Existence of a limiting pattern for a system of nonlinear equations describing interpopulation competition

Abstract

A system of arbitrarily many nonlinear ordinary differential equations which can be interpreted as describing competition between populations is studied here. It is found that limits exist for the system given specified constraints on the “signal function” which describes input-output relations at the level of single populations. Existence of limits is shown by means of a function which records which variable in the system is growing fastest at a given time i.e. who is winning the competition. Generalizations are discussed to sigmoid signal functions which appear in models of pattern discrimination by neuron populations.