Evolutionary suicide of p rey M atsuda & Abrams’ model re v isited

1. Evolutionary suicide of preyMatsuda & Abrams’ model revisited Caterina Vitale & Eva Kisdi Dept. Mathematics and Statistics, University of Helsinki Matematiikan päivät, Joensuu, 4-5 Jan 2018

2. Evolutionary suicide is a process driven by natural selection whereby a population evolves to its own extinction Gyllenberg & Parvinen (2001): Catastrophic bifurcation is necessary • saddle-node • subcritical Hopf • fold of limit cycles • homoclinic/heteroclinic bifurcation Matsuda & Abrams (1994): first example

3. Matsuda & Abrams (1994):P constant Predator-prey model c = ‘foraging effort’ (time spent active), evolving trait Rosenzweig-MacArthur model, no suicide

4. Matsuda & Abrams (1994) The nontrivial fixed points are the roots of N increasing P 0 P N 0 0

5. Matsuda & Abrams (1994) Pfixed, c varied, with c

6. Matsuda & Abrams (1994) Mutant population dynamics suicide Adaptive dynamics ESS c

7. Focal prey and alternative prey in separate habitats recover and generalize Matsuda & Abrams robustness (the alternative prey does not evolve)

8. Focal prey decoupled (α1 = 0)

9. suicide ESS Focal prey decoupled (α1 = 0) • Proposition: evolutionary suicide can happen when • the foraging effortdecreases but not when it increases • Proof: using critical function analysis

10. Focal prey decoupled (α1 = 0) (2) The predator-alternative prey system has a limit cycle: the predator is a periodic driver the focal prey’s dynamics Adaptive dynamics

11. Focal prey decoupled (α1 = 0) Evolutionary suicide via a fold bifurcation of limit cycles Construction: critical function analysis + poise the alternative prey – predator system at its Hopf bifurcation + unfold towards the limit cycle

12. Full system (α1 > 0) whenever the focal prey is present, in all examples shown below

13. Full system (α1 > 0) Karl Hadeler whenever the focal prey is present, in all examples shown below

14. evolutionary suicide † Example 1 1 – no interior equilibrium, the focal prey goes extinct 2 – stable node + saddle; stable focus-node + saddle-focus 3 – stable + unstable focus-nodes, unstable limit cycle 4 – saddle-focus + unstable focus-node, the focal prey goes extinct 5 – stable + unstable limit cycles, saddle-focus + unstable focus-node

15. GH FLC H+ H– Generalized Hopf (Bautin)

16. FLC H+ GH GH H– FLC H+ H– Generalized Hopf (Bautin)

17. evolutionary suicide † Example 1 1 – no interior equilibrium, the focal prey goes extinct 2 – stable node + saddle; stable focus-node + saddle-focus 3 – stable + unstable focus-nodes, unstable limit cycle 4 – saddle-focus + unstable focus-node, the focal prey goes extinct 5 – stable + unstable limit cycles, saddle-focus + unstable focus-node

18. evolutionary suicide † ZH † sFN H– H– sFN SN– SN+ Zero-Hopf (fold-Hopf)

19. evolutionary suicide † Example 1 high α1: the predator overhunts the prey 1 – no interior equilibrium, the focal prey goes extinct 2 – stable node + saddle; stable focus-node + saddle-focus 3 – stable + unstable focus-nodes, unstable limit cycle 4 – saddle-focus + unstable focus-node, the focal prey goes extinct 5 – stable + unstable limit cycles, saddle-focus + unstable focus-node

20. Example 2 1 – no interior equilibrium, the focal prey goes extinct 2 – stable node + saddle; stable focus-node + saddle-focus 3 – stable + unstable focus-nodes, unstable limit cycle 4 – saddle-focus + unstable focus-node, the focal prey goes extinct

21. † Example 2 H+ T T „Indirect” evolutionary suicide: the focal prey – predator system becomes vulnerable to the invasion of the alternative prey, which drives the focal prey extinct 6 – no interior stable equilibrium, (N1,0,P) and (0,N2,P) are stable 7 – boundary limit cycle through Hopf of (N1,0,P) 8 – the alternative prey invades the boundary limit cycle, focal prey goes extinct 5 – same as 8, except interior unstable equilibrium gone through (N1,0,P)

22. Summary • Critical function analysis • evolutionary suicide via a fold bifurcation of limit cycles • evolutionary suicide via SN only when evolving lower foraging effort • Embedding the model in a 3-species ecosystem • much richer dynamics, evolutionary suicide via subcritical Hopf • “indirect” evolutionary suicide: via opening the system to the invasion of a new species • (also in an intraguild predation model of Hin & de Roos)