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 s x (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. s x (y)>0 mutant population may increase. s x (y)<0 mutant population will decrease. y wins if s x (y)>0 and s y (x)<0. If s x (y)>0 and s y (x)>0 the two strategies can coexist. Fitness

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

11. Results for Explicit Model (Accelerating costly trade-off, a = 10, f''(x*)<0) 1.Graphically2.Algebraically ESS and CS –Attractor 3.Simulation

12. Results for Explicit Model (Decelerating costly trade-off, a = - 0.9, f''(x*)>0) 1.Graphically2.Algebraically Neither CS nor ESS –Repellor 3.Simulation

13. Results for Implicit Model (Accelerating costly trade-off, a = 10, f''(x*)<0) 1.Graphically2.Algebraically ESS and CS –Attractor 3.Simulation

14. Results for Implicit Model (Decelerating costly trade-off, a = - 0.9, f''(x*)>0) 1.Graphically a)Algebraically, CS not ESS – branching point. Simulation b)Algebraically, neither CS nor ESS – repellor. Simulation

15. Discussion For explicit model only attractor and repellor possible as CS and ESS conditions same. For implicit model 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.