On a Stefan-type model arising in ecology

We discuss the dispersal of a chemical, emitted from a point source, into a reacting medium. The diffusion and reaction are such that a front emerges, so that at all times the chemical remains confined to a bounded domain around the source. We focus on the final stationary state which is a non-standard Stefan-type boundary value problem for an Emden–Fowler equation. We prove its existence and uniqueness and obtain, analytically as well as numerically, bounds for the location of the front and for the concentration profile.     

Bifurcation to locked fronts in two component reaction–diffusion systems

We study invasion fronts and spreading speeds in two component reaction–diffusion systems. Using a variation of Lin's method, we construct traveling front solutions and show the existence of a bifurcation to locked fronts where both components invade at the same speed. Expansions of the wave speed as a function of the diffusion constant of one species are obtained. The bifurcation can be sub or super-critical depending on whether the locked fronts exist for parameter values above or below the bifurcation value. Interestingly, in the sub-critical case numerical simulations reveal that the spreading speed of the PDE system does not depend continuously on the coefficient of diffusion.     

Traveling wave front for a two-component lattice dynamical system arising in competition models

We study traveling front solutions for a two-component system on a one-dimensional lattice. This system arises in the study of the competition between two species with diffusion (or migration), if we divide the habitat into discrete regions or niches. We consider the case when the nonlinear source terms are of Lotka–Volterra type and of monostable case. We first show that there is a positive constant (the minimal wave speed) such that a traveling front exists if and only if its speed is above this minimal wave speed. Then we show that any wave profile is strictly monotone. Moreover, under some conditions, we show that the wave profile is unique (up to translations) for a given wave speed. Finally, we characterize the minimal wave speed by the parameters in the system.     

Turing bifurcation in a system with cross diffusion

This paper treats the conditions for the existence and stability properties of stationary solutions of reaction–diffusion equations subject to Neumann boundary data. Hence, we assume that there are two substances in a two-dimensional bounded spatial domain where they are diffusing according to Fick's law: the velocity of the flow of diffusing substance is directed opposite to the (spatial) gradient of the density and is proportional to its modulus, but the spatial flow of each substance is influenced not only by its own but also by the other one's density (cross diffusion). The domains in which the substances are diffusing are of three type: a regular hexagon, a rectangle and an isosceles rectangular triangle. It will be assumed that there is no migration across the boundary of these domains. Considering one of the relevant features of the domains as a bifurcation parameter it will be shown that at a certain critical value a diffusion driven (Turing type) instability occurs, i.e. the stationary solution stays stable with respect to the kinetic system (the system without diffusion) but becomes unstable with respect to the system with diffusion and that Turing bifurcation takes place: a spatially non-homogenous (non-constant) solution (structure or pattern) arises.     

Long-term coexistence for a competitive system of spatially varying gradient reaction–diffusion equations

Spatial distribution of interacting chemical or biological species is usually described by a system of reaction–diffusion equations. In this work we consider a system of two reaction–diffusion equations with spatially varying diffusion coefficients which are different for different species and with forcing terms which are the gradient of a spatially varying potential. Such a system describes two competing biological species. We are interested in the possibility of long-term coexistence of the species in a bounded domain. Such long-term coexistence may be associated either with a periodic in time solution (usually associated with a Hopf bifurcation), or with time-independent solutions. We prove that no periodic solution exists for the system. We also consider some steady states (the time-independent solutions) and examine their stability and bifurcations.     

一类竞争扩散系统的定态分歧与稳定性

本文研究一类竞争扩散系统,在方程所描述的模型中,两个相互竞争的物种栖息在同一有界区域内,相互制约的项是Holling-Tanner型的,在齐次Dirichlet边界条件下,应用谱分析和分歧理论的方法,研究了非负定态解的分歧及其稳定性。     

Oscillatory reaction-diffusion equations with temporally varying parameters

Periodic wave trains are the generic one-dimensional solution form for reaction-diffusion equations with a limit cycle in the kinetics. Such systems are widely used as models for oscillatory phenomena in chemistry, ecology, and cell biology. In this paper, we study the way in which periodic wave solutions of such systems are modified by periodic forcing of kinetic parameters. Such forcing will occur in many ecological applications due to seasonal variations. We study temporal forcing in detail for systems of two reaction diffusion equations close to a supercritical Hopf bifurcation in the kinetics, with equal diffusion coefficients. In this case, the kinetics can be approximated by the Hopf normal form, giving reaction-diffusion equations of λ-ω type. Numerical simulations show that a temporal variation in the kinetic parameters causes the wave train amplitude to oscillate in time, whereas in the absence of any temporal forcing, this wave train amplitude is constant. Exploiting the mathematical simplicity of the λ-ω form, we derive analytically an approximation to the amplitude of the wave train oscillations with small forcing. We show that the amplitude of these oscillations depends crucially on the period of forcing.     

Bifurcation structure of rotating wave solutions of the Fitzhugh-Nagumo equations

The FitzHugh-Nagumo (FHN) equations are model equations for nerve cell behavior. They support traveling wave solutions which depend on certain parameters. In this paper, a two parameter study of rotating wave solutions (i.e. periodic wavetrains) are considered. These solutions arise from bifurcations of stationary equilibria. The local bifurcation equations are analyzed to determine bifurcation directions as functions of the parameters. In addition, dependence on parameters is computed by numerical continuation and properties of the rotating wave solutions are summarized in parameter space. Finally, some of the biological implications are discussed.     

Stationary patterns of a predator-prey model with spatial effect

In this paper, spatial patterns of predator-prey model with cross diffusion are investigated. The Hopf and Turing bifurcation critical line in a spatial domain are obtained by using mathematical theory. Moreover, exact Turing space is given in two parameters space. Our results reveal that cross diffusion can induce stationary patterns, which may be useful to help us better understand the dynamics of the real ecosystems.     

Bifurcations in a human migration model of Scheurle-Seydel type-II: Rotating waves

This paper treats the conditions for the existence of rotating wave solutions of a system modelling the behavior of students in graduate programs at neighbouring universities near each other which is a modified form of the model proposed by Scheurle and Seydel. We assume that both types of individuals are continuously distributed throughout a bounded two-dimension spatial domain of two types (circle and annulus), across whose boundaries there is no migration, and which simultaneously undergo simple (Fickian) diffusion. We will show that at a critical value of a system-parameter bifurcation takes place: a rotating wave solution arises.     

On the dynamics of a degenerate parabolic equation: global bifurcation of stationary states and convergence

We study the dynamics of a degenerate parabolic equation with a variable, generally non-smooth diffusion coefficient, which may vanish at some points or be unbounded. We show the existence of a global branch of nonnegative stationary states, covering both the cases of a bounded and an unbounded domain. The global bifurcation of stationary states, implies-in conjuction with the definition of a gradient dynamical system in the natural phase space-that at least in the case of a bounded domain, any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium. Applications of the global bifurcation result to general degenerate semilinear as well as to quasilinear elliptic equations, are also discussed. Mathematics Subject Classification (1991) 35B40, 35B41, 35R05     

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 and numerical aspects of one-dimensional laminar flame simulation

The aim of this paper is to solve several mathematical and numerical questions related to the simulation of stationary and nonstationary premixed flat flames. Most of the results are obtained in the general context of complex chemical and diffusion mechanisms. The main mathematical results concern: (i) thea priori positivity of the mass fractions, and (ii) the sensitivity of the flame speed to the computational domain. The numerical method proposed for solving the stationary problem is a new combination of the pseudo-nonstationary approach, the Newton iterations, and the adaptive gridding. The computation of H₂-O₂-N₂ flames with various initial concentrations (including the chemical extinction zone) shows the efficiency of this method.     

Direction of Hopf bifurcation in a delayed Lotka–Volterra competition diffusion system

This paper is concerned with a delayed Lotka–Volterra two species competition diffusion system with a single discrete delay and subject to homogeneous Dirichlet boundary conditions. The main purpose is to investigate the direction of Hopf bifurcation resulting from the increase of delay. By applying the implicit function theorem, it is shown that the system under consideration can undergo a supercritical Hopf bifurcation near the spatially inhomogeneous positive stationary solution when the delay crosses through a sequence of critical values.     

Nonconstant Positive Steady States and Pattern Formation of 1D Prey-Taxis Systems

Prey-taxis is the process that predators move preferentially toward patches with highest density of prey. It is well known to have an important role in biological control and the maintenance of biodiversity. To model the coexistence and spatial distributions of predator and prey species, this paper concerns nonconstant positive steady states of a wide class of prey-taxis systems with general functional responses over 1D domain. Linearized stability of the positive equilibrium is analyzed to show that prey-taxis destabilizes prey–predator homogeneity when prey repulsion (e.g., due to volume-filling effect in predator species or group defense in prey species) is present, and prey-taxis stabilizes the homogeneity otherwise. Then, we investigate the existence and stability of nonconstant positive steady states to the system through rigorous bifurcation analysis. Moreover, we provide detailed and thorough calculations to determine properties such as pitchfork and turning direction of the local branches. Our stability results also provide a stable wave mode selection mechanism for thee reaction–advection–diffusion systems including prey-taxis models considered in this paper. Finally, we provide numerical studies of prey-taxis systems with Holling–Tanner kinetics to illustrate and support our theoretical findings. Our numerical simulations demonstrate that the \(2\times 2\) prey-taxis system is able to model the formation and evolution of various striking patterns, such as spikes, periodic oscillations, and coarsening even when the domain is one-dimensional. These dynamics can model the coexistence and spatial distributions of interacting prey and predator species. We also give some insights on how system parameters influence pattern formation in these models.     

Exhaustive study of the noise-induced phase transition in a stochastic model of self-catalyzed reactions

We completely investigate the stationary distribution density in the space of relative concentrations for the three-parameter stochastic Horsthemke–Lefever model of a binary self-catalyzed cyclic chemical reaction with perturbations produced by thermal fluctuations of reagents taken into account. This model is a stationary diffusion random process generated by a stochastic equation with the Stratonovich differential, whose marginal distribution density admits a bifurcation restructuring from the unimodal to the bimodal phase with increasing noise intensity, which is interpreted physically as a dynamical phase transition induced by fluctuations in the system.     

Modelling algal densities in harmful algal blooms (HAB) with stochastic dynamics

In this paper, we consider the growth of densities of two kinds of typical HAB algae: diatom and dianoflagellate on some coasts of China’s mainland. Since there exist many random factors that cause the change of the algae densities, we shall develop a new nonlinear dynamical model with stochastic excitations on the algae densities. Applying a stochastic averaging method on the model, we obtain a two-dimensional diffusion process of averaged amplitude and phase. Then we investigate the stability and the Hopf bifurcation of the stochastic system with FPK (Fokker Planck–Kolmogorov) theory and obtain the stationary transition probability density of the process. We obtain the critical values of parameters for the occurrences of Hopf bifurcation in terms of probability. We also investigate numerically the effects of various parameters on the stationary transition probability density of the occurrences of Hopf bifurcation. The numerical results are in good correlation with the analysis. We draw the conclusion that if the Hopf bifurcation occurs with a radius large enough, i.e., if the densities of the HAB algae reach a high value, the HAB will take place with comparatively high probability.     

Traveling fronts for a delayed reaction–diffusion system with a quiescent stage

In this paper, we study the traveling wave fronts of a delayed reaction–diffusion system with a quiescent stage for a single species population with two separate mobile and stationary states. By transforming the corresponding wave system into a scalar delayed differential equation with an integral term, we establish the existence of the minimal wave speed c_min, and the asymptotic behavior, monotonicity and uniqueness (up to a translation) of the traveling wave fronts. In particular, the effects of the delay and transfer rates on the minimal wave speed are studied.     

Modulation equations and parabolic limits of reaction random-walk systems

Reaction random-walk systems are hyperbolic models to describe spatial motion (in one dimension) with finite speed and reactions of particles. Here we present two approaches which relate reaction random-walk equations with reaction diffusion equations. First, we consider the case of high particle speeds (parabolic limit). This leads to a singular perturbation analysis of a semilinear damped wave equation. A initial layer estimate is given. Secondly, we consider the case of a transcritical bifurcation. We use techniques similar to that of the Ginzburg–Landau method to find a modulation equation for the amplitude of the first unstable mode. It turns out that the modulation equation is Fisher's equation, hence near the bifurcation point travelling wave solutions are obtained. The approximation result and the corresponding estimate is given in terms of the bifurcation parameter. Both results are based on an a priori estimate for classical solutions which follows from explicit representations of the solution of the linear telegraph equation. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Traveling wave solutions of diffusive Hindmarsh–Rose-type equations with recurrent neural feedback

From the perspective of bifurcation theory, this study investigates the existence of traveling wave solutions for diffusive Hindmarsh–Rose-type (dHR-type) equations with recurrent neural feedback (RNF). The applied model comprises two additional terms: 1) a diffusion term for the conduction process of action potentials and 2) a delay term. The delay term is introduced because if a neuron excites a second neuron, the second neuron, in turn, excites or inhibits the first neuron. To probe the existence of traveling wave solutions, this study applies center manifold reduction and a normal form method, and the results demonstrate the existence of a heteroclinic orbit of a three-dimensional vector for dHR-type equations with RNF near a fold–Hopf bifurcation. Finally, numerical simulations are presented.