Abstract
The main concern of this paper is to discuss stability and bifurcation analysis for a class of discrete predator-prey interaction with Holling type II functional response and harvesting effort. Firstly, we establish a discrete singular bioeconomic system, which is based on the discretization of a system of differential algebraic equations. It is shown that the discretized system exhibits much richer dynamical behaviors than its corresponding continuous counterpart. Our investigation reveals that, in the discretized system, two types of bifurcations (i.e., period-doubling and Neimark–Sacker bifurcations) can be studied; however, the dynamics of the continuous model includes only Hopf bifurcation. Moreover, the state delayed feedback control method is implemented for controlling the chaotic behavior of the bioeconomic model. Numerical simulations are presented to illustrate the theoretical analysis. The maximal Lyapunov exponents (MLE) are computed numerically to ensure further dynamical behaviors and complexity of the model.
Highlights
Bioeconomics is linked closely to the early development of ideas in fisheries economics due to the pioneering work of Canadian economists Gordon [1] and Anthony Scott. eir basic theories used recent developments in modeling of biological fisheries, initially the contributions made by Schaefer in 1954 and 1957 on introducing a systematic connection between fishing mechanism and growth of biological type through the implementation of mathematical modeling verified by experimental studies, and associated itself to resource protection, ecology, and the environment [2]. ese concepts were developed from the multifishing science environment in Canada at that time
Gordon in [1] suggested economic theory keeping in view the common property of resource, which was based on the effect of the harvest effort on an ecosystem by taking into account an economic perspective, assuming that x(t) and e(t) denote the density of harvested population and the harvest effort in an ecosystem, respectively; the total cost is equal to ce(t), and the total revenue is equal to pe(t)x(t), where c denotes the cost of harvest effort, and p is Discrete Dynamics in Nature and Society used for the unit price of harvested population. en, the economic interest μ for the harvest effort by the harvested population is given by μ e(t)(px(t) − c)
We discuss the dynamical behavior of a discrete-time singular bioeconomic model
Summary
Bioeconomics is linked closely to the early development of ideas in fisheries economics due to the pioneering work of Canadian economists Gordon [1] and Anthony Scott (in 1955). eir basic theories used recent developments in modeling of biological fisheries, initially the contributions made by Schaefer in 1954 and 1957 on introducing a systematic connection between fishing mechanism and growth of biological type through the implementation of mathematical modeling verified by experimental studies, and associated itself to resource protection, ecology, and the environment [2]. ese concepts were developed from the multifishing science environment in Canada at that time. Liu et al [11] analyzed the local dynamics and Hopf bifurcation for a biological economic model with Holling type II functional response and harvesting effort on prey. Babaei and Shafiee [16] reported stability analysis, bifurcation, chaotic behavior, and control for a singular bioeconomic model of prey-predator interaction governed by an algebraic equation and 3-dimensional differential equations. In [25], the authors analyzed complex dynamics of a discrete-time bioeconomic system for predator-prey interaction with the implementation of Euler approximations. Keeping in view (1) and (2), we obtain the following predator-prey biological economic model with Holling type II functional response with harvest effort:. Applying the forward Euler scheme to system (3), we obtain the discrete-time predator-prey biological economic model with Holling type II functional response as follows:. Numerical examples are presented for verification and illustration of our theoretical discussion
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