Complex dynamics of a discrete-time predator-prey system with Holling IV functional response
type: Item
instance of: scholarly article
title: Complex dynamics of a discrete-time predator-prey system with Holling IV functional response
zbMATH Open document ID: 1369.92090
author: Qianqian Cui
author: Zhipeng Qiu
author: Zengyun Hu
author: Qiang Zhang
published in: Chaos, Solitons and Fractals
publication date: 2017-02-10T00:00:00Z
review text: The paper deals with the dynamics of a discrete predator-prey system with the response function of Holling IV type, that is, a non-monotonic function with a hump. As the authors explain, to exclude existence of unrealistic negative solutions in the system, they start with a continuous system \[ \begin{aligned} \dot{x} & =rx\left( 1-\frac{x}{K}\right) -\frac{mxy}{ax^{2}+bx+1},\\ \dot{y} & =y\left( \frac{cx}{ax^{2}+bx+1}-d\right).\end{aligned} \] By applying a technique from [\textit{M. Fan} and \textit{K. Wang}, Math. Comput. Modelling 35, No. 9--10, 951--961 (2002; Zbl 1050.39022)], a system of differential equations with piecewise constant arguments is derived under the assumption that the average growth rates change regularly. The integration of a modified system over the interval \([n,n+1)\) yields the following discrete system \[ \begin{aligned} x\left( n+1\right) & =x\left( n\right) \exp\left[ r\left( 1-\frac{x\left( n\right) }{K}\right) -\frac{my\left( n\right) }{ax^{2}\left( n\right) +bx\left( n\right) +1}\right] ,\\ y\left( n+1\right) & =y\left( n\right) \exp\left[ \frac{cmx\left(n\right) }{ax^{2}\left( n\right) +bx\left( n\right) +1}-d\right] .\end{aligned}\tag{1} \] Here \(x\) and \(y\) stand for population densities of the prey and predator, respectively, \(K\) is the carrying capacity of the prey population, \(d\) is the death rate of predator population, \(c\) is a conversion rate of eaten prey into new predator abundance. Considering \(K\) as a bifurcation parameter, the authors demonstrate that the system (1) may undergo a flip bifurcation and a Neimark-Sacker bifurcation (a discrete analogue of the Andronov-Hopf bifurcation in continuous systems). Numerical simulations in Section 4 illustrate the theoretical results and reveal the complex dynamics of system (1) including orbits of period \(n,\) invariant cycles, period doubling bifurcations, and quasi-periodic orbits. The paper concludes with a brief discussion of the results.
reviewed by: Svitlana P. Rogovchenko
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 37N25
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 6683354
zbMATH Keywords: discrete predator-prey system
zbMATH Keywords: Holling IV type response function
zbMATH Keywords: flip bifurcation
zbMATH Keywords: periodic solutions
zbMATH Keywords: Neimark-Sacker bifurcation
zbMATH Keywords: invariant cycles
zbMATH Keywords: period doubling bifurcation
MaRDI profile type: Publication
full work available at URL: j.chaos.2016.04.002
OpenAlex ID: W2339190293
Wikidata QID: Q115579821
cites work: Chaotic dynamics of a discrete prey-predator model with Holling type II
cites work: Chaos
cites work: Applications of centre manifold theory
cites work: Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system
cites work: Graphical stability, enrichment, and pest control by a natural enemy
cites work: Q3895297
cites work: Predator-prey systems with group defence: The paradox of enrichment revisited
cites work: The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response
cites work: Chaotic dynamics of a discrete predator-prey system with prey refuge
cites work: Bifurcation and chaotic behavior of a discrete-time Ricardo-Malthus model
cites work: 1:2 and 1:4 resonances in a two-dimensional discrete Hindmarsh-Rose model
cites work: Complex dynamic behaviors of a discrete-time predator-prey system
cites work: Bifurcations and chaos in a two-dimensional discrete Hindmarsh-Rose model
cites work: Existence and global attractivity of positive periodic solutions of Lotka-Volterra predator-prey systems with deviating arguments
cites work: Stabilizing effects of spatial heterogeneity in predator-prey systems
cites work: Chaos in Dynamical Systems
cites work: Q4226356
cites work: Global analysis in a predator-prey system with nonmonotonic functional response
cites work: Introduction to Applied Nonlinear Dynamical Systems and Chaos
cites work: Hopf bifurcation and multiple periodic solutions in Lotka-Volterra systems with symmetries
cites work: Bifurcation Analysis of a Predator-Prey System Involving Group Defence
cites work: Bifurcation Analysis of a Predator-Prey System with Nonmonotonic Functional Response
DOI: 10.1016/J.CHAOS.2016.04.002
description: scientific article; zbMATH DE number 6683354