Turing Bifurcation Research Articles

Predator-taxis creates spatial pattern of a predator-prey model

In this paper, we investigate the spatial pattern formation to a predator-prey model with the predator-taxis under the homogeneous zero-flux boundary conditions. Firstly, we employ the predator-taxis coefficient as the potential critical value of the Turing bifurcation to perform its role in forming the spatial pattern. One finds that there are multiple thresholds of the Turing bifurcation. Hereafter, we focus on the direction of the Turing bifurcation. To this end, the technique of the weakly nonlinear analysis is employed to deduce the amplitude equation near the threshold of the Turing bifurcation in one-dimensional space. It is found that the predator-taxis can create the subcritical or the supercritical Turing bifurcation. Finally, numerical experiments check the theoretical analysis well and obtain the spatial patterns with the different predator-taxis coefficients.

SPATIOTEMPORAL DYNAMICAL ANALYSIS OF A PREDATOR–PREY SYSTEM WITH FEAR AND GROUP DEFENSE IN PREY

This paper considers a diffusive prey–predator system with fear and group defense in the prey population. Also, we consider that the mortality of predators is linear and quadratic. By using local stability analysis, we get the prerequisite of Turing instability. Using comprehensive numerical computations, we get non-Turing pattern formation in the system with linear death of predator. Turing patterns are obtained for the system with the quadratic death of the predator. The modeling technique of multiple scale analysis is used to determine amplitude equations near the Turing bifurcation origin for the model with the predator’s quadratic mortality rate. The amplitude equations stability leads to various Turing patterns such as spots, stripes, and mixed. The result focuses on changing the mortality rate linear to quadratic of a predator in the prey–predator system. The derived results support us in a more immeasurable understanding of prey–predator interaction dynamics in the actual world.

Hopf-Bifurcation and Pattern Selections in a Three Trophic Level Food Web System

The ecosystem comprises many food webs, and their existence is directly dependent upon the growth rate of primary prey; it balances the whole ecosystem. This paper studies the temporal and spatiotemporal dynamics of three trophic levels of food web system, consisting of two preys and two predators. We first obtained an equilibrium solution set and studied the system’s stability at a biological feasible equilibrium point using a Jacobian method. We show the occurrence of Hopf-bifurcation by considering the growth rate of prey as the bifurcation parameter for the temporal model. In the presence of diffusion, we study random movement in species, establish conditions for the system’s stability, and derive the Turing instability condition. A multiple-scale analysis is used to determine the amplitude equations in the neighborhood of the Turing bifurcation point. After applying amplitude equations, the system has a rich dynamical behavior. The stability analysis of these amplitude equations leads to the development of various Turing patterns. Finally, with numerical simulations, the analytical results are verified. Within this framework, our study through the dynamical behavior of the complex system and bifurcation point based on the prey growth rate can serve as a baseline for numerous researchers working on ecological models from diverse perspectives. As a result, the Hopf-bifurcation and multiple-scale analysis used in the complex food web system is particularly relevant experimentally because the linked consequences may be researched and applied to many mathematical, ecological, and biological models.

Spatiotemporal dynamics of Leslie–Gower predator–prey model with Allee effect on both populations

In the present paper, we consider a reaction–diffusion system, based on the Leslie–Gower predator–prey model with Allee effect in both the predator and prey populations. The model is a generalization of the case where the individual population is subject to the Allee effect and it is flexible such that one can easily obtain the scenario with the Allee effect only on the prey population or, predator population. The non-spatial model was analyzed by using the positivity of solutions, uniformly boundedness and local stability of the interior equilibrium. The sufficient conditions for the instability with zero-flux boundary conditions are obtained for the spatial model. We have derived the analytical conditions for Hopf and Turing bifurcation on the spatial domain. We also observe that the system exhibits complex dynamics like Bogdanov–Takens (BT) bifurcation and generalized Hopf (GH) bifurcation. Our numerical simulations reveal that depending on the strength of Allee effects, the model dynamics exhibit stripes, spots, cold–hot spots–stripes coexistence and cold–hot spots patterns. We obtain very interesting characteristics of the reaction–diffusion model which reflects the fact that the system can develop patterns both inside and outside of the Turing parameter domain. Our study may enrich the field of the Allee effect and will help us to acquire a better understanding of the predator–prey interaction in a real environment.

Bifurcations in the diffusive Bazykin model

Dangerous tipping points and catastrophic transitions in ecosystems have recently been popular for detecting early warning signals in ecology. B-tipping is induced by bifurcation such as spatial pattern formation resulting from Turing instability. As one of the most important models in predator-prey interactions, we extend the Bazykin model to incorporate diffusive movement under homogeneous Neumann boundary conditions. For the local model, we provide some preliminary analysis on stability and Hopf bifurcation. For the reaction-diffusion model, we first improve some sufficient conditions for the local and global stability of a semi-trivial constant steady state or a unique positive constant steady state in Du and Lou (2001) [11]. Next we obtain the sufficient and necessary conditions for Turing instability, show the existence of Turing bifurcation, Hopf bifurcation, Turing-Turing bifurcation, Turing-Hopf bifurcation and Turing-Turing-Hopf bifurcation, and the nonexistence of triple-Turing bifurcation. Our results reveal that the model can exhibit complex spatial, temporal and spatiotemporal patterns, including complex regime shifts and critical transitions at bifurcation points, transient states (spatially inhomogeneous periodic solutions), tristability (a pair of non-constant steady states and a spatially homogeneous periodic solution), heteroclinic orbits (connecting a spatially inhomogeneous periodic solution to a non-constant steady state or a spatially homogeneous periodic solution, connecting a spatially homogeneous periodic solution to non-constant steady states and vice versa). Finally, numerical simulations illustrate complex dynamics and verify our theoretical results.

Pattern Dynamical Behaviors of One Type of Tree-Grass Model with Cross-Diffusion

One type of tree-grass competition model with cross-diffusion is analyzed in this paper. Firstly, according to the distribution theory of the roots of the characteristic equation, the existence and stability conditions of the positive equilibrium point are obtained, and the global stability of the system is proved by using the Bendixson–Dulac theorem. Secondly, we select the fire factor as the control parameter, and obtain the critical value of Turing bifurcation and the conditions of Turing pattern. Then, we derive the amplitude equation near the Turing bifurcation point by using the standard multiscale analysis method. Finally, we do a sensitivity analysis on the tree-grass system with the fire factor, and then a series of numerical simulations based on these analysis results are carried out to verify our theoretical analysis and Turing pattern structure, which is consistent with the biological significance of the tree-grass model.

Turing Instability of a Modified Reaction–Diffusion Holling–Tanner Model Over a Random Network

Turing instability is a prominent feature of reaction–diffusion systems, which is widely investigated in many fields, such as ecology, neurobiology, chemistry. However, although the inhomogeneous diffusion between prey and predators exist in their network space, there are few considerations on how network diffusion affects the stability of prey–predator models. Therefore, in this paper we study the pattern dynamics of a modified reaction–diffusion Holling–Tanner prey–predator model over a random network. Specifically, we study the relationship between the node degrees of the random network and the eigenvalues of the network Laplacian matrix. Then, we obtain conditions under which the network system instability, Hopf bifurcation as well as Turing bifurcation occur. Also, we find an approximate Turing instability region of the diffusion coefficient and the connection probability of the network. Finally, we apply the mean-field approximation theory with numerical simulation to confirm the correctness of our results. The instability region indicates the random migration of the prey and predators among different communities.

Cross-scale dynamic interactions in compacting porous media as a trigger to pattern formation

SUMMARYPatterns in nature are often interpreted as a product of reaction-diffusion processes which result in dissipative structures. Thermodynamic constraints allow prediction of the final state but the dynamic evolution of the microprocesses is hidden. We introduce a new microphysics-based approach that couples the microscale cross-constituent interactions to the large-scale dynamic behaviour, which leads to the discovery of a family of soliton-like excitation waves. These waves can appear in hydromechanically coupled porous media as a reaction to external stimuli. They arise, for instance, when mechanical forcing of the porous skeleton releases internal energy through a phase change, leading to tight coupling of the pressure in the solid matrix with the dissipation of the pore fluid pressure. In order to describe these complex multiscale interactions in a thermodynamic consistent framework, we consider a dual-continuum system, where the large-scale continuum properties of the matrix–fluid interaction are described by a reaction-self diffusion formulation, and the small-scale dissipation of internal energy by a reaction-cross diffusion formulation that spells out the macroscale reaction and relaxes the adiabatic constraint on the local reaction term in the conventional reaction-diffusion formalism. Using this approach, we recover the familiar Turing bifurcations (e.g. rhythmic metamorphic banding), Hopf bifurcations (e.g. Episodic Tremor and Slip) and present the new excitation wave phenomenon. The parametric space is investigated numerically and compared to serpentinite deformation in subduction zones.

Analysis of Turing patterns and amplitude equations in general forms under a reaction–diffusion rumor propagation system with Allee effect and time delay

In this paper, we divide the population into three groups: susceptible individuals (S), infectious individuals (I) and removed individuals (R), and propose a rumor propagation dynamic model with Allee effect and cross-diffusion. Next, we have analyzed a general form of cross-diffusion model with time delay, and drawn a general conclusion of linear stability analysis of Turing bifurcation. However, Turing bifurcation analysis cannot give the specific shapes of the patterns under certain conditions. With the help of the “Multiple Scale Analysis” method, we derive the expression of the amplitude equation for the general form of weakly nonlinear models. Finally, we apply the above theorems to the analysis of our previously proposed model, and derive the appearance condition of the Turing bifurcation and the expression of the amplitude equation respectively. Through the numerical simulations, we have verified the correctness of the above theoretical analysis.

Spatiotemporal pattern selection in a nontoxic-phytoplankton - toxic-phytoplankton - zooplankton model with toxin avoidance effects

In this research, considering the toxin avoidance effects of zooplankton to toxic phytoplankton in space as cross-diffusion terms, a modified model is proposed on the complex spatiotemporal dynamics of non-toxic phytoplankton, toxic phytoplankton and zooplankton system. In order to explore the toxin avoidance effects on pattern formation and selection, Turing bifurcation analysis and amplitude analysis are performed. The results of Turing analysis show that patterns can be formed when diffusion coefficient is less than a critical value. Amplitude equations are calculated and amplitude analysis show that stripe patterns, hexagon patterns and mixed patterns of stripes and hexagons can be theoretically predicted. Focusing on toxin avoidance coefficient (cross-diffusion coefficient) and diffusion coefficient of zooplankton, parameter space for pattern formation and selection is plotted. Numerical simulations on pattern formation are carried out to validate the theoretical analysis with five groups of parameters. With the increase of toxin avoidance effects, simulated patterns can be obtained as hexagons, mixed patterns of stripes and hexagons, stripes, mixed patterns of stripes and hexagons, and hexagons. Finally, taking stripe pattern as an example, simulated pattern is compared with predicted pattern. Simulation results are all consistent with theoretical analysis.

Outbreak spatial pattern formation based on an SI model with the infected cross-diffusion term

This study is to discuss the pattern formations of a spatial epidemic model with cross-diffusion of the susceptible and infected groups simultaneously. The infected cross-diffusion term described the situation that the infected was allowed to move to areas with high density of the susceptible such as for work or study, especially after the pandemic. Turing analysis was applied to the model and yielded the conditions for Turing instability corresponding to the model. The amplitude equations were also given by the support of multiple-scale analysis, which then provided information about the stability of the patterns near the Turing bifurcation point. Numerical simulations revealed that there were five types of patterns, such as the spots, spots-stripes, stripes, stripes-holes, and holes. The holes indicated a disease outbreak in a region, while the spots showed non-outbreak. Furthermore, numerical simulations were carried out by varying the cross-diffusion coefficients of the susceptible and infected. The simulation results showed once the cross-diffusion coefficient of the infected was bigger than the susceptible, then an outbreak in a region was triggered. The results of this study showed that the movement of infected had a significant role in the spread of an infectious disease that could lead to another wave of pandemic. By using Turing analysis as a tool, as well as predator-prey model as the basis of movement theory, this paper tries to fill in the gaps in the discussion about the movement of infected people to areas with high density of the susceptible.

Three-dimensional pattern dynamics of a fractional predator-prey model with cross-diffusion and herd behavior

In this paper, we study the pattern dynamics in a spatial fractional predator-prey model with cross fractional diffusion, herd behavior and prey refuge. In this model, herd behavior exists in the population of predators and the prey. The spatial dynamics of the system are obtained through appropriate threshold parameters, and a series of three-dimensional patterns are observed, such as tubes, planar lamellae and spherical droplets. Specifically, linear stability analysis is applied to obtain the conditions of Hopf bifurcation and Turing instability. Then, by utilizing the central manifold reduction theory analysis, the amplitude equation near the critical point of Turing bifurcation is deduced to study the selection and stability of pattern formation. The theoretical results are verified by numerical simulation.

Bogdanov-Takens bifurcation with $ Z_2 $ symmetry and spatiotemporal dynamics in diffusive Rosenzweig-MacArthur model involving nonlocal prey competition

<p style='text-indent:20px;'>A diffusive Rosenzweig-MacArthur model involving nonlocal prey competition is studied. Via considering joint effects of prey's carrying capacity and predator's diffusion rate, the first Turing (Hopf) bifurcation curve is precisely described, which can help to determine the parameter region where coexistence equilibrium is stable. Particularly, coexistence equilibrium can lose its stability through not only codimension one Turing (Hopf) bifurcation, but also codimension two Bogdanov-Takens, Turing-Hopf and Hopf-Hopf bifurcations, even codimension three Bogdanov-Takens-Hopf bifurcation, etc., thus the concept of Turing (Hopf) instability is extended to high codimension bifurcation instability, such as Bogdanov-Takens instability. To meticulously describe spatiotemporal patterns resulting from <inline-formula><tex-math id="M2">\begin{document}$ Z_2 $\end{document}</tex-math></inline-formula> symmetric Bogdanov-Takens bifurcation, the corresponding third-order normal form for partial functional differential equations (PFDEs) involving nonlocal interactions is derived, which is expressed concisely by original PFDEs' parameters, making it convenient to analyze effects of original parameters on dynamics and also to calculate normal form on computer. With the aid of these formulas, complex spatiotemporal patterns are theoretically predicted and numerically shown, including tri-stable nonuniform patterns with the shape of <inline-formula><tex-math id="M3">\begin{document}$ \cos \omega t\cos \frac{x}{l}- $\end{document}</tex-math></inline-formula>like or <inline-formula><tex-math id="M4">\begin{document}$ \cos \frac{x}{l}- $\end{document}</tex-math></inline-formula>like, which reflects the effects of nonlocal interactions, such as stabilizing spatiotemporal nonuniform patterns.</p>

DYNAMICS ANALYSIS OF THREE-SPECIES REACTION-DIFFUSION SYSTEM VIA THE MULTIPLE SCALE PERTURBATION METHOD

In this paper, the general analysis of spatiotemporal dynamics of three-species reaction-diffusion system induced by Turing bifurcation is given. Firstly, by employing the Routh-Hurwitz criterion the conditions for Turing bifurcation in three-species reaction-diffusion equations are obtained. Secondly, through the tool of the multiple scale perturbation method the amplitude equations of Turing patterns are also given. Finally, we take a three-species predator-prey model as an example to illustrate the application of these general theoretical results, and meanwhile carry out many numerical simulations to depict spots pattern, stripes pattern and labyrinthine pattern and demonstrate the validity of these theories.

Spatiotemporal pattern formation in a prey–predator model with generalist predator

Generalist predators exploit multiple food sources and it is economical for them to reduce predation pressure on a particular prey species when their density level becomes comparatively less. As a result, a prey-predator system tends to become more stable in the presence of a generalist predator. In this article, we investigate the roles of both the diffusion and nonlocal prey consumption in shaping the population distributions for interacting generalist predator and its focal prey species. In this regard, we first derive the conditions associated with Turing instability through linear analysis. Then, we perform a weakly nonlinear analysis and derive a cubic Stuart-Landau equation governing amplitude of the resulting patterns near Turing bifurcation boundary. Further, we present a wide variety of numerical simulations to corroborate our analytical findings as well as to illustrate some other complex spatiotemporal dynamics. Interestingly, our study reveals the existence of traveling wave solutions connecting two spatially homogeneous coexistence steady states in Turing domain under the influence of temporal bistability phenomenon. Also, our investigation shows that nonlocal prey consumption acts as a stabilizing force for the system dynamics.

Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain

<p style='text-indent:20px;'>In this article, Turing instability and the formations of spatial patterns for a general two-component reaction-diffusion system defined on 2D bounded domain, are investigated. By analyzing characteristic equation at positive constant steady states and further selecting diffusion rate <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula> and diffusion ratio <inline-formula><tex-math id="M2">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> as bifurcation parameters, sufficient and necessary conditions for the occurrence of Turing instability are established, which is called the first Turing bifurcation curve. Furthermore, parameter regions in which single-mode Turing patterns arise and multiple-mode (or superposition) Turing patterns coexist when bifurcations parameters are chosen, are described. Especially, the boundary of parameter region for the emergence of single-mode Turing patterns, consists of the first and the second Turing bifurcation curves which are given in explicit formulas. Finally, by taking diffusive Schnakenberg system as an example, parameter regions for the emergence of various kinds of spatially inhomogeneous patterns with different spatial frequencies and superposition Turing patterns, are estimated theoretically and shown numerically.</p>

Analytical detection of stationary turing pattern in a predator-prey system with generalist predator

A prey-predator model with Holling type-II functional response and a generalist predator exhibits complex dynamics in response to parameter variation. Generalist predators implicitly exploiting multiple food resources reduce predation pressure on their focal prey species that causes it to become more stable compared to a prey-predator system with specialist predator. In the temporal system, bistability and tristability are observed along with various global and local bifurcations. Existence of homogeneous and heterogeneous positive steady state solutions are shown to exist for suitable ranges of parameter values in the corresponding spatio-temporal diffusive system. Weakly nonlinear analysis, using multi-scale perturbation technique, is employed to derive amplitude equation for the stationary patterns near the Turing bifurcation threshold. The analytical results of the amplitude equations are validated using exhaustive numerical simulations. We also identify bifurcation of multiple stable stationary patch solutions as well as dynamic pattern solution for parameter values in the Turing and Turing-Hopf regions.

Spatiotemporal Dynamics in a Predator–Prey Model with Functional Response Increasing in Both Predator and Prey Densities

In this paper, a diffusive predator–prey system with a functional response that increases in both predator and prey densities is considered. By analyzing the characteristic roots of the partial differential equation system, the Turing instability and Hopf bifurcation are studied. In order to consider the dynamics of the model where the Turing bifurcation curve and the Hopf bifurcation curve intersect, we chose the diffusion coefficients d1 and β as bifurcating parameters. In particular, the normal form of Turing–Hopf bifurcation was calculated so that we could obtain the phase diagram. For parameters in each region of the phase diagram, there are different types of solutions, and their dynamic properties are extremely rich. In this study, we have used some numerical simulations in order to confirm these ideas.

Two different approaches for parameter identification in a spatial–temporal rumor propagation model based on Turing patterns

The reaction–diffusion model can well describe the dynamics of groups in space and time to help us to study their behavioral mechanism. Based on the rumor propagation process, an example of reaction–diffusion model defined on continuous space is proposed in this paper to study some common problems under Turing instability. Firstly, the necessary conditions for Turing instability are studied. Furthermore, the amplitude equation of the system is derived through multi-scale analysis and weak linear analysis, and the theoretical conditions for the appearance of hexagon pattern, stripe pattern and mixed structure pattern near Turing bifurcation are given. In the numerical simulation, we verify the correctness of the theoretical analysis and find that the cross-diffusion coefficient can change the pattern type. Finally, based on a statistical method and a convolutional neural network, we have carried out a parameter identification experiment of the system respectively. The results show that the second method performs well and the first method has relatively large error in some cases.

Turing pattern analysis of a reaction-diffusion rumor propagation system with time delay in both network and non-network environments

In our modern world, the invention of Internet enables information as well as rumors to spread in an unprecedented speed. To investigate the spatio-temporal dynamics of rumor propagation, we propose a reaction-diffusion rumor spreading system with time delay as well as its variation on complex network models. Necessary conditions of the rumor-spreading equilibrium point and both diffusion and delay induced Turing bifurcation around the rumor-spreading equilibrium point are analysed. We further conduct mass numerical simulations on various network structures and non-networks models. The effects of constant and periodic diffusion terms as well as different incidence coefficients on the shapes of patterns are explored. Moreover, a type of spiral patterns on ‘LA4’ and ‘LA12’ networks with large time delay are discovered. Finally, we collect real data of information propagation on twitter, and determine values of parameters in our model to make fitting. The fitting curve matches real data well, which implies the validity of our model.