Steady state for a predator–prey cross-diffusion system with the Beddington–DeAngelis and Tanner functional response

Luo, Demou

doi:10.1186/s13661-020-01469-3

Cited by 6 publications

(5 citation statements)

References 15 publications

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“…In recent years, great attention has been paid to the predator-prey system, which has important application in biological model. Many scholars have carried out the study of different predator-prey models (see [1][2][3][4][5][6]). Among the variety of models, the predator-prey models with different Holling type functional response are refined so as to better reflect the specific characteristics of the different populations or economical needs.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics in a discrete-time predator-prey system

Liu,

Juan

2024

Preprint

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In this paper, the nonlinear dynamics of a two-dimensional discrete-time system of Leslie type with simplified Holling type IV functional response are reported. Possible codimension-two bifurcations (1:2, 1:3 and 1:4 strong resonances) are investigated under variation of two parameters for certain critical values at the positive fixed point. For each bifurcation, normal form coefficients along with its scenario are investigated thoroughly. Besides, using numerical simulations, in addition to confirming the results of our analyses, more behaviors are extracted from the model, such as fractal structure, mode-locking structure, etc. Our results generate and improve some known results and show that the discrete model has richer dynamics than the continuous one.

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Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics in a discrete-time predator-prey system

Liu,

Juan

2024

Preprint

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“…They studied the asymptotic properties of the solutions of the model. Recently, some articles have shown the role of diffusion on predator-prey interactions [13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Dynamics of a diffusive two predators- one prey system

Mukherjee

2023

Electron. J. Appl. Math.

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This paper analyses a diffusive predator-prey model consisting of a single prey species and two predator species with modified Leslie-Gower term Holling type II functional response subject to the homogeneous Neumann boundary condition. Local stability condition is derived by the application of Routh-Hurwitz criterion. Global asymptotic stability of the unique positive steady state is shown by constructing a suitable Lyapunov function when self diffusion is allowed where as non-constant positive steady states can exist due to the presence of cross-diffusion, that means, cross-diffusion can induce stationary pattern. Taking the cross diffusion as a bifurcation parameter, one can show the existence of positive non-constant solutions with the help of bifurcation theory. A brief conclusion completes the paper.

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“…The existence of all possible non‐hyperbolic positive equilibria and the dissipativity of the system are studied in this paper. Spatiotemporal dynamics of a prey–predator model with a Holling type III functional response involving Allee effect in prey and hunting cooperation in predator has been studied in Yadav et al 5 Luo 6 studied the existence of non‐constant positive steady states for a cross‐diffusion prey–predator system with the Beddington–DeAngelis and the Tanner functional response by the fixed point index theory. Cao et al 7 investigated the Turing instability and positive steady‐state solutions for a nonlinear cross‐diffusion prey–predator system under the Neumann boundary condition.…”

Section: Introductionmentioning

confidence: 99%

Steady‐state bifurcation and Hopf bifurcation in a cross‐diffusion prey–predator system with Ivlev functional response

Farshid

Jalilian

2022

Math Methods in App Sciences

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In this paper, we investigate bifurcations of stationary solutions of a cross‐diffusion prey–predator system with Ivlev functional response and Neumann boundary conditions. First, we analyze the local stability and existence of a Hopf bifurcation at a coexistence stationary solution. We show that the bifurcating periodic solutions are asymptotically orbitally stable and the bifurcation direction is supercritical when the ratio of the conversion of prey captured by predator to the death rate of predator is in the interval . Next, we derive sufficient conditions for the existence of a steady‐state bifurcation from a simple eigenvalue. We establish the existence of two intersecting curves of steady‐state solutions near the coexistence stationary solution. To illustrate our theoretical results, we give some numerical examples. From numerical simulations we observe that the coexistence stationary solution loses its stability via the Hopf bifurcation and a periodic solution emerges after a short time. Also by taking the bifurcation parameter value in the stable region, the effect of the initial condition disappears over time and the solution returns to the coexistence stationary solution.

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Steady state for a predator–prey cross-diffusion system with the Beddington–DeAngelis and Tanner functional response

Cited by 6 publications

References 15 publications

Nonlinear dynamics in a discrete-time predator-prey system

Nonlinear dynamics in a discrete-time predator-prey system

Dynamics of a diffusive two predators- one prey system

Steady‐state bifurcation and Hopf bifurcation in a cross‐diffusion prey–predator system with Ivlev functional response

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