Revisiting Host-Parasite Interaction Models: Coexistence, Extinction and the Red Queen

7/29/2019 Revisiting Host-Parasite Interaction Models: Coexistence, Extinction and the Red Queen

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Why Revisit?

Models areuseful forrepresentingtheoretical,

experimental ornatural

situations

But there is noperfect model

and,mathematical

functions shouldbe used

cautiously whenrepresenting

biological

phenomena

I

ininan


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The Math Model

, , , , =

, 1,2,

=

, , , , ,


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The Math Model

, , , , =

, 1,2,

=

, , , , ,

Population of host


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The Math Model

, , , , =

, 1,2,

=

, , , , ,

Population of parasite


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The Math Model

, , , , =

, 1,2,

=

, , , , ,

Fitness functions


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The Math Model

, , , , =

, 1,2,

=

, , , , ,

Effective growth rate

Exponential: , , , , Logistic: , , , , 1


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The Math Model

, , , , =

, 1,2,

=

, , , , ,

Parasitism


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The Math ModelFunctional Response

Representation

(let )Brief Description

Holling type I monotonic one-variable curveHolling type II

monotonic one-variable curve; h

Holling type III

monotonic one-variable curve;

Beddington-

DeAngelis (BD)

monotonic two-variable curve; involve

parasites to the activities of their c

Simplified Monod-

Haldane (MH)

non-monotonic one-variable


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1

1

1

Holling type I

Holling type III

Holling type II

MH


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The Math Model

, , , , =

, 1,2,

=

, , , , ,

Death rate


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The Math Model

, , , , =

, 1,2,

=

, , , , ,

reproduction rate due to utilihosts (numerical respon


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The Math Model

, , , , =

, 1,2,

=

, , , , ,

limiting term affecting the growthrate of parasite (e.g., inter-parasite

competition) , , , ,


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Classical Lotka-Volterra Single hostsingle parasite interaction models can be found in various books

Edelstein-Keshet 2005; de Vries et al. 2006].

;

has two equilibrium points 0,0 and , , which are unstable.

The solution (given any nonnegative initial condition and parameter values) is bou

always approaches a stable limit cycle.

This model is structurally unstable but it can be used as a groundwork for a morerepresentation.


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Generalized Lotka-Volterra

The situation where all host and parasite populations are extinct is u

It is possible for some but not all host and parasite populations to

are sure that one host and one parasite survive it is possi

generalized system reduces to the classical system)

The superior populations are more progressive; and the weak

inferior or possibly verge to vanish


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0 0 0

1, 0.5,


Risky phenomenon: popusurviving population declithe edge of extinction


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0 0.8, 0 0.5, 0 0.5 0.1, 0.5, 0.1



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Generalized Lotka-VolterraCaveat: small perturbation in the birthrate of a host species or in the death rate ofa parasite may cause divergence from thebehavior of the original system (and worstmay induce extinction in populations )

0.5

0.52,

0.48, 0.49, 0.5


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Perturbation of a parameter value can result to a bifurcation from

simple oscillatory behavior to a peculiar oscillation.

In the classical LV model, chaos is not possible, but in a generalize(with three or more species), chaos may arise.

Occurrence of such strange results constrains the use of generaliz

Volterra model in fitting experimental data.


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init P3=.69

init P3=.7

0.55 to 0.56


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Modified Rosenzweig-MacArthur Mod

=

=

, 1,2, ,

=

= , 1,2, ,

Some of the characteristics of the Lotka-Volterra system are inheriteRosenzweig-MacArthur (RM) model

The host and parasite species with the greatest fitness respectively dother host and parasite populations


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Modified Rosenzweig-MacArthur Mod Large carrying capacity induce apparent oscillation in population si

1 10

100 1000


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In the absence of parasitism, host populations never propagates unbThe size of any host (as well as any parasite) population never excee

carrying capacity.

The RM model has (0,0,,0) as an equilibrium point, denoting a syall species are extinct. However, this point is unstable. We expect thhost species survives (unlike LV, all parasite population can vanish

Most solutions to the RM model approach equilibrium points.


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Modified Rosenzweig-MacArthur Mod As more state variables and parameters are added, the model becom

challenging to analyze.

In a Holling type I system involving two or more species with equavalues, infinitely many equilibrium points may exist. This means thwith very similar characteristics tend to have different long-term psizes when their initial population are not the same.

The figure shows

initial conditions . to


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Modified Rosenzweig-MacArthur Mod RM models with Holling type I functional response may result to p

oscillations, especially when carrying capacities are large. Some of

fluctuations in the trajectory of the ODE show sign of chaos-like beh

Chaotic dynamics promptpotential unpredictabilityand inadequacy of themodels in dealing withbiological data


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Similar to the RM model with Holling type I where carrying capacirelatively small, the solution to the RM model with Holling type II

type III, BD or MH generally has damped long-term apparent osciconverges to an equilibrium point.

Stable limit cycles usually occur when host and parasite interactionunbounded parasitism (unrestrained utilization efficiency; no functresponse satiation) coupled with immense carrying capacity (unbou

growth rates). Boundless growth potential and parasitism efficiencypopulations of the interacting species to reach extreme states. Howeunboundedness is rare in nature.

Holling Type I(small carrying capacity)

Holling Type II


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(small carrying capacity)

BD

Holling type III MH

RM d d t i i ti L tti d


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In situations where host and parasite populations share common environmental carry

(e.g., continuous deterministic version of the corresponding lattice model) then we can

following:

= = , 1,2, ,

= = , 1,2, ,.

Let and to represent population densities. By algebraic manipulatioparasite model becomes similar to the original model where . The solution tgenerally converges to an equilibrium point.

RM and deterministic Lattice mode

I t d i d i


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Unpredictability of population sizes may not only result from possi

dynamics but also from demographic and environmental un(except probably when one knows the underlying probability distrib

It is always a good practice to identify how much the solutions to

are affected by variability. A model that is so sensitive to random

useless for fitting empirical data and predicting biological phenomuncertainties in biological and ecological processes are prevalent.

Introducing random noise


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0 .5 =

0.5

=

0.5 =

0.5

=

An example of different Ito SDE sample paths (generatedusing Euler-Maruyama method) given demographicvariability

Randomness in parameter values caused by


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collective environmental variability.

We suppose that some parameters follow mean-reverting

Ornstein-Uhlenbeck process. For example, the random

parameter becomes a state variable with dynamics .

The SDE for the random parameter depictsenvironmental randomness influencing to vary aboutthe mean . As , becomes normally distributedwith mean and variance

.

Model with co evolutionary dynamics


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Consider cases where there are co-evolving species due to competiti

Define

- mean quantitative trait of the i-th host population specific fwith the -th parasite population

- mean quantitative trait of the -th parasite population sdealing with the -th host population

1 and 1 - slow constant genetical changes representevalues of the speeds of evolutionary adaptation

Model with co-evolutionary dynamics



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1,2, , ; 1,2, ,.

Model with co-evolutionary dynamicsEvolving parameters:

or

, for example,

1 ()

>

<



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1,2, , ; 1,2, ,.

Model with co-evolutionary dynamicsEvolving parameters:

Birth rates: and Progressive evolution has a trade-off since evoand an indefinitely advancing trait is unlikely. inferior trait to a stronger trait results to a declithe evolving population.

We use rational functions + a

Note: Polynomial, such as 1 = but will destroy the system. In contrast to the usuafunctions, the rational trade-off functions assure th are always in the interval [0,] and [0,], rpositive trait value.



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1,2, , ; 1,2, ,.


A positive selection gradient (value of the paderivative) drives the population to climb a

value, and a negative gradient drives the pohave a lower trait value.

For example, when decreases due to the inthen the value of should be reduced for tthe species.

On the other hand, when increases due toin , then the value of should be improv

For example:


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1 1

1 1

1 1

2

1 1

21

1

21 1 1

21 1

1

The species have to balance the benefitand cost of evolution. We can noticethis phenomenon by observing thedynamics of the trait value.

More pressure is given to the parasitespecies since parasite species canvanish if it cannot run after theevolving host, yet it cannot overrunthe host too much to avoid extinctionof host. Remember that parasitescannot live without a host.

All species do not evolve except the tra(bl k) i t it 1 ( d)


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(black) against parasite 1 (red)

Parasites need to evolve more to highcompared to the host speciesParasite trait

Host trait



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Oscillations in the model with co-evolution are probably caused by the inherent dynamics inmodel without co-evolution. We can see here that oscillations, that some interpret as Red Queprobably because of inherent interaction type and not solely by co-evolution.

To minimize misinterpretation of oscillating population sizes as result of the Red Queen dynarecommend observing the Red Queen dynamics beyond the inherent oscillations.


Without evolution With evolution



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Oscillations in the model with co-evolution are probably caused by the inherent dynamics inmodel without co-evolution. We can see here that oscillations, that some interpret as Red Queprobably because of inherent interaction type and not solely by co-evolution.

To minimize misinterpretation of oscillating population sizes as result of the Red Queen dynarecommend observing the Red Queen dynamics beyond the inherent oscillations.


Without evolution With evolution



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=0.1, =0.

Theres an ideal speed of evolution (between very fast and very slow) that will keep the pa

thousands of years.

Fast and very slow speeds of evolutionary adaptation of parasites can be detrimental to the pa



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=0.1, =0.0


thousands of years.




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=0.1, =0.


thousands of years.




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thousands of years.



=0.1, =0.

The Red Queen


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The parasite species in the model with co-evolution continuously change trait value to pursattaining (even just for a finite time) the population size equal to the equilibrium state of thmodel without co-evolution.

This phenomenon is probably an example of the running considered by Van Valen (1973Red Queen dynamics. Species keeps running to attain same state.

The Red Queen

Original system without

co-evolution

The Red Queen This phenomenon is dependent on the numberf i i l d d t l


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The parasite species in the model with co-evolution continuously change trait value to pursattaining (even just for a finite time) the population size equal to the equilibrium state of thmodel without co-evolution.

This phenomenon is probably an example of the running considered by Van Valen (1973Red Queen dynamics. Species keeps running to attain same state.

The Red Queen

Original system with

co-evolution

Evolving traits

of species involved and parameter values.

The Red Queen manifested in this exam

interaction with host and competition w

Concluding Remarks


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There are many possible model representations (not just those discussed) for the host

predator system, justified using either mechanistic or empirical explanations. Paramet

estimated using several techniques, such as by curve fitting or by machine learning, yet minim

To have biological relevance, we should always ensure that for any finite time, a uniq

model exists and that state variables should always be non-negative. Sensitivity analysis sho

because chaotic dynamics and uncertainties prompt potential unpredictability and inadequac

dealing with biological data.

Keen investigation is important in determining the robustness of the model and if empi

match the behavior of the theoretical model to be used in representing biological phenomena

the dynamics of co-evolving species when speeds of evolutionary adaptation are not anymore

Concluding Remarks


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Documents

Revisiting Host-Parasite Interaction Models: Coexistence, Extinction and the Red Queen