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Predator Prey Systems with an Alternative Food Source By Karuna Batcha and Victoria Nicolov By Karuna Batcha and Victoria Nicolov

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Introduction Extension Of Lotka-Volterra Realistic Application of the model Question: Under what conditions is stability likely to occur because of food source switching? Results: Differed significantly from Lotka Volterra, and adjustments to parameters lead to various long term solutions. Extension Of Lotka-Volterra Realistic Application of the model Question: Under what conditions is stability likely to occur because of food source switching? Results: Differed significantly from Lotka Volterra, and adjustments to parameters lead to various long term solutions.

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Lotka-Volterra dR/dt = aR - bRF dF/dt = -cF + dRF R=rabbits F=foxes a=birth rate of rabbits c=death rate of foxes b=effect of the interaction between rabbits and foxes on rabbits d=benefit of the interaction between rabbits and foxes for foxes dR/dt = aR - bRF dF/dt = -cF + dRF R=rabbits F=foxes a=birth rate of rabbits c=death rate of foxes b=effect of the interaction between rabbits and foxes on rabbits d=benefit of the interaction between rabbits and foxes for foxes

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Geometric Representation

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Predator Prey Systems with an Alternative Food Source dR/dt = aR – f R (R,A)F dF/dt =-cF + C R f R (R,A)F + C A f A (R,A)F New Terms: f R (R,A)=the consumption rate of the prey, with respect to the prey and the alternative food source f A (R,A)=the consumption rate of the alternative food source with respect to the prey and the alternative food source C R =nutritional value of the prey C A =nutritional value of the alternative food source dR/dt = aR – f R (R,A)F dF/dt =-cF + C R f R (R,A)F + C A f A (R,A)F New Terms: f R (R,A)=the consumption rate of the prey, with respect to the prey and the alternative food source f A (R,A)=the consumption rate of the alternative food source with respect to the prey and the alternative food source C R =nutritional value of the prey C A =nutritional value of the alternative food source

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Functional Response Equations f R (R,A) = R/(1+T R R +pT A A) f A (R,A) =pA/(1+T R R +pT A A) A=the amount of the alternative food source R=the amount of rabbits T R =the time it takes for a predator to kill a prey T A =the time it takes to handle the alternative food source p=the probability the predator will consume the alternative food source upon encountering it f R (R,A) = R/(1+T R R +pT A A) f A (R,A) =pA/(1+T R R +pT A A) A=the amount of the alternative food source R=the amount of rabbits T R =the time it takes for a predator to kill a prey T A =the time it takes to handle the alternative food source p=the probability the predator will consume the alternative food source upon encountering it

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Simplifications T R =T A p=1 A=b (the amount of the alternative food source is equal to some parameter b) f R (R,A) = R/(1+b+R) f A (R,A) = b/(1+b+R) Final Equation: dR/dt = aR – (b/(b+1+R))F dF/dt =-cF + C R (R/(b+1+R))F + C A (b/(b+1+R))F T R =T A p=1 A=b (the amount of the alternative food source is equal to some parameter b) f R (R,A) = R/(1+b+R) f A (R,A) = b/(1+b+R) Final Equation: dR/dt = aR – (b/(b+1+R))F dF/dt =-cF + C R (R/(b+1+R))F + C A (b/(b+1+R))F

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Example 1 a=3 c=1 C R =C A =1 b=1 a=3 c=1 C R =C A =1 b=1 dR/dt = 3R – (1/(1+1+R))F dF/dt =-1F + 1(R/(1+1+R))F +1(1/(1+1+R))F

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a=3 c=1 C R =1 C A =5 b=1 a=3 c=1 C R =1 C A =5 b=1 Example 2 dR/dt = 3R – (1/(1+1+R))F dF/dt =-1F + 1(R/(1+1+R))F + 5(1(1+1+R))F

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Example 3 a=3 c=1 C R =3 C A =1 b=1 a=3 c=1 C R =3 C A =1 b=1 dR/dt = aR – (b/(b+1+R))F dF/dt =-cF + C R (R/(b+1+R))F + C A (b/(b+1+R))F

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Conclusion Nutritional values drastically changed the system No stable equilibrium (except (0,0)) Nullclines predicted the appearance of the graphs Nutritional values drastically changed the system No stable equilibrium (except (0,0)) Nullclines predicted the appearance of the graphs

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