Phase Relations in the Ten-Year Cycle

Phase Relations in the Ten-Year CycleAuthor(s): M. G. BulmerSource: Journal of Animal Ecology, Vol. 44, No. 2 (Jun., 1975), pp. 609-621Published by: British Ecological SocietyStable URL: http://www.jstor.org/stable/3614 .

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609

PHASE RELATIONS IN THE TEN-YEAR CYCLE

BY M. G. BULMER

Department of Biomathematics, Pusey Street, Oxford

INTRODUCTION

A method of analysing data on the 10-year wildlife cycle was described in a previous paper (Bulmer 1974) and was used to obtain estimates of the amplitude and phase of the cycle in Canadian fur-bearing mammals. The results to a large extent confirmed the conclus- ions reached in a less quantitative way by other authors (Seton 1912; Hewitt 1921; Elton 1927; Butler 1953; Keith 1963). One of the more interesting results is that there are large differences in phase between different species (see Table 1). It was argued previously that the cycle in all other species is driven by the cycle in the snowshoe rabbit.* Several cyclic predators, such as the lynx, depend largely for food on the snowshoe rabbit. In other cases it is necessary to look for indirect links in the food chain. Thus it was suggested that the muskrat cycle is driven by the mink cycle, which might in turn be driven by cyclical fluctuations in the horned owl which eats both mink and rabbit.

Under this theory the cycle in any species, except the rabbit, is driven by another cyclic species which either eats it or is eaten by it. The phase and amplitude in the driven species will depend only on the phase and amplitude in the driving species and on the way in which the driven species responds to the oscillations in the driving species. The purpose of this paper is to investigate the theoretical phase difference between predator and prey, taking into account factors such as density-dependence and age at first breeding, and then to try to explain the observed differences in the light of the theoretical findings.

This theory does not, of course, explain the underlying cycle in the rabbit, which must be considered separately. The snowshoe rabbit cycle has been studied by Green & Evans (1940) and by Meslow & Keith (1968). Both authors agree that the most important factor in causing the cycle is survival of the young in the first few months of life; Meslow & Keith (1968) found that juvenile survival varied from 3o during a population decline to 24% during a period of increase. Meslow & Keith (1968) also found changes in litter size and adult survival, though they were of much smaller magnitude and were not found by Green & Evans (1940). The impact of lynx predation on a snowshoe rabbit population has been studied by Nellis, Wetmore & Keith (1972), who found that lynx removed about 6o of the rabbit population between December and March just before a rabbit low and 2% just after the low. Thus lynx predation may have some effect on the rabbit cycle, but is too small to be a major determinant of it.

The above facts suggest that the rabbit cycle is not caused by the rabbit-lynx interaction. If it is due to a prey-predator interaction, the predator must be one which eats mainly young rabbit. It was suggested by Lack (1954) that the rabbit cycle is due to periodic food shortage and is thus part of a plant-herbivore interaction. This is consistent with the dominant role of juvenile survival since food shortage would be expected to have a greater effect on juvenile survival, presumably affecting the young through the mother's

* Scientific names are given in the Appendix of Bulmer (1974).



milk, than on adult survival. This suggestion can only be regarded as tentative, however, since there is no information on vegetational changes. Fortunately the argument of the present paper is unaffected by the reason for the rabbit cycle; it need only be assumed that this cycle exists.

THEORETICAL RESULTS

A continuous time model

We shall first consider a model in which population size changes continuously; this is clearly unrealistic since it ignores the discreteness of the breeding season, but it will be useful as a standard for comparison with more realistic models. It is likely to provide a more realistic model for prey oscillations which are driven by predator oscillations than for the converse case since predator numbers act on the mortality of the prey throughout the year whereas prey numbers are more likely to act on the fertility of the predator during the breeding season.

If x and y denote the logarithms of the sizes of the driving and the driven species, the growth of the driven species can be represented by the equation

dy = r, (1)

where r is the net reproductive rate (the difference between the birth and death rates), which will in general be a function of both x and y and will therefore oscillate if x oscillates. In the steady state, when x = x and y = y, r must be zero. If we write rx (a Dr/lx) and ry (= arlay) for the partial derivatives of r evaluated at the steady state (x,y) and expand r in a Taylor series about this steady state, we find to a first approximation that

r = rX(x- ) + r(y-). (2)

Suppose now that the driving population oscillates about its steady state value in such a way that

x-x = p sin 27not, (3)

where ,f is the amplitude and co the frequency of the oscillations. (The amplitude is half the difference between peak and trough; the frequency is the reciprocal of the period between successive peaks or troughs.) Then

dy = rx, sin 27rot+ry(y-y). (4)

The solution of this differential equation is a sinusoidal oscillation about 9 with amplitude

1rx,p[(2c)2 + r- (5)

and with phase difference between x and y given by

- + arctan - /2r(o. (6) 4co \ 2nro

The + sign is to be taken if r> 0, the - sign if r< <0. In the first case y lags behind x; in the second, x lags behind y.

The term rx represents the effect of a change in the size of the driving species on the net

610 Ten-year cycles



reproductive rate of the driven species. It will be positive for prey driving predator and negative for predator driving prey. In both cases the predator cycle will lag behind the prey cycle. The term ry represents the effect of a change in the size of the driven species on its own reproductive rate. For prey driving predator it represents density-dependent factors in the predator, such as competition for breeding sites, which are due to the density of predators, together with any effect of the prey which is due to the number of prey per predator rather than to the absolute density of prey. Both factors will depress the net reproductive rate of the predator so that ry < 0. (The second factor will also make a contribution of the same magnitude, but of opposite sign, to rx.) For predator driving prey ry represents density dependent factors in the prey, due either to resource limitation or to competition for space, together with any change in the predation rate caused by a change in the density of prey. The first set of factors will depress the net reproductive rate of the prey and will make a negative contribution to ry. The second factor depends on the functional response of the predator; the contribution to ry will be positive or negative according as the response curve is concave or convex at the steady state position. The resultant value of ry may thus be either positive or negative, though it seems more likely to be negative.

4 -

3-

o ti l -I -0-5 0 05 1

Density dependence (r,)

FIG. 1. Phase difference between predator and prey in the 10-year cycle under a continuous time model.

The predator cycle always lags behind the prey cycle whether prey are driving predator or vice versa. The phase lag of the predator is a quarter of a period when there is no density-dependence (r. = 0). As density-dependence becomes more pronounced, that is to say as ry decreases, the phase lag decreases for prey driving predator (r,>O) but increases for predator driving prey (r.<O). Numerical values are shown in Fig. 1 when the period, l/w, is 9-63 years, which is the best estimate of the period for the 10-year cycle in Canada.

A discrete time model

We shall now consider a driven species which breeds once a year and which breeds first at age 1. If Y(t) is the total population size in year t, counted just before the breeding season, we may write

M. G. BULMER 611



Y(t + 1=RY(t) (7)

where R = f+ s,f being the fertility in year t, defined as the average number of offspring which survive until the following year, and s the annual adult survival rate. Bothf and s are assumed to be independent of age, but may depend on the sizes of the driving and the driven species. Taking logarithms, and writing in lower case the logarithm of the corres- ponding upper case quantity in (7),

y(t+ 1) = y(t)+r. (8)

Using (2), which remains valid, and supposing that the driving species oscillates about its steady state value as in (3), we find that

(y(t+ 1)-y)-(1 + ry)(y(t) -) = rf,B sin 2iot. (9)

The solution of this difference equation is a sinusoidal oscillation about y with amplitude

Irl/1[2(1 -cos 27r))(1 + ry)+ r]- (10) and with phase difference between x and y given by

1 +arctan( + ry - cos 2r2n . (11) 4c \ sin 2c )/2

(See Theorem 3 in the Appendix.) As before, the + sign is to be taken if rx>0 (prey driving predator), the - sign if rx<0 (predator driving prey); the predator cycle lags behind the prey cycle in both cases.

When there is no density-dependence (ry = 0) the phase lag of the predator is half a year more than a quarter of a period for prey driving predator and half a year less for predator driving prey. When ry = - 1, the phase lag is exactly 1 year in the first case and 1 year less than half a period in the second. The phase lag is exactly a quarter of a period in both cases when ry = cos 2nrc)-1. Numerical values for a period of 9-63 years are shown in Fig. 2.

So far we have considered only the oscillations in the logarithm of the total population

4-

a)- 2 -

0reyr

o I 1 -1 -0-5 0 0-5

Density dependence (r,)

FIG. 2. Phase difference between predator and prey in the 10-year cycle under a discrete time model when animals breed at age 1.

612 Ten-year cycles



size. If the driving species acts on the mortality of the driven species at all ages, as is likely to be the case for predator driving prey, the resulting oscillations will be approxi- mately synchronous in all age classes. (It is shown in the Appendix that this is exactly true if there is no density-dependence and if the driving species acts equally on the adult survival rate of the driven species and on its fertility because of the element of juvenile mortality in the latter.) For prey driving predator, however, changes in the abundance of prey are likely to have a much greater effect on the fertility than on the adult survival rate of the predator; among cyclic predators this has been documented for the coyote by Knowlton (1972) and Clark (1972) and for the lynx by Nellis et al. (1972). In this case it is shown in the Appendix that the cycle in 1-year olds is one year ahead of that in 2-year olds, which is one year ahead of that in 3-year olds, and so on. The same relationship holds for the cumulative totals (animals aged 1 year or more, 2 years or more, and so on). The cycle in 1-year olds is ahead of that in the total population by an amount depending

2-0-

1-5 - 0.5-

I'0 /- a) /

5 1-0- /

05- /

0 0-2 0-4 0-6 0-8 1-0 Annual survival rate (s)

FIG. 3. Phase advance of 1-year old animals compared with all animals in the 10-year cycle when the driving species acts on the fertility of the driven species.

on the adult survival rate (see (A10) in the Appendix) which is shown in Fig. 3 when the period is 9-63 years. For many cyclic predators the annual survival rate in adults, judged from the age distribution of trapped animals, is about 60%. (See Knowlton (1972) for the coyote, Eadie & Hamilton (1958) for the fisher, Rausch & Pearson (1972) for the wolverine.) One would, therefore, expect the cycle in 1-year olds to be about 1 year ahead of the cycle in the total population.

Age atfirst breeding Some animals do not breed until they are two or more years old. If Yi(t) denotes the

number of animals of the driven species of age i in year t, we may write 0oo

Y,(t+ 1) = fZ Yi(t) i=k (12)

Yi+ (t+1) = sY,(t), i>l,

where f is the fertility of animals aged k or more, defined as the average number of

M. G. BULMER 613



offspring per individual which survive until the following year, k is the age at first breeding and s is the annual survival rate, assumed to be the same in all animals aged 1 or more.

If the driving species acts on mortality, it is shown in the Appendix that, at least in the absence of density-dependence, the resulting oscillations are unaffected by the delay in the age at first breeding; the results of the previous subsection, therefore, remain un- changed. However, most cyclic species which do not breed at age 1 are predators in which the fertility is more likely to be affected than the survival rate by changes in the abundance of prey. The case of a prey oscillation driving a predator oscillation by causing changes only in its fertility is considered in the Appendix; it is assumed that density-dependence acts through competition between breeding individuals so that the fertility may also depend on the size of the breeding population. The phase relationships between the different age groups are the same as those described in the previous subsection, but the phase lag in the logarithm of the total population size depends on k, s and the amount of

s=0-9 s=0-75 s=075 s=025

s=0'5 2 - -- s=0.25

s=0 K25

- /

a-

K=2 K=3 -1-

-2 -2 I I II -1 -0-5 0 -I -0-5 0

Density dependence (F,)

FIG. 4. Effect of age of first breeding on the phase difference between predator and prey in the 10-year cycle when prey abundance acts on the fertility of the predator.

density-dependence. The latter is most conveniently measured by Fy, where F = fsk- is the average number of offspring which survive to breed and Fy is the derivative of F with respect to the logarithm of the size of the breeding population evaluated at the steady state. Numerical results (calculated from (A14) in the Appendix) are shown in Fig. 4. When there is no density-dependence (Fy = 0) the phase lag decreases slightly as the age at first breeding increases, but the effect is small unless the annual survival rate is low. The phase lag decreases as density-dependence becomes more pronounced, and may even become negative in extreme cases so that the predator oscillates ahead of the prey.

In some species, such as the coyote, the age at first breeding is variable, a proportion of the population breeding at age k and the remainder not breeding until age k+ 1. The proportion breeding at age k as well as the fertility of breeding animals may vary with the

Ten-year cycles 614



M. G. BULMER

> 2 6-

8 \.

2-4 -

L \\

2-2 -

202- 1.0 1-5 2.0 2-5 3'0

Average age at first breeding (years)

FIG. 5. Effect of variable age at first breeding on the phase difference between predator and prey in the 10-year cycle in the absence of density dependence when prey abundance acts on

the fertility of the predator.

food supply. (See Knowlton (1972) and Clark (1972) for data on the coyote.) This situation is considered in the Appendix under the simplifying assumption that there is no density-dependence. The basic equation governing the predator oscillations is given in (A17). The phase lag of the logarithm of the total predator population size depends only on the average age at first breeding and the survival rate. Numerical values are shown in Fig. 5. It will be seen that the phase lag decreases smoothly as the average age at first breeding increases.

OBSERVED PHASE RELATIONSHIPS

We shall now try to explain the observed phase relations in the 10-year cycle in the light of the preceeding theoretical results. Table 1 shows estimates of the phase lag, relative to the lynx, and other relevant information for nine cyclic mammals in Canada. The phase estimates for the red fox, marten, mink, muskrat and skunk are taken from Table 3

Table 1. Phase lag and related information about cyclic mammals in Canada

Phase lag Age at first Delayed Importance of Species (relative to lynx) breeding (female) implantation rabbit in diet

Coyote 1.1 1-2 No + Fisher 1-2 1 Yes + Red fox -01 1 No + Lynx 0.0 1 No + + Marten -2-0 2 Yes - Mink -1-7 1 No - Muskrat -4-0 1 No - Skunk 0-2 1 No - Wolverine 2.1 1-2 Yes +

615



in Bulmer (1974) and are based on nineteenth century data which are considered to be the most reliable since several species have ceased to be cyclic during the present century, probably due to habitat disturbance and hunting pressure. However, an advantage of twentieth century data is that a geographical breakdown by province is available in the annual reports on Fur Production published by the Canadian Bureau of Statistics since 1920. (These data were unfortunately not known to me when I wrote my previous paper.) It is, therefore, possible to estimate the phase lag, relative to the lynx, separately for each province in which the species is trapped in appreciable numbers and then to average these estimates. This method eliminates a bias which might be introduced into the estimate of the phase in species with a limited geographical range because of geographical variations in the phase of the cycle; it has been used for the coyote, fisher and wolverine. (For the coyote only the period 1920-44 was used since this species ceased to be cyclic after then. The wolf also appeared to be cyclic in the period 1920-44 but the provincial figures for this species are too erratic to allow a firm conclusion to be drawn.) Un- fortunately, data on the numbers of snowshoe rabbit are rather unsatisfactory, but the best evidence suggests that it cycles about two years, or perhaps slightly less, ahead of the lynx.

The snowshoe rabbit is a dominant item in the food of the coyote, fisher, red fox, lynx and wolverine, and there is no difficulty in attributing the cycles in these species directly to the rabbit. The red fox and the lynx cycle together about 2 years behind the rabbit in the absence of density-dependence, but evidence of quite strong density-dependence in both species has been presented elsewhere (Bulmer, in press); this seems to be the most likely explanation for the reduction in the phase lag.

The coyote cycles about 3 years behind the rabbit. The female breeds first at age 1 or 2, depending largely on the food supply (Clark 1972; Knowlton 1972). Litter size is also affected by food supply, and the annual adult survival rate is about 600. Taking the average age at first breeding as 1? years, we find from Fig. 5 that the phase lag in the absence of density-dependence should be 2-8 years, which is in good agreement with the observed lag. It is therefore suggested that density-dependent factors are considerably less important in the coyote than in the lynx or the red fox. It should be remembered that density-dependence in this context includes competition for space and regulation by the number of prey per predator but excludes regulation by the absolute density of prey.

The situation in the fisher and the wolverine is complicated by delayed implantation. The fisher mates for the first time at age 1 and the wolverine at age 1 or 2 but they do not give birth to the young until the year after mating. (For the fisher, see Eadie & Hamilton (1958) and Wright & Coulter (1967); for the wolverine, Rausch & Pearson (1972).) It has been suggested by Butler (1953) that this will cause the cycle to be a year later than it would otherwise have been, but this is not necessarily the case. If shortage of food acts at the time of mating, for example by affecting the number of ova shed or, in the case of the wolverine, the proportion of females which mate at age 1, then delayed implantation will cause the cycle to be one year later; in the absence of density-dependence it should, therefore, lag 3-9 and 3-8 years behind the rabbit in the fisher and the wolverine re- spectively. On the other hand, if shortage of food acts at the time of birth, for example by affecting the number of ova which implant or the mortality of the young, or if it acts on adult mortality, then the predator will behave like a species without delayed implantation which breeds for the first time at age 2 for the fisher or at age 2-3 for the wolverine. Thus if shortage of food acts on juvenile mortality, assuming that there is no density- dependence and that the adult survival rate is about 60% per year, the phase lag in the

616 Ten-year cycles



fisher should be about 2-6 years and in the wolverine about 2-5 years. If shortage of food acts equally on juvenile and adult mortality the phase lag in both species should be about 2-9 years in the absence of density-dependence since age at first breeding makes no difference in this case. The observed phase lag behind the rabbit is about 3 years for the fisher and 4 years for the wolverine. For the wolverine it seems likely that delayed implantation has delayed the cycle by a year, and it is suggested that shortage of food acts either on the number of ova shed or on the proportion of females which mate at age 1. (Alternatively, it might be suggested that the wolverine, which is a well-known scavenger, is more likely to be trapped when rabbit are scarce and that the cycle in this species is an artefact due to trapping factors.) For the fisher it is not necessary to postulate that delayed implantation has caused an appreciable delay in the cycle, and it is suggested that food shortage acts mainly on juvenile or adult mortality.

We turn now to the marten, mink and muskrat which cycle either in phase with or before the rabbit. It has been suggested previously (Bulmer 1974) that the cycles in these species are induced by cyclic predators. In the case of predator driving prey, the prey cycle should be ahead of the predator by either 2-4 or 1-9 years in the absence of density- dependence, according as a continuous time or a discrete time model is adopted. Neither a delay in age at first breeding nor delayed implantation, which are both found in the marten (Jonkel & Weckworth 1963), will affect the phase. Density-dependence reduces the phase difference between predator and prey. In this context both resource limitation and competition for space are density-dependent factors, but a change in the predation rate caused by a change in the density of prey will usually act as an inverse density- dependent factor.

It was suggested previously (Bulmer 1974) that the muskrat cycle is due to predation by mink. The muskrat cycle is 2-3 years ahead of the mink cycle, which is in good agreement with this theory. It was also tentatively suggested that the marten and mink cycles might be due to cycles in their predators, such as the horned owl and possibly lynx and fisher, which themselves depend on snowshoe rabbit. The phase lag in the marten and mink is consistent with the theory that their cycles are due to predators which are either in phase with or slightly later than the lynx.

The reason for the cyclic behaviour of the skunk in the nineteenth, though not in the twentieth, century is obscure. The phase of the cycle suggests that it may be due to a cyclic prey, and it was remarked previously (Bulmer 1974) that the skunk is a minor predator on the ruffed grouse, which cycles in phase with or slightly before the snowshoe rabbit, but this explanation must be regarded as very tentative since the skunk is pri- marily insectivorous.

Bird populations are not as well documented as mammals, but there is strong evidence of a 10-year cycle in Canada both in birds of prey, such as the horned owl and the goshawk, which prey on snowshoe rabbit, and in some gallinaceous birds, of which the ruffed grouse is the best known (Lack 1954; Keith 1963). The ruffed grouse is preyed on by several cyclic predators including lynx, horned owl and goshawk (Rusch & Keith 1971) and cycles either with or slightly before the rabbit (Keith 1963), that is to say 2-3 years before its predators. This is consistent with the theory that the grouse cycle is directly due to the cycle in its predators. It has been suggested that predator switching from rabbit to grouse when rabbit are scarce may be an important factor in the grouse cycle (Lack 1954), but this would cause the grouse cycle to be later than the rabbit cycle. It is, therefore, suggested that the number of predators is more important than predator switching in determining the amount of predation on grouse.

M. G. BULMER 617



Ten-year cycles

SUMMARY

It is suggested that the 10-year cycle in any cyclic species, except the snowshoe rabbit, is driven by another cyclic species which either eats it or is eaten by it. A model is set up to investigate the theoretical phase difference between predator and prey when one of them drives a cycle in the other, taking into account factors such as continuous or discrete generations, age at first breeding, density-dependence and delayed implantation. If there is no density-dependence in the driven species the predator should cycle about a quarter of a period behind the prey. Density-dependence decreases the phase difference for prey driving predator but increases it for predator driving prey. The observed phase relations between cyclic species in Canada are discussed in the light of the theoretical findings.

REFERENCES

Bulmer, M. G. (1974). A statistical analysis of the 10-year cycle in Canada. J. Anim. Ecol. 43, 701-18. Bulmer, M. G. (in press). The statistical analysis of density dependence. Biometrics. Butler, L. (1953). The nature of cycles in populations of Canadian mammals. Can. J. Zool. 31, 242-62. Clark, F. W. (1972). Influence of jackrabbit density on coyote population change. J. Wildl. Mgmt, 36,

343-56. Eadie, W. R. & Hamilton, W. J. (1958). Reproduction in the fisher in New York. N. Y. Fish Game J. 5,

77-83. Elton, C. (1927). Animal Ecology. Sidgwick & Jackson, London. Green, R. G. & Evans, C. A. (1940). Studies on a population cycle of snowshoe hares on the Lake Alex-

ander area. J. Wildl. Mgmt, 4, 220-38, 267-78, 347-58. Hewitt, C. G. (1921). The Conservation of the WildLife of Canada. Charles Scribner's Sons, New York. Jonkel, C. J. & Weckworth, R. P. (1963). Sexual maturity and implantation of blastocysts in the wild

pine marten. J. Wildl. Mgmt, 27, 93-8. Keith, L. B. (1963). Wildlife's Ten-year Cycle. University of Wisconsin Press, Madison, Wisconsin. Knowlton, F. F. (1972). Preliminary interpretations of coyote population mechanics with some manage-

ment implications. J. Wildl. Mgmt, 36, 369-82. Lack, D. (1954). The Natural Regulation of Animal Numbers. Clarendon Press, Oxford. Meslow, E. C. & Keith, L. B. (1968). Demographic parameters of a snowshoe hare population. J.

Wildl. Mgmt, 32, 812-34. Nellis, C. H., Wetmore, S. P. & Keith, L. B. (1972). Lynx-prey interactions in central Alberta. J. Wildl.

Mgmt, 36, 320-9. Rausch, R. A. & Pearson, A. M. (1972). Notes on the wolverine in Alaska and the Yukon territory. J.

Wildl. Mgmt, 36, 249-68. Rusch, D. H. & Keith, L. B. (1971). Seasonal and annual trends in numbers of Alberta ruffed grouse. J.

Wildl. Mgmt, 35, 803-22. Seton, E. T. (1912). The Arctic Prairies. Constable, London. Wright, P. L. & Coulter, M. W. (1967). Reproduction and growth in Maine fishers. J. Wildl. Mgmt, 31

70-87.

(Received 8 August 1974)

MATHEMATICAL APPENDIX

The purpose of this Appendix is to derive the phase relations between prey and predator oscillations under the discrete time model discussed in the text. For the sake of concrete- ness it will be assumed that a prey oscillation is driving a predator oscillation. Exactly the same arguments will go through, where appropriate, in the converse case if the words 'prey' and 'predator' are interchanged, provided it is remembered that the quantities F. and y are negative instead of positive. We first state three theorems which will be required; they all follow from the addition formulae of trigonometry.

618



Theorem 1 oa sin 2710t+02 cos 2rcot = o sin 2co(t- ))

where a= (a2+a4)b ) =- arctan (c2/ocl)/27ro, if oc > 0.

If otc <0, then signum (o2)/2co must be subtracted from ).

Theorem 2 Suppose that

x(t) = a sin 27rcot

y(t) = Yak x(t+k). Write

A = ak cos 2nrok B = ak sin 2r(ok.

Then y(t) = aA sin 2nrot + aB cos 2rcot = ap sin 2nco(t- 4))

where =(A2+B2)?

4 = -arctan (BIA)/2nro, if A >0.

If A <0, then signum (B)/20 must be subtracted from .

Theorem 3 Suppose that

ak y(t + k) = c sin 2rcot.

Define A and B as in Theorem 2. Then the steady state solution of this difference equation is

y(t) = oa sin 2nco(t-O) where

p=(A2 B2)-

4) = arctan (B/A)/2nco, if A>0.

If A <0, then signum (B)/2co must be added to (). We shall now consider the model

Y(t+ 1) = -fi Y(t) (Al)

Yi+ (t 1) = sYi(t), i>1. (A2)

It is assumed for the time being that the survival rate, s, is constant but that the age- specific fertility, fi, may depend on the numbers of both prey and predators. We shall write

00

Y*(t) = Z j(t) (A3) j=i

for the number of animals aged i or more (in particular Y1(t) is the total population size), and yi(t) and y*(t) for the natural logarithms of Y(t) and Y,*(t). If we write i andfi for the steady state values, then

M. G. BULMER 619



9P = Si-1 (A4) i = 1. (A5)

It will also be convenient to write A Yi(t), Afi and so on for departures from the steady state values.

Since s is constant it follows from (A2) that oscillations in yi+1(t) have the same amplitude but are one year later than oscillations in y#(t); this is a consequence of the assumption that the oscillations are driven by oscillations in fertility. Furthermore

Y*+(t+ 1) sY*(t) (A6)

so that the same conclusion holds for the relationship between y*+ (t) and y*(t). We can also find the relation between the oscillations in y1(t) (one-year olds) and y*(t) (the total population). For

Y1(t) = Y (t)- Y2(t) = Y (t)-sY (t-1). (A7)

Taking logarithms and expanding the right-hand side about its steady state value in a Taylor series, we find to a first approximation that

Ayl(t) = (1 -s)-l(/y (t)-sAy(t- 1)). (A8)

If Ay*(t) is oscillating sinusoidally with frequency Wo, it follows from Theorem 2 that Ay1(t) is also sinusoidal with the same frequency, with amplitude multiplied by

(2(1 -s cos 27rw)))/(1 - s) (A9)

and with phase in advance by

arctan (s sin 2io/(l -s cos 21ir))/2not. (A10)

We now suppose that animals first breed at age k and that fertility is independent of age for animals aged k or more, so that f = 0 (i<k), fi =f (i k). It is convenient to write F = fsk- 1 for the average number of offspring which survive to breed. We suppose that F depends both on x(t), the logarithm of the prey population size, and on yk(t), the logarithm of the number of breeding predators. We also suppose that x(t) oscillates sinusoidally about its steady state

x(t)-X = 3 sin 2C(ot. (A11)

To a first approximation we may write

F = Ft+ FxAx(t) + Fy,Ay(t), (A12)

where FX = aF/ax and Fy = aF/ay*, both derivatives being evaluated at the steady state. From (A5) it follows that F = (1-s). From (Al) and (A2) we find that

Ykt(t+k) = sYk(t+k- 1)+FYk*(t). (A13)

Taking logarithms and expanding the right-hand side about its steady state value, we find that

Ayk(t+k)-sAyk(t+k- )-(1-s+Fy)Ayk'(t) = Fxp sin 2rcot. (A14)

The solution of this difference equation is given by Theorem 3. The logarithm of the total population size oscillates with the same amplitude but is (k- 1) years earlier.

We now consider the case when a proportion of the population first breeds at age k

620 Ten-year cycles



and the remainder at age k + 1, so that f = 0 (i< k),fk = pf,fi = f(i> k). For simplicity we suppose that p andf depend only on x(t). We write

F1 = fsk- lp F2 = fsk(1-p) (A15) F = F+F2.

As before, we find that

Yk(t+k) = sYk(t+k-l)+F1 Yk(t)+F2 Y(t-1). (A16)

This leads to the equation

Ay*(t+ k)-sAy*(t + k- 1)-(s +p(1 -s)) [p(l -s)Ay (t)-s(1 -p)(l -s)Ay (t- 1)] = FX, sin 27cot (A17)

whose solution is given by Theorem 3. We finally consider the case when the scarcity or abundance of prey affects the mortality

rather than the fecundity of the predator. Since the fertility as defined here also includes juvenile mortality it will also be affected by the number of prey; it will be assumed that aln flax = aln s/ax = y, both derivatives being evaluated at the steady state. It will be assumed that there are no density-dependent effects so thatf and s are not functions of the predator population size. From (A6) we find that

Ay*+ 1(t + 1) = Ay*(t)+fly sin 2nrot. (A18)

If animals first breed at age k, the analogue of (A13) is k-1

Yk (t+k) = s(t+k-)Y(tk- 1 ) +f( Yk*(t f (t) Hs(t+j). (A19) j=1

Taking logarithms and expanding about the steady state we find that

Ay1k(t + k) - Ay (t + k- 1)- (1 - )Ay(t) k-2

= fy(sin 27(o(t+k-1)+(1 - s) Z sin 2nco(t+j)). (A20) j=o

This equation is equivalent to

Ayk(t + 1)-Ay (t) = fly sin 2Inrt. (A21)

This is the same as (9) in the text. Furthermore, it follows from (A18) that (A21) holds for any age class. Thus all age classes oscillate with the same amplitude and phase which are independent of k, the age at first breeding.

M. G. BULMER 621



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Phase Relations in the Ten-Year Cycle