Pattern formation driven by cross-diffusion in a 2D domain

In this work we investigate the process of pattern formation in a two dimensional domain for a reaction–diffusion system with nonlinear diffusion terms and the competitive Lotka–Volterra kinetics. The linear stability analysis shows that cross-diffusion, through Turing bifurcation, is the key mechanism for the formation of spatial patterns. We show that the bifurcation can be regular, degenerate non-resonant and resonant. We use multiple scales expansions to derive the amplitude equations appropriate for each case and show that the system supports patterns like rolls, squares, mixed-mode patterns, supersquares, and hexagonal patterns.     

Mathematical analysis and numerical simulation of pattern formation under cross-diffusion

Cross-diffusion driven instabilities have gained a considerable attention in the field of population dynamics, mainly due to their ability to predict some important features in the study of the spatial distribution of species in ecological systems. This paper is concerned with some mathematical and numerical aspects of a particular reaction–diffusion system with cross-diffusion, modeling the effect of allelopathy on two plankton species. Based on a stability analysis and a series of numerical simulations performed with a finite volume scheme, we show that the cross-diffusion coefficient plays a important role on the pattern selection.     

Turing patterns induced by cross-diffusion in a predator-prey system in presence of habitat complexity

In this paper, we have investigated the phenomena of Turing pattern formation in a predator-prey model with habitat complexity in presence of cross diffusion. Using the linear stability analysis, the conditions for the existence of stationary pattern and the existence of Hopf bifurcation are obtained. It is shown analytically that the presence of cross diffusion in the system supports the formation of Turing pattern. Two parameter bifurcation analysis are done analytically and corresponding bifurcation diagrams are presented numerically. A series of simulation results are plotted for different biologically meaningful parameter values. Effects of variation of habitat complexity and the predator mortality rate and birth rate of prey on pattern formation are also reported. It is shown that cross-diffusion can lead to a wide variety of spatial and spatiotemporal pattern formation. It is found that the model exhibits spot and stripe pattern, and coexistence of both spot and strip patterns under the zero flux boundary condition. It is observed that cross-diffusion, habitat complexity, birth rate of prey and predator’s mortality rate play a significant role in the pattern formation of a distributed population system of predator-prey type.     

Global behavior of solutions in a Lotka–Volterra predator–prey model with prey-stage structure

In this paper, the global behavior of solutions is investigated for a Lotka–Volterra predator–prey system with prey-stage structure. First, we can see that the stability properties of nonnegative equilibria for the weakly coupled reaction–diffusion system are similar to that for the corresponding ODE system, that is, linear self-diffusions do not drive instability. Second, using Sobolev embedding theorems and bootstrap arguments, the existence and uniqueness of nonnegative global classical solution for the strongly coupled cross-diffusion system are proved when the space dimension is less than 10. Finally, the existence and uniform boundedness of global solutions and the stability of the positive equilibrium point for the cross-diffusion system are studied when the space dimension is one. It is found that the cross-diffusion system is dissipative if the diffusion matrix is positive definite. Furthermore, cross diffusions cannot induce pattern formation if the linear diffusion rates are sufficiently large.     

Turing instability and coexistence in an extended Klausmeier model with nonlocal grazing

In this paper, we study the coexistence of an extended Klausmeier model with cross-diffusion and nonlocal sustained grazing. First, we analyze a saddle–node bifurcation of spatially homogeneous system. Second, we focus on the reaction–diffusion system with nonlocal sustained grazing. Our main result is that nonlocal terms promote linear stability, and the system may produce pattern under the influences of self-diffusion and cross-diffusion. Moreover, both the grazing parameter and rainfall rate can induce transitions among bare soil state, vegetation pattern state and homogeneous vegetation state. Finally, we address the nonlocal reaction–diffusion system as a bifurcation problem, and analyze the existence and stability of bifurcation solutions. Furthermore, numerical simulations have been illustrated to verify our theoretical findings.     

Cross-diffusion induced instability and stability in reaction-diffusion systems

In a reaction-diffusion system, diffusion can induce the instability of a uniform equilibrium  which is stable with respect to a constant perturbation, as shown by Turing in 1950s. We show that cross-diffusion can destabilize  a uniform equilibrium  which is stable for the kinetic and self-diffusion reaction systems; on the other hand, cross-diffusion can also stabilize  a uniform equilibrium which is stable for the kinetic system but unstable for the self-diffusion reaction system. Application is given to predator-prey system with preytaxis and vegetation pattern formation in a water-limited ecosystem.     

Stability and cross-diffusion-driven instability in a diffusive predator–prey system with hunting cooperation functional response

This paper presents a qualitative study of a diffusive predator–prey system with the hunting cooperation functional response. For the system without diffusion, the existence, stability and Hopf bifurcation of the positive equilibrium are explicitly determined. It is shown that the hunting cooperation affects not only the existence of the positive equilibrium but also the stability. For the diffusive system, the stability and cross-diffusion driven Turing instability are investigated according to the relationship of the self-diffusion and the cross-diffusion coefficients. Stability and cross-diffusion instability regions are theoretically determined in the plane of the cross-diffusion coefficients. The technique of multiple time scale is employed to deduce the amplitude equation of Turing bifurcation and then pattern dynamics driven by the cross-diffusion is also investigated by the corresponding amplitude equation.     

Complex patterns in a predator–prey model with self and cross-diffusion

In this paper, we present a theoretical analysis of processes of pattern formation that involves organisms distribution and their interaction of spatially distributed population with self as well as cross-diffusion in a Beddington–DeAngelis-type predator–prey model. The instability of the uniform equilibrium of the model is discussed, and the sufficient conditions for the instability with zero-flux boundary conditions are obtained. Furthermore, we present novel numerical evidence of time evolution of patterns controlled by self as well as cross-diffusion in the model, and find that the model dynamics exhibits a cross-diffusion controlled formation growth not only to stripes-spots, but also to hot/cold spots, stripes and wave pattern replication. This may enrich the pattern formation in cross-diffusive predator–prey model.     

Turing pattern formation in a three species model with generalist predator and cross-diffusion

In a natural ecosystem, specialist predators feed almost exclusively on one species of prey. But generalist predators feed on many types of species. Consequently, their dynamics is not coupled to the dynamics of a specific prey population. However, the defense of prey formed by congregating made the predator tend to move in the direction of lower concentration of prey species. This is described by cross-diffusion in a generalist predator–prey model. First, the positive equilibrium solution is globally asymptotically stable for the ODE system and for the reaction–diffusion system without cross-diffusion, respectively, hence it does not belong to the classical Turing instability scheme. But it becomes linearly unstable only when cross-diffusion also plays a role. This implies that cross–diffusion can lead to the occurrence and disappearance of the instability. Our results exhibit some interesting combining effects of cross-diffusion, predations and intra-species interactions. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions.     

Stationary pattern of a diffusive prey–predator model with trophic intersections of three levels

The paper is concerned with a diffusive prey–predator model subject to the homogeneous Neumann boundary condition, which models the trophic intersections of three levels. We will prove that under certain assumptions, even though the unique positive constant steady state is globally asymptotically stable for the dynamics with diffusion, the non-constant positive steady state can exist due to the emergence of cross-diffusion. We demonstrate that the cross-diffusion can create stationary pattern. Moreover, we treat the cross-diffusion parameter as a bifurcation parameter and discuss the existence of non-constant positive solutions to the system with cross-diffusion.     

Cross-diffusion induced Turing instability for a three species food chain model

In this paper, we study a strongly coupled reaction–diffusion system describing three interacting species in a food chain model, where the third species preys on the second one and simultaneously the second species preys on the first one. We first show that the unique positive equilibrium solution is globally asymptotically stable for the corresponding ODE system. The positive equilibrium solution remains linearly stable for the reaction–diffusion system without cross-diffusion, hence it does not belong to the classical Turing instability scheme. We further proved that the positive equilibrium solution is globally asymptotically stable for the reaction–diffusion system without cross-diffusion by constructing a Lyapunov function. But it becomes linearly unstable only when cross-diffusion also plays a role in the reaction–diffusion system, hence the instability is driven solely from the effect of cross-diffusion. Our results also exhibit some interesting combining effects of cross-diffusion, intra-species competitions and inter-species interactions.     

Spatial patterns of the Holling–Tanner predator–prey model with nonlinear diffusion effects

In this article, we study the Holling–Tanner predator–prey model with nonlinear diffusion terms under homogeneous Neumann boundary condition. The nonlinear diffusion terms here mean that the prey runs away from the predator, and the predator chases the prey. Nonexistence and existence of nonconstant positive steady states are obtained, which reveal that cross-diffusion can create spatial patterns even when the random diffusion fails to do so. Moreover, asymptotic behaviour of positive solutions as the cross-diffusion tends to ∞ is shown.     

Cross-Diffusion Driven Instability in a Predator-Prey System with Cross-Diffusion

In this work we investigate the process of pattern formation induced by nonlinear diffusion in a reaction-diffusion system with Lotka-Volterra predator-prey kinetics. We show that the cross-diffusion term is responsible of the destabilizing mechanism that leads to the emergence of spatial patterns. Near marginal stability we perform a weakly nonlinear analysis to predict the amplitude and the form of the pattern, deriving the Stuart-Landau amplitude equations. Moreover, in a large portion of the subcritical zone, numerical simulations show the emergence of oscillating patterns, which cannot be predicted by the weakly nonlinear analysis. Finally, when the pattern invades the domain as a travelling wavefront, we derive the Ginzburg-Landau amplitude equation which is able to describe the shape and the speed of the wave.     

Cross-diffusion effects on a morphochemical model for electrodeposition

We analyze the effects of cross-diffusion on pattern formation in a PDE reaction-diffusion system introduced in Bozzini et al. 2013 to describe metal growth in an electrodeposition process. For this morphochemical model - which refers to the physico-chemical problem of coupling of growth morphology and surface chemistry - we have found that negative cross-diffusion in the morphological elements as well as positive cross-diffusion in the surface chemistry produce larger Turing parameter spaces and favor a tendency to stripeness that is not found in the case without cross-diffusion. The impact of cross-diffusion on pattern selection has been also discussed by the means of a stripeness index. Our theoretical findings are validated by an extensive gallery of numerical simulations that allow to better clarify the role of cross-diffusion both on Turing parameter spaces and on pattern selection. Experimental evidence of cross-diffusion in electrodeposition as well as a physico-chemical discussion of the expected impact of cross diffusion-controlled pattern formation in alloy electrodeposition processes complete the study.     

Turing pattern formation in a predator–prey system with cross diffusion

The paper explores the impacts of cross-diffusion on the formation of spatial patterns in a ratio-dependent predator–prey system with zero-flux boundary conditions. Our results show that under certain conditions, cross-diffusion can trigger the emergence of spatial patterns which is however impossible under the same conditions when cross-diffusion is absent. We give a rigorous proof that the model has at least one spatially heterogenous steady state by means of the Leray–Schauder degree theory. In addition, numerical simulations are performed to visualize the complex spatial patterns.     

Finite-time blowup and global-in-time unbounded solutions to a parabolic–parabolic quasilinear Keller–Segel system in higher dimensions

In this paper we consider quasilinear Keller–Segel type systems of two kinds in higher dimensions. In the case of a nonlinear diffusion system we prove an optimal (with respect to possible nonlinear diffusions generating explosion in finite time of solutions) finite-time blowup result. In the case of a cross-diffusion system we give results which are optimal provided one assumes some proper non-decay of a nonlinear chemical sensitivity. Moreover, we show that once we do not assume the above mentioned non-decay, our result cannot be as strong as in the case of nonlinear diffusion without nonlinear cross-diffusion terms. To this end we provide an example, interesting by itself, of global-in-time unbounded solutions to the nonlinear cross-diffusion Keller–Segel system with chemical sensitivity decaying fast enough, in a range of parameters in which there is a finite-time blowup result in a corresponding case without nonlinear cross-diffusion.     

Study of cross-diffusion induced Turing patterns in a ratio-dependent prey-predator model via amplitude equations

Cross-diffusion models the situation where the presence, absence or abundance of one species of population affects the movement of other species of population in the domain under consideration and vice versa. Inclusion of cross-diffusion terms makes the modeling approach more realistic and shows significant impact on the spatio-temporal pattern formation scenario. In this paper, cross-diffusion is considered in a prey-predator model with ratio-dependent functional response, in addition to self-diffusion. Weakly nonlinear analysis is used near the Turing bifurcation boundary to derive the amplitude equations. From the stability analysis of the amplitude equations, conditions for emergence of Turing patterns such as cold spot, hot spot, mixture of spots and stripes and labyrinthine are identified. The analytical results are then verified with the help of numerical simulations. Results are general in nature and can be used to study the effect of cross-diffusion on other prey predator models both analytically and numerically.     

Strategies for the existence of spatial patterns in predator–prey communities generated by cross-diffusion

Fear of predators is an important drive for predator–prey interactions, which increases survival probability but cost the overall population size of the prey. In this paper, we have extended our previous work spatiotemporal dynamics of predator–prey interactions with fear effect by introducing the cross-diffusion. The conditions for cross-diffusion-driven instability are obtained using the linear stability analysis. The standard multiple scale analysis is used to derive the amplitude equations for the excited modes near Turing bifurcation threshold by taking the cross-diffusion coefficient as a bifurcation parameter. From the stability analysis of amplitude equations, the conditions for the emergence of various ecologically realistic Turing patterns such as spot, stripe, and mixture of spots and stripes are identified. Analytical results are verified with the help of numerical simulations. Turing bifurcation diagrams are plotted taking diffusion coefficients as control parameters. The effect of the cross-diffusion coefficients on the homogeneous steady state and pattern structures of the self-diffusive model is illustrated using the simulation techniques. It is also observed that the level of fear has stabilizing effect on the cross-diffusion induced instability and spot patterns change to stripe, then a mixture of spots and stripes and finally to the labyrinthine type of patterns with an increase in the level of fear.     

The global stability and pattern formations of a predator–prey system with consuming resource

The paper deals with a model that describes a predator–prey system with a common consuming resource. We use Lyapunov functions to prove the global stability of the kinetic system and the diffusive system. The existence of non-constant positive steady state solutions is shown to identify the range of parameters of spatial pattern formation for the cross-diffusion system.     

Existence of weak solutions to the Keller–Segel chemotaxis system with additional cross-diffusion

In this paper, we consider the Keller–Segel chemotaxis system with additional cross-diffusion term in the second equation. This system is consisting of a fully nonlinear reaction–diffusion equations with additional cross-diffusion. We establish the existence of weak solutions to the considered system by using Schauder’s fixed point theorem, a priori energy estimates and the compactness results.