A predator–prey model with generic birth and death rates for the predator

• We analyse a predator–prey model with Holling type II predator functional response.
• The predator birth rate is increasing in the predator functional response.
• The predator death rate is decreasing in the predator functional response.
• The solution can converge to a periodic orbit or a co-existence fixed point.
• Numerical simulations corroborate our analysis for a particular example.

We propose and study a predator–prey model in which the predator has a Holling type II functional response and generic per capita birth and death rates. Given that prey consumption provides the energy for predator activity, and that the predator functional response represents the prey consumption rate per predator, we assume that the per capita birth and death rates for the predator are, respectively, increasing and decreasing functions of the predator functional response. These functions are monotonic, but not necessarily strictly monotonic, for all values of the argument. In particular, we allow the possibility that the predator birth rate is zero for all sufficiently small values of the predator functional response, reflecting the idea that a certain level of energy intake is needed before a predator can reproduce. Our analysis reveals that the model exhibits the behaviours typically found in predator–prey models – extinction of the predator population, convergence to a periodic orbit, or convergence to a co-existence fixed point. For a specific example, in which the predator birth and death rates are constant for all sufficiently small or large values of the predator functional response, we corroborate our analysis with numerical simulations. In the unlikely case where these birth and death rates equal the same constant for all sufficiently large values of the predator functional response, the model is capable of structurally unstable behaviour, with a small change in the initial conditions leading to a more pronounced change in the long-term dynamics.