A predator-prey system involving group defense: a connection matrix approach

Abstract

THE PURPOSE of this paper is to give a concrete demonstration of how the connection matrix can be used to analyse a l-parameter family of differential equations. In particular we have four goals: (1) To show how to compute connection matrices. (2) To use these matrices to classify the structure of solutions to our set of differential equations for various parameter values. (3) To prove the existence of local and global bifurcations. (4) To demonstrate how to ignore certain bifurcations. Strange as it may seem, we want to emphasize the importance of this last goal. Even in seemingly simple systems it is possible for global bifurcations to occur which are difficult to detect or exclude (in our case, global bifurcations which lead to the existence of multiple periodic orbits). Nevertheless, one wants to be able to make significant statements about the general structure of the flow for all parameter values. Thus it is important to have techniques which are able to ignore subtle or difficult to detect phenomenon and still work. The set of equations which we have chosen to study arise from a predator-prey model in which the prey exhibits group defense, i.e. the more prey the better their chances for protecting themselves from the predator. This set of equations has been studied before (see Freedman and Wolkowicz [3], Wolkowicz [13], and Mischaikow and Wolkowicz [9]) and we do not claim any new results, rather it is our techniques which are novel. To be more specific, the system we consider is two dimensional and hence phase plane techniques can be employed to obtain the results which we shall present herein. This is the approach taken in [13]. Of course, phase plane techniques are more difficult, if not impossible, to apply in the case of higher dimensional systems. The advantage of our approach is that the connection matrix is dimension independent and hence the tehniques we are describing here are in principle applicable to higher dimensional