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Predation, Mutualism & Competition.

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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

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Predation Model where

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

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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

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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

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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)

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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)

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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

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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

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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!

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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.

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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

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

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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!

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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!

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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

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