Lions and Wildebeest: A Predator-Prey Model

ABSTRACT In this article, we develop the classical Lotka-Volterra predator-prey equations to model the interactions between lions and wildebeest based on field data from the Kruger National Park in South Africa. From the data, coefficients for the intrinsic growth rate of both lions and wildebeest are estimated as well as the predator efficiency to convert food into fertility. We modify this model by considering three possible lion cropping strategies: cropping at a constant rate, cropping proportional to the population present, and cropping proportional to the square of the population present. By doing so, we see dramatic changes in the nature of the critical points of the model and obtain interesting interpretations of centers, repelling spiral points, and attracting spiral points. While we do include the mathematical theory for our models, we concentrate on model interpretation which students generally find more difficult than performing the calculations. Our discussion is elementary and suitable for students in a second semester course in elementary differential equations or in a first course in ecology or wildlife management. INTRODUCTION Predator-prey systems are being studied with greater interest in beginning differential equations courses. This is largely due to advances in technology which permits a shift from the traditional emphasis on specialized techniques for solving differential equations to a more qualitative analysis, modeling, and investigation of nonlinear systems of equations. There are real world concerns which make the study of such systems important, not the least of which is conservation management of wildlife. In this field, the interactions between predators and prey are highly complex and it is generally accepted that these relationships should be disturbed as little as possible. However, often interaction between wildlife and humans results in the necessity for control of wildlife populations (e.g., cropping lions) and thus predator-prey models are important tools for the prediction of the effect of such controls. In this article, we discuss the classical Lotka-Volterra predator-prey model applied to lions (Panthera leo) and wildebeest (Connochaetas taurinus) populations. We indicate how the coefficients of the terms in this system of nonlinear first order differential equations can be estimated from field data. This field data [6], collected over a six year period in the Kruger National Park in South Africa, consists of lion and wildebeest population estimates, kill rates, cropping numbers, and calf survival rates. We show how to introduce cropping of the predator into the model and not surprisingly that it has a significant effect upon the way the populations behave. We are able to predict extinctions, cyclical behavior, and eventual stable populations depending on the nature of the critical values of the model. We also give some indications of how to "tune" the model and thus gain insight to what happens to the two populations, with and without cropping. 2. THE LOTKA-VOLTERRA MODEL IMAGE FORMULA8 3. BIOLOGICAL BACKGROUND In this section, we give a brief synopsis of the more detailed account of the status of the populations of lions and wildebeests that is given in [6]. We recommend this reference (particularly chapters 2 and 4), as it makes excellent background reading for students. They will get to think about a wide variety of non-mathematical considerations necessary in designing a mathematical model. The central grasslands of the Kruger National Park has always supported large herds of zebra (Equus burchell) and wildebeest. During the decade of the sixties, the rainfall was unusually low and while the zebra and wildebeest both thrived during this period, by 1965 there was evidence of range over-utilization. This led to a management decision to initiate a cropping program of both herds. …