Predator Prey Systems with an Alternative Food Source

1. Predator Prey Systems with an Alternative Food Source 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.

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

4. Geometric Representation

5. Predator Prey Systems with an Alternative Food Source dR/dt = aR – fR(R,A)F dF/dt =-cF + CRfR(R,A)F + CAfA(R,A)F New Terms: • fR(R,A)=the consumption rate of the prey, with respect to the prey and the alternative food source • fA(R,A)=the consumption rate of the alternative food source with respect to the prey and the alternative food source • CR=nutritional value of the prey • CA=nutritional value of the alternative food source

6. Functional Response Equations fR(R,A) = R/(1+TRR +pTAA) fA(R,A) =pA/(1+TRR +pTAA) • A=the amount of the alternative food source • R=the amount of rabbits • TR=the time it takes for a predator to kill a prey • TA=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 • TR=TA • p=1 • A=b (the amount of the alternative food source is equal to some parameter b) fR(R,A) = R/(1+b+R) fA(R,A) = b/(1+b+R) Final Equation: dR/dt = aR –(b/(b+1+R))F dF/dt =-cF + CR(R/(b+1+R))F + CA(b/(b+1+R))F

8. Example 1 • dR/dt = 3R –(1/(1+1+R))F • dF/dt =-1F + 1(R/(1+1+R))F +1(1/(1+1+R))F • a=3 • c=1 • CR=CA=1 • b=1

9. Example 2 dR/dt = 3R –(1/(1+1+R))FdF/dt =-1F + 1(R/(1+1+R))F + 5(1(1+1+R))F • a=3 • c=1 • CR=1 • CA=5 • b=1

10. Example 3 • dR/dt = aR –(b/(b+1+R))F • dF/dt =-cF + CR(R/(b+1+R))F + CA(b/(b+1+R))F • a=3 • c=1 • CR=3 • CA=1 • b=1

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