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.
Highlights
Predator-prey population dynamics are often modeled with a set of nonlinear differential equations
The Lotka-Volterra predator-prey model is a pair of first-order nonlinear differential equations ax − bxy cxy − dy where x, y represent the numbers of the prey and predator respectively, and a,b, c, d are positive parameters that describe the following dynamic interaction of the two species
We introduce two control schemes to reduce the risk of extinction
Summary
Predator-prey population dynamics are often modeled with a set of nonlinear differential equations. (2014) Control Schemes to Reduce Risk of Extinction in the Lotka-Volterra Predator-Prey Model. Without the predator ( y = 0) , the prey growth rate is directly proportional to the population size. Without the prey ( x = 0) , the predator growth rate is negative and directly proportional to the population size. We introduce exogenous control to change the dynamics so as to achieve certain desired characteristics. It is well known [5] [6] that there is a high risk that the prey and/or predator may end up with extinction in the Lotka-Volterra model. We study the robustness of the two control schemes against parameter mismatch that often arises in practice
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