Diffusion-induced Instability Research Articles

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.

Hybrid control of Turing instability and bifurcation for spatial-temporal propagation of computer virus

ABSTRACT In this era of information technology, information leakage and file corruption due to computer virus intrusion have been serious issues. How to detect and prevent the spread of the computer virus is the major challenge we are facing now. To target this problem, a class of virus propagation models with hybrid control scheme are formulated to investigate the dynamic evolution and prevention from a spatial-temporal perspective in this paper. Diffusion-induced Turing instability is detected in response to the computer virus propagation. The introduction of hybrid control scheme can effective suppress Turing instability and turn the propagation system back to a stable state. And then, the time delay is selected as the bifurcation parameter. If the time delay exceeds the bifurcation threshold, the propagation will be destabilised and a Hopf bifurcation will occur. The hybrid control tactic can not only regulate the occurrence of Hopf bifurcation well, but also optimise the properties of bifurcating period solutions. In the end, the correctness and validity of the theoretical results are verified via numerical simulations.

Pattern formation and qualitative analysis for a vegetation-water model with diffusion

In this paper, a diffusive vegetation-water model under Neumann boundary conditions is considered. Firstly, the stability and the diffusion-induced Turing instability are studied. Then, some a priori estimates of positive steady-state solutions are obtained by the maximum principle. Moreover, the bifurcations at both simple and double eigenvalues are investigated in detail. Finally, numerical simulations are shown to support and supplement theoretical analysis results. In particular, the evolution processes of vegetation patterns are depicted under different parameters.

Impacts of prey-taxis and nonconstant mortality on a spatiotemporal predator–prey system

Prey-taxis, which plays an important role in biological control and ecological balance, is the phenomenon that predators shift directly toward regions with the highest density of prey. In this paper, we consider a spatiotemporal predator–prey system involving prey-taxis and nonconstant mortality. By structuring Lyapunov functional, we obtain the global stability of spatially homogeneous steady states. The minimal mortality of predator will determine whether they can survive. In addition, we identify the parameter ranges that spatially homogeneous steady state remains stable or becomes unstable, and discuss steady state bifurcation by choosing prey-taxis coefficient. It is found that a diffusion-induced instability can occur under nonconstant mortality, and prey-taxis can annihilate the spatial patterns induced by diffusion. Numerical simulations illustrate that prey-taxis and nonconstant mortality can result in rich pattern formations including spots pattern, spots–strips pattern, labyrinth pattern, strips-like pattern. It is worth noting that the weak prey-taxis is disadvantageous in terms of the number of two species.

Instabilities in hyperbolic reaction–diffusion system with cross diffusion and species-dependent inertia

The hyperbolic reaction–diffusion (HRD) equation may overcome the physical shortcomings of the parabolic reaction–diffusion (PRD) equation, where the initially localized disturbance propagates infinitely fast through space. Instead, species often exhibit inertia, resulting in delayed effect in their spatial movement. Incorporating such response time for the onset of species flow due to a concentration gradient leads to an HRD equation with inertia. In this paper, we develop the general theory for pattern-forming instabilities in a two-species HRD system, which becomes a PRD system in the limiting case, with cross-diffusion and species-dependent inertia to explore how they play a role in the pattern forming instabilities. In particular, we determine various criteria for diffusion-induced instabilities (like Turing, wave, wave–Turing) and Hopf-induced instabilities (like pure Hopf, Hopf–wave, Hopf–Turing, and Hopf–wave–Turing) arise due to the cross-diffusion and inertial time. The theoretical results are demonstrated with an example where the Brusselator system represents the local interaction.

COMPLEX FRACTIONAL-ORDER HIV DIFFUSION MODEL BASED ON AMPLITUDE EQUATIONS WITH TURING PATTERNS AND TURING INSTABILITY

A weakly nonlinear analysis provides a system constituting amplitude equations and its related analysis is capable of predicting parameter regimes with different patterns expected to co-exist in dynamical circumstances that exhibit complex fractional-order system characteristics. The Turing mechanism of pattern formation as a result of diffusion-induced instability of the homogeneous steady state is concerned with unpredictable conditions. The Turing instability caused by fractional diffusion in a Human Immunodeficiency Virus model has been addressed in this study. It is important that the effect of the Human Immunodeficiency Virus to the immune system can be modeled by the interaction of uninfected cells, unhealthy cells, virus particles and antigen-specific. Initially, all potential equilibrium points are defined and the stability of the interior equilibrium point is then evaluated using the Routh–Hurwitz criteria. The conditions for Turing instability are obtained by local equilibrium points with stability analysis. In the neighborhood of the Turing bifurcation point, weakly nonlinear analysis is employed to deduce the amplitude equations. After applying amplitude equations, it is observed that this system has a very rich dynamical behavior. The constraints for the formation of the patterns like a hexagon, spot, mixed and stripe patterns are identified for the amplitude equations by dynamic analysis. Furthermore, by using the numerical simulations, the theoretical results are verified. Within this framework, this study through the dynamical behavior of the complex system perspective and bifurcation point based on the viral death rate can provide the basis for several researchers working on Human Immunodeficiency Virus model through various aspects. Accordingly, the Turing bifurcation point and weakly nonlinear analysis employed within the complex fractional-order dynamics addressed herein are highly relevant experimentally since the related effects can be studied and applied concerning different mathematical, physical, engineering and biological models.

SPATIAL PATTERN FORMATIONS IN DIFFUSIVE PREDATOR-PREY SYSTEMS WITH NON-HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS

A reaction-diffusion predator-prey system with non-homogeneous Dirichlet boundary conditions describes the persistence of predator and prey species on the boundary. Compared with homogeneous Neumann boundary conditions, the former conditions may prompt or prevent the spatial patterns produced through diffusion-induced instability. The spatial pattern formation induced by non-homogeneous Dirichlet boundary conditions is characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Furthermore, transient spatiotemporal behaviors are observed through numerical simulations.

The Influence of Dirichlet Boundary Conditions on the Dynamics for a Diffusive Predator–Prey System

A reaction–diffusion predator–prey system with homogeneous Dirichlet boundary conditions describes the lethal risk of predator and prey species on the boundary. The spatial pattern formations with the homogeneous Dirichlet boundary conditions are characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Compared with homogeneous Neumann boundary conditions, we see that the homogeneous Dirichlet boundary conditions may depress the spatial patterns produced through the diffusion-induced instability. In addition, the existence of semi-trivial steady states and the global stability of the trivial steady state are characterized by the comparison technique.

Dynamics and pattern formation of a diffusive predator–prey model with predator-taxis

We propose a new reaction–diffusion predator–prey model system with predator-taxis in which the preys could move in the opposite direction of predator gradient. A similar situation also occurs when susceptible population avoids the infected ones in epidemic spreading. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and any predator-taxis sensitivity coefficient are proved. It is also shown that such predator-taxis does not qualitatively affect the existence and stability of coexistence steady state solutions in many cases. For diffusive predator–prey system with diffusion-induced instability, it is shown that the presence of predator-taxis may annihilate the spatial patterns.

Delay-induced wave instabilities in single-species reaction-diffusion systems.

The Turing (wave) instability is only possible in reaction-diffusion systems with more than one (two) components. Motivated by the fact that a time delay increases the dimension of a system, we investigate the presence of diffusion-driven instabilities in single-species reaction-diffusion systems with delay. The stability of arbitrary one-component systems with a single discrete delay, with distributed delay, or with a variable delay is systematically analyzed. We show that a wave instability can appear from an equilibrium of single-species reaction-diffusion systems with fluctuating or distributed delay, which is not possible in similar systems with constant discrete delay or without delay. More precisely, we show by basic analytic arguments and by numerical simulations that fast asymmetric delay fluctuations or asymmetrically distributed delays can lead to wave instabilities in these systems. Examples, for the resulting traveling waves are shown for a Fisher-KPP equation with distributed delay in the reaction term. In addition, we have studied diffusion-induced instabilities from homogeneous periodic orbits in the same systems with variable delay, where the homogeneous periodic orbits are attracting resonant periodic solutions of the system without diffusion, i.e., periodic orbits of the Hutchinson equation with time-varying delay. If diffusion is introduced, standing waves can emerge whose temporal period is equal to the period of the variable delay.

Pattern formations of an epidemic model with Allee effect and time delay

Allee effect widely exists for endangered plants and animals in ecosystem, which indicates that the minimum population density or size is necessary for population survival, namely, Allee threshold. In this paper, a delayed reaction-diffusion epidemic model with respect to Allee effect is investigated. The instability of the positive constant steady state is induced by two mechanisms, one is diffusion-induced instability, the other is delay-induced instability. The first case gives rise to Turing patterns. Moreover, Turing region becomes narrow as incubation delay being increased. We further observe that the range of Turing mode is enlarged with the increase of Allee threshold. The numerical simulations verify our theoretical results. The combined effects of Allee effect and disease on the spatial distributions of endangered species are studied, which provides new insights for human intervention in conservation management of these species.

Vegetation pattern formation of a water-biomass model

In this paper, a mathematical model with diffusion and cross-diffusion is proposed to describe the interaction between the vegetation and the soil water. Based on the view of Turing pattern, we discuss the conditions of the diffusion-induced instability and the cross-diffusion-induced instability of a homogenous uniform steady state. We find that either a fast diffusion speed of water or a great hydraulic diffusivity due to the suction of roots may drive the instability of the homogenous steady state. Furthermore, we find that both the rain-fall rate and the infiltration feedback parameter can induce the transitions among the vegetation state, pattern formation and bare soil state. It is also found that the “terrain slope” may cause the instability of the homogenous steady state and drive the formation of periodic stripe pattern. Consequently, the diversity of dryland vegetation in reality can be explained as a result of pattern solutions of the model.

Pattern formation in multiplex networks.

The advances in understanding complex networks have generated increasing interest in dynamical processes occurring on them. Pattern formation in activator-inhibitor systems has been studied in networks, revealing differences from the classical continuous media. Here we study pattern formation in a new framework, namely multiplex networks. These are systems where activator and inhibitor species occupy separate nodes in different layers. Species react across layers but diffuse only within their own layer of distinct network topology. This multiplicity generates heterogeneous patterns with significant differences from those observed in single-layer networks. Remarkably, diffusion-induced instability can occur even if the two species have the same mobility rates; condition which can never destabilize single-layer networks. The instability condition is revealed using perturbation theory and expressed by a combination of degrees in the different layers. Our theory demonstrates that the existence of such topology-driven instabilities is generic in multiplex networks, providing a new mechanism of pattern formation.

Logarithmic Lipschitz norms and diffusion-induced instability

This paper proves that ordinary differential equation systems that are contractive with respect to Lp norms remain so when diffusion is added. Thus, diffusive instabilities, in the sense of the Turing phenomenon, cannot arise for such systems, and in fact any two solutions converge exponentially to each other. The key tools are semi inner products and logarithmic Lipschitz constants in Banach spaces. An example from biochemistry is discussed, which shows the necessity of considering non-Hilbert spaces. An analogous result for graph-defined interconnections of systems defined by ordinary differential equations is given as well.

Emergence of Spatial Patterns in a Simple Model of Wetland Ecosystem: Application to Keoladeo National Park (KNP), India

Emergence of Spatial Patterns in a Simple Model of Wetland Ecosystem: Application to Keoladeo National Park (KNP), India An attempt has been made to understand the spatiotemporal dynamics of good biomass, bad biomass and bird population in the wetland part of KNP. Spatiotemporal distributions of species biomass are simulated using the diffusivity assumptions realistic for natural wetland systems. Stability analysis, Hopf-bifurcation and diffusion-induced instability of the model system have been carried out. Computer simulations performed to identify various parameters for the good health of the wetland ecosystem. We observed that growth rate and carrying capacity for the bad biomass, carrying capacity and half-saturation constant of the good biomass are responsible for the good health of the wetland ecosystem. The spatial movements of good biomass acquires stable stationary patterns in the presence of bad biomass which performs swinging motion and selects a steady state spatial pattern, thereby ensuring the persistence as well as extinction of multiple species system in space and time.

Spiral waves with superstructures in a mixed-mode oscillatory medium

Diverse spatiotemporal patterns are generated in a three-variable reaction-diffusion model that supports 1(1) mixed-mode oscillations. Diffusion-induced instability results in spatiotemporal patterns such as amplitude-modulated overtargets (circular super-waves superimposed on spiral waves) and superspirals. The types of superstructure waves are determined by the ratio of diffusion coefficients, which controls the interaction and competition between two local oscillatory modes, one of which is the original homogeneous 1(1) mixed-mode oscillation, resulting in periodic amplitude modulation in space. Variation of the control parameter can reverse the chirality and radial propagation direction (outward or inward rotation) of a superspiral pattern. These amplitude-modulated patterns may provide insight into mechanisms of pattern development in some living systems.

Potential Turing instability and application to plant–insect models

We show in this paper that the analysis of diffusion-induced instability in spatially extended models can be performed by separating local dynamics from diffusion. This is possible not only in the case studied by Turing, namely models with two interacting variables, but also in the general case of three or more variables. The advantage of this decomposition, based on the notion of potential Turing instability, is illustrated through the analysis of two spatially extended plant–insect models.

The spatial dimension in environmental and resource economics

ABSTRACTMechanisms generating patterns in spatial domains have been extensively studied in biology, but also in economics in the context of new economic geography. The Turing mechanism or Turing diffusion-induced instability has been central to the understanding of forces which endogenously generate spatial patterns, but in a context where agents do not explicitly optimize an objective. The present paper reviews tools to study, in the spirit of Turing's analysis, a mechanism generating optimal diffusion-induced instability where optimizing agents generate optimal agglomerations. By extending the maximum principle to the optimal control of partial differential equations, it is shown how under certain conditions it will be optimal to design controls so that the price-quantity system implied by the costate–state functions of the optimal control of distributed parameter systems induces optimal spatial patterns. These methods might be useful in studying pattern formation both in problems of resource management and of economic development.

Nutrient flows between ecosystems can destabilize simple food chains

Dispersal of organisms has large effects on the dynamics and stability of populations and communities. However, current metacommunity theory largely ignores how the flows of limiting nutrients across ecosystems can influence communities. We studied a meta-ecosystem model where two autotroph–consumer communities are spatially coupled through the diffusion of the limiting nutrient. We analyzed regional and local stability, as well as spatial and temporal synchrony to elucidate the impacts of nutrient recycling and diffusion on trophic dynamics. We show that nutrient diffusion is capable of inducing asynchronous local destabilization of biotic compartments through a diffusion-induced spatiotemporal bifurcation. Nutrient recycling interacts with nutrient diffusion and influences the susceptibility of the meta-ecosystem to diffusion-induced instabilities. This interaction between nutrient recycling and transport is further shown to depend on ecosystem enrichment. It more generally emphasizes the importance of meta-ecosystem theory for predicting species persistence and distribution in managed ecosystems.

Diffusion-induced modulational instability in magnetised semiconductor plasmas: effect of carrier heating

The present paper is aimed to the exploration of modulational amplification in transversely magnetised diffusive semiconductors. By considering that the origin of modulational interaction lies in the third-order optical susceptibility \(\chi^{(3)}\) arising from the nonlinear diffusion current density and using coupled mode theory, an analytical investigation of frequency modulational interaction between co-propagating laser beams and internally generated acoustic mode is presented. We have studied the steady state and transient amplification characteristics of modulated waves arising in the transversely magnetised semiconductor plasmas. The effect of carrier heating adds new dimensions to the present study. The heating effect reduces the required threshold amplitude of wave and enhances steady-state as well as transient gain of the generated acoustic mode.