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

2. 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.

3. 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

4. Geometric Representation

5. 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

6. 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

7. 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

8. 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

9.  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

10. 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

11. 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