Pattern formation by fractional cross-diffusion in a predator–prey model with Beddington–DeAngelis type functional response

2019 ◽  
Vol 33 (25) ◽  
pp. 1950296
Author(s):  
Naveed Iqbal ◽  
Ranchao Wu
Keyword(s):  
Stability Analysis ◽  
Functional Response ◽  
Equilibrium Points ◽  
Turing Instability ◽  
Turing Patterns ◽  
Scaling Analysis ◽  
Amplitude Equations ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
The Stability

In this paper, we explore the emergence of patterns in a fractional cross-diffusion model with Beddington–DeAngelis type functional response. First, we explore the stability of the equilibrium points with or without fractional cross-diffusion. Instability of equilibria can be induced by cross-diffusion. We perform the linear stability analysis to obtain the constraints for the Turing instability. It is found by theoretical analysis that cross-diffusion is an important mechanism for the appearance of Turing patterns. For the dynamics of pattern, the weakly nonlinear multi-scaling analysis has been performed to obtain the amplitude equations. Finally, we ensure the existence of Turing patterns such as squares, spots and stripes by using the stability analysis of the amplitude equations. Moreover, with the assistance of numerical simulations, we verify the theoretical results.

scholarly journals Pattern Formation in Predator-Prey Model with Delay and Cross Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Xinze Lian ◽  
Shuling Yan ◽  
Hailing Wang
Keyword(s):  
Time Delay ◽  
Pattern Formation ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model ◽  
Diffusion Controlled ◽  
The Stability ◽  
Self Replication

We consider the effect of time delay and cross diffusion on the dynamics of a modified Leslie-Gower predator-prey model incorporating a prey refuge. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: the model dynamics exhibits a delay and diffusion controlled formation growth not only to spots, stripes, and holes, but also to spiral pattern self-replication. The results indicate that time delay and cross diffusion play important roles in pattern formation.

Stripe and Spot Patterns for General Gierer–Meinhardt Model with Common Sources

2017 ◽  
Vol 27 (02) ◽  
pp. 1750018 ◽  
Author(s):  
You Li ◽  
Jinliang Wang ◽  
Xiaojie Hou
Keyword(s):  
Periodic Solution ◽  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Theoretical Analysis ◽  
Stable Equilibrium ◽  
Turing Instability ◽  
Turing Patterns ◽  
Stability Conditions ◽  
Ode System ◽  
The Stability

This paper focuses on the Turing patterns in the general Gierer–Meinhardt model of morphogenesis. The stability analysis of the equilibrium for the associated ODE system is carried out and the stability conditions are obtained. Furthermore, we perform a detailed Hopf bifurcation analysis for this system. The results show that the equilibrium undergoes a supercritical Hopf bifurcation in certain parameter range and the bifurcated limit cycle is stable. With added diffusions, we then show that both the stable equilibrium and the Hopf periodic solution experience Turing instability with unequal spatial diffusions and obtain the instability conditions. Numerical simulations are given to illustrate the theoretical analysis, which show that the Turing patterns are of either spot or stripe type.

TURING PATTERNS OF A PREDATOR–PREY MODEL WITH A MODIFIED LESLIE–GOWER TERM AND CROSS-DIFFUSIONS

2012 ◽  
Vol 05 (06) ◽  
pp. 1250060 ◽  
Author(s):  
GUANG-PING HU ◽  
XIAO-LING LI
Keyword(s):  
Turing Instability ◽  
Turing Patterns ◽  
Positive Equilibrium ◽  
Asymptotically Stable ◽  
Strongly Coupled ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey System ◽  
Positive Steady States

In this paper, a strongly coupled diffusive predator–prey system with a modified Leslie–Gower term is considered. We will show that under certain hypotheses, even though the unique positive equilibrium is asymptotically stable for the dynamics with diffusion, Turing instability can produce due to the presence of the cross-diffusion. In particular, we establish the existence of non-constant positive steady states of this system. The results indicate that cross-diffusion can create stationary patterns.

Cross-Diffusion Induced Turing Instability and Amplitude Equation for a Toxic-Phytoplankton–Zooplankton Model with Nonmonotonic Functional Response

2017 ◽  
Vol 27 (06) ◽  
pp. 1750088 ◽  
Author(s):  
Renji Han ◽  
Binxiang Dai
Keyword(s):  
Stability Analysis ◽  
Functional Response ◽  
Spatiotemporal Distribution ◽  
Spatiotemporal Pattern ◽  
Amplitude Equations ◽  
Toxic Phytoplankton ◽  
Cross Diffusion ◽  
The Cross ◽  
Nonmonotonic Functional Response ◽  
The Impact

The spatiotemporal pattern induced by cross-diffusion of a toxic-phytoplankton–zooplankton model with nonmonotonic functional response is investigated in this paper. The linear stability analysis shows that cross-diffusion is the key mechanism for the formation of spatial patterns. By taking cross-diffusion rate as bifurcation parameter, we derive amplitude equations near the Turing bifurcation point for the excited modes in the framework of a weakly nonlinear theory, and the stability analysis of the amplitude equations interprets the structural transitions and stability of various forms of Turing patterns. Furthermore, we illustrate the theoretical results via numerical simulations. It is shown that the spatiotemporal distribution of the plankton is homogeneous in the absence of cross-diffusion. However, when the cross-diffusivity is greater than the critical value, the spatiotemporal distribution of all the plankton species becomes inhomogeneous in spaces and results in different kinds of patterns: spot, stripe, and the mixture of spot and stripe patterns depending on the cross-diffusivity. Simultaneously, the impact of toxin-producing rate of toxic-phytoplankton (TPP) species and natural death rate of zooplankton species on pattern selection is also explored.

Turing Instability and Bifurcation in a Diffusion Predator–Prey Model with Beddington–DeAngelis Functional Response

2018 ◽  
Vol 28 (09) ◽  
pp. 1830029 ◽  
Author(s):  
Wei Tan ◽  
Wenwu Yu ◽  
Tasawar Hayat ◽  
Fuad Alsaadi ◽  
Habib M. Fardoun
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Steady State Solution ◽  
Turing Instability ◽  
Rigorous Analysis ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Global Positive Solution ◽  
The Stability

In this paper, we consider a predator–prey model with Beddington–DeAngelis functional response with or without diffusion. For this system, we give a complete and rigorous analysis of the dynamics including the existence of a global positive solution, the stability/Turing instability and the Hopf bifurcation. In the meanwhile, we show, via numerical simulations, that there appears Hopf bifurcation, steady state solution and Turing–Hopf bifurcation with the changes of some parameters of the system.

Spatial patterns created by cross-diffusion for a three-species food chain model

2014 ◽  
Vol 07 (02) ◽  
pp. 1450013 ◽  
Author(s):  
Canrong Tian ◽  
Zhi Ling ◽  
Zhigui Lin
Keyword(s):  
Stability Analysis ◽  
Numerical Simulations ◽  
Spatial Patterns ◽  
Food Chain ◽  
Turing Instability ◽  
Steady States ◽  
Chain Model ◽  
Cross Diffusion ◽  
The Stability ◽  
Food Chain Model

This paper deals with the stability analysis to a three-species food chain model with cross-diffusion, the results of which show that there is no Turing instability but cross-diffusion makes the model instability possible. We then show that the spatial patterns are spotted patterns by using numerical simulations. In order to understand why the spatial patterns happen, the existence of the nonhomogeneous steady states is investigated. Finally, using the Leray–Schauder theory, we demonstrate that cross-diffusion creates nonhomogeneous stationary patterns.

EMERGENCE OF OSCILLATORY TURING PATTERNS INDUCED BY CROSS DIFFUSION IN A PREDATOR–PREY SYSTEM

2012 ◽  
Vol 26 (31) ◽  
pp. 1250193 ◽  
Author(s):  
AN-WEI LI ◽  
ZHEN JIN ◽  
LI LI ◽  
JIAN-ZHONG WANG
Keyword(s):  
Turing Instability ◽  
Turing Patterns ◽  
Wave Instability ◽  
Predator Prey ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Oscillating Solutions

In this paper, we presented a predator–prey model with self diffusion as well as cross diffusion. By using theory on linear stability, we obtain the conditions on Turing instability. The results of numerical simulations reveal that oscillating Turing patterns with hexagons arise in the system. And the values of the parameters we choose for simulations are outside of the Turing domain of the no cross diffusion system. Moreover, we show that cross diffusion has an effect on the persistence of the population, i.e., it causes the population to run a risk of extinction. Particularly, our results show that, without interaction with either a Hopf or a wave instability, the Turing instability together with cross diffusion in a predator–prey model can give rise to spatiotemporally oscillating solutions, which well enrich the finding of pattern formation in ecology.

scholarly journals Bifurcation and Chaos of a Discrete Predator-Prey Model with Crowley–Martin Functional Response Incorporating Proportional Prey Refuge

2020 ◽  
Vol 2020 ◽  
pp. 1-18 ◽  
Author(s):  
P. K. Santra ◽  
G. S. Mahapatra ◽  
G. R. Phaijoo
Keyword(s):  
Functional Response ◽  
Control Method ◽  
Equilibrium Points ◽  
Period Doubling ◽  
Phase Portraits ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Feedback Control Method ◽  
And Control

The paper investigates the dynamical behaviors of a two-species discrete predator-prey system with Crowley–Martin functional response incorporating prey refuge proportional to prey density. The existence of equilibrium points, stability of three fixed points, period-doubling bifurcation, Neimark–Sacker bifurcation, Marottos chaos, and Control Chaos are analyzed for the discrete-time domain. The time graphs, phase portraits, and bifurcation diagrams are obtained for different parameters of the model. Numerical simulations and graphics show that the discrete model exhibits rich dynamics, which also present that the system is a chaotic and complex one. This paper attempts to present a feedback control method which can stabilize chaotic orbits at an unstable equilibrium point.

scholarly journals Stability and Bifurcation of a Prey-Predator-Scavenger Model in the Existence of Toxicant and Harvesting

2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Huda Abdul Satar ◽  
Raid Kamel Naji
Keyword(s):  
Numerical Simulation ◽  
Stability Analysis ◽  
Food Web ◽  
Equilibrium Points ◽  
Local Bifurcation ◽  
Persistence Conditions ◽  
Stability And Bifurcation ◽  
Periodic Dynamics ◽  
The Stability ◽  
Food Web Model

In this paper a prey-predator-scavenger food web model is proposed and studied. It is assumed that the model considered the effect of harvesting and all the species are infected by some toxicants released by some other species. The stability analysis of all possible equilibrium points is discussed. The persistence conditions of the system are established. The occurrence of local bifurcation around the equilibrium points is investigated. Numerical simulation is used and the obtained solution curves are drawn to illustrate the results of the model. Finally, the nonexistence of periodic dynamics is discussed analytically as well as numerically.

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