DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

IMPULSIVE CONTROL OF THE POPULATION DYNAMICS
M. U. AKHMET, D. ARUĞASLAN, AND M. BEKLIOĞLU
We investigate the dynamics of the Lotka-Volterra system with variable time of impulses.
Sufficient conditions are obtained for the existence of focus in the noncritical case. The
focus-center problem in the critical case and the Hopf bifurcation are considered.
Copyright © 2006 M. U. Akhmet et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The Lotka-Volterra system describes the interaction of two species in an ecosystem, a prey
and a predator. Since there are two species, this system involves two equations,
dx
dt = x′ = ax − bxy,
dy
dt = y′ =−cy+ dxy,
(1.1)
where x and y denote the prey and predator population densities, respectively, a (the
growth rate of prey), b (the rate at which predators consume prey), c (the death rate of
predator), and d (the rate at which predators increase by consuming prey) are positive
constants. This system has only one positive equilibrium that is (c/d,a/b) asacenter.
However, having the equilibrium as a center, this system is ecologically undesirable. In
other words, the hypothesis of (1.1) does not seem to be in accordance with the observations
[8].
The system (1.1) describes populations whose members can respond immediately to
any change in the environment. But, in real populations, both prey and predator require
reaction time lags. By introducing a time lag into system (1.1), instead of a center, the
point of equilibrium may be either a stable focus or a stable node. Moreover, this point
may be an unstable focus surrounded by a stable limit cycle [10].
The Lotka-Volterra population growth model does not assume human activities at all.
We introduce human intervention by impulsive perturbation. In general, the appearence
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Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 21–29