About a mathematical predator-prey model, with intraspecific competition

Abstract

This article is a continuation of the study of the predator-prey system and the Ferhulst equation, considered in our previous works. In this work, we considered one of the many disadvantages of the basic predator-prey system, namely, we considered the fact that with zero initial values of the predator, the number of prey populations should not grow all the time, because such growth is not observed in real situations, since food reserves are not infinite and the habitats of any populations are limited. The same restrictions, therefore, must be considered for predators. Therefore, in order to consider, the limited food reserves and the limited habitat, we introduce terms from the Ferhulst model -cx², -ry² into the system of equations. They will consider the lack of food and habitat in the predator-prey model. We also give practical advice on how to distinguish the basic model from the model with intraspecific competition in real conditions. In this paper, solutions of a non-zero stationary state are considered. Depending on the parameters of the equations, the system has a special point, a stable focus or a stable node. Here we examine only the steady focus state. Next, we show how to determine the parameters of the predator-prey system with intraspecific competition based on statistical data, without analyzing the reserves of the food base for the victim, which significantly reduces the cost of research. A partial solution and a phase portrait of this system were obtained on Phyton. We think that our practical advice will be useful for controlling the number of some pests of agricultural crops.