:l;!jl,llljjljljj

FIG.9. ModelC-density dependentconstraintson parasitepopulationgrowth;The solidlines enclose unshadedregionsin which the parametervalues lead to parasite regulatedhostpopulationequilibriawhicharegloballystable.Theseboundaries reshown ntermsof/Iand a for variouspatternsof dispersionof parasitenumbersper host; the lightlyshadedregions ndicate heparameter alueswhichgiverise to unregulated ostpopulationgrowth.In the darklyshadedareasthe parasiteregulateshost populationgrowthbut the equilibriaproducedareunstable. a) Positivebinomial,k' = 6-0; (b) positivebinomial,k = 20-0;(c)

Poisson; (d) negativebinomial,k = 4.0; (e) negativebinomial,k = 1.0; (a = 3.0, b = 1.0,Ho = 10-0,A = 6-0).

FIG.9. ModelC-density dependentconstraintson parasitepopulationgrowth;The solidlines enclose unshadedregionsin which the parametervalues lead to parasite regulatedhostpopulationequilibriawhicharegloballystable.Theseboundaries reshown ntermsof/Iand a for variouspatternsof dispersionof parasitenumbersper host; the lightlyshadedregions ndicate heparameter alueswhichgiverise to unregulated ostpopulationgrowth.In the darklyshadedareasthe parasiteregulateshost populationgrowthbut the equilibriaproducedareunstable. a) Positivebinomial,k' = 6-0; (b) positivebinomial,k = 20-0;(c)

Poisson; (d) negativebinomial,k = 4.0; (e) negativebinomial,k = 1.0; (a = 3.0, b = 1.0,Ho = 10-0,A = 6-0).

500

250

500

250

(a)a)

0 10 2'0 3'0 0 8 16 24oa

FIG.10. ModelC-density dependentconstraintson parasitepopulationgrowth. (a) Theinfluenceof the parameter on the numerical ize of the host populationequilibriumH*(parasitesrandomlydistributed-Poisson model); the shaded region defines parametervalues which lead to unregulatedhost populationgrowth,while in the unshadedregionsthe parasiteregulatedhost equilibriaare globallystable. (b) The influenceof the rate ofparasitereproduction,, on H*; the shaded and unshadedregionsare as defined or (a);

(a = 3'0, b = 1.0, , = 0-1, Ho = 10-0, A= 6.0, a = 0'5).

0 10 2'0 3'0 0 8 16 24oa

FIG.10. ModelC-density dependentconstraintson parasitepopulationgrowth. (a) Theinfluenceof the parameter on the numerical ize of the host populationequilibriumH*(parasitesrandomlydistributed-Poisson model); the shaded region defines parametervalues which lead to unregulatedhost populationgrowth,while in the unshadedregionsthe parasiteregulatedhost equilibriaare globallystable. (b) The influenceof the rate ofparasitereproduction,, on H*; the shaded and unshadedregionsare as defined or (a);

(a = 3'0, b = 1.0, , = 0-1, Ho = 10-0, A= 6.0, a = 0'5).

k11

23737

:':i*i j:"'i.

t

i if

:i j:

I0.6 i-Fot

:':i*i j:"'i.

t

i if

:i j:

I0.6 i-Fot

I

I - ---- ---



238 Regulationand stability of host-parasite nteractions.I

dispersion is high (small k). But if the equilibriumcan exist, for given k, it is more stablein model C than in model A.

These remarksare furtherbourne out by Figs 9 and 10. The stable domain in the u,c

parameterspace increases in size when the pattern of dispersion of the parasites withinthe host population moves from regular to random (Fig. 9(a)-(c)), and then decreasesas the patternchanges from random to aggregated (Fig. 9 (c)-(e)). When overdispersionis marked (k small), as is the case for many parasitic species (Table 2), the region of

parameter space in which the parasiteregulatesthe host population is rathersmall (Fig.9(e)). The model's predictions therefore suggest that when the parasite population is

tightly controlled by densitydependentconstraints (i.e. when # is large) and parasitesare

overdispersed within the host population, other factors than parasite induced hostmortalities will tend to stabilise and regulate host population growth.

O-(a)a120-(b)

0 \

103 60 -

>.0 ~ ~ ~ ~ ~0

0.

o II IIa 0 300 600 0 30000 60000

15

1 ^-(0

(7 -

0 200 400

Parasite population density

FIG. 11. Someexamples

f functionalelationships

etweenhe rateofproduction

fparasitetransmission tages (A)and the parasite populationdensity(i) within individualhosts; hedotsareobservedointswhilehesolid inesare ittedbyeye. a)Laboratoryiceinfectedwith hetapeworm ymenolepisanaSiebold)dataromGhazal&Avery1974);(b)calves nfectedwithOstertagiastertagidata romMichel1969); c) sheep nfected

with heflukeFasciola epaticadata romBoray1969).

Mammalian hosts which possess well developed immunological responses to parasiticinvasion, creating tight density dependent controls on parasite population growth, maywell fall into this category.

When parasites are independentlyrandomly distributed, however, the main effect ofthe density dependence of model C is to help stabilize the system. In the case of Poisson

distributed parasites,a furtherinsight, concerning the influence of a upon H*, is illustra-ted in Fig. 10(a): too high or too low values of a lead to exponential runawayof the host

population.The above discussion has been for densitydependencein the naturalparasitemortality

238 Regulationand stability of host-parasite nteractions.I

dispersion is high (small k). But if the equilibriumcan exist, for given k, it is more stablein model C than in model A.

These remarksare furtherbourne out by Figs 9 and 10. The stable domain in the u,c

parameterspace increases in size when the pattern of dispersion of the parasites withinthe host population moves from regular to random (Fig. 9(a)-(c)), and then decreasesas the patternchanges from random to aggregated (Fig. 9 (c)-(e)). When overdispersionis marked (k small), as is the case for many parasitic species (Table 2), the region of

parameter space in which the parasiteregulatesthe host population is rathersmall (Fig.9(e)). The model's predictions therefore suggest that when the parasite population is

tightly controlled by densitydependentconstraints (i.e. when # is large) and parasitesare

overdispersed within the host population, other factors than parasite induced hostmortalities will tend to stabilise and regulate host population growth.

O-(a)a120-(b)

0 \

103 60 -

>.0 ~ ~ ~ ~ ~0

0.

o II IIa 0 300 600 0 30000 60000

15

1 ^-(0

(7 -

0 200 400

Parasite population density

FIG. 11. Someexamples

f functionalelationships

etweenhe rateofproduction

fparasitetransmission tages (A)and the parasite populationdensity(i) within individualhosts; hedotsareobservedointswhilehesolid inesare ittedbyeye. a)Laboratoryiceinfectedwith hetapeworm ymenolepisanaSiebold)dataromGhazal&Avery1974);(b)calves nfectedwithOstertagiastertagidata romMichel1969); c) sheep nfected

with heflukeFasciola epaticadata romBoray1969).

Mammalian hosts which possess well developed immunological responses to parasiticinvasion, creating tight density dependent controls on parasite population growth, maywell fall into this category.

When parasites are independentlyrandomly distributed, however, the main effect ofthe density dependence of model C is to help stabilize the system. In the case of Poisson

distributed parasites,a furtherinsight, concerning the influence of a upon H*, is illustra-ted in Fig. 10(a): too high or too low values of a lead to exponential runawayof the host

population.The above discussion has been for densitydependencein the naturalparasitemortality




rate. For many host-parasiteassociations, the 'fecundity'Ais also known to be a functionof the parasitepopulation density due, as in the case of the death rate, to either immuno-

logical processes or intraspecific competition. Some quantitative examples of such

responsesare shown in

Fig.11 for helminth

parasitesof vertebrates. The inclusion of such

processes in the basic model leads to qualitativelysimilar dynamical properties to thoseoutlined above for density dependentparasite death rates.

CONCLUSIONS

Our population models of host-parasite interactions are all characterizedby the central

assumption that parasites cause host mortalities. In particular, we have assumed thatthe net rate of such mortalities is relatedto the averageparasiteburden of the members ofa host population, and therefore to the statistical distribution of the parasites within ahost population. We regardthe inducement of host mortalities and/or reduction in host

reproductivepotential as a necessary condition for the classification of an organism asparasitic.

Specieswhich exhibit such characteristicsmay in certain circumstancesplay an impor-tant role in regulatingor controlling the growth of their host population. In such cases,the parasite will play an analogous role to a predatorwhich suppresses the growth of its

prey population. We have demonstrated that three specific categories of populationprocesses are of particularsignificance in stabilizing the dynamical behaviour of a host-

parasite interaction and enhancing the regulatoryinfluence of the parasite.Thefirst of these categories of population processes concerns the functional relation-

ships between the rate of parasite induced host deaths and the parasite burden of indi-

vidual hosts. It is interesting to compare the significance of these relationships on thedynamical properties of a given host-parasite association, with the role played byfunctional responses in predator-prey,and host-insect parasitoid interactions (Solomon1949; Holling 1959; Hassell & May 1973). The presence of a complex sigmoid function

response between the attack rate of a predatorand the density of prey (type III responsein the terminology of predator-preyinteractions as recently reviewed by Murdoch &Oaten 1975) may, or may not, have a stabilizing influence on the dynamics of suchinteractions.There will be a rangeof prey densitiesover which the death rate of the preyimposed by the predator is an increasing function of prey density and hence such a

response will be stabilizing since it acts in a density dependent manner. However, incontrast, if the

preydeath rate is constant over a wide

rangeof

preydensities

(asin

typeII responses), then the predator will not play a major stabilizing role in the dynamics ofthe prey. In such cases other constraints, such as intraspecific competition for finiteresourcesamongst the prey population, will stabilise the interaction.

We have demonstrated, in the case of host-parasite associations, that when the para-sites are randomly distributed within the host population, a linear relationship betweenhost death rate and parasite burden gives rise to the pathological condition of neutral

stability (Basic model). If however, the rate of parasite induced host mortalities is an

increasing function of parasite burden, perhaps following a power law or exponentialform, random distributions of parasites may lead to globally stable equilibria where the

parasite is effectivelyregulatingthe growth of its host population (model B). Thus, for a

given distribution patternof parasites, certaintypes of functional response may stabilizethe dynamics, while others lead to instabilities as in the case of predator-preyassocia-tions.

rate. For many host-parasiteassociations, the 'fecundity'Ais also known to be a functionof the parasitepopulation density due, as in the case of the death rate, to either immuno-

logical processes or intraspecific competition. Some quantitative examples of such

responsesare shown in

Fig.11 for helminth

parasitesof vertebrates. The inclusion of such

processes in the basic model leads to qualitativelysimilar dynamical properties to thoseoutlined above for density dependentparasite death rates.

CONCLUSIONS

Our population models of host-parasite interactions are all characterizedby the central

assumption that parasites cause host mortalities. In particular, we have assumed thatthe net rate of such mortalities is relatedto the averageparasiteburden of the members ofa host population, and therefore to the statistical distribution of the parasites within ahost population. We regardthe inducement of host mortalities and/or reduction in host

reproductivepotential as a necessary condition for the classification of an organism asparasitic.

Specieswhich exhibit such characteristicsmay in certain circumstancesplay an impor-tant role in regulatingor controlling the growth of their host population. In such cases,the parasite will play an analogous role to a predatorwhich suppresses the growth of its

prey population. We have demonstrated that three specific categories of populationprocesses are of particularsignificance in stabilizing the dynamical behaviour of a host-

parasite interaction and enhancing the regulatoryinfluence of the parasite.Thefirst of these categories of population processes concerns the functional relation-

ships between the rate of parasite induced host deaths and the parasite burden of indi-

vidual hosts. It is interesting to compare the significance of these relationships on thedynamical properties of a given host-parasite association, with the role played byfunctional responses in predator-prey,and host-insect parasitoid interactions (Solomon1949; Holling 1959; Hassell & May 1973). The presence of a complex sigmoid function

response between the attack rate of a predatorand the density of prey (type III responsein the terminology of predator-preyinteractions as recently reviewed by Murdoch &Oaten 1975) may, or may not, have a stabilizing influence on the dynamics of suchinteractions.There will be a rangeof prey densitiesover which the death rate of the preyimposed by the predator is an increasing function of prey density and hence such a

response will be stabilizing since it acts in a density dependent manner. However, incontrast, if the

preydeath rate is constant over a wide

rangeof

preydensities

(asin

typeII responses), then the predator will not play a major stabilizing role in the dynamics ofthe prey. In such cases other constraints, such as intraspecific competition for finiteresourcesamongst the prey population, will stabilise the interaction.

We have demonstrated, in the case of host-parasite associations, that when the para-sites are randomly distributed within the host population, a linear relationship betweenhost death rate and parasite burden gives rise to the pathological condition of neutral

stability (Basic model). If however, the rate of parasite induced host mortalities is an

increasing function of parasite burden, perhaps following a power law or exponentialform, random distributions of parasites may lead to globally stable equilibria where the

parasite is effectivelyregulatingthe growth of its host population (model B). Thus, for a

given distribution patternof parasites, certaintypes of functional response may stabilizethe dynamics, while others lead to instabilities as in the case of predator-preyassocia-tions.

23939



Regulationand stability of host-parasiteinteractions.Iegulationand stability of host-parasiteinteractions.I

The second category of population processes, which influence the regulatory role of

parasites, concerns the statistical distribution of parasite numbers per host. When therate of parasite induced host mortalities is linearly dependent on the number of parasitesharboured, we have demonstrated that parasites may regulate host population growth

providedthey exhibit an overdispersedpatternof distributionwithin the host population(model A). The accumulating information in the literature on the overdispersedcharac-ter of parasitedistributionsin naturalpopulations of hosts (videTable 2) lends supporttothe importance of this insight.The stabilizing influenceof parasite contagion is somewhat

analogous to the influenceof aggregateddistributions of insect parasitoidson the stabilityof host-parasitoidassociations (Hassell & May 1974). Parsitoid aggregationwas shown,in certain circumstances, to stabilize models which otherwise would have been quiteunstable.

The thirdcategory of processeswhich enhance the regulatory role played by a parasiteact within each individual parasite population (or subpopulation) harboured in each

member of the host population. Model C demonstrated that density dependent con-straints on parasite population growth, caused either by intraspecific competition for

finite resourcesor immunological attack by the host, can in the absence of overdispersionor non-linear functional responsesstabilise the dynamics of an interaction.

On the other hand, there are features of host-parasite interactions which tend to

destabilize the system. These are discussed in the following paper (May & Anderson

1978), and a general review of pertinent field and laboratory data on host parasiteassociations is then presented.

ACKNOWLEDGMENTS

We thank the Natural Environment Research Council for support of R.M.A. and theNational Science Foundation for support of R.M.M. (Grant No DEB77 01563).

REFERENCES

Ackert,J. E., Graham,G. L., Nolf, L. O. &Porter,D. A. (1931). Quantitative tudies on the administra-tion of variable numbersof nematode eggs (Ascaridia ineata) to chickens. Transactions f theAmericanMicroscopicalSociety,50, 206-14.

Askew,R. R. (1971).ParasiticInsects.Heinemann,LondonAnderson,R. M. (1974a).Populationdynamicsof the cestodeCaryophyllaeusaticeps Pallas,1781) n the

bream AbramisbramaL.). Journalof AnimalEcology,43, 305-21.Anderson,R. M. (1974b).An analysisof the influenceof host morphometriceatureson the population

dynamicsof Diplozoon aradoxumNordmann,1832).Journal f AnimalEcology,43, 873-87.

Anderson,R. M. (1976a). Seasonal variationin the populationdynamicsof Caryophyllaeusaticeps.Parasitology,72, 281-305.Anderson,R. M. (1976b). Some simple models of the population dynamics of eucaryotic parasites

MathematicalModels in Medicine Ed. by J. Berger,W. Buhler,R. Repgesand P. Tautu).LectureNotes in Biomathematics.1, 16-57. Springer-Verlag, erlin.

Anderson,R. M. (1976c). Dynamic aspectsof parasitepopulation Ecology. EcologicalAspectsof Para-sitology(Ed. by C. R. Kennedy)pp. 431-62 North-HollandPublishingCompany,Amsterdam.

Anderson,R. M. & MichelJ. F. (1977).Densitydependent urvival npopulationsof Osteragiaostertagi.International ournalof Parasitology,7, 321-9.

Anderson,R. M., Whitfield,P. J. &Mills,C. A. (1977).An experimentaltudyof thepopulationdynam-ics of anectoparasitic igenean,Transversotremapatialense(Soparker):hecercarialand adultstages.Journalof AnimalEcology,46, 555-80.

Beddington, . R., Hassell,M. P. & Lawton,J. H. (1976).The componentsof arthropodpredation.II.The predator ate of increase.Journalof AnimalEcology,45, 165-86.

Boray,J. C. (1969).Experimental ascioliasis nAustralia.AdvancesnParasitology, , 95-210.Boswell,M. T. &Patil, G. P. (1971).Chancemechanismsgenerating he negativebinomialdistribution.RandomCountsn ModelsandStructuresEd.byG. P. Patil)pp.21-38. PennsylvaniaStateUniversityPress,London.

The second category of population processes, which influence the regulatory role of

parasites, concerns the statistical distribution of parasite numbers per host. When therate of parasite induced host mortalities is linearly dependent on the number of parasitesharboured, we have demonstrated that parasites may regulate host population growth

providedthey exhibit an overdispersedpatternof distributionwithin the host population(model A). The accumulating information in the literature on the overdispersedcharac-ter of parasitedistributionsin naturalpopulations of hosts (videTable 2) lends supporttothe importance of this insight.The stabilizing influenceof parasite contagion is somewhat

analogous to the influenceof aggregateddistributions of insect parasitoidson the stabilityof host-parasitoidassociations (Hassell & May 1974). Parsitoid aggregationwas shown,in certain circumstances, to stabilize models which otherwise would have been quiteunstable.

The thirdcategory of processeswhich enhance the regulatory role played by a parasiteact within each individual parasite population (or subpopulation) harboured in each

member of the host population. Model C demonstrated that density dependent con-straints on parasite population growth, caused either by intraspecific competition for

finite resourcesor immunological attack by the host, can in the absence of overdispersionor non-linear functional responsesstabilise the dynamics of an interaction.

On the other hand, there are features of host-parasite interactions which tend to

destabilize the system. These are discussed in the following paper (May & Anderson

1978), and a general review of pertinent field and laboratory data on host parasiteassociations is then presented.

ACKNOWLEDGMENTS

We thank the Natural Environment Research Council for support of R.M.A. and theNational Science Foundation for support of R.M.M. (Grant No DEB77 01563).

REFERENCES

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Askew,R. R. (1971).ParasiticInsects.Heinemann,LondonAnderson,R. M. (1974a).Populationdynamicsof the cestodeCaryophyllaeusaticeps Pallas,1781) n the

bream AbramisbramaL.). Journalof AnimalEcology,43, 305-21.Anderson,R. M. (1974b).An analysisof the influenceof host morphometriceatureson the population

dynamicsof Diplozoon aradoxumNordmann,1832).Journal f AnimalEcology,43, 873-87.

Anderson,R. M. (1976a). Seasonal variationin the populationdynamicsof Caryophyllaeusaticeps.Parasitology,72, 281-305.Anderson,R. M. (1976b). Some simple models of the population dynamics of eucaryotic parasites

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Anderson,R. M. & MichelJ. F. (1977).Densitydependent urvival npopulationsof Osteragiaostertagi.International ournalof Parasitology,7, 321-9.

Anderson,R. M., Whitfield,P. J. &Mills,C. A. (1977).An experimentaltudyof thepopulationdynam-ics of anectoparasitic igenean,Transversotremapatialense(Soparker):hecercarialand adultstages.Journalof AnimalEcology,46, 555-80.

Beddington, . R., Hassell,M. P. & Lawton,J. H. (1976).The componentsof arthropodpredation.II.The predator ate of increase.Journalof AnimalEcology,45, 165-86.

Boray,J. C. (1969).Experimental ascioliasis nAustralia.AdvancesnParasitology, , 95-210.Boswell,M. T. &Patil, G. P. (1971).Chancemechanismsgenerating he negativebinomialdistribution.RandomCountsn ModelsandStructuresEd.byG. P. Patil)pp.21-38. PennsylvaniaStateUniversityPress,London.

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Rocha,U. F.

(1958).On the statistical

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Pennycuik,L. (1971).Frequencydistributionsof parasites n a populationof three-spined ticklebacks,

24141




GasterosteusculeatusL., withparticulareference o thenegativebinomialdistribution. arasitology,63, 389-406.

Randolph,S. E. (1975). Patternsof distributionsof the tick Ixodes triangulicepsBirula, on its host.Journalof AnimalEcology,44, 451-74.

Schmid,W.D. &Robinson,E. J. (1972).Thepatternof a host-parasite istribution. ournal fParasitology,57, 907-10.

Solomon,M. E. (1949).Thenaturalcontrolof animalpopulations.Journal f AnimalEcology,18, 1-35.Starr,M. P. (1975).A generalisedcheme or classifyingorganismicassociations.Symbiosis Ed.byP.H.

Jenningsand D. L. Lee),pp. 1-20. CambridgeUniversityPress,London.Stromberg,P. C. & Crites,J. L. (1974). Survival,activityand penetrationof the first stage larvaeof

Cammallanusxycephalus,WardandMagath,1916.Internationalournal fParasitology, , 417-21.Watkins,C. V. &Harvey,L. A. (1942).On the parasitesof silver oxes on somefarms n the South West.

Parasitology,34, 155-79.Williams, . C. (1963).Theinfestationsof the redfishSebastesmarinusL)andS. mentellaTravin Sclero-

parei; Scorpaenidae)by the copeopodsPeniculusclavatus Muller),Sphyrion umpi(Kroyer)andChondracanthopsisodosus Muller) n the easternNorth Atlantic.Parasitology, 3, 501-26.

Winfield,G. F. (1932). Quantitative tudies on the rat nematodeHeterakisspumosa,Schneider,1866.AmericanJournalof Hygiene, 17, 168-228.

(Received16 May 1977)

APPENDIX 1

This appendixlists the probability generatingfunctions (p.g.f.'s), n(Z), of the probability

p(i) of observing (i) parasites in a host, for three discrete probability distributions. In

addition, the expectations of i2 and i3 are outlined for each of the distributionsin terms of

a few parameters such as the mean number of parasites per host, m, and measures of

overdispersion k and underdispersionk'.

(a) Positive binomial

I(Z) = (q pZ)k' (1.1)

wherep is the probability of a successful infection occurring, and q = 1-p. The para-meter k' representsthe maximum number of parasitesa host can harbour (see Anderson

1974b).

The mean and variance of this distribution are

m E(i) = k'p (1.2)

var(i) = k'pq. (1.3)

The expectations of i2 and i3 are

E(i2) = (k'p)2+ k'pq. (1.4)

that is,

E(i2) = m2(k'-l)/k' + m. (1.4a)

E(i3) = (k'p)3(k'-l) (k'-2)/k'2 + 3(k'p)2(k'-l)/k' + k'p. (1.5)

(b)Poisson

nI(Z)= exp {m(Z-1)} (1.6)

where m is the sole parameterof the distributionand

GasterosteusculeatusL., withparticulareference o thenegativebinomialdistribution. arasitology,63, 389-406.

Randolph,S. E. (1975). Patternsof distributionsof the tick Ixodes triangulicepsBirula, on its host.Journalof AnimalEcology,44, 451-74.

Schmid,W.D. &Robinson,E. J. (1972).Thepatternof a host-parasite istribution. ournal fParasitology,57, 907-10.

Solomon,M. E. (1949).Thenaturalcontrolof animalpopulations.Journal f AnimalEcology,18, 1-35.Starr,M. P. (1975).A generalisedcheme or classifyingorganismicassociations.Symbiosis Ed.byP.H.

Jenningsand D. L. Lee),pp. 1-20. CambridgeUniversityPress,London.Stromberg,P. C. & Crites,J. L. (1974). Survival,activityand penetrationof the first stage larvaeof

Cammallanusxycephalus,WardandMagath,1916.Internationalournal fParasitology, , 417-21.Watkins,C. V. &Harvey,L. A. (1942).On the parasitesof silver oxes on somefarms n the South West.

Parasitology,34, 155-79.Williams, . C. (1963).Theinfestationsof the redfishSebastesmarinusL)andS. mentellaTravin Sclero-

parei; Scorpaenidae)by the copeopodsPeniculusclavatus Muller),Sphyrion umpi(Kroyer)andChondracanthopsisodosus Muller) n the easternNorth Atlantic.Parasitology, 3, 501-26.

Winfield,G. F. (1932). Quantitative tudies on the rat nematodeHeterakisspumosa,Schneider,1866.AmericanJournalof Hygiene, 17, 168-228.

(Received16 May 1977)

APPENDIX 1

This appendixlists the probability generatingfunctions (p.g.f.'s), n(Z), of the probability

p(i) of observing (i) parasites in a host, for three discrete probability distributions. In

addition, the expectations of i2 and i3 are outlined for each of the distributionsin terms of

a few parameters such as the mean number of parasites per host, m, and measures of

overdispersion k and underdispersionk'.

(a) Positive binomial

I(Z) = (q pZ)k' (1.1)

wherep is the probability of a successful infection occurring, and q = 1-p. The para-meter k' representsthe maximum number of parasitesa host can harbour (see Anderson

1974b).

The mean and variance of this distribution are

m E(i) = k'p (1.2)

var(i) = k'pq. (1.3)


E(i2) = (k'p)2+ k'pq. (1.4)

that is,

E(i2) = m2(k'-l)/k' + m. (1.4a)

E(i3) = (k'p)3(k'-l) (k'-2)/k'2 + 3(k'p)2(k'-l)/k' + k'p. (1.5)

(b)Poisson

nI(Z)= exp {m(Z-1)} (1.6)

where m is the sole parameterof the distributionand

24242



ROYM. ANDERSONNDROBERT . MAY 243

E(i) m (1.7)

var(i) = m. (1.8)

The expectationsof i2 and i3 are

E(i2) = m+m (1.9)

E(i3) = m3 + 3m2 + m. (1.10)

The Poisson distribution corresponds to the limit k'-+oo(positive binomial), or k-oo

(negative binomial).

(c) Negative binomial

n(z) = (q -p)-k (1.11)

wherep is as defined for the positive binomial, q = 1+p, and the parameterk is a meas-ure which varies inversely with the degree of aggregation of the parasites within the

host population.The mean and varianceare given by

m E(i)= kp (1.12)

var(i) =kpq. (1.13)


E(i2) = (kp)2(k+ l)/k + kp. (1.14)

That is,

E(i2) = m2(k+ l)/k + m (1.14a)

E(i3) = (kp)3(k+ 1)(k+ 2)/k2+ 3(kp)2(k+ l)/k + kp. (1.15)

Notice that (as follows from comparison of the defining functions (1.1) and (1.11)) the

negative binomial results can all be obtained from the positive binomial ones by the

simple transformation p- -p and k'-+-k, or equivalently, m-m and k'- k. In

particular,positive and negative binomial expressionsfor E(i2) are interrelatedby k' -

k.

APPENDIX 2

This appendix gives a global account of the stability properties of the Basic model

defined by eqns (7) and (9).The expressionsfor the equilibrium populations H* and P* are given by eqns (10) and

(11), and the criterion that this point be biologically sensible (H*>0) is expressed by

eqn (12).

For a fully nonlinear discussion of the stabilitypropertiesof this model, it is sufficient

to notice that the function

V(H, P) eaPIH (Ho + H) H - (b ++) p(b-a) > 0 (2.1)

is a Lyapunovpotential for this system (Minorsky 1962),and furthermorethat


E(i) m (1.7)

var(i) = m. (1.8)

The expectationsof i2 and i3 are

E(i2) = m+m (1.9)

E(i3) = m3 + 3m2 + m. (1.10)

The Poisson distribution corresponds to the limit k'-+oo(positive binomial), or k-oo

(negative binomial).

(c) Negative binomial

n(z) = (q -p)-k (1.11)

wherep is as defined for the positive binomial, q = 1+p, and the parameterk is a meas-ure which varies inversely with the degree of aggregation of the parasites within the

host population.The mean and varianceare given by

m E(i)= kp (1.12)

var(i) =kpq. (1.13)


E(i2) = (kp)2(k+ l)/k + kp. (1.14)

That is,

E(i2) = m2(k+ l)/k + m (1.14a)

E(i3) = (kp)3(k+ 1)(k+ 2)/k2+ 3(kp)2(k+ l)/k + kp. (1.15)

Notice that (as follows from comparison of the defining functions (1.1) and (1.11)) the

negative binomial results can all be obtained from the positive binomial ones by the

simple transformation p- -p and k'-+-k, or equivalently, m-m and k'- k. In

particular,positive and negative binomial expressionsfor E(i2) are interrelatedby k' -

k.

APPENDIX 2

This appendix gives a global account of the stability properties of the Basic model

defined by eqns (7) and (9).The expressionsfor the equilibrium populations H* and P* are given by eqns (10) and

(11), and the criterion that this point be biologically sensible (H*>0) is expressed by

eqn (12).

For a fully nonlinear discussion of the stabilitypropertiesof this model, it is sufficient

to notice that the function

V(H, P) eaPIH (Ho + H) H - (b ++) p(b-a) > 0 (2.1)

is a Lyapunovpotential for this system (Minorsky 1962),and furthermorethat

dV/dt = 0V/dt = 0 (2.2)2.2)




for all t. Thus V is a positive constant, determined by the initial values of H and P; the

population values H(t) and P(t) will endlessly cycle around some closed trajectory. In

other words, the Basic model correspondsto a neutrallystable or conservative dynamical

system, with the consequences discussed in the main text.Although the potential V(t) of eqn (2.1) was found by analytical trickery,ratherthan

being revealedin a vision, simply writingit down saves space while preservingrigour. Its

properties may be verified by differentiating to get dV/dt in terms of dH/dt and dP/dt,

whereuponuse of eqns (7) and (9) lead to the central result (2.2).

APPENDIX 3

This appendix outlines the analysis of the stability properties of the various host-parasitemodels presented in this paper; similar analysis pertains to those in the tollowing paper

(May & Anderson 1978).With the exception of model B (which involves E(i2) in the equation for dH/dt, and

E(i3) in that for dP/dt), and model F in the following paper (which involves time delays),the models underconsideration all have the mathematical form

dH/dt = c1 H-c2P (3.1)

dP/dt = P{)H/(Ho + H)-c3 -c P/H}. (3.2)

Here ci (i = 1,2,3,4,) are constants which depend upon the particularbiological assump-tions in a given model.

Specifically,using eqn (1.14a) to expressE(i2) in eqns (8) and (20) in terms of P/H and

k, for a negativebinomial distribution of parasiteswe may tabulateci for models A and Cas follows:

Model A Model C

c1 =a-b c1 =a-b (3.3)

c2 = a C2= a (3.4)

c3 =b++oa c3 = b+p+ (3.5)

c4 = {k+ l}/k c4 = {ca+}{k+1}/k. (3.6)

The corresponding expressions for a Poisson distribution of parasites are obtained by

putting k- oo, and for a positive binomial by putting k-- k'.

Fromeqns (3.1)

and(3.2),

theequilibriumpopulations

H* andP* are

P*IH* = cl/c2 (3.7)

H*= Ho(C3 + C C4/C2)/(/l-C3-C1 C4/C2). (3.8)

Thisprovides thefirst importantconstraint,namely thatfor a biologically sensible equilib-rium to be possible it is requiredthat

i > c3 + C /C2. (3.9)

A linearized stability analysis of this equilibrium is carried out along standard lines

(see, e.g. May 1975). Writing H(t) = H* +x(t) and P(t) = P* +y (t), and linearizing by

neglectingterms of orderx2, xy and y2, we getfrom

eqns (3.1)and

(3.2)dx/dt = c1x-c2y1 (3.10)

for all t. Thus V is a positive constant, determined by the initial values of H and P; the

population values H(t) and P(t) will endlessly cycle around some closed trajectory. In

other words, the Basic model correspondsto a neutrallystable or conservative dynamical

system, with the consequences discussed in the main text.Although the potential V(t) of eqn (2.1) was found by analytical trickery,ratherthan

being revealedin a vision, simply writingit down saves space while preservingrigour. Its

properties may be verified by differentiating to get dV/dt in terms of dH/dt and dP/dt,

whereuponuse of eqns (7) and (9) lead to the central result (2.2).

APPENDIX 3

This appendix outlines the analysis of the stability properties of the various host-parasitemodels presented in this paper; similar analysis pertains to those in the tollowing paper

(May & Anderson 1978).With the exception of model B (which involves E(i2) in the equation for dH/dt, and

E(i3) in that for dP/dt), and model F in the following paper (which involves time delays),the models underconsideration all have the mathematical form

dH/dt = c1 H-c2P (3.1)

dP/dt = P{)H/(Ho + H)-c3 -c P/H}. (3.2)

Here ci (i = 1,2,3,4,) are constants which depend upon the particularbiological assump-tions in a given model.

Specifically,using eqn (1.14a) to expressE(i2) in eqns (8) and (20) in terms of P/H and

k, for a negativebinomial distribution of parasiteswe may tabulateci for models A and Cas follows:

Model A Model C

c1 =a-b c1 =a-b (3.3)

c2 = a C2= a (3.4)

c3 =b++oa c3 = b+p+ (3.5)

c4 = {k+ l}/k c4 = {ca+}{k+1}/k. (3.6)

The corresponding expressions for a Poisson distribution of parasites are obtained by

putting k- oo, and for a positive binomial by putting k-- k'.

Fromeqns (3.1)

and(3.2),

theequilibriumpopulations

H* andP* are

P*IH* = cl/c2 (3.7)

H*= Ho(C3 + C C4/C2)/(/l-C3-C1 C4/C2). (3.8)

Thisprovides thefirst importantconstraint,namely thatfor a biologically sensible equilib-rium to be possible it is requiredthat

i > c3 + C /C2. (3.9)

A linearized stability analysis of this equilibrium is carried out along standard lines

(see, e.g. May 1975). Writing H(t) = H* +x(t) and P(t) = P* +y (t), and linearizing by

neglectingterms of orderx2, xy and y2, we getfrom

eqns (3.1)and

(3.2)dx/dt = c1x-c2y1 (3.10)

24444

dy/dt = c5x-(ClC4/c2)y.y/dt = c5x-(ClC4/c2)y. (3.11)3.11)




Herefor notationalconvenience, 5 is definedas

c5 = ( oo H*/(Ho + H*)2 + C1C4/c2)Cl/c2). (3.12)

The temporalbehaviourof x(t) and y(t) then goes as exp (At), where the stability-determining ampingrates(or eigenvalues)A aregivenfromeqns (3.10)and 3.11)bythe quadratic quation

A2+AA+B=0. (3.13)Here

A- (c/lc2)(c4-c2) (3.14)

B C2C5-cc4/c2- cAHoH*/(Ho + H*)2. (3.15)

The requirementfor neighbourhood stability is that the real part of both eigen values A

be negative; the necessaryand sufficientcondition for this to be so is given by the Routh-

Hurwitz criterion A>0 and B>0. From eqn (3.15) we see that B is always positive.Thereforethe equilibriumpoint will be locally stable if A > 0, that is if

c4-c2 > 0. (3.16)

Thisprovides the second importantconstraint,determining the stability characterof the

equilibriumpoint (if it exists).Since the above models are modifications of the globally neutrally stable Basic model,

it is reasonable to assume that if they are locally stable, they are globally stable; and

converselythat if they are locally unstable, they are globally unstable. A rigorous prooffollows from the observation that, for the general pair of eqns (3.1) and (3.2), the func-

tionV = {exp(c2P/H)} (Ho+ H)P- c~H{(c 3 -4/C2

is a global Lyapunov function. That is, V>0 (providedthe initial host and parasite pop-ulations are positive), and

dV/dt = (c2-C4) (c - C2P/H)2 C2 1

as maybe verifiedby differentiating V(H,P), andusing eqns(3.1)and(3.2)to expressdH/dt

and dP/dt. Local stability requires c4>c2 (see eqn (3.16)), whence dV/dt<O, connoting

global stability. Conversely local instability requires c4<c2, whence dV/dt > 0, connoting

global instability. The structurally unstable razor's edge of c4 = c2 gives both local(eqn (3.16)) and global (dV/dt = 0) neutralstability, as discussed more fully in Appendix2.

In brief, local and global stability propertiesmarch together in these models.We now proceed to indicate how the basic criteria (3.9) and (3.16) may be applied to

the various models to derive the dynamical properties discussed in the main text, andillustrated in Figs 5, 6, 7, 9 and 10.

Model A

For a negative binomial distribution of parasites, the expressions (3.3) through (3.6)may be substituted in eqn (3.9) to arrive at the criterion for an equilibriumsolution to be

possible. This expression is given and discussed as eqn (16) in the main text. The stabilitycriterion (3.16) here reads.


Herefor notationalconvenience, 5 is definedas

c5 = ( oo H*/(Ho + H*)2 + C1C4/c2)Cl/c2). (3.12)

The temporalbehaviourof x(t) and y(t) then goes as exp (At), where the stability-determining ampingrates(or eigenvalues)A aregivenfromeqns (3.10)and 3.11)bythe quadratic quation

A2+AA+B=0. (3.13)Here

A- (c/lc2)(c4-c2) (3.14)

B C2C5-cc4/c2- cAHoH*/(Ho + H*)2. (3.15)

The requirementfor neighbourhood stability is that the real part of both eigen values A

be negative; the necessaryand sufficientcondition for this to be so is given by the Routh-

Hurwitz criterion A>0 and B>0. From eqn (3.15) we see that B is always positive.Thereforethe equilibriumpoint will be locally stable if A > 0, that is if

c4-c2 > 0. (3.16)

Thisprovides the second importantconstraint,determining the stability characterof the

equilibriumpoint (if it exists).Since the above models are modifications of the globally neutrally stable Basic model,

it is reasonable to assume that if they are locally stable, they are globally stable; and

converselythat if they are locally unstable, they are globally unstable. A rigorous prooffollows from the observation that, for the general pair of eqns (3.1) and (3.2), the func-

tionV = {exp(c2P/H)} (Ho+ H)P- c~H{(c 3 -4/C2

is a global Lyapunov function. That is, V>0 (providedthe initial host and parasite pop-ulations are positive), and

dV/dt = (c2-C4) (c - C2P/H)2 C2 1

as maybe verifiedby differentiating V(H,P), andusing eqns(3.1)and(3.2)to expressdH/dt

and dP/dt. Local stability requires c4>c2 (see eqn (3.16)), whence dV/dt<O, connoting

global stability. Conversely local instability requires c4<c2, whence dV/dt > 0, connoting

global instability. The structurally unstable razor's edge of c4 = c2 gives both local(eqn (3.16)) and global (dV/dt = 0) neutralstability, as discussed more fully in Appendix2.

In brief, local and global stability propertiesmarch together in these models.We now proceed to indicate how the basic criteria (3.9) and (3.16) may be applied to

the various models to derive the dynamical properties discussed in the main text, andillustrated in Figs 5, 6, 7, 9 and 10.

Model A

For a negative binomial distribution of parasites, the expressions (3.3) through (3.6)may be substituted in eqn (3.9) to arrive at the criterion for an equilibriumsolution to be

possible. This expression is given and discussed as eqn (16) in the main text. The stabilitycriterion (3.16) here reads.

o/k > 0/k > 0 (3.17)3.17)




which is always satisfied; the equilibrium point is necessarily stable.A Poisson distribution of parasites(as in the Basic model) is obtained as the limit koo

-, thus producing eqn (12) as the criterionfor an equilibrium point to exist. In this case,

c4 = c2 = o, so that A=0 and the eigen value equation (3.13) becomes A2+B = 0, or

A=1-B. (3.18)

That is, both eigen values are purely imaginary, leading to neutral stability in the linear-ised analysis; this, of course, is the local version of the global result established in

Appendix 3.For a positive binomial distribution,the stability criterion (3.16) becomes

-o/k' >0 (3.19)

which cannot be satisfied. This model is ineluctablyunstable.

Model C

For an overdispersed distribution, we substitute the appropriate expressions (3.3)through(3.6) into eqn (3.9) to arriveat eqn (21) for the criterion for an equilibriumpointto be possible. The stabilitycriterion (3.16) is here

{( + (k + 1)}/k > 0 (3.20)

which is always satisfied.For a Poisson distribution, putting k-+oo in eqn (21) produces the criterion for an

equilibriumto exist. Similarly, putting k-oo in eqn (3.20) gives the stability conditiont>0, which always holds.For an underdisperseddistributions, we put k -k' in eqn (21) to get the condition

for equilibriumto exist; see Fig. 10. By likewise putting k - k' in the stability criterion

(3.20), we see that such anequilibrium will be stable if

#k'-y -ac > 0. (3.21)

Stability is helped by high natural parasite mortality compared with parasite inducedhost mortality (i.e. by t large compared with x), and hindered by high underdispersion(i.e. by small k').

Model BIn this model, the equation for dH/dt involves E(i2) and thence a term in P2/H, and

the equation for dP/dt involves E(i3) and thence a term in P3/H2. Consequently these

host-parasite equations are not exactly of the form of eqns (3-1) and (3-2), and requireseparate treatment.

For a negative binomial distribution of parasites, use of the expressions (1.14) and

(1.15) for E(i3) and E(i2) in eqns (18) and (19), respectively,leads to

dH/dt = (a- b)H- P- a{(k + 1)/k}P21H, (3.22)

dP/dt = P{IHI(Ho + H) - ( + b+ o) - 3c{k(+ 1)/k}PH - a{(k + 1)(k+ 2)/k2}P2/H2}.

(3.23)As ever, the corresponding equations for a Poisson distribution are obtained as the

limit k- oo, and for a positive binomial by the substitution k-- k'.

which is always satisfied; the equilibrium point is necessarily stable.A Poisson distribution of parasites(as in the Basic model) is obtained as the limit koo

-, thus producing eqn (12) as the criterionfor an equilibrium point to exist. In this case,

c4 = c2 = o, so that A=0 and the eigen value equation (3.13) becomes A2+B = 0, or

A=1-B. (3.18)

That is, both eigen values are purely imaginary, leading to neutral stability in the linear-ised analysis; this, of course, is the local version of the global result established in

Appendix 3.For a positive binomial distribution,the stability criterion (3.16) becomes

-o/k' >0 (3.19)

which cannot be satisfied. This model is ineluctablyunstable.

Model C

For an overdispersed distribution, we substitute the appropriate expressions (3.3)through(3.6) into eqn (3.9) to arriveat eqn (21) for the criterion for an equilibriumpointto be possible. The stabilitycriterion (3.16) is here

{( + (k + 1)}/k > 0 (3.20)

which is always satisfied.For a Poisson distribution, putting k-+oo in eqn (21) produces the criterion for an

equilibriumto exist. Similarly, putting k-oo in eqn (3.20) gives the stability conditiont>0, which always holds.For an underdisperseddistributions, we put k -k' in eqn (21) to get the condition

for equilibriumto exist; see Fig. 10. By likewise putting k - k' in the stability criterion

(3.20), we see that such anequilibrium will be stable if

#k'-y -ac > 0. (3.21)

Stability is helped by high natural parasite mortality compared with parasite inducedhost mortality (i.e. by t large compared with x), and hindered by high underdispersion(i.e. by small k').

Model BIn this model, the equation for dH/dt involves E(i2) and thence a term in P2/H, and

the equation for dP/dt involves E(i3) and thence a term in P3/H2. Consequently these

host-parasite equations are not exactly of the form of eqns (3-1) and (3-2), and requireseparate treatment.

For a negative binomial distribution of parasites, use of the expressions (1.14) and

(1.15) for E(i3) and E(i2) in eqns (18) and (19), respectively,leads to

dH/dt = (a- b)H- P- a{(k + 1)/k}P21H, (3.22)

dP/dt = P{IHI(Ho + H) - ( + b+ o) - 3c{k(+ 1)/k}PH - a{(k + 1)(k+ 2)/k2}P2/H2}.

(3.23)As ever, the corresponding equations for a Poisson distribution are obtained as the

limit k- oo, and for a positive binomial by the substitution k-- k'.

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The equilibrium mean parasite burden per host, m* = P*/H*, follows from eqn

(3.22):

m* = {k/(k + 1)}[{1 + 4(a - b)(k+ l)/(kc)} - 1]. (3.24)

It then follows from eqn (3.23), after some tedious algebraic manipulations, that the

equilibriumhost population is

H* = HoH/( - ), (3.25)

with Cdefinedfor convenience as

C= , + a + o+ 2(a - b)lk + {(2k+1)/k}xm* (3.26)

A biologically sensible equilibriumis possible only if

A> . (3.27)

This relation among the parametersis illustrated in Fig. 7; remember that for a Poissondistribution the above equations are to be read with k-+ oo, and for a positive binomialwith k- - k'.

The stability of such an equilibrium point is determined by a linearized stabilityanalysis of the kind described above (and, e.g. in May 1975). Linearizing eqns (3.22) and

(3.23) about the equilibrium point defined by eqns (3.24) and (3.25), we eventuallyobtain the quadratic eigenvalue equation (3.13), where the coefficientsA and B are

A = om*{k(2k + 3) +4(k + )m*}/k2, (3.28)

B = oc{1 2m*(k + 1)/k}{HoP*/(Ho + H*)2}. (3.29)

From eqn (3.24) it follows that B >0 for all positive m*.Clearly A>0 for overdispersedor Poisson (k-+oo) distributions, so their equilibrium

points are stable. For underdispersed distributions (positive binomial, k-+-k'), the

stability condition A>0 leads to the requirementthat k'>1.5 and

4m* < k'(2k'- 3)/(k'- 1). (3.30)

Thus underdisperseddistributions can have stable equilibria,provided the mean parasiteburden is relatively low (small m*) and the underdispersion is not pronounced (largek'): Fig. 7 bears out these remarks.

REFERENCES TO APPENDICES

Anderson,R. M. (1974).An analysisof the influenceof host morphometriceatureson the populationdynariusof Diplozoonparadoxom Nordmann, 1832).Journalof AnimalEcology,43, 873-87.

May,R. M. (1975).StabilityandComplexityn ModelEcosystems 2ndedn).PrincetonUniversityPress,Princeton.

May,R. M. &Anderson,R. M. (1978).Regulationandstabilityof host-parasitenteractions: I. Destabil-izing processes.Journalof AnimalEcology,47, 249-67.

Minorsky,N. (1962). NonlinearOscillations.Van Nostrand, Princeton.

The equilibrium mean parasite burden per host, m* = P*/H*, follows from eqn

(3.22):

m* = {k/(k + 1)}[{1 + 4(a - b)(k+ l)/(kc)} - 1]. (3.24)

It then follows from eqn (3.23), after some tedious algebraic manipulations, that the

equilibriumhost population is

H* = HoH/( - ), (3.25)

with Cdefinedfor convenience as

C= , + a + o+ 2(a - b)lk + {(2k+1)/k}xm* (3.26)

A biologically sensible equilibriumis possible only if

A> . (3.27)

This relation among the parametersis illustrated in Fig. 7; remember that for a Poissondistribution the above equations are to be read with k-+ oo, and for a positive binomialwith k- - k'.

The stability of such an equilibrium point is determined by a linearized stabilityanalysis of the kind described above (and, e.g. in May 1975). Linearizing eqns (3.22) and

(3.23) about the equilibrium point defined by eqns (3.24) and (3.25), we eventuallyobtain the quadratic eigenvalue equation (3.13), where the coefficientsA and B are

A = om*{k(2k + 3) +4(k + )m*}/k2, (3.28)

B = oc{1 2m*(k + 1)/k}{HoP*/(Ho + H*)2}. (3.29)

From eqn (3.24) it follows that B >0 for all positive m*.Clearly A>0 for overdispersedor Poisson (k-+oo) distributions, so their equilibrium

points are stable. For underdispersed distributions (positive binomial, k-+-k'), the

stability condition A>0 leads to the requirementthat k'>1.5 and

4m* < k'(2k'- 3)/(k'- 1). (3.30)

Thus underdisperseddistributions can have stable equilibria,provided the mean parasiteburden is relatively low (small m*) and the underdispersion is not pronounced (largek'): Fig. 7 bears out these remarks.

REFERENCES TO APPENDICES

Anderson,R. M. (1974).An analysisof the influenceof host morphometriceatureson the populationdynariusof Diplozoonparadoxom Nordmann, 1832).Journalof AnimalEcology,43, 873-87.

May,R. M. (1975).StabilityandComplexityn ModelEcosystems 2ndedn).PrincetonUniversityPress,Princeton.

May,R. M. &Anderson,R. M. (1978).Regulationandstabilityof host-parasitenteractions: I. Destabil-izing processes.Journalof AnimalEcology,47, 249-67.

Minorsky,N. (1962). NonlinearOscillations.Van Nostrand, Princeton.

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Regulation and Stability of Host-Parasite Population Interactions