The evolution of host resistance to microparasites

1. The evolution of host resistance to microparasites Roger G. Bowers1, Andrew Hoyle1 & Michael Boots2 1Department of Mathematical Sciences, The University of Liverpool 2 Department of Animal and Plant Sciences, The University of Sheffield Andrew White – Heriot Watt University,

2. Contents • Fitness and invasion boundaries • Reciprocal invasion plots • Trade-offs • Acceleratingly/deceleratingly costly • Trade-off and invasion plots (TIPs) • The singular TIP • The geometric characterisation of the singularity • TIPs and simulations compared • Host-pathogen TIPs – what dynamics, trade-off types and trade-off ‘strengths’ lead to • attractors, repellors and branching points (speciation) • Discussion/Summary

3. Fitness and invasion boundaries • Two strains labelled x = (x1, x2) and y = (y1,y2) • Example A host species for a microparasitex1’ y1 rate of transmission of microparasitex2, y2 intrinsic growth rate of host • Trade-off region y1< x1 & y2< x2 A gain in resistance to a parasite is bought at a cost in intrinsic growth rate • Fitness – per capita growth rate of strainsx(y) fitness of invader y with resident xs y(x) fitness of invader x with resident y

4. Fitness and invasion boundaries • y2 = f1(x,y1) sx(y)=0.

5. Fitness and invasion boundaries • y2 = f2(x,y1) sy(x)=0.

6. Reciprocal Invasion Plots • y2 = f1(x,y1) sx(y)=0. • y2 = f2(x,y1) sy(x)=0.

7. Trade-offs and cost types • y2 = f(y1),x2 = f(x1)The trade-off represents the cost (of resistance) and defines the set of feasible strains • f′ is of fixed sign

8. Trade-offs: accelerating/deceleratingly costly • Each improvement comes at an ever… • increasing cost – acceleratingly costly trade-off.

9. Trade-offs: accelerating/deceleratingly costly • Each improvement comes at an ever… • decreasing cost – deceleratingly costly trade-off.

10. Trade-offs: accelerating/deceleratingly costly • Each improvement comes at an ever… • increasing cost – acceleratingly costly trade-off. • decreasing cost – deceleratingly costly trade-off.

11. Trade-off and invasion plots (TIPs)

12. The singular TIP

13. The singular TIP

14. The singular TIP • The singular behaviour depends on the relative curvatures of the trade-off and invasion boundaries

15. The geometric characterisation of the singularity Attractor – curvature of f is less than that of f1.

16. The geometric characterisation of the singularity Repellor – curvature of f is greater than the mean curvature.

17. The geometric characterisation of the singularity Branching points – curvature of f is between that of f1 and the mean curvature (and sx(y)>0 and sy(x)>0 there).

18. The geometric characterisation of the singularity

19. TIPs and simulations compared

20. An applications of TIPs • Study a range of host-microparasite models • What type and ‘strength’ of trade-off/cost is associated with • Attractors • Repellors • Branching points (these particularly since they may indicate speciation)

21. Host-parasite system – without recovery. Trade-off – r vs. β(avoidance) • Attractors … acceleratingly costly trade-offs • Branching points…weakly deceleratingly costly trade-offs • Repellors… strongly deceleratingly costly trade-offs

22. Calculation of fitness Dynamics of rare mutant Fitness Residence times Growth rates So … where the equilibrium resident densities are known in terms of the resident parameters

23. Host-parasite system – with recovery Trade-off – 1) r vs. β(avoidance) • Attractors … acceleratingly costly trade-offs • Branching points…weakly deceleratingly costly trade-offs • Repellors… strongly deceleratingly costly trade-offs

24. Host-parasite system – with recovery Trade-off – 2) r vs. γ(recovery) • Attractors … acceleratingly and weakly deceleratinglycostly trade-offs • Branching points…moderately deceleratingly costly trade-offs • Repellors… strongly deceleratingly costly trade-offs

25. Host-parasite system – with recovery Trade-off - 3) r vs. α(tolerance) • Attractors … strongly acceleratingly costly trade-offs • Garden of Eden repellors…moderately acceleratingly trade-offs • Repellors… weakly acceleratingly and all deceleratingly costly trade-offs

26. Host-parasite system – with reproduction from infecteds Previously Now • Branching points can occur with weakly acceleratingly costly trade-offs.

27. Discussion/Summary • Trade-off and invasion plots allow a convenient geometric characterisation of evolutionary behaviour • Host-pathogen systems with a single reproductive class • Branching points with intermediately deceleratingly costly trade-offs • Host-pathogen systems with a multiple reproductive classes • Branching points can occur with weakly acceleratingly costly trade-offs.