Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect

2016 ◽  
Vol 275 ◽  
pp. 1-12 ◽  
Author(s):  
Yahong Peng ◽  
Tonghua Zhang
Keyword(s):  
Allee Effect ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Prey System

scholarly journals Neimark-Sacker-Turing Instability and Pattern Formation in a Spatiotemporal Discrete Predator-Prey System with Allee Effect

2018 ◽  
Vol 2018 ◽  
pp. 1-17
Author(s):  
Huayong Zhang ◽  
Xuebing Cong ◽  
Tousheng Huang ◽  
Shengnan Ma ◽  
Ge Pan
Keyword(s):  
Pattern Formation ◽  
Allee Effect ◽  
Initial Conditions ◽  
Turing Instability ◽  
Spatiotemporal Chaos ◽  
Closed Curves ◽  
Predator Prey ◽  
Prey System ◽  
Route To Chaos ◽  
The Rich

A spatiotemporal discrete predator-prey system with Allee effect is investigated to learn its Neimark-Sacker-Turing instability and pattern formation. Based on the occurrence of stable homogeneous stationary states, conditions for Neimark-Sacker bifurcation and Turing instability are determined. Numerical simulations reveal that Neimark-Sacker bifurcation triggers a route to chaos, with the emergence of invariant closed curves, periodic orbits, and chaotic attractors. The occurrence of Turing instability on these three typical dynamical behaviors leads to the formation of heterogeneous patterns. Under the effects of Neimark-Sacker-Turing instability, pattern evolution process is sensitive to tiny changes of initial conditions, suggesting the occurrence of spatiotemporal chaos. With application of deterministic initial conditions, transient symmetrical patterns are observed, demonstrating that ordered structures can exist in chaotic processes. Moreover, when local kinetics of the system goes further on the route to chaos, the speed of symmetry breaking becomes faster, leading to more fragmented and more disordered patterns at the same evolution time. The rich spatiotemporal complexity provides new comprehension on predator-prey coexistence in the ways of spatiotemporal chaos.

scholarly journals Turing Patterns in a Predator-Prey System with Self-Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Hongwei Yin ◽  
Xiaoyong Xiao ◽  
Xiaoqing Wen
Keyword(s):  
Spatial Patterns ◽  
Nonlinear Diffusion ◽  
Turing Instability ◽  
Turing Patterns ◽  
Predator Prey ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Prey System ◽  
Diffusion Parameters ◽  
Normal Diffusion

For a predator-prey system, cross-diffusion has been confirmed to emerge Turing patterns. However, in the real world, the tendency for prey and predators moving along the direction of lower density of their own species, called self-diffusion, should be considered. For this, we investigate Turing instability for a predator-prey system with nonlinear diffusion terms including the normal diffusion, cross-diffusion, and self-diffusion. A sufficient condition of Turing instability for this system is obtained by analyzing the linear stability of spatial homogeneous equilibrium state of this model. A series of numerical simulations reveal Turing parameter regions of the interaction of diffusion parameters. According to these regions, we further demonstrate dispersion relations and spatial patterns. Our results indicate that self-diffusion plays an important role in the spatial patterns.

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.

scholarly journals Spatiotemporal Pattern in a Self- and Cross-Diffusive Predation Model with the Allee Effect

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Feng Rao
Keyword(s):  
Allee Effect ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Reaction Diffusion Model ◽  
Flux Boundary Conditions ◽  
Prey Interaction

This paper proposes and analyzes a mathematical model for a predator-prey interaction with the Allee effect on prey species and with self- and cross-diffusion. The effect of diffusion which can drive the model with zero-flux boundary conditions to Turing instability is investigated. We present numerical evidence of time evolution of patterns controlled by self- and cross-diffusion in the model and find that the model dynamics exhibits a cross-diffusion controlled formation growth to spotted and striped-like coexisting and spotted pattern replication. Moreover, we discuss the effect of cross-diffusivity on the stability of the nontrivial equilibrium of the model, which depends upon the magnitudes of the self- and cross-diffusion coefficients. The obtained results show that cross-diffusion plays an important role in the pattern formation of the predator-prey model. It is also useful to apply the reaction-diffusion model to reveal the spatial predation in the real world.

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 Characterizing the Effects of Self- and Cross-Diffusion on Stationary Patterns of a Predator–Prey System

2020 ◽  
Vol 30 (03) ◽  
pp. 2050041
Author(s):  
Jia-Fang Zhang ◽  
Hong-Bo Shi ◽  
Anotida Madzvamuse
Keyword(s):  
Coupled System ◽  
Turing Instability ◽  
Diffusion System ◽  
Positive Equilibrium ◽  
Predator Prey ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
Prey System ◽  
Nonlinear Coupled System ◽  
The Stability

In this paper, we study a nonlinear coupled predator–prey diffusion system which widely exists in ecosystem. It is found that the self-diffusion and cross-diffusion do not change the stability of the semi-equilibrium point of the corresponding predator–prey system. However, the two kinds of diffusion play an important role on the positive equilibrium, in virtue of which Turing instability of the corresponding diffusion system either continues to exist or disappears and becomes stable. On the stationary patterns of the nonlinear coupled system, we find some interesting results which differ from the phenomenon found in corresponding diffusion system. Strong cross-diffusion can make the corresponding system generate stationary patterns. Finally, numerical simulation is also done to verify the existence of the effects of self-diffusion and cross-diffusion.

Turing Instability and Pattern Formation in a Strongly Coupled Diffusive Predator–Prey System

2020 ◽  
Vol 30 (08) ◽  
pp. 2030020 ◽  
Author(s):  
Guangping Hu ◽  
Zhaosheng Feng
Keyword(s):  
Dynamical Behavior ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Wave Pattern ◽  
Strongly Coupled ◽  
Predator Prey ◽  
Reaction Diffusion Systems ◽  
Cross Diffusion ◽  
Prey System ◽  
Turing Instabilities

We are concerned with the Turing instability and pattern caused by cross-diffusion in a strongly coupled spatial predator–prey system. We explore how cross-diffusion destabilizes the spatially uniform steady state which is stable in reaction–diffusion systems, and explicitly describe the Turing space under certain conditions. Particularly, when the parameter values are taken in the Turing–Hopf domain, in which the spatiotemporal dynamical behavior is influenced by both Hopf and Turing instabilities, we investigate the formation of all possible patterns, including non-Turing structures such as wave pattern, competing dynamics as well as stationary Turing pattern. Furthermore, numerical simulations are illustrated to verify our theoretical findings.

scholarly journals Complex Dynamical Behavior of a Predator-Prey System with Group Defense

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jianglin Zhao ◽  
Min Zhao ◽  
Hengguo Yu
Keyword(s):  
Numerical Simulation ◽  
Theoretical Basis ◽  
Dynamical Behavior ◽  
Turing Instability ◽  
Equilibrium Density ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
Group Defense ◽  
Complex Dynamical Behavior

A diffusive predator-prey system with prey refuge is studied analytically and numerically. The Turing bifurcation is analyzed in detail, which in turn provides a theoretical basis for the numerical simulation. The influence of prey refuge and group defense on the equilibrium density and patterns of species under the condition of Turing instability is explored by numerical simulations, and this shows that the prey refuge and group defense have an important effect on the equilibrium density and patterns of species. Moreover, it can be obtained that the distributions of species are more sensitive to group defense than prey refuge. These results are expected to be of significance in exploration for the spatiotemporal dynamics of ecosystems.

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