Conditions For Diffusion-driven Instability Research Articles

Hopf bifurcation and patterns formation in a diffusive two prey-one predator system with fear in preys and help

Recognizing the relationship between the spatial patterns in species concentrations and ecological heterogeneity is crucial for understanding demographics and species governance in a given domain, as ecological patterning processes are believed to be imitated in real ecosystems. In this present article, we have considered a two-prey-one-predator system with Holling type-II functional response where both the preys have fear of predator and compete in the absence of predator. Additionally, we have assumed that both the prey help each other when the predator attacks them. We have incorporated self-diffusion in this system under Neumann boundary conditions. The existence and stability conditions of different equilibria are discussed. We have studied the Hopf bifurcation around positive offset and derived the stability and direction of periodic solution. We have done a series of numerical simulations to ensure our theoretical finding. Diffusion-driven instability conditions are obtained. Different instability regions and fascinating patterns, such as spots, mixtures, and stripes, are depicted. Spatiotemporal dynamics also reflects that prey populations are located in isolated areas of low population concentration with increasing level of fear, whereas an increase in handling time forms lower-density stripes. But with increasing help, prey becomes more concentrated in some stripe regions. In the Hopf-Turing region, it is observed that the diffusion coefficient of predators can stabilize the system. Overall, pattern creation in predator-prey systems can help anticipate long-term system behaviour and analyze the influence of ecological factors on system dynamics.

Spatio-temporal patterns resulting from a predator-based disease with immune prey

Propagation of a disease through a spatially varying population poses complex questions about disease spread and population survival. We consider a spatio-temporal predator–prey model in which a disease only affects the predator. Diffusion-driven instability conditions are analytically derived for the spatio-temporal model. We perform numerical simulation using experimental data given in previous studies and demonstrate that travelling waves, periodicity and chaotic patterns are possible. We show that the introduction of disease in the predator species makes the standard Rosenzweig–MacArthur model capable of producing Turing patterns, which is not possible without disease. However, in the absence of infection, both species can coexist in spiral non-Turing patterns. It follows that disease persistence may be predictable, while eradication may not be.

Turing Patterning in Stratified Domains

Reaction–diffusion processes across layered media arise in several scientific domains such as pattern-forming E. coli on agar substrates, epidermal–mesenchymal coupling in development, and symmetry-breaking in cell polarization. We develop a modeling framework for bilayer reaction–diffusion systems and relate it to a range of existing models. We derive conditions for diffusion-driven instability of a spatially homogeneous equilibrium analogous to the classical conditions for a Turing instability in the simplest nontrivial setting where one domain has a standard reaction–diffusion system, and the other permits only diffusion. Due to the transverse coupling between these two regions, standard techniques for computing eigenfunctions of the Laplacian cannot be applied, and so we propose an alternative method to compute the dispersion relation directly. We compare instability conditions with full numerical simulations to demonstrate impacts of the geometry and coupling parameters on patterning, and explore various experimentally relevant asymptotic regimes. In the regime where the first domain is suitably thin, we recover a simple modulation of the standard Turing conditions, and find that often the broad impact of the diffusion-only domain is to reduce the ability of the system to form patterns. We also demonstrate complex impacts of this coupling on pattern formation. For instance, we exhibit non-monotonicity of pattern-forming instabilities with respect to geometric and coupling parameters, and highlight an instability from a nontrivial interaction between kinetics in one domain and diffusion in the other. These results are valuable for informing design choices in applications such as synthetic engineering of Turing patterns, but also for understanding the role of stratified media in modulating pattern-forming processes in developmental biology and beyond.

Bifurcation and turing instability for a predator-prey model with nonlinear reaction cross-diffusion

A nonlinear reaction cross-diffusion predator-prey system under Neumann boundary condition is considered. Negative diffusion coefficients with local accumulation effect of prey are introduced. Firstly, the criteria for local asymptotic stability of the positive homogeneous steady state with or without cross-diffusion are discussed. Moreover, the conditions for diffusion-driven instability are obtained and the Turing regions in the plane of cross-diffusion coefficients is achieved. Secondly, the existence and multiplicity of spatially nonhomogeneous/homogeneous steady-state solutions are studied by virtue of the Lyapunov–Schmidt reduction. Finally, to clarify the theoretical results, some numerical simulations are carried out. One of the most interesting finding is that Turing instability in the model is induced by the negative diffusion coefficients.

Positive solutions and pattern formation in a diffusive tritrophic system with Crowley–Martin functional response

In the present work, we have studied a diffusive tritrophic food chain model in which the species at each trophic level interact in accordance with Crowley–Martin functional response under mixed boundary conditions. Using degree theory and fixed point index-based methods, we have proved the existence of the positive solutions of the proposed system. We have proved the permanence of the positive solutions and existence of global attractor. The conditions for diffusion-driven instability have been obtained analytically. Moreover, the pattern formation due to diffusion-driven instability has been investigated numerically. We have shown the existence of the positive solutions both analytically and numerically.

Pattern formation in reaction-diffusion systems with piecewise kinetic modulation: An example study of heterogeneous kinetics.

The study of pattern emergence together with exploration of the exemplar Turing model is enjoying a renaissance both from theoretical and experimental perspective. Here, we implement a stability analysis of spatially dependent reaction kinetics by exploring the effect of a jump discontinuity within piecewise constant kinetic parameters, using various methods to identify and confirm the diffusion-driven instability conditions. Essentially, the presence of stability or instability in Turing models is a local property for piecewise constant kinetic parameters and, as such, may be analyzed locally. In particular, a local assessment of whether parameters are within the Turing space provides a strong indication that for a large enough region with these parameters, an instability can be induced.

Turing patterns in a predator–prey model with seasonality

Many ecological systems show striking non-homogeneous population distributions. Diffusion-driven instabilities are commonly studied as mechanisms of pattern formation in many fields of biology but only rarely in ecology, in part because some of the conditions seem quite restrictive for ecological systems. Seasonal variation is ubiquitous in temperate ecosystems, yet its effect on pattern formation has not yet been explored. We formulate and analyze an impulsive reaction-diffusion system for a resource and its consumer in a two-season environment. While the resource grows throughout the 'summer' season, the consumer reproduces only once per year. We derive conditions for diffusion-driven instability in the system, and we show that pattern formation is possible with a Beddington-DeAngelis functional response. More importantly, we find that a low overwinter survival probability for the resource enhances the propensity for pattern formation: diffusion-driven instability occurs even when the diffusion rates of prey and predator are comparable (although not when they are equal).

Analysis and Simulations of Coupled Bulk-surface Reaction-Diffusion Systems on Exponentially Evolving Volumes

In this article we present a system of coupled bulk-surface reaction-diffusion equations on exponentially evolving volumes. Detailed linear stability analysis of the homogeneous steady state is carried out. It turns out that due to the nature of the coupling (linear Robin-type boundary conditions) the characterisation of the dispersion relation in the absence and presence of spatial variation (i.e. diffusion), can be decomposed as a product of the dispersion relation of the bulk and surface models thereby allowing detailed analytical tractability. As a result we state and prove the conditions for diffusion-driven instability for systems of coupled bulk-surface reaction-diffusion equations. Furthermore, we plot explicit evolving parameter spaces for the case of an exponential growth. By selecting parameter values from the parameter spaces, we exhibit pattern formation in the bulk and on the surface in complete agreement with theoretical predictions.

Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion

This article presents stability analytical results of a two component reaction-diffusion system with linear cross-diffusion posed on continuously evolving domains. First the model system is mapped from a continuously evolving domain to a reference stationary frame resulting in a system of partial differential equations with time-dependent coefficients. Second, by employing appropriately asymptotic theory, we derive and prove cross-diffusion-driven instability conditions for the model system for the case of slow, isotropic domain growth. Our analytical results reveal that unlike the restrictive diffusion-driven instability conditions on stationary domains, in the presence of cross-diffusion coupled with domain evolution, it is no longer necessary to enforce cross nor pure kinetic conditions. The restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Reaction-cross-diffusion models with equal diffusion coefficients in the principal components as well as those of the short-range inhibition, long-range activation and activator-activator form can generate patterns only in the presence of cross-diffusion coupled with domain evolution. To confirm our theoretical findings, detailed parameter spaces are exhibited for the special cases of isotropic exponential, linear and logistic growth profiles. In support of our theoretical predictions, we present evolving or moving finite element solutions exhibiting patterns generated by a short-range inhibition, long-range activation reaction-diffusion model with linear cross-diffusion in the presence of domain evolution.

DIFFUSION-DRIVEN INSTABILITY AND WAVE PATTERNS OF LESLIE–GOWER COMPETITION MODEL

In this paper, the diffusion-driven instability of the Leslie–Gower competition model with the periodic boundary conditions is investigated. By using the linearization method and the inner product techniques, the instability conditions of this model at the coexistence fixed point and the competitive exclusion fixed points are obtained, respectively. As an example, the diffusion-driven instability conditions of a symmetric Leslie–Gower competition model at the coexistence fixed point is obtained when the diffusion coefficients are equal. Under these instability conditions, various patterns, including spirals, traveling waves and disorders, are observed in the numerical simulations. On the other hand, we also numerically investigate the effects of diffusion coefficient and the strength of the interspecific competition on the wave patterns.

Stability analysis and simulations of coupled bulk-surface reaction-diffusion systems.

In this article, we formulate new models for coupled systems of bulk-surface reaction–diffusion equations on stationary volumes. The bulk reaction–diffusion equations are coupled to the surface reaction–diffusion equations through linear Robin-type boundary conditions. We then state and prove the necessary conditions for diffusion-driven instability for the coupled system. Owing to the nature of the coupling between bulk and surface dynamics, we are able to decouple the stability analysis of the bulk and surface dynamics. Under a suitable choice of model parameter values, the bulk reaction–diffusion system can induce patterning on the surface independent of whether the surface reaction–diffusion system produces or not, patterning. On the other hand, the surface reaction–diffusion system cannot generate patterns everywhere in the bulk in the absence of patterning from the bulk reaction–diffusion system. For this case, patterns can be induced only in regions close to the surface membrane. Various numerical experiments are presented to support our theoretical findings. Our most revealing numerical result is that, Robin-type boundary conditions seem to introduce a boundary layer coupling the bulk and surface dynamics.

Investigating the Turing conditions for diffusion-driven instability in the presence of a binding immobile substrate.

Turing's diffusion-driven instability for the standard two species reaction-diffusion system is only achievable under well-known and rather restrictive conditions on both the diffusion rates and the kinetic parameters, which necessitates the pairing of a self-activator with a self-inhibitor. In this study we generalize the standard two-species model by considering the case where the reactants can bind to an immobile substrate, for instance extra-cellular matrix, and investigate the influence of this dynamics on Turing's diffusion-driven instability. Such systems have been previously studied on the grounds that binding of the self-activator to a substrate may effectively reduce its diffusion rate and thus induce a Turing instability for species with equal diffusion coefficients, as originally demonstrated by Lengyel and Epstein (1992) under the assumption that the bound state dynamics occurs on a fast timescale. We, however, analyse the full system without any separation of timescales and demonstrate that the full system also allows a relaxation of the standard constraints on the reaction kinetics for the Turing instability, increasing the type of interactions that could give rise to spatial patterning. In particular, we show that two self-activators can undertake a diffusively driven instability in the presence of a binding immobile substrate, highlighting that the interactions required of a putative biological Turing instability need not be associated with a self-activator-self-inhibitor morphogen pair.

Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations

By introducing linear cross-diffusion for a two-component reaction-diffusion system with activator-depleted reaction kinetics (Gierer and Meinhardt, Kybernetik 12:30-39, 1972; Prigogine and Lefever, J Chem Phys 48:1695-1700, 1968; Schnakenberg, J Theor Biol 81:389-400, 1979), we derive cross-diffusion-driven instability conditions and show that they are a generalisation of the classical diffusion-driven instability conditions in the absence of cross-diffusion. Our most revealing result is that, in contrast to the classical reaction-diffusion systems without cross-diffusion, it is no longer necessary to enforce that one of the species diffuse much faster than the other. Furthermore, it is no longer necessary to have an activator-inhibitor mechanism as premises for pattern formation, activator-activator, inhibitor-inhibitor reaction kinetics as well as short-range inhibition and long-range activation all have the potential of giving rise to cross-diffusion-driven instability. To support our theoretical findings, we compute cross-diffusion induced parameter spaces and demonstrate similarities and differences to those obtained using standard reaction-diffusion theory. Finite element numerical simulations on planary square domains are presented to back-up theoretical predictions. For the numerical simulations presented, we choose parameter values from and outside the classical Turing diffusively-driven instability space; outside, these are chosen to belong to cross-diffusively-driven instability parameter spaces. Our numerical experiments validate our theoretical predictions that parameter spaces induced by cross-diffusion in both the [Formula: see text] and [Formula: see text] components of the reaction-diffusion system are substantially larger and different from those without cross-diffusion. Furthermore, the parameter spaces without cross-diffusion are sub-spaces of the cross-diffusion induced parameter spaces. Our results allow experimentalists to have a wider range of parameter spaces from which to select reaction kinetic parameter values that will give rise to spatial patterning in the presence of cross-diffusion.

Existence of spatial patterns in a predator–prey model with self- and cross-diffusion

In this work, we investigate the spatiotemporal dynamics of reaction–diffusion equations subject to cross-diffusion in the frame of a two-dimensional ratio-dependent predator–prey model. The conditions for diffusion-driven instability are obtained and the Turing space in the parameters space is achieved. Moreover, the criteria for local and global asymptotic stability of the unique positive homogeneous steady state without diffusion are discussed. Numerical simulations are carried out in order to validate the feasibility of the obtained analytical findings. Different types of spatial patterns through diffusion-driven instability of the proposed model are portrayed and analysed. Lastly, the paper finishes with an external discussion of biological relevance of the analysis regarding cross-diffusion and pattern issues.

Spatial pattern formation of a ratio-dependent predator prey model

This paper presents a theoretical analysis of evolutionary process that involves organisms distribution and their interaction of spatially distributed population with diffusion in a Holling-III ratio-dependent predator–prey model, the sufficient conditions for diffusion-driven instability with Neumann boundary conditions are obtained. Furthermore, it presents novel numerical evidence of time evolution of patterns controlled by diffusion in the model, and finds that the model dynamics exhibits complex pattern replication, and the pattern formation depends on the choice of the initial conditions. The ideas in this paper may provide a better understanding of the pattern formation in ecosystems.

Turing instability conditions for growing domains with divergence free mesh velocity

Questions have been asked from a theoretical point of view about the effects of domain growth on Turing [A.M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Lond. B 237 (1952) 37–72] diffusion-driven instability analysis for reaction–diffusion systems on continuously growing domains. The increase in computational power and tools has given the scientific community an almost immediate vast amount of computational data on pattern transitions (peak splitting, peak insertion, period doubling, circular patterns etc) on growing domains but with little knowledge of the theoretical justifications of the role played by domain growth in pattern formation. This paper derives the conditions for Turing diffusion-driven instability on growing domains whose domain velocity is divergence free. In particular we establish the result that the conditions for diffusion-driven instability growing domains whose domain velocity is divergence free are equivalent to those derived on a fixed domain. For this type of growth, the advection terms have no effect on the standard diffusion-driven instability conditions. Under these conditions, travelling wave solutions are obtained. This result has implications in the selection of parameter values as these are selected such that they belong to the Turing space. Our results do not however generalise to arbitrary growth protocols such as exponential, linear or logistic growths since the divergence of the mesh velocity is time-dependent.

Stability analysis of Reaction-Diffusion Systems with constant coefficients on growing domains

We present stability theory for reaction-diffusion systems with constant coefficients on growing domains. The model equations on growing domains are transformed to fixed domains at each time yielding a conservative system. We derive and show that the diffusion-driven instability conditions for an exponentially growing domain depend on the domain growth rate. By looking at the eigenvalues, we show that the shifting of the Turing space is equivalent to the standard Turing space on a fixed domain but with eigenvalues shifted to the left of the complex plane by a constant factor given by the divergence of the domain velocity.