3 Interacting populations 2: predator-prey
A simple predator-prey model
In the previous chapter we studied a first example of interacting populations with two species who compete with one another largely indirectly. Another common interaction that is much more direct is predation. Many species rely on others as a primary food source: foxes and rabbits; lions and zebras; ladybirds and aphids to name just a few. Such predator-prey interactions will have a similar structure to the competition models we considered last time, except that now one species (the predator) benefits from the interaction.
We will not start with the logistic model this time, but instead think of the simplest possible way we can model the dynamics of a prey with density [latex]N[/latex] and predator with density [latex]P[/latex],
[latex]\begin{align} \frac{dN}{dt} & =N(a-b P)\\ \frac{dP}{dt} &=P(-c+dN). \end{align}[/latex]
The prey has a birth rate [latex]a[/latex], no intraspecific competition and its death rate is only due to predation. This death from predation depends on the predator density (with parameter [latex]b[/latex]) with more predators leading to higher prey death and no saturation of predation at higher densities. In contrast, the predator’s birth rate depends on its ability to predate, and so depends on the prey density (with parameter [latex]d[/latex]). It has a simple death rate however at per-capita rate, [latex]c[/latex]. We might expect the parameters [latex]b[/latex] and [latex]d[/latex] to be linked, since one controls the amount of predation and the other the ability to reproduce which itself depends on predation. In some models you will see the predator birth term written as something like [latex]\theta bPN[/latex], where [latex]\theta[/latex] is the ‘conversion’ of energy from feeding into new births.
We have already identified a few simplifications we have made here even compared to our first few models – no intraspecific competition and continually increasing predation. A good approach to modelling is always to start with the simplest reasonable model and then build increasing complexity as needed. This is also the ‘classic’ predator-prey model you will see elsewhere, so it seems sensible to cover it here.
Let’s begin our analysis by finding the equilibria of the model. There is an extinction equilibrium at [latex](N,P)=(0,0)[/latex], and a coexistence equilibrium at [latex](N,P)=(c/d,a/b)[/latex]. The stability of these steady states is calculated, as before, by finding the Jacobian of the system.
Exercises
Click for solution
Recall that to write down the Jacobian we go through the ODEs in turn and form a matrix of the partial derivatives. For this model that gives us,
[latex]\begin{align} J=&\left( \begin{array}{cc} a-bP^* & -bN^*\\ dP^* & -c+dN^* \end{array} \right), \end{align}[/latex]
We can quickly see that the extinction equilibrium, [latex](N,P)=(0,0)[/latex], is always unstable (since [latex]a>0[/latex]). What about the coexistence equilibri