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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 8713651, 17 pages
https://doi.org/10.1155/2018/8713651
Research Article

Neimark-Sacker-Turing Instability and Pattern Formation in a Spatiotemporal Discrete Predator-Prey System with Allee Effect

Research Center for Engineering Ecology and Nonlinear Science, North China Electric Power University, China

Correspondence should be addressed to Huayong Zhang; nc.ude.upecn@sneecr

Received 20 January 2018; Accepted 19 April 2018; Published 26 June 2018

Academic Editor: Guang Zhang

Copyright © 2018 Huayong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A spatiotemporal discrete predator-prey system with Allee effect is investigated to learn its Neimark-Sacker-Turing instability and pattern formation. Based on the occurrence of stable homogeneous stationary states, conditions for Neimark-Sacker bifurcation and Turing instability are determined. Numerical simulations reveal that Neimark-Sacker bifurcation triggers a route to chaos, with the emergence of invariant closed curves, periodic orbits, and chaotic attractors. The occurrence of Turing instability on these three typical dynamical behaviors leads to the formation of heterogeneous patterns. Under the effects of Neimark-Sacker-Turing instability, pattern evolution process is sensitive to tiny changes of initial conditions, suggesting the occurrence of spatiotemporal chaos. With application of deterministic initial conditions, transient symmetrical patterns are observed, demonstrating that ordered structures can exist in chaotic processes. Moreover, when local kinetics of the system goes further on the route to chaos, the speed of symmetry breaking becomes faster, leading to more fragmented and more disordered patterns at the same evolution time. The rich spatiotemporal complexity provides new comprehension on predator-prey coexistence in the ways of spatiotemporal chaos.