Nontrivial large-time behaviour in bistable reaction–diffusion equations

Bistable reaction–diffusion equations are known to admit one-dimensional travelling waves which are globally stable to one-dimensional perturbations—Fife and McLeod [7]. These planar waves are also stable to two-dimensional perturbations—Xin [30], Levermore-Xin [19], Kapitula [16]—provided that these perturbations decay, in the direction transverse to the wave, in an integrable fashion. In this paper, we first prove that this result breaks down when the integrability condition is removed, and we exhibit a large-time dynamics similar to that of the heat equation. We then apply this result to the study of the large-time behaviour of conical-shaped fronts in the plane, and exhibit cases where the dynamics is given by that of two advection–diffusion equations.      

Cubic autocatalysis in a reaction–diffusion annulus: semi-analytical solutions

Semi-analytical solutions for cubic autocatalytic reactions are considered in a circularly symmetric reaction–diffusion annulus. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for this novel geometry. Singularity theory is used to determine the regions of parameter space in which the different types of steady-state diagram occur. The region of parameter space, in which Hopf bifurcations can occur, is found using a degenerate Hopf bifurcation analysis. A novel feature of this geometry is the effect, of varying the width of the annulus, on the static and dynamic multiplicity. The results show that for a thicker annulus, Hopf bifurcations and multiple steady-state solutions occur in a larger portion of parameter space. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with numerical solutions of the governing partial differential equations.     

Exact solutions of a reaction–diffusion system for Proteus mirabilis bacterial colonies

A mathematical model for Proteus mirabilis colonies is considered in the framework of transformation groups. New solutions via classical and non-classical symmetries are obtained.     

Space–time chaos in a system of reaction–diffusion equations

We find conditions for the bifurcation of periodic spatially homogeneous and spatially inhomogeneous solutions of a three-dimensional system of nonlinear partial differential equations describing a soil aggregate model. We show that the transition to diffusion chaos in this model occurs via a subharmonic cascade of bifurcations of stable limit cycles in accordance with the universal Feigenbaum–Sharkovskii–Magnitskii bifurcation theory.     

The global existence and asymptotic stability of solutions for a reaction–diffusion system

This paper studies the solutions of a reaction–diffusion system with nonlinearities that generalize the Lengyel–Epstein and FitzHugh–Nagumo nonlinearities. Sufficient conditions are derived for the global asymptotic stability of the solutions. Furthermore, we present some numerical examples.     

Stability of a nonlocal delayed reaction–diffusion equation with a non-monotone bistable nonlinearity

The present work is devoted to the stability and attractivity analysis of a nonlocal delayed reaction–diffusion equation (DRDE) with a non-monotone bistable nonlinearity that describes the population dynamics for a two-stage species with Allee effect. By the idea of relating the dynamics of the nonlinear term to the DRDE and some stability results for the monostable case, we describe some basin of attractions for the DRDE. Additionally, existence of heteroclinic orbits and periodic oscillations are also obtained. Numerical simulations are also given at last to verify our theoretical results.     

Null controllability of a degenerated reaction–diffusion system in cardiac electro-physiology

This Note is devoted to the analysis of the null controllability of a nonlinear reaction–diffusion system, approximating a parabolic–elliptic system, modeling electrical activity in the heart. The uniform, with respect to the degenerating parameter, null controllability of the approximating system by a single control force acting on a subdomain is shown. The proof needs a precise estimate with respect to the degenerating parameter and it is done combining Carleman estimates and energy inequalities.     

Steady-state solutions of a reaction–diffusion system arising from intraguild predation and internal storage

Intraguild predation is added to a mathematical model of competition between two species for a single nutrient with internal storage in the unstirred chemostat. At first, we established the sharp a priori estimates for nonnegative solutions of the system, which assure that all of nonnegative solutions belong to a special cone. The selection of this special cone enables us to apply the topological fixed point theorems in cones to establish the existence of positive solutions. Secondly, existence for positive steady state solutions of intraguild prey and intraguild predator is established in terms of the principal eigenvalues of associated nonlinear eigenvalue problems by means of the degree theory in the special cone. It turns out that positive steady state solutions exist when the associated principal eigenvalues are both negative or both positive.     

Exact solutions for logistic reaction–diffusion equations in biology

Reaction–diffusion equations with a nonlinear source have been widely used to model various systems, with particular application to biology. Here, we provide a solution technique for these types of equations in N-dimensions. The nonclassical symmetry method leads to a single relationship between the nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent solutions for the Kirchhoff variable are exponential in time (either growth or decay) and satisfy the linear Helmholtz equation in space. Example solutions are given in two dimensions for particular parameter sets for both quadratic and cubic reaction terms.     

Dynamics of a reaction–diffusion–ODE system with quiescence

We consider a reaction–diffusion–ODE quiescent model in which the species can switch between mobile and immobile categories. We assume that the population inhabits a bounded region and study how its dynamics depend on the parameters describing switching rates and local population dynamics. Our results suggest that the transfer displays a stabilizing effect and inhibits the generation of spatial periodic solutions. A new method to obtain global stability and dissipative structure is also explored by constructing Lyapunov functionals to overcome the loss of compactness.     

Spreading speeds of a partially degenerate reaction–diffusion system in a periodic habitat

This paper is devoted to the study of the spreading speeds of a partially degenerate reaction–diffusion system with monostable nonlinearity in a periodic habitat. We first obtain sufficient conditions for the existence of principal eigenvalues in the case where solution maps of the associated linear systems lack compactness, and prove a threshold type result on the global dynamics for the periodic initial value problem. Then we establish the existence and computational formulae of spreading speeds for the general initial value problem. It turns out that the spreading speed is linearly determinate.     

Periodic solutions in reaction–diffusion equations with time delay

Spatial diffusion and time delay are two main factors in biological and chemical systems. However, the combined effects of them on diffusion systems are not well studied. As a result, we investigate a nonlinear diffusion system with delay and obtain the existence of the periodic solutions using coincidence degree theory. Moreover, two numerical examples confirm our theoretical results. The obtained results can also be applied in other related fields.     

Existence and uniqueness of forced waves in a delayed reaction–diffusion equation in a shifting environment

The existence and uniqueness of forced waves in a general reaction–diffusion equation with time delay under climate change is concerned in this paper. By using upper and lower solutions method, monotone iteration scheme combined with the strong maximum principle, we show that there exists a nondecreasing and unique wave front with the speed consistent with the habitat shifting speed. Our results indicate the propagation of both the leading and trailing edges of the comoving population wavefront lag behind the climate envelope, which drives the species to extinction. Three examples and their corresponding numerical simulations are also given to illustrate the universality of analytical conclusions.     

A self-organized mesh generator using pattern formation in a reaction–diffusion system

A new type of mesh generator is developed by using a self-organized pattern in a reaction–diffusion system. The system is the Gray–Scott model, which creates a spot pattern in a specific parameter region. The spots correspond to nodes of a mesh. The mesh generator has several advantages: the algorithm is simple and processes to improve the mesh, such as smoothing, (locally) addition, and removal of nodes, are automatically performed by the system.     

Nonnegative solutions to reaction–diffusion system with cross-diffusion and nonstandard growth conditions

We establish the existence of nonnegative weak solutions to nonlinear reaction–diffusion system with cross-diffusion and nonstandard growth conditions subject to the homogeneous Neumann boundary conditions. We assume that the diffusion operators satisfy certain monotonicity condition and nonstandard growth conditions and prove that the existence of weak solutions using Galerkin's approximation technique.