Predation, Mutualism

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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 rf(c) and
  setting gives
• where

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(No Transcript)
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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 rf(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 Xj on a resident
strain Xi with Y

• Eigenvalue needing investigation is
• Algebraic manipulation, trade-off rf(c) and
  setting gives
• where

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Attempted invasion by strain Xk on a resident
strains Xi and Xj with Y

• Eigenvalue needing investigation is
• Algebraic manipulation, trade-off rf(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 Xj on a resident
strain Xi with Y

• Eigenvalue needing investigation is
• Algebraic manipulation, trade-off rf(c) and
  setting gives
• where
• Which is identical to predation case!

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Attempted invasion by strain Xk on a resident
strains Xi and Xj with Y

• Eigenvalue needing investigation is
• Algebraic manipulation, trade-off rf(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