A SIMPLE MATHEMATICAL MODEL FOR ONE-PREDATOR : ONE-PREY SYSTEM LEADING TO EXTINCTION OF BOTH SPECIES

Abstract

A simple mathematical model originating from SHINOZAKI'S model is proposed to simulate one-predator-one-prey system. To construct the model the following assumptions are made : the prey-density (or biomass) follows the logistic curve and rate of its loss is proportional to the predator-density (or biomass), the predator-density follows the logistic curve whose asymptore is proportional to the prey-density, the intrinsic growth rates of both species are equal to each other and furthermore they equal the product of the proportional constants relating to the rate of loss and the asymtore of predator. The analytical solutions of the present non-linear model distinguish between various patterns of time courses of predator and prey densities, and they show that both species will become extinct after a long time. Despite the simplicity of the model, the solutions simulate the data of predator-prey dynamics of mites observed by HUFFAKER (1958) well.