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 X j on a resident strain X i with Y Eigenvalue needing investigation is Algebraic manipulation, trade-off r=f(c) and setting gives where

12. Attempted invasion by strain X k on a resident strains X i and X j 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 X j on a resident strain X i 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 X k on a resident strains X i and X j 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