TITLE:
Control Schemes to Reduce Risk of Extinction in the Lotka-Volterra Predator-Prey Model
AUTHORS:
Jessica Li
KEYWORDS:
Lotka-Voterra, Predator-Prey Model, Extinction Control, Feedback Control, Stability
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.2 No.7,
June
20,
2014
ABSTRACT: The Lotka-Volterra predator-prey model is widely used in many disciplines
such as ecology and economics. The model consists of a pair of first-order
nonlinear differential equations. In this paper, we first analyze the dynamics,
equilibria and steady state oscillation contours of the differential equations
and study in particular a well-known problem of a high risk that the prey
and/or predator may end up with extinction. We then introduce exogenous control
to reduce the risk of extinction. We propose two control schemes. The first
scheme, referred as convergence guaranteed scheme, achieves very fine granular
control of the prey and predator populations, in terms of the final state and
convergence dynamics, at the cost of sophisticated implementation. The second
scheme, referred as on-off scheme, is very easy to implement and drive the
populations to steady state oscillation that is far from the risk of
extinction. Finally we investigate the robustness of these two schemes against
parameter mismatch and observe that the on-off scheme is much more robust.
Hence, we conclude that while the convergence guaranteed scheme achieves
theoretically optimal performance, the on-off scheme is more attractive for
practical applications.