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.

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.

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 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.

scholarly journals Exploring Spatiotemporal Complexity of a Predator-Prey System with Migration and Diffusion by a Three-Chain Coupled Map Lattice

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-19 ◽  
Author(s):  
Tousheng Huang ◽  
Huayong Zhang ◽  
Xuebing Cong ◽  
Ge Pan ◽  
Xiumin Zhang ◽  
...  
Keyword(s):  
Turing Instability ◽  
Coupled Map Lattice ◽  
Self Organization ◽  
Turing Patterns ◽  
Predator Prey ◽  
Migration Rates ◽  
Self Organized ◽  
Prey System ◽  
And Migration ◽  
And Diffusion

The topic of utilizing coupled map lattice to investigate complex spatiotemporal dynamics has attracted a lot of interest. For exploring the spatiotemporal complexity of a predator-prey system with migration and diffusion, a new three-chain coupled map lattice model is developed in this research. Based on Turing instability analysis, pattern formation conditions for the predator-prey system are derived. Via numerical simulation, rich Turing patterns are found with subtle self-organized structures under diffusion-driven and migration-driven mechanisms. With the variation of migration rates, the predator-prey system exhibits a gradual dynamical transition from diffusion-driven patterns to migration-driven patterns. Moreover, new results, the self-organization of non-Turing patterns, are also revealed. We find that even in the cases where the nonspatial predator-prey system reaches collapse, the migration can still drive pattern self-organization. These non-Turing patterns suggest many new possible ways for the coexistence of predator and prey in space, under the effects of migration and diffusion.

SPATIAL PATTERN IN A PREDATOR-PREY SYSTEM WITH BOTH SELF- AND CROSS-DIFFUSION

2009 ◽  
Vol 20 (01) ◽  
pp. 71-84 ◽  
Author(s):  
GUI-QUAN SUN ◽  
ZHEN JIN ◽  
YI-GUO ZHAO ◽  
QUAN-XING LIU ◽  
LI LI
Keyword(s):  
Spatial Patterns ◽  
Temporal Dynamics ◽  
Spatial Scales ◽  
Natural Populations ◽  
Spatial Dynamics ◽  
Density Variation ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Prey System ◽  
Dynamics Of Population

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. And spatial patterns are ubiquitous in nature, which can modify the temporal dynamics and stability properties of population densities at a range of spatial scales. Thus, in this paper, a predator-prey system with Michaelis-Menten-type functional response and self- and cross-diffusion is investigated. Based on the mathematical analysis, we obtain the condition of the emergence of spatial patterns through diffusion instability, i.e., Turing pattern. A series of numerical simulations reveal that the typical dynamics of population density variation is the formation of isolated groups, i.e., stripe-like or spotted or coexistence of both. The obtained results show that the interaction of self-diffusion and cross-diffusion plays an important role on the pattern formation of the predator-prey system.

Turing instability in a diffusive SIS epidemiological model

2015 ◽  
Vol 08 (01) ◽  
pp. 1550006 ◽  
Author(s):  
Shaban Aly ◽  
Houari B. Khenous ◽  
Fatma Hussien
Keyword(s):  
Infectious Diseases ◽  
Modeling And Simulation ◽  
Spatial Patterns ◽  
Disease Model ◽  
Turing Instability ◽  
Nonlinear Incidence Rate ◽  
Self Diffusion ◽  
Cross Diffusion ◽  
The Stability ◽  
Theoretical Results

Modeling and simulation of infectious diseases help to predict the likely outcome of an epidemic. In this paper, a spatial susceptible-infective-susceptible (SIS) type of epidemiological disease model with self- and cross-diffusion are investigated. We study the effect of diffusion on the stability of the endemic equilibrium with disease-induced mortality and nonlinear incidence rate. In the absence of diffusion the stationary solution stays stable but becomes unstable with respect to diffusion and that Turing instability takes place. We show that a standard (self-diffusion) system may be either stable or unstable, cross-diffusion response can stabilize an unstable standard system or decrease a Turing space (the space which the emergence of spatial patterns is holding) compared to the Turing space with self-diffusion, i.e. the cross-diffusion response is an important factor that should not be ignored when pattern emerges. Numerical simulations are provided to illustrate and extend the theoretical results.

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 Pattern Formation in a Semi-Ratio-Dependent Predator-Prey System with Diffusion

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Hunki Baek ◽  
Dong Ick Jung ◽  
Zhi-wen Wang
Keyword(s):  
Numerical Simulations ◽  
Initial Conditions ◽  
Random Perturbation ◽  
Turing Instability ◽  
Wave Pattern ◽  
Spatiotemporal Dynamics ◽  
Turing Patterns ◽  
Predator Prey ◽  
Prey System ◽  
Ratio Dependent

We investigate spatiotemporal dynamics of a semi-ratio-dependent predator-prey system with reaction-diffusion and zero-flux boundary. We obtain the conditions for Hopf, Turing, and wave bifurcations of the system in a spatial domain by making use of the linear stability analysis and the bifurcation analysis. In addition, for an initial condition which is a small amplitude random perturbation around the steady state, we classify spatial pattern formations of the system by using numerical simulations. The results of numerical simulations unveil that there are various spatiotemporal patterns including typical Turing patterns such as spotted, spot-stripelike mixtures and stripelike patterns thanks to the Turing instability, that an oscillatory wave pattern can be emerged due to the Hopf and wave instability, and that cooperations of Turing and Hopf instabilities can cause occurrence of spiral patterns instead of typical Turing patterns. Finally, we discuss spatiotemporal dynamics of the system for several different asymmetric initial conditions via numerical simulations.

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