Rich dynamics and anticontrol of extinction in a prey-predator system

TL;DR

This paper explores complex dynamics in a prey-predator model, demonstrating how anticontrol methods can prevent species extinction and revealing rich bifurcation phenomena including chaos and strange nonchaotic attractors.

Contribution

It introduces an anticontrol algorithm to prevent species extinction and analyzes the system's bifurcations and attractors using both analytical and numerical methods.

Findings

System exhibits quasiperiodic, stable, chaotic, and hyperchaotic orbits.

Anticontrol successfully prevents species extinction.

Presence of strange nonchaotic attractors in certain parameter ranges.

Abstract

This paper reveals some new and rich dynamics of a two-dimensional prey-predator system and to anticontrol the extinction of one of the species. For a particular value of the bifurcation parameter, one of the system variable dynamics is going to extinct, while another remains chaotic. To prevent the extinction, a simple anticontrol algorithm is applied so that the system orbits can escape from the vanishing trap. As the bifurcation parameter increases, the system presents quasiperiodic, stable, chaotic and also hyperchaotic orbits. Some of the chaotic attractors are Kaplan-Yorke type, in the sense that the sum of its Lyapunov exponents is positive. Also, atypically for undriven discrete systems, it is numerically found that, for some small parameter ranges, the system seemingly presents strange nonchaotic attractors. It is shown both analytically and by numerical simulations that the original system and the anticontrolled system undergo several Neimark-Sacker bifurcations. Beside the classical numerical tools for analyzing chaotic systems, such as phase portraits, time series and power spectral density, the 0-1 test is used to differentiate regular attractors from chaotic attractors.