The adaptive dynamics of the evolution of host resistance to indirectly transmitted microparasites
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The Adaptive Dynamics of the Evolution of Host Resistance to Indirectly Transmitted Microparasites. By Angela Giafis & Roger Bowers. Introduction. Aim

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The adaptive dynamics of the evolution of host resistance to indirectly transmitted microparasites

The Adaptive Dynamics of the Evolution of Host Resistance to Indirectly Transmitted Microparasites.

By

Angela Giafis & Roger Bowers


Introduction

Introduction

Aim

Using an adaptive dynamics approach we investigate the evolutionary dynamics of host resistance to microparasitic infection transmitted via free stages.

Contents

  • Fitness

  • Evolutionary Outcomes

  • Trade-off Function

  • Results

  • Discussion


Fitness

Fitness

  • Resident individuals, x.

  • Mutant individuals, y.

  • If x>y then the resident individuals are less resistant to infection than the mutant individuals.

  • Mutant fitness function sx(y)is the growth rate of y in the environment where x is at its population dynamical attractor.

    • Point equilibrium…leading eigenvalue of appropriate Jacobian.


Fitness1

Fitness

  • sx(y)>0 mutant population may increase.

  • sx(y)<0 mutant population will decrease.

  • y wins if sx(y)>0 and sy(x)<0.

  • If sx(y)>0 and sy(x)>0 the two strategies can coexist.


Properties of x

Properties of x*

  • Local fitness gradient

  • Local fitness gradient=0 at evolutionary singular strategy, x*.

  • Evolutionary stable strategy (ESS)

  • Convergence stable (CS)


Evolutionary outcomes

Evolutionary Outcomes

  • An evolutionary attractor is both CS and ESS.

  • An evolutionary repellor is neither CS nor ESS.

  • An evolutionary branching point is CS but not ESS.


Models

Models

Explicit Model

Implicit Model


Trade off function

Trade-off function

For a>0 we have an

acceleratingly costly

trade-off.

For -1<a<0 we have a deceleratingly costly trade-off.


Fitness functions

Fitness Functions

  • From the Jacobian representing the point equilibrium of the resident strain alone with the pathogen we find:

  • Explicit Model

  • Implicit Model


Results

Explicit Model

ESS

CS

Implicit Model

ESS

CS

Results

Recall f(x) denotes the trade-off


Results for explicit model accelerating costly trade off a 10 f x 0

Graphically

Algebraically

ESS and CS

Attractor

Simulation

Results for Explicit Model(Accelerating costly trade-off, a = 10, f''(x*)<0)


Results for explicit model decelerating costly trade off a 0 9 f x 0

Graphically

Algebraically

Neither CS nor ESS

Repellor

Simulation

Results for Explicit Model(Decelerating costly trade-off, a = - 0.9, f''(x*)>0)


Results for implicit model accelerating costly trade off a 10 f x 0

Graphically

Algebraically

ESS and CS

Attractor

Simulation

Results for Implicit Model(Accelerating costly trade-off, a = 10, f''(x*)<0)


Results for implicit model decelerating costly trade off a 0 9 f x 0

Graphically

Algebraically, CS not ESS – branching point. Simulation

Algebraically, neither CS nor ESS – repellor. Simulation

Results for Implicit Model(Decelerating costly trade-off, a = - 0.9, f''(x*)>0)


Discussion

Discussion

  • For explicitmodel only attractor and repellor possible as CS and ESS conditions same.

  • For implicitmodel CS and ESS conditions differ. CS gives us weak curvature condition so branching point is possible.

  • Shown there is a relationship between type of evolutionary singularity and form of trade-off function.


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