Abstract:
In biological systems Lotka-Volterra predator-prey model describes the population
dynamics of two interacting species of predators and its preys. Classical predatorprey
model is a primitive deterministic model governed by the two differential
equations, namely,
���� = (������ − ��������) ���� and ���� = (�������� − ������) ����
where �� and �� denote prey and predator respectively, and ����, ����, ���� and ���� are
parameters.
This model can be improved by introducing stochasticity that accounts for the
random fluctuations of a realistic predator-prey dynamical system. In this research
work, we use Stochastic Differential Equation (SDE) approach. There are various
ways, based on various assumptions, to incorporate SDE. One common approach is
to use equations of the following form:
���� = (������ − ��������) ���� + ��(���� + ����)�� ������
���� = (�������� − ������) ���� + ��(���� + ����)�� ������
These types of Stochastic Differential Equations (SDE) can be simulated in Matlab
using numerical methods such as Euler-Maruyama method. Phase planes of the
deterministic and stochastic models are carried out to demonstrate the behavior of
this modified model.
Our initial goal is to compare different stochastic models with the original
deterministic model through simulations. The deterministic model has a positive
equilibrium which is globally stable for positive values of the parameters.
Nevertheless, in the stochastic model, the predator and prey populations may tend to
extinction. Extinction percentages of predator or prey population are summarized
and analyzed through this research work.
