Complex Dynamics on the Routes to Chaos in a Discrete Predator-Prey System with Crowley-Martin Type Functional Response

Abstract

We present in this paper an investigation on a discrete predator-prey system with Crowley-Martin type functional response to know its complex dynamics on the routes to chaos which are induced by bifurcations. Via application of the center manifold theorem and bifurcation theorems, occurrence conditions for flip bifurcation and Neimark-Sacker bifurcation are determined, respectively. Numerical simulations are performed, on the one hand, verifying the theoretical results and, on the other hand, revealing new interesting dynamical behaviors of the discrete predator-prey system, including period-doubling cascades, period-2, period-3, period-4, period-5, period-6, period-7, period-8, period-9, period-11, period-13, period-15, period-16, period-20, period-22, period-24, period-30, and period-34 orbits, invariant cycles, chaotic attractors, sub-flip bifurcation, sub-(inverse) Neimark-Sacker bifurcation, chaotic interior crisis, chaotic band, sudden disappearance of chaotic dynamics and abrupt emergence of chaos, and intermittent periodic behaviors. Moreover, three-dimensional bifurcation diagrams are utilized to study the transition between flip bifurcation and Neimark-Sacker bifurcation, and a critical case between the two bifurcations is found. This critical bifurcation case is a combination of flip bifurcation and Neimark-Sacker bifurcation, showing the nonlinear characteristics of both bifurcations.Hindawi

Global Stability of Prey-Predator Model with Crowley- Martin type Functional Response and Stage Structure for Predator

“…where 𝜶𝜶, 𝝈𝝈, 𝝑𝝑 are positive parameters that indicate the maximum level of predator predation, handling time, and the magnitude of disturbance among predators with variables x and y being the population density of the prey and predator (Dubey et al, 2018) (Mortoja et al, 2018. The Crowley Martin-type prey-predator model of response function is a development of the Holling-type II predator-prey model which focuses more on efforts to stabilize prey-predator interactions (Li et al, 2015) (Chen, 2017) (Zhang et al, 2018).…”

Global Stability of Prey-Predator Model with Crowley-Martin type Functional Response and Stage Structure for Predator

“…where 𝜶𝜶, 𝝈𝝈, 𝝑𝝑 are positive parameters that indicate the maximum level of predator predation, handling time, and the magnitude of disturbance among predators with variables x and y being the population density of the prey and predator (Dubey et al, 2018) (Mortoja et al, 2018. The Crowley Martin-type prey-predator model of response function is a development of the Holling-type II predator-prey model which focuses more on efforts to stabilize prey-predator interactions (Li et al, 2015) (Chen, 2017) (Zhang et al, 2018).…”

Bifurcation, chaos analysis and control in a discrete-time predator–prey system

“…In general, bifurcation is often a precursor to chaos, and chaos can cause the population to run a higher risk of extinction due to the unpredictability, so they are harmful for the breeding of biological population [19]. However, in some existing literature about biological systems [20][21][22], only the phenomena of bifurcation and chaos were presented. But, to the best of our knowledge, there has been much less work on the problem of bifurcation and chaos control in biological systems.…”

“…We not only give the theoretical analysis of objectively existing bifurcation and chaos phenomena for this biological system, but also we try to design two control strategies to delay the appearance of bifurcation and stabilize chaotic orbit if system is chaotic, respectively. Our work can be considered as the continuation and development of the work in [3], and the control methods in this paper can be applicable for [20][21][22] etc.…”