The Study of the Modification of the Lotka - Volterra Model: Influence of External Factors on the Evolution of the System of Competing Processes

Abstract

In the article the modified Lotka - Volterra model - it is called the «predator-prey» model as well - is studied. Solutions of the system of differential equations which simulate the nonlinear influence of both external and internal factors on an ecosystem are obtained. The choice of the additional terms in this work is caused by the following reasons:

1) the nonlinear character of the reproduction rate of the prey;

2) the existence of competition between the prey for food;

3) the nonlinear mortality rate of the prey due to the ecological catastrophes;

4) the nonlinear character of the rate of eating the prey by predators;

5) the existence of the competition between predators for eating the prey;

6) the nonlinear character of the reproduction rate of predators.

Various parameters of the system of differential equations which cover practically the whole set of possible beneficial and adverse effect on evolutionary trajectories are considered.

In this article we study the model of the competing populations with the amendments  and  both for the prey and predators respectively, and the following global cases: the model of equal ratio of the prey and predators due to reproduction of the prey and death of predators (k = l), and due to eating the prey by predators (a = b); the model of the fast decrease of the prey abundance with respect to the increase of predator abundance due to eating the prey by predators.

Let us draw the conclusions on the basis of the carried out numerical calculations and the analysis of results of the solution of the modified Lotka – Volterra system with various parameters. The following conditions are necessary for survival of an ecosystem: enough high the reproduction rate of the prey; the reproduction rate of the prey must be greater than the mortality rate of predators; the existence of positive nonlinear character of reproduction rate of the prey. Only in this case, we observe the oscillatory nature of evolution of a biocenosis.

In the case of ecological catastrophes the biocenosis inevitably perishes. For its restoration the reproduction rate of the prey needs to be significantly increased. Then the ecosystem restoration is still possible even at insignificant negative nonlinear terms.