Bifurcation and chaotic behavior of a discrete singular biological economic system

Abstract

In this paper, the complex dynamics of a discrete singular biological economic system are first investigated. Firstly, we establish a discrete singular biological economic system, which is based on the discretization of a differential–algebraic equations that is described by a ratio-dependent predator–prey system with harvesting and economic factor. Then it is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation in the interior of R+3, by using the new normal form of discrete singular systems, the center manifold theorem and the bifurcation theory, as varying the economic profit μ in some range. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamic behaviors, such as cascades of period-doubling bifurcation in orbits of period 2, 4, 8, and chaotic sets. These results reveal far richer dynamics of the discrete model compared with the continuous model. The Lyapunov exponents are numerically computed to confirm further the complexity of the dynamical behaviors.