Project II : Ecological Modeling

where αu models the birth of the prey, βuv models death of the prey by being captured by predators, ρβuv models the rate of birth of the predators (which depends on how many prey are captured), and γv models the death of the predators. The basic model exhibits solutions which have periodic oscillating predator and prey populations. There is one non-trivial steady-state which is a center which is stable, but is not asymptotically stable. Consequently, the initial conditions for the predator and prey population size play an important role in determining the overall dynamics. To obtain the basic Lotka-Volterra Predator-Prey Model a number of assumptions were made. In particular, the predators are assumed to die at a constant per capita rate and the prey are assumed to reproduce at a constant per capita rate. The predator birth rate is assumed to depend only linearly on the rate at which predators catch prey. While the basic model may be a good starting point in modeling predator-prey populations, it neglects many features of populations which may be potentially important in practice. For example, the effect of the different seasons of the year on the birth rate of the prey and the death rate of the predators was not taken into account. For instance, in the winter we might expect the birth rate of the prey population to be greatly reduced and the death rate of the predators to increase. An interesting question is how these periodic variations effect the population dynamics, which is already prone to its own intrinsic periodic oscillations through the predator-prey interactions. In this project you will explore modifications of the basic predator-prey model using both analytical and computational techniques. One way to incorporate seasonality into the model is to modify the constant rates above, say β, to be β(1 + ǫ sin(2πωt)) where ǫ controls the influence of the seasonality and ω controls the frequency of the seasons. Use matlab codes to simulate the model and explore the influence of seasonality separately for each rate constant, then try combinations. In your report carefully address the following questions: