By Angela Giafis & Roger Bowers

1. The Adaptive Dynamics of the Evolution of Host Resistance to Indirectly Transmitted Microparasites. By Angela Giafis & Roger Bowers

2. 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

3. 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.

4. 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.

5. Properties of x* • Local fitness gradient • Local fitness gradient=0 at evolutionary singular strategy, x*. • Evolutionary stable strategy (ESS) • Convergence stable (CS)

6. 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.

7. Models Explicit Model Implicit Model

8. Trade-off function For a>0 we have an acceleratingly costly trade-off. For -1<a<0 we have a deceleratingly costly trade-off.

9. Fitness Functions • From the Jacobian representing the point equilibrium of the resident strain alone with the pathogen we find: • Explicit Model • Implicit Model

10. Explicit Model ESS CS Implicit Model ESS CS Results Recall f(x) denotes the trade-off

11. Graphically Algebraically ESS and CS Attractor Simulation Results for Explicit Model(Accelerating costly trade-off, a = 10, f''(x*)<0)

12. Graphically Algebraically Neither CS nor ESS Repellor Simulation Results for Explicit Model(Decelerating costly trade-off, a = - 0.9, f''(x*)>0)

13. Graphically Algebraically ESS and CS Attractor Simulation Results for Implicit Model(Accelerating costly trade-off, a = 10, f''(x*)<0)

14. 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)

15. 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.