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.     

Spreading speed and traveling waves for a nonmonotone reaction–diffusion model with distributed delay and nonlocal effect

In this paper, spreading speed and traveling waves for reaction–diffusion model with distributed delay and nonlocal effect without monotonicity are investigated. It is shown that there exists the spreading speed c^∗ which coincides with the minimal wave speed, and its limiting integral equation has an unique traveling wave with speed c > c^∗, and no traveling wave with c < c^∗. Moreover, the dependence of the spreading speed on the delay and the nonlocal effect is considered.     

Doubly nonlocal reaction–diffusion equations and the emergence of species

The paper is devoted to a reaction–diffusion equation with doubly nonlocal nonlinearity arising in various applications in population dynamics. One of the integral terms corresponds to the nonlocal consumption of resources while another one describes reproduction with different phenotypes. Linear stability analysis of the homogeneous in space stationary solution is carried out. Existence of traveling waves is proved in the case of narrow kernels of the integrals. Periodic traveling waves are observed in numerical simulations. Existence of stationary solutions in the form of pulses is shown, and transition from periodic waves to pulses is studied. In the applications to the speciation theory, the results of this work signify that new species can emerge only if they do not have common offsprings. Thus, it is shown how Darwin’s definition of species as groups of morphologically similar individuals is related to Mayr’s definition as groups of individuals that can breed only among themselves.     

The delay effect on reaction–diffusion equations

In this work we study a reaction–diffusion problem with delay and we make an analysis of the stability of solutions by means of bifurcation theory. We take the delay constant as a parameter. Special conditions on the vector field assure existence of a spatially nonconstant positive equilibrium U_k , which is stable for small values of the delay. An increase of the delay destabilizes the equilibrium of U_k and leads to super or subcritical Hopf bifurcation.     

Stability of traveling wave fronts for nonlocal reaction diffusion equations with delay北大核心CSCD

This paper is concerned with the stability of traveling wave fronts for reaction diffusion equations with nonlocal delay. We prove that, in the appropriate weighted L∞ spaces, the non-critical traveling wave fronts are globally exponentially stable, and the critical traveling wave fronts are globally algebraically stable. Moreover, we obtain the rates of convergence by weighted energy estimates. We apply these results to a host-vector disease model, the generalized Nicholson blowflies equation, and a modified vector disease model. ©, 2015, Chinese Academy of Sciences. All right reserved.     

Traveling waves in delayed predator–prey systems with nonlocal diffusion and stage structure

This paper is concerned with the existence of traveling wave solutions of a delayed predator–prey system with stage structure and nonlocal diffusion. By introducing the partial quasi-monotone condition and cross-iteration scheme, we first consider a class of delayed systems with nonlocal diffusion and deduce the existence of traveling wave solutions to the existence of a pair of upper–lower solutions. When the result is applied to the predator–prey system, we establish the existence of traveling wave solutions, as well as its precisely asymptotic behavior. Our result implies that there is a transition zone moving from the steady state with no species to the steady state with the coexistence of both species.     

Asymptotic speed of spread and traveling fronts for a nonlocal reaction–diffusion model with distributed delay

In this paper, a nonlocal reaction–diffusion model with distributed delay is studied. The asymptotic speed of spread is established for this model, and its coincidence with the minimal wave speed for traveling wave fronts is proved. Moreover, the dependence of the asymptotic speed of spread on delay and the nonlocal effect is considered. Our main finding here is that the delay can induce slow asymptotic speed of spread while the nonlocal effect can increase fast asymptotic speed of spread.     

Long-term behavior of reaction–diffusion equations with nonlocal boundary conditions on rough domains

We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction–diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Hölder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, for the well-known reaction–diffusion equation in smooth domains, the framework we develop also makes possible a number of new results for all diffusion models in other non-smooth settings.     

Finite difference reaction–diffusion equations with nonlinear diffusion coefficients

Summary. A monotone iterative method for numerical solutions of a class of finite difference reaction-diffusion equations with nonlinear diffusion coefficient is presented. It is shown that by using an upper solution or a lower solution as the initial iteration the corresponding sequence converges monotonically to a unique solution of the finite difference system. It is also shown that the solution of the finite difference system converges to the solution of the continuous equation as the mesh size decreases to zero. Received February 18, 1998 / Revised version received April 21, 1999 / Published online February 17, 2000     

Stability and bifurcation in a two-species reaction–diffusion–advection competition model with time delay

In this paper, we are concerned with the dynamics of a class of two-species reaction–diffusion–advection competition models with time delay subject to the homogeneous Dirichlet boundary condition or no-flux boundary condition in a bounded domain. The existence of steady state solution is investigated by means of the Lyapunov–Schmidt reduction method. The stability and Hopf bifurcation at the spatially nonhomogeneous steady-state are obtained by analyzing the distribution of the associated eigenvalues. Finally, the effect of advection on Hopf bifurcation is explored, which shows that with the increase of convection rate, the Hopf bifurcation phenomenon is more likely to emerge.     

Non-simultaneous blow-up in localized reaction–diffusion equations

This article deals with blow-up solutions in reaction–diffusion equations coupled via localized exponential sources, subject to null Dirichlet conditions. The optimal and complete classification is obtained for simultaneous and non-simultaneous blow-up solutions. Moreover, blow-up rates and blow-up sets are also discussed. It is interesting that, in some exponent regions, blow-up phenomena depend sensitively on the choosing of initial data, and the localized nonlinearities play important roles in the blow-up properties of solutions.     

Numerical simulation of blowup in nonlocal reaction–diffusion equations using a moving mesh method

In this paper we implement the moving mesh PDE method for simulating the blowup in reaction–diffusion equations with temporal and spacial nonlinear nonlocal terms. By a time-dependent transformation, the physical equation is written into a Lagrangian form with respect to the computational variables. The time-dependent transformation function satisfies a parabolic partial differential equation — usually called moving mesh PDE (MMPDE). The transformed physical equation and MMPDE are solved alternately by central finite difference method combined with a backward time-stepping scheme. The integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach. The monitor function in MMPDEs plays a key role in the performance of the moving mesh PDE method. The dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle. A variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests.     

Spatially discrete reaction–diffusion equations with discontinuous hysteresis

We address the question: Why may reaction–diffusion equations with hysteretic nonlinearities become ill-posed and how to amend this? To do so, we discretize the spatial variable and obtain a lattice dynamical system with a hysteretic nonlinearity. We analyze a new mechanism that leads to appearance of a spatio-temporal pattern called rattling: the solution exhibits a propagation phenomenon different from the classical traveling wave, while the hysteretic nonlinearity, loosely speaking, takes a different value at every second spatial point, independently of the grid size. Such a dynamics indicates how one should redefine hysteresis to make the continuous problem well-posed and how the solution will then behave. In the present paper, we develop main tools for the analysis of the spatially discrete model and apply them to a prototype case. In particular, we prove that the propagation velocity is of order $a{t}^{?1/2}$ as $t\to \infty$ and explicitly find the rate a.     

Blocking and invasion for reaction–diffusion equations in periodic media

We investigate the large time behavior of solutions of reaction–diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous bistable equation set in a domain with periodic holes, and specifically on the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one.     

On one-dimensional forward–backward diffusion equations with linear convection and reaction

We study the initial-Neumann boundary value problem for a class of one-dimensional forward–backward diffusion equations with linear convection and reaction. The diffusion flux function is assumed to contain two forward-diffusion phases. We prove that for all smooth initial data with derivative value lying in certain phase transition regions one can construct infinitely many Lipschitz solutions that exhibit instantaneous phase transitions between the two forward phases. Furthermore, we introduce a notion of transition gauge for such solutions and prove that the transition gauge of all such constructed solutions can be arbitrarily close to a certain fixed constant. The results are new even for the pure forward–backward diffusion problem without convection and reaction. Our primary approach relies on a combination of the convex integration and Baire's category methods for a related nonlocal differential inclusion.