Stability and bifurcation analysis in a predator–prey system with Michaelis–Menten type predator harvesting

Hu, Dongpo; Cao, Hongjun

doi:10.1016/j.nonrwa.2016.05.010

Cited by 177 publications

(82 citation statements)

References 31 publications

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“…Substituting the eigenvalues (14) into the characteristic equation (8), and separating the real and imaginary parts, then we have…”

Section: Hopf Bifurcationmentioning

confidence: 99%

“…Understanding the interactions between prey and predator is a significant topic of longstanding interest in biology. The prey-predator models with various functional responses, such as Holling I, II, III, and IV, Leslie-Gower, Michaelis-Menten, ratio-dependent, Beddington-DeAngelis, and so on, have been extensively discussed [7][8][9][10]. Many other ecological mechanisms have been also included and investigated in the prey-predator models.…”

Section: Introductionmentioning

confidence: 99%

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Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

Zhang

Kang

Huang

et al. 2018

Mathematical Problems in Engineering

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Complex dynamics of a four-species food web with two preys, one middle predator, and one top predator are investigated. Via the method of Jacobian matrix, the stability of coexisting equilibrium for all populations is determined. Based on this equilibrium, three bifurcations, i.e., Hopf bifurcation, Hopf-Hopf bifurcation, and period-doubling bifurcation, are analyzed by center manifold theorem, bifurcation theorem, and numerical simulations. We reveal that, influenced by the three bifurcations, the food web can exhibit very complex dynamical behaviors, including limit cycles, quasiperiodic behaviors, chaotic attractors, route to chaos, period-doubling cascade in orbits of period 2, 4, and 8 and period 3, 6, and 12, periodic windows, intermittent period, and chaos crisis. However, the complex dynamics may disappear with the extinction of one of the four populations, which may also lead to collapse of the food web. It suggests that the dynamical complexity and food web stability are determined by the food web structure and existing populations.

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“…Substituting the eigenvalues (14) into the characteristic equation (8), and separating the real and imaginary parts, then we have…”

Section: Hopf Bifurcationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

Zhang

Kang

Huang

et al. 2018

Mathematical Problems in Engineering

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“…The majority of the results in this topic is mainly concentrated on the study of the qualitative aspects of the continuous systems described by systems of differential equations. It is worth to mention some recent typical works such as [16,17,20,29,33,34]. .…”

Section: Introductionmentioning

confidence: 99%

Nonstandard finite difference schemes for a general predator–prey system

Dang

Hoang

2019

Journal of Computational Science

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In this paper we transform a continuous-time predator-prey system with general functional response and recruitment for both species into a discrete-time model by nonstandard finite difference scheme (NSFD). The NSFD model shows complete dynamic consistency with its continuous counterpart for any step size. Especially, the global stability of a non-hyperbolic equilibrium point in a particular case of parameters is proved by the Lyapunov stability theorem. The performed numerical simulations confirmed the validity of the obtained theoretical results.

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“…However, there are some specific classes among them, called the Gause type models [1,2]. The research of predator-prey model and infectious disease model has always been a hot topic in biomathematics [1][2][3][4][5][6][7][8][9]. In 1931, Allee discovered that the living state of the cluster is conducive to the growth of the population, but the density is too high and will inhibit the growth of the population and even become extinct due to resource competition.…”

Section: Introductionmentioning

confidence: 99%

Dynamic study of a predator-prey model with Allee effect and Holling type-I functional response

Hua

Wei

et al. 2019

Adv Differ Equ

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In this paper, a prey-predator model with Allee effect and Holling type-I functional response is established, and its dynamical behaviors are studied in detail. The existence, boundedness and stability of the model are qualitatively discussed. Hopf bifurcation analysis is also taken into account. We further illustrate our theoretical analysis by means of numerical simulation. Using computer simulation, we found the position of each equilibrium point in the phase diagram that we drew. We found the threshold for undergoing a Hopf bifurcation in the bifurcation diagram. One of the interesting questions is which model with strong Allee effect is a bistable system.

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Stability and bifurcation analysis in a predator–prey system with Michaelis–Menten type predator harvesting

Cited by 177 publications

References 31 publications

Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

Nonstandard finite difference schemes for a general predator–prey system

Dynamic study of a predator-prey model with Allee effect and Holling type-I functional response

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