In this lab, we study a simple predator-prey model which contains an isoperimetric constraint. It is partially based on the work by Goh, Leitmann, and Vincent . We start with the standard Lotka-Volterra model
N ′1(t) = (α1 − β1N2(t))N1(t), N1(0) = N10, N ′2(t) = (β2N1(t)− α2)N2(t), N2(0) = N20,
where N1(t) is the prey population at time t, and N2(t) is the predator population. Here, we have scaled time to some arbitrary unit. Also, α1, α2, β1, and β2 are positive constants, subject to this time scaling. We wish to consider a situation where the prey act as a pest, such as an
insect population. The goal should be to reduce the pest population with the use of a chemical or biological agent, or pesticide. An ideal pesticide is one that affects only the pests (not the predators), leaves no residue, and kills in a density dependent manner. In practice, none of these is usually true. For simplicity, we study a pesticide which adheres to the last two assumptions. More information on models with less ideal pesticides can be found in [70, 81, 180]. Suppose the application of a pesticide kills both the pest/prey and predator
in a density dependent manner, with density parameters d1 > 0 and d2 > 0 respectively. Let u(t) be the rate of application at time t. Then, our model becomes
N ′1(t) = (α1 − β1N2(t))N1(t)− d1N1(t)u(t), N1(0) = N10 > 0, N ′2(t) = (β2N1(t)− α2)N2(t)− d2N2(t)u(t), N2(0) = N20 > 0.