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

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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