Bifurcations, Complex Behaviors, and Dynamic Transition in a Coupled Network of Discrete Predator-Prey System

Abstract

The nonlinear dynamics of predator-prey systems coupled into network is an important issue in recent biological advances. In this research, we consider each node of the coupled network represents a discrete predator-prey system, and the network dynamics is investigated. By applying Jacobian matrix, center manifold theorem and bifurcation theorems, stability of fixed points, flip bifurcation and Neimark-Sacker bifurcation of the discrete predator-prey system are analyzed. Via the method of Lyapunov exponents, the nonchaos-chaos transition of the coupled network along the routes to chaos induced by bifurcations is determined. Numerical simulations are performed to demonstrate the bifurcations, various attractors and dynamic transitions of the coupled network. Via comparison, we find that the coupled network exhibits far richer and more complex behaviors than single predator-prey system, including period-doubling cascades in orbits of period-2, period-4, period-8, invariant closed curves, dynamic windows for periodic orbits and invariant curves, quasiperiodic orbits, tori, and chaotic sets. Moreover, the attractors of the coupled network show more diverse and complicated structures. These results may provide a new perspective on the predator-prey dynamics in complex networks.

Dynamics of a Conformable Fractional Order Generalized Richards Growth Model on Star Network with N=20 Nodes

“…Networks are named according to the shape of the connections such as globally coupled network, star network, Erdos-Renyi network and are used to understand the origin and complexity of the dynamical system. Studying on dynamical analysis of different types of complex networks can be found in the studies [22][23][24][25][26][27].…”

Transition chaos in fractional order Cournot duopoly game model on scale-free network

“…In the mathematical modeling of this complex structures networks that consists of nodes and edges are used.Therefore, it is not surprising that researchers have shown so much interest to networks. In the literature, network that is the extension of graph theory has applications in many fields such as engineering 18 , biology 19,20,21,22,23,24,25 , economics 26 , social science 27 , physics 28 , chemistry 29 , computer science 30 . One of the most widely known and used types of networks is scale-free networks.…”

“…In the study 20,21 , transition chaos with respect to coupling strength parameter has been reported for logistic map on both scale free and Erdos Renyi random network with 𝑁 = 1000 nodes. Huang et al 22 investigated dynamical behavior discrete time predator prey model on globally coupled network and observed rich dynamical behavior such as stable equilibrium point, Flip and Neimark-Sacker bifurcation and chaos in the complex network.…”

Dynamical Analysis for the Hybrid Network Model of Delayed Predator-Prey Gompertz Systems with Impulsive Diffusion between Two Patches

“…Furthermore, in the study of population dynamical systems, due to the universality and importance of the predator-prey relationship, the dynamics of the predator-prey system has been widely concerned. In recent decades, the dynamical behaviors of the predator-prey model defined on the network have enjoyed remarkable progress [4][5][6][7][8]. In [6], each node of the coupled network represents a discrete predator-prey system, and the network dynamics are investigated.…”

“…In recent decades, the dynamical behaviors of the predator-prey model defined on the network have enjoyed remarkable progress [4][5][6][7][8]. In [6], each node of the coupled network represents a discrete predator-prey system, and the network dynamics are investigated. In [7], Chang studied instability induced by time delay for a predator-prey model on complex networks and instability conditions were obtained via linear stability analysis of network organized systems.…”