Trade-off & invasion plots, accelerating/decelerating costs and evolutionary branching points.

1. Trade-off & invasion plots, accelerating/decelerating costs and evolutionary branching points. By Andy Hoyle & Roger Bowers. (In collaboration with Andy White & Mike Boots.)

2. Outline of Talk. • Adaptive dynamics & TIPs: • Evolution in the adaptive dynamics world, • Possible evolutionary outcomes, • Trade-off and invasion plots, • Accelerating/decelerating costs. • Examples of interactions: • Single species, • Competition, • Predator-prey, • Host-parasite.

3. The evolutionary cycle in adaptive dynamics. • Resident Population (x) existing at equilibrium.

4. The evolutionary cycle in adaptive dynamics. • Resident Population (x) existing at equilibrium. • Mutation in a few individuals ( y=x±ε).

5. The evolutionary cycle in adaptive dynamics. • Resident Population (x) existing at equilibrium. • Mutation in a few individuals ( y=x±ε). • Fitness of y given by sx(y), if sx(y)<0 y will die out.

6. The evolutionary cycle in adaptive dynamics. • Resident Population (x) existing at equilibrium. • Mutation in a few individuals ( y=x±ε). • Fitness of y given by sx(y), if sx(y)<0 y will die out. if sx(y)>0y may invade x. • y spreads becoming the new resident.

7. Co-existence. • When sx(y)>0 AND sy(x)>0…

8. Evolutionary outcomes. Attractor

9. Evolutionary outcomes. Attractor Repellor

10. Evolutionary outcomes. Attractor Repellor Branching point

11. Where a TIP exists. • Trade-off f,y1 vs. y2 (defines feasible strains).

12. Where a TIP exists. • Trade-off f,y1 vs. y2 (defines feasible strains). • Fixed strain x on f.

13. Where a TIP exists. • Trade-off f,y1 vs. y2 (defines feasible strains) • Fixed strain x on f. • Axes of the TIP (strain y varies).

14. The invasion boundaries. • y2 = f1(x,y1) sx(y)=0.

15. The invasion boundaries. • y2 = f2(x,y1) sy(x)=0.

16. The invasion boundaries. • y2 = f1(x,y1) sx(y)=0. • y2 = f2(x,y1) sy(x)=0.

17. The singular TIP.

18. The singular TIP.

19. The singular TIP.

20. The singular TIP. Attractor – curvature of f is less than that of f1.

21. The singular TIP. Repellor – curvature of f is greater than the mean curvature.

22. The singular TIP. If sx(y)>0and sy(x)>0, then branching points occur if curvature of f is between that of f1 and the mean curvature.

23. Accelerating/decelerating costs. • Each improvement comes at an ever…

24. Accelerating/decelerating costs. • Each improvement comes at an ever… • increasing cost – acceleratingly costly trade-off.

25. Accelerating/decelerating costs. • Each improvement comes at an ever… • decreasing cost – deceleratingly costly trade-off.

26. Accelerating/decelerating costs. • Each improvement comes at an ever… • increasing cost – acceleratingly costly trade-off. • decreasing cost – deceleratingly costly trade-off.

27. Applications of TIPs. • Study a range of biological models. • Primarily to investigate potential branching points. • Type, and magnitude, of costs necessary.

28. Single species – single stage.

29. Single species – single stage. Fitness: sx(y)= -Asy(x)  f1 = f2. No possibility of branching points.

30. Single species - Maturation.

31. Single species - Maturation. Carrying capacity tied to births  q’= q’’=0 sx(y)= -Asy(x)  f1 = f2  No branching points.

32. Single Species - Maturation. Carrying capacity tied to births  q’= q’’=0 sx(y)= -Asy(x)  f1 = f2  No branching points. Carrying capacity tied to deaths  q=0  No branching points.

33. Competition.

34. Competition. Competition relation: czx=g(cxz). Trade-off: rvs. c.

35. Competition. Competition relation: czx=g(cxz). Trade-off: rvs. c. Branching points iff g’(cxz)<0, with (gentle) deceleratingly costly trade-offs. eg. red/grey squirrels czx=1/cxz

36. Predator-prey.

37. Predator-prey. Branching points with (gentle) deceleratingly costly trade-offs.

38. Host-parasite – without recovery. Trade-off – r vs. β

39. Host-parasite – without recovery. Trade-off – r vs. β Branching points with (gentle) deceleratingly costly trade-offs.

40. Host-parasite – with recovery. Trade-offs 1) r vs. β 2)r vs. γ 3)r vs. α

41. Host-parasite – with recovery. 1) r vs. β Branching points with (gentle) deceleratingly costly trade-offs.

42. Host-parasite – with recovery. 2) r vs. γ Branching points with (moderately) deceleratingly costly trade-offs. Attractors with (gentle) deceleratingly costly trade-offs.

43. Host-parasite – with recovery. 3) r vs. α No possibility of branching points.

44. Conclusion. • Single Species – • No branching points. • Two Species + Single Class – • Branching points with (gentle) deceleratingly costly trade-offs. • Two Species + Two Classes – • Branching points and attractors with deceleratingly costly trade-offs.