Conditions For Turing Instability Research Articles

Exploring cooperative hunting dynamics and PRCC analysis: insights from a spatio-temporal mathematical model

The proposed mathematical model explores the intricate dynamics of a predator-prey system involving prey infection and cooperative hunting of predators. The model incorporates habitat complexity, emphasizing its influence on ecological interactions. The well-posedness of the system has rigorously been examined in a temporal setting and also conducted stability analysis. The bifurcation analysis reveals the existence of several local bifurcations on the system, namely transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation. Furthermore, these investigations delineate the two-dimensional bifurcations including Bogdanov–Takens and cusp bifurcations for different parametric combinations. With suitable choices of parameter values, the proposed model exhibits diverse dynamic phenomena, including bistable and tri-stable behavior. Latin hypercube sampling is utilized to conduct uncertainty analysis on input parameters, aiming to observe their effects on population dynamics. Subsequently, Kendall’s tau and Spearman’s rank correlation coefficients are also computed to investigate the impact of these uncertainties on the population. In the later part, a spatio-temporal system is proposed with two-dimensional diffusion terms to obtain the conditions for Turing instability. Numerical simulations have been conducted to observe the emergence of spatial patterns and the impact of predator cooperation in these patterns. The study provides valuable insights into the dynamics of complex ecological systems, emphasizing the interplay of spatial and temporal factors in shaping population dynamics and predator-prey interactions.

Nonlinear lattices from the physics of ecosystems: The Lefever–Lejeune nonlinear lattice in ℤ2$$ {\mathrm{\mathbb{Z}}}^2 $$

We argue that the spatial discretization of the strongly nonlinear Lefever–Lejeune partial differential equation defines a nonlinear lattice that is physically relevant in the context of the nonlinear physics of ecosystems, modelling the dynamics of vegetation densities in dry lands. We study the system in the lattice , which is especially relevant because of its natural dimension for the emergence of pattern formation. Theoretical results identify parametric regimes for the system that distinguish between extinction and potential convergence to nontrivial states. Importantly, we analytically identify conditions for Turing instability, detecting thresholds on the discretization parameter for the manifestation of this mechanism. Numerical simulations reveal the sharpness of the analytical conditions for instability and illustrate the rich potential for pattern formation even in the strongly discrete regime, emphasizing the importance of the interplay between higher dimensionality and discreteness.

Turing instability analysis and parameter identification based on optimal control and statistics method for a rumor propagation system.

This study establishes a reaction-diffusion system to capture the dynamics of rumor propagation, considering two possibilities of contact transmission. The sufficient and necessary conditions for a positive equilibrium point are provided, and the Turing instability conditions for this equilibrium point are derived. Furthermore, utilizing variational inequalities, a first-order necessary condition for parameter identification based on optimal control is established. During the numerical simulation process, the correctness of the Turing instability conditions is verified, and optimal control-based parameter identification is applied to the target pattern. Additionally, statistical methods are employed for pattern parameter identification. The identification results demonstrate that optimal control-based parameter identification exhibits higher efficiency and accuracy. Finally, both theories' parameter identification principles are extended to a small-world network, yielding consistent conclusions with continuous space.

Complex dynamic analysis of a reaction–diffusion predator–prey model in the network and non-network environment

This paper considers an improved two-dimensional reaction–diffusion predator–prey model with time delay. Firstly, the distribution of the equilibrium points of the system and the existence conditions are discussed. Secondly, under the assumption of the existence of equilibrium points, the linear approximations of the system in both network and non-network settings are derived. Thirdly, the Turing instability conditions for both non-delay and delay systems are investigated, including cases of diffusion-induced and delay-induced instabilities. Fourthly, amplitude equations are derived based on the non-delay system. Finally, extensive numerical simulations are conducted to validate and illustrate the theoretical results, and to analyze and explain their practical significance. The above results effectively demonstrate that the theoretical findings, simulations, and natural reality are consistent. The numerical modeling conducted on the network structure based on the Monte Carlo method exhibits a more diverse range of dynamic behaviors in the parameter of the Holling III functional response function.

How to regulate pattern formations for malware propagation in cyber-physical systems.

Malware propagation can be fatal to cyber-physical systems. How to detect and prevent the spatiotemporal evolution of malware is the major challenge we are facing now. This paper is concerned with the control of Turing patterns arising in a malware propagation model depicted by partial differential equations for the first time. From the control theoretic perspective, the goal is not only to predict the formation and evolution of patterns but also to design the spatiotemporal state feedback scheme to modulate the switch of patterns between different modes. The Turing instability conditions are obtained for the controlled malware propagation model with cross-diffusion. Then, the multi-scale analysis is carried out to explore the amplitude equations near the threshold of Turing bifurcation. The selection and stability of pattern formations are determined based on the established amplitude equations. It is proved that the reaction-diffusion propagation model has three types of patterns: hexagonal pattern, striped pattern, and mixed pattern, and selecting the appropriate control parameters can make the pattern transform among the three patterns. The results of the analysis are numerically verified and provide valuable insights into dynamics and control of patterns embedded in reaction-diffusion systems.

Turing Instabilities are Not Enough to Ensure Pattern Formation.

Symmetry-breaking instabilities play an important role in understanding the mechanisms underlying the diversity of patterns observed in nature, such as in Turing's reaction-diffusion theory, which connects cellular signalling and transport with the development of growth and form. Extensive literature focuses on the linear stability analysis of homogeneous equilibria in these systems, culminating in a set of conditions for transport-driven instabilities that are commonly presumed to initiate self-organisation. We demonstrate that a selection of simple, canonical transport models with only mild multistable non-linearities can satisfy the Turing instability conditions while also robustly exhibiting only transient patterns. Hence, a Turing-like instability is insufficient for the existence of a patterned state. While it is known that linear theory can fail to predict the formation of patterns, we demonstrate that such failures can appear robustly in systems with multiple stable homogeneous equilibria. Given that biological systems such as gene regulatory networks and spatially distributed ecosystems often exhibit a high degree of multistability and nonlinearity, this raises important questions of how to analyse prospective mechanisms for self-organisation.

Cross-diffusion mediated Spatiotemporal patterns in a predator–prey system with hunting cooperation and fear effect

Cooperation during hunting is a common predation plan of action among several large predators to increase their biomass raising capturing potential which can also produce fear in the prey and dispersal-influenced pattern creation is also relevant from both a fundamental and an applied standpoint. Considering these, we present a modified Leslie–Gower predator–prey model incorporating fear, intra-specific competition and hunting cooperation of predators following Berec’s encounter-driven functional response in presence of cross-diffusion. In the non-spatial system, biologically feasible equilibria and their stability conditions are derived. Taking the level of fear as bifurcation parameter, occurrence of limit cycle via Hopf bifurcation and the stability of limit cycle computing Lyapunov coefficient are discussed. In diffusive model, spacial focus on Turing instability conditions and to determine the relevant intervals at which various patterns will begin to appear. In the two dimensional spatial domain, under Neumann boundary condition, various Turing patterns such as spot, stripe, mixture of spot and stripe, labyrinth patterns are emerged varying fear, hunting cooperation and cross-diffusion coefficient. Hopf–Turing region is also a centre for several complex patterns. The numerical simulations indicate that the patterns are in same or opposite phase according to the sign of cross-diffusion coefficient. Intra-specific competition among predators tend the system towards stability. Cross-diffusion causes a paradox to population density with raising fear and hunting cooperation.

Turing patterns in domains with periodic inhomogeneities; a homogenization approach

In this work, we study reaction–diffusion equations with space-dependent, differentiable, and ɛ-periodic diffusion coefficients, using the asymptotic homogenization technique. The importance of this problem is that experimental evidence suggests that in some biological systems, spatial inhomogeneities are important to regulate spatial patterns, and several pattern-forming systems are regulated by a stratified domain that determines the transport properties. Here, we consider the case when the stratification is periodic with a period ɛ; consequently, the diffusion coefficient is a space-dependent ɛ-periodic function. A methodology based on the asymptotic homogenization technique was developed to solve this problem, and the conditions for Turing instability were found in terms of the effective coefficients of the equivalent homogenized domain. Additionally, extensive numerical calculations were performed to validate the results predicted by the homogenization procedure; one- and two-dimensional examples are presented. We conclude that the homogenization method is useful in the field of pattern formation to handle problems involving spatial inhomogeneities or stratified domains. In addition, the proposed generalization of Turing analysis for a homogenized problem enriches the field of pattern formation.

Turing instability analysis of a reaction–diffusion system for rumor propagation in continuous space and complex networks

Research on the propagation of online rumors necessitates careful consideration of rumor-spreading mechanisms. Media correction, self-correction, and the time delays associated with contact propagation are expressly introduced into the established reactive–diffusion rumor-spreading model in this paper. The conditions for a unique positive equilibrium point in the model are analyzed, laying the necessary groundwork for the occurrence of Turing instability in reaction–diffusion rumor propagation systems. A linearization approach is employed to derive the conditions for Turing instability in scenarios involving small delay to address the challenges posed by models with time delay. Furthermore, in order to anticipate the Turing pattern formation in advance, we employed a multiscale analysis approach to derive the amplitude equation within this model. The validity of the theoretical results is verified through numerical simulations, while the propagation dynamics of the model were analyzed within two complex network models, namely small-world networks and scale-free networks. Simulation results indicate that increasing time delay decelerates rumor propagation, while higher degrees of cross-diffusion, contact spreading, media correction, and self-correction expedite the spread of rumors. In small-world networks, the trust level in rumors has a narrower range, while in scale-free networks, the stability of rumors is faster. This indicates how to effectively adjust network structures for controlling or managing the spread of rumors.

A Study on Spatial Pattern of a Class of SI Models Driven by Cross Diffusion

Currently, most research on the spatial patterns of the SI model, both domestically and internationally, focuses on the impacts of self-diffusion and system parameters on pattern formation. However, there is a little study on how cross-diffusion influences the evolution of spatial patterns. In this paper, a spatial epidemic model with both self-diffusion and cross-diffusion is established. The study investigates the effects of cross-diffusion on the stability, rate of stability, and pattern structure of the SI model, while considering scenarios where self-diffusion either induces system instability or it does not. The stability of the non-diffusive system is analyzed, and the conditions for Turing instability in the presence of diffusion terms are elucidated. It is found that when the system is stable under self-diffusion-driven conditions, the introduction of cross-diffusion can change the system's local stability, and can produce Turing patterns. Furthermore, different cross-diffusion coefficients can generate patterns with different structures. When the system is unstable under self-diffusion-driven conditions, the introduction of cross-diffusion can change the pattern structure. Specifically, when the cross-diffusion coefficient <inline-formula><tex-math id="M1">\begin{document}$D_1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M1.png"/></alternatives></inline-formula> for the susceptible individuals is negative, the pattern structure transitions from spot-stripe patterns to spot patterns, and when positive, it transitions from spot-stripe patterns to labyrinthine patterns, and eventually to a uniform solid color distribution. When the cross-diffusion coefficient <inline-formula><tex-math id="M2">\begin{document}$D_2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M2.png"/></alternatives></inline-formula> for the infected individuals is positive, the pattern transformation is similar to when the cross-diffusion coefficient <inline-formula><tex-math id="M3">\begin{document}$D_1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M3.png"/></alternatives></inline-formula> for susceptible individuals is negative, gradually changing to spot patterns, while when <inline-formula><tex-math id="M4">\begin{document}$D_2$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M4.png"/></alternatives></inline-formula> is negative, the pattern structure exhibits a porous structure, eventually transitioning to a uniform solid color distribution. Regarding the rate of stability of the SI model, in the case of a stable self-diffusion system, the introduction of cross-diffusion may alter the rate of system stability, and the larger the cross-diffusion coefficient <inline-formula><tex-math id="M5">\begin{document}$D_1$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20231877_M5.png"/></alternatives></inline-formula> for the susceptible individuals, the faster the system stabilizes. When the self-diffusion-driven system is unstable, cross-diffusion causes the system to transition from an unstable state to a locally stable state, and the smaller the susceptible individuals' cross-diffusion coefficient, the slower the rate of system stabilization. Therefore, cross-diffusion has a significant impact on the stability, rate of stability, and pattern structure of the SI model.

Modeling the Bifurcation Dynamics of Rumor Propagation in the Spatial Environment

The harm caused by rumors is immeasurable. Studying the dynamic characteristics of rumors can help control their spread. In this paper, we propose a nonsmooth rumor model with a nonlinear propagation rate. First, we utilize the positive invariant regions to prove the boundedness of solutions. Second, we analyze the conditions for the existence of equilibrium points in both the left and right systems. Additionally, we confirm the occurrence of saddle-node bifurcation in the left system. Next, by considering the influence of spatial diffusion, we establish the conditions for Turing instability. Then we discuss the conditions for spatial homogeneous and inhomogeneous Hopf bifurcations in the left and right systems, respectively. We differentiate between supercritical and subcritical bifurcations using the Lyapunov coefficient. Furthermore, we examine the conditions for the existence of discontinuous Hopf bifurcation at the demarcation point. Finally, in the numerical simulation section, we validate our theorems on Turing patterns. We also investigate the impact of parameter changes on rumor propagation and conclude that an increase in the psychological inhibitory factor significantly reduces the rate of rumor propagation, providing an effective strategy for curbing rumors. To that end, we fit actual data to our system and the results are excellent, confirming the validity of the system.

Parameter estimation for network-organized Turing system based on convolution neural networks

Turing dynamics mainly focuses on non-homogeneous self-adaptive spatial patterns of reaction diffusion systems in a continuous space. If defining the diffusion environment as network structure, then network patterns are available under Turing instability conditions. In allusion to parameter estimation for network Turing-instable systems, this paper proposes a new recognition method using artificial neural networks. When diffusion occurs on quadrilateral lattice networks, a deep convolutional neural network (CNN) is built to invert the unknown parameters. The results on training set and validation set indicate that the CNN model exhibits great robustness without overfitting and gradient explosion. Meanwhile, the prediction performance on two test sets is excellent, with average relative errors of the estimators ending up at 0.68% and 1.04%, respectively. When diffusion occurs on non-regular networks, a spatial domain graph convolutional network (GCN) is established, which is slightly less robust than the CNN. The average relative error of the estimators on both test sets ends up between 1.1% and 2.8%, which is still in the ideal range.

Spatiotemporal patterns of reaction–diffusion systems with advection mechanisms on large-scale regular networks

This paper primarily investigates the dynamics behaviour of reaction–diffusion systems with advection mechanisms on network structures. Introducing the advection mechanism from continuous space to network structure, this study uses the Laplacian matrix of a directed network to describe it and sets up a network-organized reaction–diffusion system with directed migration mechanism based on this. Firstly, we derived the Turing instability conditions of the system under certain restrictions. We then focus on the Turing dynamical of the system on a type of composite circular networks and a type of composite torus networks. Furthermore, through numerical methods, we reveal the existence of travelling Turing patterns in the system under various circumstances, including the significant case where the system only has diffusion behaviours but no directed migration. Simultaneously, the result of numerical simulations shows that on the network structures presented in this paper, increasing the global directed migration strength promotes homogeneity among the connected nodes; increasing the difference of directed migration strengths between two groups is conducive to the occurrence of Turing instability.

Spatiotemporal dynamics of a multi-delayed prey-predator system with variable carrying capacity.

This paper presents the temporal and spatiotemporal dynamics of a delayed prey-predator system with a variable carrying capacity. Prey and predator interact via a Holling type-II functional response. A detailed dynamical analysis, including well-posedness and the possibility of coexistence equilibria, has been performed for the temporal system. Local and global stability behavior of the co-existence equilibrium is discussed. Bistability behavior between two coexistence equilibria is demonstrated. The system undergoes a Hopf bifurcation with respect to the parameter β, which affects the carrying capacity of the prey species. The delayed system exhibits chaotic behavior. A maximal Lyapunov exponent and sensitivity analysis are done to confirm the chaotic dynamics. In the spatiotemporal system, the conditions for Turing instability are derived. Furthermore, we analyzed the Turing pattern formation for different diffusivity coefficients for a two-dimensional spatial domain. Moreover, we investigated the spatiotemporal dynamics incorporating two discrete delays. The effect of the delay parameters in the transition of the Turing patterns is depicted. Various Turing patterns, such as hot-spot, coldspot, patchy, and labyrinth, are obtained in the case of a two-dimensional spatial domain. This study shows that the parameter β and the delay parameters significantly instigate the intriguing system dynamics and provide new insights into population dynamics. Furthermore, extensive numerical simulations are carried out to validate the analytical findings. The findings in this article may help evaluate the biological revelations obtained from research on interactions between the species.

A study of a spatiotemporal delayed predator–prey model with prey harvesting: Constant and periodic diffusion

This study investigates the emergence of Turing patterns in a predator–prey system with time delay at variable carrying capacity and prey harvesting in the presence of cross-diffusion. The impact of time delay and periodic diffusion on the stability of the equilibrium point and corresponding Turing pattern is primarily explored. Theoretical implications of Turing instability conditions caused by time delay and periodic diffusion are examined, followed by numerical simulations for ecologically meaningful control parameter values. The effects of carrying capacity specification parameters, time delay, and prey-harvesting effort on the emerging pattern are reported. Cross-diffusion results in a wide range of spatiotemporal pattern creation, including spot and stripe patterns, as well as both coexist under the zero flux border condition in unharvested or harvested dynamics. Furthermore, an intertwined pattern emerges under certain parameters, including time delay. The final comparison highlights the significant roles played by periodic diffusion, time delay, and harvesting effort in the evolution of the Turing pattern.

Spatiotemporal dynamics in a diffusive predator-prey model with multiple Allee effect and herd behavior

In this paper, we investigated a diffusive predator-prey model with multiple Allee effect, herd behavior. Moreover, we consider the quadratic mortality on predator species. We found that the dynamics of system near the positive equilibria is quite rich. We are more concerned with the spatial dynamics near the positive equilibrium. The necessary conditions for Turing instability occurring are obtained. And the stability and direction of Hopf and steady state bifurcations are explored by using the normal form method. Furthermore, some numerical simulations are presented to support our theoretical analysis. We found that Allee effect does not alter the local stability of the boundary equilibrium and the transcritical bifurcation occurs at the highest intensity of Allee effect. The biomass conversion rate does affect the stability of the system and the occurrence of Turing instability. This indicates that the biomass conversion rate is essentially significant for the predator-prey system, and we can control the biological conversion rate to achieve the coexistence of the predator and prey. Finally, we summarize our findings in the conclusion.

Analytical detection of stationary and dynamic patterns in a prey-predator model with reproductive Allee effect in prey growth.

Allee effect in population dynamics has a major impact in suppressing the paradox of enrichment through global bifurcation, and it can generate highly complex dynamics. The influence of the reproductive Allee effect, incorporated in the prey's growth rate of a prey-predator model with Beddington-DeAngelis functional response, is investigated here. Preliminary local and global bifurcations are identified of the temporal model. Existence and non-existence of heterogeneous steady-state solutions of the spatio-temporal system are established for suitable ranges of parameter values. The spatio-temporal model satisfies Turing instability conditions, but numerical investigation reveals that the heterogeneous patterns corresponding to unstable Turing eigenmodes act as a transitory pattern. Inclusion of the reproductive Allee effect in the prey population has a destabilising effect on the coexistence equilibrium. For a range of parameter values, various branches of stationary solutions including mode-dependent Turing solutions and localized pattern solutions are identified using numerical bifurcation technique. The model is also capable to produce some complex dynamic patterns such as travelling wave, moving pulse solution, and spatio-temporal chaos for certain range of parameters and diffusivity along with appropriate choice of initial conditions. Judicious choices of parametrization for the Beddington-DeAngelis functional response help us to infer about the resulting patterns for similar prey-predator models with Holling type-II functional response and ratio-dependent functional response.

Spatiotemporal dynamics of a predator-prey system with fear effect

Recent field experiments on vertebrates show that though mere presence of a predator causes a dramatic change in prey demography, the fear of predators increases the survival probability of prey leading to a cost of prey production. Based on the experimental findings, we proposed and analyzed a mathematical model that incorporates the fear-induced birth reduction in the prey population due to presence of predator. A modified and more realistic fear function is proposed in this study. Qualitative behavior of the model is performed including positivity and boundedness of solutions, existence of critical points and their local stability analysis, existence of transcritical and Hopf bifurcation. We analyzed Hopf bifurcation with respect to the prey growth rate and the level of fear. Transcritical bifurcation is analyzed by varying the prey growth rate. Distribution of the population of interacting species in a large scale natural system is heterogeneous and subject to alter for different reasons. Thus, we investigate how behavioral modification in prey population due to fear for predators and mutual interference among predator species can create various spatiotemporal pattern formation in population distribution. In the spatially extended system, we provide a detailed stability analysis and obtain the conditions for Turing instability. Numerical simulations are performed to validate analytical results for both non-spatial and spatial models. Warm spot patterns are obtained by considering three different types of initial data and discussed the biological significance of these patterns for the two-dimensional spatial model. Our numerical simulation demonstrates that the fear effect in a diffusive predator-prey system with mutual interference between predators may exhibit more complicated dynamics.

Effect of random movement and cooperative hunting in the prey–predator system: A dynamical approach

Self-diffusion prerequisite is obtained as the spreading approach of biological populations. Cooperative hunting is a common behavior in predator populations that promotes predation and the coexistence of the prey–predator system. On the other side, the Allee effect among prey may cause the system to become unstable. In this paper, a diffusive prey–predator system with cooperative hunting and the weak Allee effect in prey populations is discussed. The linear stability and Hopf-bifurcation analysis had been used to examine the system’s stability. From the spatial stability of the system, the conditions for Turing instability have been derived. The multiple-scale analysis has been used to derive the amplitude equations of the system. The stability analysis of these amplitude equations leads to the formation of Turing patterns. Finally, numerical simulations are used to analyze spatial patterns forming in 1-D and 2-D. The studies indicate that the model can generate a complex pattern structure and that self-diffusion has a drastic impact on species distribution.

Strong stability with impact of maturation delay and diffusion on a toxin producing phytoplankton–zooplankton model

Plankton species is backbone for any healthy aquatic system, but a global increment in toxin phytoplankton blooms is a major threat to aquatic systems. As toxic phytoplankton Microcystis aeruginosa becomes too high, Artemia salina a zooplankton species avoids to consume it and more inclines towards a non-toxic phytoplankton Chaetoceros gracilis for food. Considering this fact, additional food is provided to zooplankton species to survive when non-toxic species is insufficient for the zooplankton and toxin phytoplanktons are harvested linearly to control toxicity level in the system. Also, increasing of phytoplankton may increase competitive behaviour between toxic and non-toxic phytoplanktons due to limited resources. Keeping these in mind, we propose a three populations reaction–diffusion delay model. In this study, existence of feasible equilibria has been studied. Conditions for Turing instability of delayed and non-delayed system have been discussed analytically. Strong stability analysis with respect to both delay and diffusion is performed and excitable nature through Hopf bifurcation is also examined. It has been investigated that if the delay spatial system shows diffusion driven instability, then it is also shown by the delay free system for the same parameters. The study explores that the model depicts strong stability with respect to both delay and diffusion. Movements of the species are responsible for the occurrence of planktonic bloom via Hopf bifurcation due to diffusion and affect the critical value of the time delay. This study suggests that a specific amount of harvesting ensures the long-term sustainability. It is also inspected that considerable amount of alternative food supply supports the zooplankton for its growth. Turing patterns like spots are observed. The behaviour of the system depends on the both variation of delay and choice of initial population. This study says that maturation delay stabilizes the system more quickly in the presence of diffusion than in the absence of diffusion.