Abstract
Bifurcation and chaotic behavior of a discrete-time singular bioeconomic system are investigated. First, the traditional catch equation is modified after accounting for the handling time of the catch in a singular bioeconomic system. To discover the richer dynamics compared with the continuous form, the proposed system is considered difference scheme. Specially, the tangent space local parameterization condensed method for DAEs is generalized. The new local parameterization method is sufficiently general to be applicable to this type of discrete singular system. Also the dynamic behaviors of the system are investigated, by using normal form theory, center manifold theorem and bifurcation theory, it is shown that the system undergoes a Neimark–Sacker bifurcation and a flip bifurcation, on varying step-size in some range. In addition, numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors.
Highlights
The study of bioeconomics has a very long history
The main objective of this paper is to investigate the dynamic behaviors of a discretetime economic predator–prey model
6 Conclusion The present paper is concerned with the dynamics of a discrete-time economic predator– prey system in the presence of a type of nonlinear harvesting function
Summary
Bioeconomics is closely related to the early development of theories in fishery economics. Is called the Schaefer catch–effort relation [1]. E is the harvesting effort, in the fishery setting, which is typically specified as the number of (standard) vessels. The units of harvesting effort would be Standard Vessel Units (SVU). C is the cost of effort, and p is the ex-vessel price of fish. The parameter q is a constant, called catchability, units of q are SVU–1 × (time unit)–1. Q represents the proportion of the current stock x caught by one standard vessel in one time unit (to normalize units, one could let q = 1). G(x) represents the population growth per day and h is the daily harvesting rate, both being measured in tonnes per day
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