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

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.

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

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

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

PIPs for Explicit Model

ESS and CS –Attractor Acceleratingly costly trade-off, a = 10

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

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

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

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

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.