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Recent research suggests that the presence of predator may affect the physical characteristics of prey species to the level that it is more effective compared to direct predation in reducing the prey biomass. Furthermore, such nonlethal impacts might persist across seasons or generations. This research work centers on a predator–prey interaction considering the predator’s fear and its carryover effects (COEs) on prey as well as the Allee effect on predator species. This model incorporates both self- and cross-diffusions to explore the species distribution due to the movement of the species subject to the Neumann boundary condition. First, we study the positivity, boundedness, feasible equilibria and their local stability; different bifurcations appear in the nonspatial system. Global dynamics of the system are discussed, showing that coexistence equilibrium can never be globally stable. Both fear and its COEs destabilize the system, but the Allee effect in predators, once destabilized, again stabilizes the system. COEs of fear can influence the effect of Allee and destabilize the system even when the Allee effect is high, but fear has no such impact on the system dynamics, coupling with the Allee parameter. The system exhibits bistability, and the combined influence of fear and its COEs results in the “paradox of enrichment”. For the spatial model, Turing instability conditions, wave number range for instability and different instability regions are derived. The system dynamics permits mainly the spot stationary patterns. Diffusion causes an increase in gatherings with rising COEs and the Allee effect in predators, but an increase in the fear effect diminishes the prey gatherings in specific spots. Hopf instability leads to oscillation in the system. The delicate dynamics of fear and its COEs, the predator’s Allee effect along with diffusion reveal insights into the harmony in nature.

This paper is concerned with a classical two-species prey–predator reaction–diffusion system with ratio-dependent functional response and subject to homogeneous Neumann boundary condition in a two-dimensional rectangle domain. By analyzing the associated eigenvalue problem, the spatially homogeneous Hopf bifurcation curve and Turing bifurcation curve of system at the constant coexistence equilibrium are established. Then when the bifurcation parameter is in the interior of range for Turing instability and near Turing bifurcation curve, the amplitude equations of the original system near the constant coexistence equilibrium are obtained by multiple-scale time perturbation analysis. On the basis of the obtained amplitude equations, the stability and classifications of spatiotemporal patterns of the original system at the constant coexistence equilibrium are discussed. Finally, to verify the validity of the obtained theoretical results, numerical simulations are also carried out.

In this paper, we discuss the stability and pattern formation issues of a spatiotemporal discrete system based on the modified Klausmeier model. We begin by constructing the corresponding coupled map lattices model. Then the existence and stability analysis is employed to derive the prerequisites for a stable homogeneous stationary state. Through the center manifold theorem and bifurcation theory, the threshold parameter values for flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are individually determined. Based on the analysis of bifurcation, four pattern formation mechanisms are presented. Finally, we simulate the corresponding results numerically. The simulations exhibit rich dynamical behaviors, such as period-doubling cascades, invariant cycles, periodic windows, chaos, and rich Turing patterns. Four pattern formation mechanisms give rise to rich and complex patterns, including mosaics, spots, circles, spirals and cyclic fragmentation. The analysis and findings from this study enhance our comprehension of the intricate relationships among bifurcation, chaos and pattern formation for the spatiotemporal discrete Klausmeier model.

We report a noise assisted and optimized resonant transition behavior between two metastable Turing pattern states in a piecewise FitzHu–Nagumo model. The response of the bistable Turing system to a weak periodic signal which modulates the attraction domain of the piecewise function is enhanced with the addition of noise. The quantity that measures the resonant transition between the bistable states and the periodicity of the Turing pattern shows a tent-shape dependence on the intensity of noise, which is the fingerprint of stochastic resonance. As the frequency of the signal is increased to be high enough, the coherent transition disappears.

Spatial epidemiology is the study of spatial variation in disease risk or incidence, including the spatial patterns of the population. Thus, an epidemic model with spatial structure based on the cellular automata method, which is different from deterministic and probabilistic CA models, is investigated. The construction of the cellular automata is based on the work of Bussemaker et al. [Phys. Rev. Lett.78, 5018–5021 (1997)]. For the appropriately chosen parameters, Turing pattern formation can emerge from a randomly perturbed uniform state, which is shown by numerical simulations. The results obtained confirm that diffusion can form the disease being in high density and the population being more stable.

Pattern estimation and selection in media can give important clues to understand the collective response to external stimulus by detecting the observable variables. Both reaction–diffusion systems (RDs) and neuronal networks can be treated as multi-agent systems from molecular level, intrinsic cooperation, competition. An external stimulus or attack can cause collapse of spatial order and distribution, while appropriate noise can enhance the consensus in the spatiotemporal systems. Pattern formation and synchronization stability can bridge isolated oscillators and the network by coupling these nodes with appropriate connection types. As a result, the dynamical behaviors can be detected and discussed by developing different spatial patterns and realizing network synchronization. Indeed, the collective response of network and multi-agent system depends on the local kinetics of nodes and cells. It is better to know the standard bifurcation analysis and stability control schemes before dealing with network problems. In this review, dynamics discussion and synchronization control on low-dimensional systems, pattern formation and synchronization stability on network, wave stability in RDs and neuronal network are summarized. Finally, possible guidance is presented when some physical effects such as polarization field and electromagnetic induction are considered.

Turing instability and pattern formation in the Lengyel–Epstein (L–E) model with superdiffusion are investigated in this paper. The effects of superdiffusion on the stability of the homogeneous steady state are studied in detail. In the presence of superdiffusion, instability will occur in the stable homogeneous steady state and more complex dynamics will exist. As a result of Turing instability, some patterns are formed. Through stability analysis of the system at the equilibrium point, conditions ensuring Turing and Hopf bifurcations are obtained. To further explore pattern selection, the weakly nonlinear analysis and multiple scale analysis are employed to derive amplitude equations of the stationary patterns. Then complex dynamics of amplitude equations, such as the existence of homogeneous solutions, stripe and hexagon patterns, mixed structure patterns, their stability, interaction and transition between them, are analyzed. Then different patterns occur immediately. Finally, the numerical simulations are presented to show the effectiveness of theoretical analysis and patterns are identified numerically. Whereas in the existing results of such model with normal diffusion, no amplitude equations are derived and patterns are only identified through numerical simulations.

This paper focuses on the dynamical behavior of a Lotka–Volterra competitive system with nonlocal delay. We first establish the conditions of Turing bifurcation occurring in the system. According to it and by using multiple scale method, the amplitude equations of the different Turing patterns are obtained. Then, we observe when these patterns (spots pattern and stripes pattern) arise in the Lotka–Volterra competitive system. Finally, some numerical simulations are given to verify our theoretical analysis.

In this paper, an attempt has been made to understand the role of predator’s interference and additional food on the dynamics of a diffusive population model. We have studied a predator–prey interaction system with mutually interfering predator by considering additional food and Crowley–Martin functional response (CMFR) for both the reaction–diffusion model and associated spatially homogeneous system. The local stability analysis ensures that as the quantity of alternative food decreases, predator-free equilibrium stabilizes. Moreover, we have also obtained a condition providing a threshold value of additional food for the global asymptotic stability of coexisting steady state. The nonspatial model system changes stability via transcritical bifurcation and switches its stability through Hopf-bifurcation with respect to certain ranges of parameter determining the quantity of additional food. Conditions obtained for local asymptotic stability of interior equilibrium solution of temporal system determines the local asymptotic stability of associated diffusive model. The global stability of positive equilibrium solution of diffusive model system has been established by constructing a suitable Lyapunov function and using Green’s first identity. Using Harnack inequality and maximum modulus principle, we have established the nonexistence of nonconstant positive equilibrium solution of the diffusive model system. A chain of patterns on increasing the strength of additional food as spots$\to$stripes$\to$spots has been obtained. Various kind of spatial-patterns have also been demonstrated via numerical simulations and the roles of predator interference and additional food are established.

A diffusive predator–prey model with Allee effect and constant stocking rate for predator is investigated and it is shown that Allee effect is the decisive factor driving the formation of Turing pattern. Furthermore, it is observed that Turing pattern appears only when the diffusion rate of the prey is faster than that of the predator, which is just opposite to the condition of Turing pattern in the classical predator–prey system. Some sufficient conditions are obtained to ensure the asymptotical stability of a spatially homogeneous steady-state solution. The existence and nonexistence of positive nonconstant steady-state solutions are investigated to understand the mechanisms of generating spatiotemporal patterns. Furthermore, Hopf and steady-state bifurcations are analyzed in detail by using Lyapunov–Schmidt reduction.

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.

Phytoplankton patterns have been observed widely in aquatic systems. Although pattern formation has been investigated based on many PDEs, discrete models on aquatic systems can provide more complex dynamics. A discrete toxic-phytoplankton–zooplankton model is studied in this paper, with the consideration of Allee effect and cross-diffusion. Focusing on Allee effect coefficient, flip and Neimark–Sacker bifurcation analyses are carried out. And focusing on cross-diffusion coefficient, Turing bifurcation analyses are carried out. Parameter conditions and bifurcation diagram of these bifurcations are obtained correspondingly. Numerical simulations are then performed which are consistent with results of theoretical analysis. Irregular patterns can be formed by flip bifurcation. Spirals can be formed by Neimark–Sacker bifurcation. Spots and stripes can be formed by Turing bifurcation. When Turing and flip, or Turing and Neimark–Sacker bifurcations both occur, special patterns can be obtained.

In prey–predator interaction, many factors, such as the fear effect, Allee effect, cooperative hunting, and group behavior, can influence the population dynamics. Hence, studying these factors in prey–predator makes the model more realistic. In this paper, we have proposed the prey–predator model having herd and Allee effect in prey population, where predators follow hunting cooperation. We have employed temporal analysis to examine the role of the Allee effect and hunting cooperation. Furthermore, we have extended the analysis to spatiotemporal analysis to examine the role of dispersal and the type of spatial structure formed by the population due to random movement. We first discuss the proposed model’s existence and positivity, then the stability of the existing equilibrium points through Routh–Hurwitz criteria. The temporal analysis is carried out through Hopf-bifurcation at the coexistence equilibrium point by considering the Allee threshold ($\alpha$), hunting cooperation ($\gamma$), and attack rate ($\beta$) as controlled parameters. With the addition of diffusion to the model, we examine the spatial model stability and derive the Turing instability condition, which will give rise to various Turing patterns. Finally, numerical simulations are performed to validate the analytical results. The theoretical study and numerical simulation results demonstrate that the Allee effect, hunting cooperation, and diffusion coefficient are sensitive parameters to the model’s stability.

In this research paper, we consider a Leslie–Gower Reaction–Diffusion (RD) model with a predator-driven Allee term in the prey population. We derive conditions for the existence of nontrivial solutions, uniform boundedness, local stability at co-existing equilibrium points, and Hopf bifurcation criteria from the temporal system. We identify sufficient conditions for Turing instability with no-flux boundary condition for the spatial system. Our investigation delves into the analysis of diffusion-induced Turing instability, incorporating stability conditions for the constant steady-state in the spatial model. We also investigate the conditions for the existence and nonexistence of nonconstant steady states in the diffusion-induced model. During numerical simulations, we observe that the predator-driven Allee term is essential for the model to generate Turing structures. Our findings reveal intriguing properties within the RD system, demonstrating its ability to produce patterns within the Turing domain. The simulation confirms that cold–hot spots and stripes-like patterns (a mixture of spots and strips) arises for different strengths of the predation parameter and Allee parameter. In contrast, we observe that for the above threshold value of the Allee parameter, the above-mentioned patterns may disappear from the system. Interestingly, we also observe that the stationary system produces patterns for both large and small amplitudes of perturbation in the vicinity of the Turing boundary. Our research may contribute valuable insights into the Allee effect and enhance our understanding of predator–prey interactions in naturalistic environments.

In the ecological scenario, predators often risk their lives pursuing dangerous prey, potentially reducing their chances of survival due to injuries. Prey, on the other hand, try to strike a balance between reproduction rates and safety. In our study, we introduce a two-dimensional prey–predator model inspired by Tostowaryk’s work, specifically focusing on the domed-shaped functional response observed in interactions between pentatomid predators and neo-diprionid sawfly larvae. To account for the varying effectiveness of larval group defense, we incorporate a new component into the response equation. Our investigation delves into predator trade-off dynamics by adjusting the predator’s mortality rate to reflect losses incurred during encounters with dangerous prey and prey’s trade-off between safety and reproduction rate incorporating this domed-shaped functional response. Our model demonstrates bistability and undergoes various bifurcations, including transcritical, saddle-node, Hopf, Bogdanov–Takens, and Homoclinic bifurcations. Critical parameters impact both predator and prey populations, potentially leading to predator extinction if losses due to dangerous prey encounters become excessive, highlighting the risks predators face for their survival. Furthermore, the efficacy of group defense mechanisms can further endanger predators. Expanding our analysis to a spatially extended model under different perturbations, we explore Turing instability to explain the relationship between diffusion and encounter parameters through both stationary and dynamic pattern formation. Sensitivity to initial conditions uncovers spatiotemporal chaos. These findings provide valuable insights into comprehending the intricate dynamics of prey–predator interactions within ecological systems.

In this paper, we scrutinize the dynamics of a temporal and spatiotemporal prey–predator model incorporating the fear effect on prey and team hunting by the predator. Additionally, we explore the influence of delayed anti-predation response. The analysis includes discussions on well-posedness, local stability, and various bifurcations such as saddle-node, transcritical, Hopf and Bogdanov–Takens bifurcations. The impact of fear cost and delay parameters on model dynamics is investigated by considering them as bifurcation parameters. We investigate how bifurcation values change with varying parameters by exploring different bi-parameter planes. It is observed that the system transitions into chaotic behavior through Hopf bifurcation for significant anti-predation response delay. The positivity of the maximal Lyapunov exponent indicates the confirmed characteristics of chaotic behavior. Furthermore, within the spatiotemporal model framework, a thorough analysis of local and global stability is provided, including the establishment of criteria for identifying Turing instability in cases of self- and cross-diffusion. Various stationary and dynamic patterns are elucidated as diffusion coefficients vary, showcasing the diverse dynamics of the spatiotemporal model. In order to illustrate the dynamic characteristics of the system, a series of comprehensive numerical simulations are conducted. The discoveries outlined in this paper could prove advantageous for understanding the biological implications resulting from the examination of predator–prey relationships.

In this paper, we consider a sex-structured predator–prey model with strongly coupled nonlinear reaction diffusion. Using the Lyapunov functional and Leray–Schauder degree theory, the existence and stability of both homogenous and heterogenous steady-states are investigated. Our results demonstrate that the unique homogenous steady-state is locally asymptotically stable for the associated ODE system and PDE system with self-diffusion. With the presence of the cross-diffusion, the homogeneous equilibrium is destabilized, and a heterogenous steady-state emerges as a consequence. In addition, the conditions guaranteeing the emergence of Turing patterns are derived.

The study of rumor propagation dynamics is of great significance to reduce false news and ensure the authenticity of news information. In this paper, a SI reaction-diffusion rumor propagation model with nonlinear saturation incidence is studied. First, through stability analysis, we obtain the conditions for the existence and local stability of the positive equilibrium point. By selecting suitable variable as the control parameter, the critical value of Turing bifurcation and the existence theorem of Turing bifurcation are obtained. Then, using the above theorem and multi-scale standard analysis, the expression of amplitude equation around Turing bifurcation point is obtained. By analyzing the amplitude equation, different types of Turing pattern are divided such as uniform steady-state mode, hexagonal mode, stripe mode and mixed structure mode. Further, in the numerical simulation part, by observing different patterns corresponding to different values of control variable, the correctness of the theory is verified. Finally, the effects of different network structures on patterns are investigated. The results show that there are significant differences in the distribution of users on different network structures.

There are many interesting patterns observed in activator-inhibitor systems. A well-known model is the FitzHugh-Nagumo system in which the reaction terms are in coupled with a skew-gradient structure. In conjunction with variational methods, there is a close relation between the stability of a steady state and its relative Morse index. We give a sufficient condition in diffusivity for the existence of standing wavefronts joining with Turing patterns.