Bio 481 - Problem Set 8 - Predator-Prey Dynamics

General Note: a^b stands for a raised to the b power.

1. Start with the classic Lotka-Volterra predator/prey system. Use some reasonable values for the parameters. Graph the isoclines for this system. Next, using different starting points, project the system in P/V phase-space.
   
   The general equations for the system are: dV/dt=r*V-b*V*P and dP/dt=c*V*P-d*P. I've used r=3.0, b=0.1, c=0.01, and d=.1. You probably want c less than b since most predators have fewer than 1 offspring for each prey that they eat (c=b*conversion rate). Below you can see the isoclines (V=d/c, P=r/b) and three different projections.
   
   
   
   
2. Analyze problem 1 analytically. Find equations for each of the isoclines and solve for the equilbrium point in the system.
   
   The equilibrium occurs when the isoclines cross, so just set them equal to one another. V=d/c and P=r/b are the two values. For my choice of constants, the equilibrium point is (10.0, 30.0) in V,P space.
   
   
3. Adapt your system from problem 1 to include density dependence in the prey. Use the basic logistic form to do this. Graph the isoclines and again do some projections in phase-space.
   
   The general equations for the system are: dV/dt=r*V*(1-V/K)-b*V*P and dP/dt=c*V*P-d*P. I've used K=100, r=3.0, b=0.1, c=0.01, and d=.1. Below you can see the isoclines (V=d/c, P=r*(1-V/K)/b) and three different projections. The equilibrium occurs when the isoclines cross, so just set them equal to one another. V=d/c and P=r*(1-d/(K*c))/b are the two values. For my choice of constants, the equilibrium point is (10.0, 27.0) in V,P space.
   
   
   
   
4. What happens if you add density dependence to the predator equation in the previous problem?
   
   Now the equations become dV/dt=r*V*(1-V/K)-b*V*P and dP/dt=c*V*P*(1-P/L)-d*P. I've used K=100, L=100?, r=3.0, b=0.1, c=0.01, and d=.1. Below you can see the isoclines (V=d/(c*(1-P/L)), P=r*(1-V/K)/b) and a single projection. The isoclines will always cross each other at exactly one point. Depending on the value of L, the system may have oscillations or not. In any case, the system has one point attractor.
   
   
   
   
5. Use a type II functional response in a classic Lotka-Volterra system. How have the isoclines changed? What is the dynamics now in phase-space? Can you see these dynamics easily in a time series plot?
   
   Now the equations become dV/dt=r*V-b*V*P/(e+V) and dP/dt=c*V*P/(e+V)-d*P. I've used e=1.0, r=3.0, b=0.5, c=0.2, and d=.1. Below you can see the isoclines (V=d*e/(c-d), P=r*(e+V)/b) and one projection. As you can see, the system is unstable and leads to exponential growth.
   
   
   
   
6. Combine the type II functional response and density dependence in the prey. Again, look at the isoclines and dynamics.
   
   The only difference now is that the isocline has become P=r*(e+V)*(1-V/K)/b. This is enough to stabilize the system to persistent oscillations or a stable point attractor (note it will depend where the two curves cross).
   
   
   
   
7. Pick some other characteristic that you might think is important in biological systems (eg. type III functional response, refuge effect, seasonality in prey growth, another trophic level, etc.) and analyze this new system the way you have these previous ones. Make sure you understand fully how the parameters affect the system.