Pattern Dynamics of Prey–Predator Model with Swarm Behavior via Turing Instability and Amplitude Equation

2021 ◽  
pp. 275-285
Author(s):  
Teekam Singh ◽  
Shivam ◽  
Mukesh Kumar ◽  
Vrince Vimal
Keyword(s):  
Turing Instability ◽  
Amplitude Equation ◽  
Pattern Dynamics ◽  
Swarm Behavior ◽  
Predator Model

scholarly journals Turing Instability and Amplitude Equation of Reaction-Diffusion System with Multivariable

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Qianqian Zheng ◽  
Jianwei Shen
Keyword(s):  
Reaction Diffusion ◽  
Turing Instability ◽  
Amplitude Equation ◽  
Diffusion System ◽  
Matrix Analysis ◽  
Turing Bifurcation ◽  
Pattern Dynamics ◽  
Novel Approach ◽  
Multivariate Pattern ◽  
Theoretical Results

In this paper, we investigate pattern dynamics with multivariable by using the method of matrix analysis and obtain a condition under which the system loses stability and Turing bifurcation occurs. In addition, we also derive the amplitude equation with multivariable. This is an effective tool to investigate multivariate pattern dynamics. The example and simulation used in this paper validate our theoretical results. The method presented is a novel approach to the investigation of specific real systems based on the model developed in this paper.

Pattern Dynamics in a Spatial Predator–Prey Model with Nonmonotonic Response Function

2018 ◽  
Vol 28 (06) ◽  
pp. 1850077 ◽  
Author(s):  
Xiaoling Li ◽  
Guangping Hu ◽  
Zhaosheng Feng
Keyword(s):  
Response Function ◽  
Turing Instability ◽  
Spatiotemporal Pattern ◽  
Weakly Nonlinear ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Pattern Dynamics ◽  
Spatial Systems ◽  
Different Types ◽  
Selection Of

In this paper, we study a diffusive predator–prey system with the nonmonotonic response function. The conditions on Hopf bifurcation and Turing instability of spatial systems are obtained. Near the Turing bifurcation point we apply the weakly nonlinear analysis to derive the amplitude equations of stationary pattern, to investigate the selection of spatiotemporal pattern. It shows that different types of patterns will occur in the model under various conditions. Numerical simulations agree well with our theoretical analysis when control parameters are in the Turing space. This study may provide some deep insights into the formation and selection of patterns for diffusive predator–prey systems.

scholarly journals Pattern Dynamics of Nonlocal Delay SI Epidemic Model with the Growth of the Susceptible following Logistic Mode

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Zun-Guang Guo ◽  
Jing Li ◽  
Can Li ◽  
Juan Liang ◽  
Yiwei Yan
Keyword(s):  
Time Delay ◽  
Epidemic Model ◽  
Turing Instability ◽  
Average Density ◽  
Linear Stability Theory ◽  
Susceptible Population ◽  
Nonlocal Delay ◽  
Pattern Dynamics ◽  
And Control ◽  
Theoretical Results

In this paper, we investigate pattern dynamics of a nonlocal delay SI epidemic model with the growth of susceptible population following logistic mode. Applying the linear stability theory, the condition that the model generates Turing instability at the endemic steady state is analyzed; then, the exact Turing domain is found in the parameter space. Additionally, numerical results show that the time delay has key effect on the spatial distribution of the infected, that is, time delay induces the system to generate stripe patterns with different spatial structures and affects the average density of the infected. The numerical simulation is consistent with the theoretical results, which provides a reference for disease prevention and control.

scholarly journals Bifurcation and Patterns Analysis for a Spatiotemporal Discrete Gierer-Meinhardt System

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 243
Author(s):  
Biao Liu ◽  
Ranchao Wu
Keyword(s):  
Bifurcation Theory ◽  
Center Manifold ◽  
Turing Instability ◽  
Flip Bifurcation ◽  
Center Manifold Theory ◽  
Pattern Dynamics ◽  
Instability Theory ◽  
Pattern Formations ◽  
Manifold Theory ◽  
Theoretical Results

The Gierer-Meinhardt system is one of the prototypical pattern formation models. The bifurcation and pattern dynamics of a spatiotemporal discrete Gierer-Meinhardt system are investigated via the couple map lattice model (CML) method in this paper. The linear stability of the fixed points to such spatiotemporal discrete system is analyzed by stability theory. By using the bifurcation theory, the center manifold theory and the Turing instability theory, the Turing instability conditions in flip bifurcation and Neimark–Sacker bifurcation are considered, respectively. To illustrate the above theoretical results, numerical simulations are carried out, such as bifurcation diagram, maximum Lyapunov exponents, phase orbits, and pattern formations.

Pattern Dynamics in a Predator–Prey Model with Schooling Behavior and Cross-Diffusion

2019 ◽  
Vol 29 (11) ◽  
pp. 1950146
Author(s):  
Wen Wang ◽  
Shutang Liu ◽  
Zhibin Liu ◽  
Da Wang
Keyword(s):  
Spatial Dynamics ◽  
Multiple Scale ◽  
Turing Instability ◽  
Predator Prey ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Schooling Behavior ◽  
Pattern Dynamics ◽  
Prey Model ◽  
Pattern Formations

In this paper, a diffusive predator–prey model is considered in which the predator and prey populations both exhibit schooling behavior. The system’s spatial dynamics are captured via a suitable threshold parameter, and a sequence of spatiotemporal patterns such as hexagons, stripes and a mixture of the two are observed. Specifically, the linear stability analysis is applied to obtain the conditions for Hopf bifurcation and Turing instability. Then, employing the multiple-scale analysis, the amplitude equations near the critical point of Turing bifurcation are derived, through which the selection and stability of pattern formations are investigated. The theoretical results are verified by numerical simulations.

scholarly journals Delay-Induced Self-Organization Dynamics in a Prey-Predator Network with Diffusion

2021 ◽  
Author(s):  
Qing Hu ◽  
Jianwei Shen
Keyword(s):  
Time Delays ◽  
Random Network ◽  
Turing Instability ◽  
Self Organization ◽  
Network System ◽  
Turing Pattern ◽  
Network Connection ◽  
Pattern Dynamics ◽  
Connection Probability ◽  
The Stability

Abstract Time delays can induce the loss of stability and degradation of performance. In this paper, the pattern dynamics of a prey-predator network with diffusion and delay are investigated, where the inhomogeneous distribution of species in space can be viewed as a random network, and delay can affect the stability of the network system. Our results show that time delay can induce the emergence of Hopf and Turing bifurcations, which are independent of the network, and the conditions of bifurcation are derived by linear stability analysis. Moreover, we find that the Turing pattern can be related to the network connection probability. The Turing instability region involving delay and network connection probability is obtained. Finally, the numerical simulation verifies our results.

Pattern Dynamics in a Diffusive Gierer–Meinhardt Model

2020 ◽  
Vol 30 (12) ◽  
pp. 2030035
Author(s):  
Mengxin Chen ◽  
Ranchao Wu ◽  
Liping Chen
Keyword(s):  
Dimensional Space ◽  
Turing Instability ◽  
Spatiotemporal Chaos ◽  
Wave Number ◽  
Positive Equilibrium ◽  
Stripe Pattern ◽  
Weakly Nonlinear ◽  
Pattern Dynamics ◽  
Inhibitor System ◽  
Eckhaus Instability

The purpose of the present paper is to investigate the pattern formation and secondary instabilities, including Eckhaus instability and zigzag instability, of an activator–inhibitor system, known as the Gierer–Meinhardt model. Conditions on the Hopf bifurcation and the Turing instability are obtained through linear stability analysis at the unique positive equilibrium. Then, the method of weakly nonlinear analysis is used to derive the amplitude equations. Especially, by adding a small disturbance to the Turing instability critical wave number, the spatiotemporal Newell–Whitehead–Segel equation of the stripe pattern is established. It is found that Eckhaus instability and zigzag instability may occur under certain conditions. Finally, Turing and non-Turing patterns are obtained via numerical simulations, including spotted patterns, mixed patterns, Eckhaus patterns, spatiotemporal chaos, nonconstant steady state solutions, spatially homogeneous periodic solutions and spatially inhomogeneous solutions in two-dimensional or one-dimensional space. Theoretical analysis and numerical results are in good agreement for this diffusive Gierer–Meinhardt model.

scholarly journals Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 101 ◽  
Author(s):  
Kalyan Manna ◽  
Vitaly Volpert ◽  
Malay Banerjee
Keyword(s):  
Prey Species ◽  
Population Distribution ◽  
Chaotic Oscillation ◽  
Turing Instability ◽  
Transient Dynamics ◽  
Nonlocal Interaction ◽  
Interaction Terms ◽  
Population Distributions ◽  
Predator Model ◽  
Spatiotemporal Models

Investigation of interacting populations is an active area of research, and various modeling approaches have been adopted to describe their dynamics. Mathematical models of such interactions using differential equations are capable to mimic the stationary and oscillating (regular or irregular) population distributions. Recently, some researchers have paid their attention to explain the consequences of transient dynamics of population density (especially the long transients) and able to capture such behaviors with simple models. Existence of multiple stationary patches and settlement to a stable distribution after a long quasi-stable transient dynamics can be explained by spatiotemporal models with nonlocal interaction terms. However, the studies of such interesting phenomena for three interacting species are not abundant in literature. Motivated by these facts here we have considered a three species prey–predator model where the predator is generalist in nature as it survives on two prey species. Nonlocalities are introduced in the intra-specific competition terms for the two prey species in order to model the accessibility of nearby resources. Using linear analysis, we have derived the Turing instability conditions for both the spatiotemporal models with and without nonlocal interactions. Validation of such conditions indicates the possibility of existence of stationary spatially heterogeneous distributions for all the three species. Existence of long transient dynamics has been presented under certain parametric domain. Exhaustive numerical simulations reveal various scenarios of stabilization of population distribution due to the presence of nonlocal intra-specific competition for the two prey species. Chaotic oscillation exhibited by the temporal model is significantly suppressed when the populations are allowed to move over their habitat and prey species can access the nearby resources.

scholarly journals Turing Instability of Brusselator in the Reaction-Diffusion Network

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Yansu Ji ◽  
Jianwei Shen
Keyword(s):  
Diffusion Coefficient ◽  
Random Network ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spontaneous Generation ◽  
Brusselator Model ◽  
Network Connection ◽  
Pattern Dynamics ◽  
Connection Probability ◽  
And Diffusion

Turing instability constitutes a universal paradigm for the spontaneous generation of spatially organized patterns, especially in a chemical reaction. In this paper, we investigated the pattern dynamics of Brusselator from the view of complex networks and considered the interaction between diffusion and reaction in the random network. After a detailed theoretical analysis, we obtained the approximate instability region about the diffusion coefficient and the connection probability of the random network. In the meantime, we also obtained the critical condition of Turing instability in the network-organized system and found that how the network connection probability and diffusion coefficient affect the reaction-diffusion system of the Brusselator model. In the end, the reason for arising of Turing instability in the Brusselator with the random network was explained. Numerical simulation verified the theoretical results.

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