Abstract
In this paper, local dynamics, bifurcations and chaos control in a discrete-time predator-prey model have been explored in ℝ + 2 . It is proved that the model has a trivial fixed point for all parametric values and the unique positive fixed point under definite parametric conditions. By the existing linear stability theory, we studied the topological classifications at fixed points. It is explored that at trivial fixed point model does not undergo the flip bifurcation, but flip bifurcation occurs at the unique positive fixed point, and no other bifurcations occur at this point. Numerical simulations are performed not only to demonstrate obtained theoretical results but also to tell the complex behaviors in orbits of period-4, period-6, period-8, period-12, period-17, and period-18. We have computed the Maximum Lyapunov exponents as well as fractal dimension numerically to demonstrate the appearance of chaotic behaviors in the considered model. Further feedback control method is employed to stabilize chaos existing in the model. Finally, existence of periodic points at fixed points for the model is also explored.
Highlights
In the mid-1920s, an Italian Biologist U
Selachians depend on food fish for their survival as selachians are predators and food fish are prey
Volterra would come up with a mathematical model of the growth of the selachians and food fish, their prey, and his model would provide the answer to D’Ancona’s question. Volterra started his analysis on this problem by separating all the fish into the prey population x(t) and the predator population y(t). en, he reasoned that the food fish do not compete very fast among themselves for their food supply since this is very plentiful, and the density of fish population is not very much
Summary
In the mid-1920s, an Italian Biologist U. D’Ancona was surprised by a very large increase in the percentage of selachians during the World War I He reasoned the increase in the percentage of selachians was because of decline in fishing during this period. Volterra would come up with a mathematical model of the growth of the selachians and food fish, their prey, and his model would provide the answer to D’Ancona’s question. Volterra started his analysis on this problem by separating all the fish into the prey population x(t) and the predator population y(t).
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