Predation, Mutualism & Competition.

1. Predation, Mutualism & Competition.

2. Predation • the interaction between species in which one species, the predator, attacks and feeds upon the other, the prey • 2 species: one strain of one species (predator) and n strains of other species (prey) • cost/benefit relationship

3. Predation Model where

4. Equilibrium Solutions • origin • each prey strain alone at its carrying capacity • predator alone • n monomorphic states • dimorphic states • no others possible

5. Attempted invasion by prey strain j on a resident prey strain i with the predator Eigenvalue needing investigation is Algebraic manipulation, trade-off r=f(c) and setting gives where

7. Attempted invasion by prey strain k on a resident prey strains i and j with the predator Eigenvalue needing investigation is Algebraic manipulation, trade-off r=f(c) and setting gives

8. Concave down trade-off: Invasion can only occur when the invading strain is between the two residents • Concave up trade-off: Invasion can only occur when the invading strain is at extreme ends of the strain distribution • Thus, with each successful invasion: the strains will either diverge (concave up) or converge (concave down)

9. Mutualism • the interaction between two species where both species benefit • 2 species: one strain of one mutualist and n strains of the other • benefit/benefit relationship • mutualism can be obligatory or facultative (non-obligatory)

10. Facultative Model Equilibrium points are: the origin, each single strain of species X alone, species Y alone, n monomorphic states, and dimorphic states Only monomorphic & dimorphic states are stable

11. Attempted invasion by strain Xj on a resident strain Xi with Y Eigenvalue needing investigation is Algebraic manipulation, trade-off r=f(c) and setting gives where

12. Attempted invasion by strain Xk on a resident strains Xi and Xj with Y Eigenvalue needing investigation is Algebraic manipulation, trade-off r=f(c) and setting gives Which is identical to predation case!

13. Obligatory Mutualism Model Equilibrium points are: the origin, each single strain of species X alone, species Y alone, n monomorphic states, and dimorphic states Only origin is stable! Monomorphic feasibility and stability conditions contradict each other. Therefore dimorphism cannot be stable either.

14. Competition • the act of striving against each other to ensure success • 2 species: one strain of species Y and n strains of species X • cost/cost relationship • “competitive exclusion principle” states that if two species are too similar they cannot co-exist

15. Competition Model Each strain of species X alone, species Y alone, monomorphic states and dimorphic states can all be stable when feasible.

16. Attempted invasion by strain Xj on a resident strain Xi with Y Eigenvalue needing investigation is Algebraic manipulation, trade-off r=f(c) and setting gives where Which is identical to predation case!

17. Attempted invasion by strain Xk on a resident strains Xi and Xj with Y Eigenvalue needing investigation is Algebraic manipulation, trade-off r=f(c) and setting gives Which is identical to predation & mutualism!

18. Conclusions/Discussion • Invasions on monomorphic states can occur for predation, competition and facultative mutualism but not for obligatory mutualism • All invasions on dimorphic states have identical results