Overview of: Introducing rabbit predators
Project summary
Evaluate the strategy of introducing foxes to Foxless Island as a way to control the rabbit population by analyzing a mathematical model of the rabbit-fox system, the Lotka-Volterra equations of equation (5).
Requirements for project report
Your project report should be no longer than three pages. It should contain the following sections.
The uncoupled system: Solve the uncoupled rabbit equation (1) and the uncoupled fox equation (2) for any positive values of the parameters $\alpha$ and $\gamma$ and any initial conditions $r(0)=r_0$ and $f(0)=f_0$. What would happen to the rabbit population if there were no foxes? What would happen to the fox population if there were no rabbits?
The model: Write down the general form of the predator-prey (or Lotka-Volterra) equations, i.e., equations (3) and (4) in terms of the four parameters $\alpha$, $\beta$, $\gamma$, and $\delta$ (don't give them numerical values yet). Explain the meaning of the parameters and the four terms from the system of equations. Explain why might one expect this model to exhibit better control of the rabbit population than a model where you continually harvest a fixed fraction of rabbits. The latter model might look like $$\diff{r}{t} = \alpha r - \beta r.$$ (Hint: how does the fact the the fox population size responds to fluctuations in the rabbit population size relieve some of the problems of the fixed fraction harvesting model.)
Model solution: Use the applet to explore the solution of the model with the given parameters, i.e., equation (5), starting with the initial conditions of 3000 rabbits and 70 foxes. Describe the behavior of the system as time evolves, and, using the model equation, explain why this occurs. Evaluate whether or not this type of behavior makes sense for the evolution of the rabbit and fox populations. Explore whether or not the behavior changes with different initial conditions and slightly different values of the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$. Determine if the oscillations are robust to small changes in parameters and initial conditions. In your presentation, include some plots of $r$ and $p$ versus time $t$ as well as plots of the solution curve $(r(t),p(t))$ through the phase plane. Since the phase plane curve does not show time, indicate where in the phase plane the state $(r(t),p(t))$ moves faster or slower.
Equilibria: Show that the system (5) has two equilibria: the trivial equilibrium $(0,0)$ with no rabbits or foxes and another non-zero equilibrium. Calculate the values of the equilibria analytically from the equations. From simulating the solution with nearby initial conditions using the applet, determine it the equilibria seem to be stable or unstable. Justify your conclusions.
Model stupidity: Describe the model prediction for the case if you introduced only two foxes rather than 70, i.e., with initial conditions $(r(0),p(0))=(3000,2)$. Explain what doesn't make sense about the prediction of the model for these initial conditions.
Conclusion: Explain what can you conclude from your model analysis. Can you make a convincing case that introducing foxes to Foxless Island will help control the rabbit population? Even if it doesn't show that one will get good results in all cases, can you use the model to develop a strategy that will lead to a more or less stable population of around a thousand rabbits? In particular, if we estimate that Foxless currently has around 3000 rabbits, can you get to a more or less stable population of a thousand rabbits just by introducing an appropriate number of foxes? Can you improve your result if you help the foxes out by initially harvesting some rabbits yourself? Does the model help you address the two objections that you envisioned people making (at the end of the Background section)? Why or why not?