Tutorial 8: Rosenzweig-MacArthur predator-prey model
Model
The Rosenzweig-MacArthur predator-prey model assumes that the density of prey, \(N\), grows logistically in the absence of predators, with per capita growth rate \(r\) and carrying capacity \(k\). Predators, with density \(P\), consume prey at a per prey per predator rate of \(a/(1+b N)\). The form of this per capita consumption rate implies that the rate of prey eaten per predator asymptotes to \(a/b\) as \(N\) gets large; essentially, we assume it takes some time for a predator to eat a prey, limiting the rate prey are consumed even when there are many of them. Finally, we assume each consumed prey is converted into a fraction \(c\) new predators (i.e., consuming \(1/c\) prey is required to make a new predator), and that predators die at per capita rate \(d\). Putting all this together in a continuous-time model we have
\(\frac{\mathrm{d}N}{\mathrm{d}t} = r N (1 - N / k) - \frac{a}{1 + b N} N P\)
\(\frac{\mathrm{d}P}{\mathrm{d}t} = c\frac{a}{1 + b N} N P - d P\).
Based on the model description we can assume all parameters are positive.
Problem
Solve for the three equilibria. When are they biologically valid?
Determine the conditions for local stability for each equilibria, assuming they are biologically valid. (Hint: use the Routh-Hurwtiz criteria.)
Show that the equilibrium with both species present is unstable when the prey's carrying capacity is large enough, \(k>\frac{ac+bd}{b(ac-bd)}\). (Hint: use the trace.) This is called the paradox of enrichment, because enriching the resources available for prey (increasing \(k\)) might be (naively) expected to lead to more prey and hence more stable predator-prey coexistence.