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

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 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.

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

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

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 Explicit Model Implicit Model

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 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 Recall f(x) denotes the trade-off

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

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

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

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

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.