Abstract
The paper investigates the dynamical behaviors of a two-species discrete predator-prey system with Crowley–Martin functional response incorporating prey refuge proportional to prey density. The existence of equilibrium points, stability of three fixed points, period-doubling bifurcation, Neimark–Sacker bifurcation, Marottos chaos, and Control Chaos are analyzed for the discrete-time domain. The time graphs, phase portraits, and bifurcation diagrams are obtained for different parameters of the model. Numerical simulations and graphics show that the discrete model exhibits rich dynamics, which also present that the system is a chaotic and complex one. This paper attempts to present a feedback control method which can stabilize chaotic orbits at an unstable equilibrium point.
Highlights
Fixed Points and Stability Analysis of the Prey-Predator SystemFixed points of system (3) are determined by solving the following nonlinear system of equations: c(1 − b)xy x ax(1 −
A certain degree of protection can be offered by nature to a set number of prey populations by providing refuges. e effects of prey refuges on the population dynamics are intricate and complex in nature; it can be considered to be constituted by two components for modeling purposes
Bifurcation Analysis is section carries out an investigation of the conditions for the existence of Neimark–Sacker bifurcation (NSB) and period-doubling bifurcation (PDB) at the positive fixed point P2(x2, y2) of proposed system (3). ere is an emergence of different kinds of bifurcations from the fixed point in dynamical systems, when a particular parameter passes through its critical value
Summary
Fixed points of system (3) are determined by solving the following nonlinear system of equations: c(1 − b)xy x ax(1 −. E prey-predator model (3) undergoes period-doubling bifurcation at P2(x2, y2), when parameters vary in a small neighborhood of PD1 or PD2. Model (3) undergoes Neimark–Sacker bifurcation at P2(x2, y2), when parameters vary in a small neighborhood of NS1 or NS2. 4. Bifurcation Analysis is section carries out an investigation of the conditions for the existence of Neimark–Sacker bifurcation (NSB) and period-doubling bifurcation (PDB) at the positive fixed point P2(x2, y2) of proposed system (3). We discuss NSB and PDB for the positive fixed point P2(x2, y2) of the prey-predator system (3) taking b as a bifurcation parameter. It is clear that one of the eigenvalues of the positive fixed point P2(x2, y2) is λ1 − 1 and the other λ2 is neither 1 nor − 1 if parameters of the model are located in the following set:. P2(x2, y2) is a snap-back repeller of Map (3), and Map (3) is chaotic in the sense of Marotto
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