Understanding Global Elephant and Tree Dynamics with Ode's

Abstract

1. INTRODUCTION Teaching differential equations is more meaningful to most students when applications to real world problems are found. However, real world problems usually involve systems of equations that are often nonlinear without analytical solutions. Today these problems can be used in courses on differential equations because of modern computing hardware and sophisticated software. In this article, we examine a simple model to describe the global characteristics of an ecosystem inhabited by elephants. In such a system the elephants have a significant effect because each elephant eats hundreds of kilograms of trees or grass per day. Thus, understanding the dynamics between elephants and their food source is crucial. In particular, South Africa has experienced a rapid change from traditional livestock farming to game ranch farming driven by a change in the costs and revenues of these farming methods. Many farmers have introduced elephants to promote tourism, but face the problem of a significant reduction in tree densities to an extent that can adversely affect their game ranches. Here we show how to investigate the global dynamics of an elephant tree system with a simple set of ordinary differential equations. For simplification we consider savannah regions which are dry, have trees but almost no grass. This example results in a further simplification in that there is a reduced probability of fire. Examples of such systems are relatively common in southern Africa. Caughley [2] was the first person to suggest the use of a predator-prey ordinary differential equation model to describe the dynamics between elephants and trees. Here we describe the elephant-tree system using a different variation of predator-prey model first suggested by Holling [8]. We suggest that this version of the predator-prey model appears applicable to analyze elephant-tree dynamics. First we investigate the stability properties of the system and discuss the importance of those properties for an understanding of the global system dynamics. We then investigate actual solutions to the equations using the software package Matlab employing the Runge-Kutta numerical routine. Many other software packages such as Mathemaica or Maple could also be used. 2. THE MODEL Basic predator-prey models were first proposed by Lotka in 1925 [13] and then independentcy by Voltera in 1927 [14]. The approach is based on the logistic model for single species population growth given by where N denotes population density at time t, r is the intrinsic relative growth rate and K is the carrying capacity of the environment for the species. A version of the original model for two interacting species due to Holling [8] is used here for the interaction between elephant and trees and is given by wher T and E denote the densities of the tree and elephant populations at time t; r^sup T^ and r^sup E^ are the respective growth rates and K is the carrying capacity for the trees in the absence of elephant. The parameter g is the density of trees required for elephant equilibrium. It is expected that tree removal will be a function of tree density. However, elephants cannot remove trees at rates beyond their own capacity and so this function is expected to drop off with density. The relationship between prey removal and prey is called the functional response and given by f(T) in (2.2). Functional response in predator-prey models, was considered in detail by Holling [8], who described three broad categories of respose (Tyupe I, TypeII, TypeIII). Type I shows a linear increase to a plateau. In TypeII the predation rate is sigmoidal with an increase in prey. In Type III the predation rate is sigmodial with an increase in prey density. We have unpublished data that would suggest the elephant and trees show a Type II functional response. Holling [8] suggested a rectangular hyperbola to describe the TypeII functional response given here for elephant and trees as where d^sub T^ is the death rate of trees dues to elephants, and h is the average time spent by each elephant on each tree. …

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