Adaptive Dynamics, Indirectly Transmitted Microparasites and the Evolution of Host Resistance.

1. Adaptive Dynamics, Indirectly Transmitted Microparasites and the Evolution of Host Resistance. 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 • Adaptive Dynamics • Fitness • Evolutionary Outcomes • Trade-off Function • Pairwise Invadability Plots (PIPs) • Summary and Discussion

3. Adaptive Dynamics • Looks at long term effects of small mutations on a system. • Can be applied to various ecological settings. • Gives information about the evolution of the system. • Shows whether or not a mutant’s invasion of an initially monomorphic population is successful. • Distinguishes various evolutionary outcomes associated with attractors, repellors or branching points.

4. 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.

5. 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.

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

7. 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.

8. Models Explicit Model Implicit Model

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

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

11. Results • Explicit Model • ESS • CS • Implicit Model • ESS • CS

12. What are PIPs? • These represent the spread of mutants in a given population. • Indicate the sign of sx(y) for all possible values of x and y. • Along main diagonal sx(y) is zero.

13. + above and – below main diagonal indicates positive fitness gradient. - above and + below main diagonal indicates negative fitness gradient. Contains another line where sx(y)=0 and intersection of this with main diagonal corresponds to singular strategy. What are PIPs?

14. PIPs for Explicit Model

15. PIPs for Explicit Model • ESS and CS • Attractor Acceleratingly costly trade-off, a = 10

16. Neither CS nor ESS Repellor PIPs for Explicit Model Deceleratingly costly trade-off, a = -0.9

17. PIPs for Implicit Model Acceleratingly costly trade-off, a = 10 • ESS and CS • Attractor

18. CS not ESS Branching Point Neither CS nor ESS Repellor PIPs for Implicit Model Deceleratingly costly trade-off, a = -0.9

19. Summary Explicit Model • Determined evolutionary outcomes • Algebraically • From PIPs • Using Simulations • Attractor and repellor Implicit Model • Determined evolutionary outcomes • Algebraically • From PIPs • Using Simulations • Attractor, repellor and branching point

20. 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.