Delayed Reaction Diffusion Equations Research Articles

Existence and uniqueness result for reaction-diffusion model of diffusive population dynamics

The present work investigates a convolution type nonlinear integrodifferential equation with diffusion. This type of equations represents not only pure mathematical interest, but also is widely used in various applications, especially in wide range of problems on population dynamics arising in biology. The existence of a parametric family of travelling wave solutions as well as the uniqueness of the solution in certain class of bounded continuous functions on R \mathbb {R} is proved. The study investigates also some important properties as well as asymptotic behaviour of constructed solutions. These results are then used to derive a new uniform estimate for the deviation between successive iterations, which provides us with a strong tool to control the number of iterations on our way of computing the desired numerical approximation of the exact solution. Finally, we apply our theoretical results to two well-known population problems modelled by delayed reaction-diffusion equation: Diffusion model for spatial-temporal spread of epidemics and stage structured population model.

Exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain

The main objective of this paper is to investigate exponential attractors for a nonlocal delayed reaction-diffusion equation on an unbounded domain. We first obtain the existence of a globally attractive absorbing set for the dynamical system generated by the equation under the assumption that the nonlinear term is bounded. Then, we construct exponential attractors of the equation directly in its natural phase space, i.e., a Banach space with explicit fractal dimension by combining squeezing properties of the system as well as a covering lemma of finite dimensional subspaces of a Banach space. Our result generalizes the methods established in Hilbert spaces and weighted spaces, and the fractal dimension of the obtained exponential attractor does not depend on the entropy number but only depends on some inner characteristic of the studied equation.

Time-dependent uniform upper semicontinuity of pullback attractors for non-autonomous delay dynamical systems: Theoretical results and applications

In this paper we provide general results on the uniform upper semicontinuity of pullback attractors with respect to the time parameter for non-autonomous delay dynamical systems. Namely, we establish a criteria in terms of the multi-index convergence of solutions for the delay system to the non-delay one, locally pointwise convergence and local controllability of pullback attractors. As an application, we prove the upper semicontinuity of pullback attractors for a non-autonomous delay reaction-diffusion equation to the corresponding nondelay one over any bounded time interval as the delay parameter tends to zero.

Long-term dynamics of fractional stochastic delay reaction–diffusion equations on unbounded domains

Long-term dynamics of fractional stochastic delay reaction–diffusion equations on unbounded domains

Comments to ‘Stability of traveling waves for deterministic and stochastic delayed reaction-diffusion equation based on phase shift’

In Liu et al. (2023), the authors set out to prove orbital stability for traveling waves in a (stochastic) delay reaction–diffusion equation. For the deterministic proof, they follow a phase-tracking technique as developed by Zumbrun and Howard (1998) [1] and for the stochastic version, they use a stochastic phase-tracking technique as developed by Hamster and Hupkes (2019, 2020a,b) [2,3] (which in turn is based on the aforementioned technique by Howard and Zumbrun). Note that the authors do not acknowledge the origins of their techniques.In the deterministic case, stability results are known (Mei, 2009) [4], but the application of phase-tracking to this problem appears to be new. However, there is a very subtle interplay between the time-dependent phase and the delay in the delay equation which is not accounted for, nor is explained or proved if the chosen phase has the same properties in the general framework of delay semigroups as for regular semigroups.The proof of the stochastic case of course has the same issues, but let us assume that the issues can be resolved, then there are still several issues with the proof. Especially, the phase tracking introduces terms that depend on the gradient of the solution which must be bounded in the H1 norm. This introduces singularities in the bounds that do not show in these proofs.

Spreading speeds of a nonmonotonic delayed equation in a shifting environment

This article studies the spreading speeds of a nonlocal delayed reaction–diffusion equation in a shifting environment. The equation does not satisfy the quasimonotone property. Some auxiliary equations are given to estimate its spreading speeds. Applying the theory to an SIR model, we may observe different propagation terraces.

Traveling waves in nonlocal delayed reaction–diffusion bistable equations and applications

In this paper, we are concerned with traveling waves in a class of reaction–diffusion equations with nonlocal delays. By introducing new variables, we transform equations with nonlocal delays to non‐delayed system; the existence of traveling waves is obtained by means of Routh–Hurwitz criterion, and by considering the regularity of the Cauchy problem and using spectral theory, we investigate the global stability of traveling waves and then the uniqueness of wave speeds with the help of upper and lower solution method. Finally, our results are applied to two nonlocal delayed reaction–diffusion equations.

Stability of traveling waves for deterministic and stochastic delayed reaction–diffusion equation based on phase shift

In this paper, we establish the nonlinear orbital stability of the traveling wave solution of deterministic and stochastic delayed reaction–diffusion equation. Employing the deterministic phase shift and establishing a delayed-integral inequality, we obtain the exponential stability of the traveling wave solution for the deterministic delayed reaction–diffusion equation. Applying a stochastic phase shift and time transformation, we then verify that the traveling wave solution of the deterministic equation retain the nonlinear orbital stability when the noise intensity is small enough and the initial value sufficiently closes the traveling wave.

Random attractors for a stochastic nonlocal delayed reaction–diffusion equation on a semi-infinite interval

Abstract The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic non-local delayed reaction–diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition at the finite end. This equation models the spatial–temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions. The theoretical results are illustrated by application to the stochastic non-local delayed Nicholson’s blowfly equation.

The effect of grazing intensity on pattern dynamics of the vegetation system

The evolution of vegetation system in arid and semi-arid grazing areas is a complex dynamical system which depends not only on the rainfall, but also grazing intensity. Currently, most of the research focuses on the influence of rainfall, but the effects of grazing have not been fully understood. Simultaneously, the intraspecific competition delay widely exists in vegetation system. In this project, we develop a vegetation model coupled with the intraspecific competition delay, discuss the vegetation pattern caused by the combination of grazing intensity, rainfall and competition delay, by analyzing conditions of the occurrence of Turing instability. In particular, we propose a new theoretical method to deal with the Turing instability of delayed diffusion system when the system exists a zero eigenvalue and the corresponding transversity condition is not zero. Using the theory of dynamics, we can show: (i) a bistable region in which vegetation-existence and vegetation-extinction states coexist for relatively large grazing rate, in the corresponding ordinary differential system, while the vegetation-existence state is replaced by vegetation pattern state for non-delayed diffusion system, and the range of bistable region is extended as the increase of rainfall; (ii) spatial distribution structure and spatially mean density of vegetation patterns reveal that degradation of the vegetation becomes more and more obvious with the increase of grazing intensity; (iii) the bistable region corresponding to the delayed diffusion system is narrowed, and vegetation system undergoes regime shift from the pattern state to bare-soil state before reaching the threshold of grazing rate. Overall, this study yields a new theoretical perspective for pattern dynamics of delayed reaction–diffusion equation, and provides valuable insights into the study of vegetation system in grazing ecosystem.

Dynamics of Fractional Delayed Reaction-Diffusion Equations.

The long-term behavior of the weak solution of a fractional delayed reaction-diffusion equation with a generalized Caputo derivative is investigated. By using the classic Galerkin approximation method and comparison principal, the existence and uniqueness of the solution is proved in the sense of weak solution. In addition, the global attracting set of the considered system is obtained, with the help of the Sobolev embedding theorem and Halanay inequality.

Symmetry-breaking bifurcations in a delayed reaction–diffusion equation

Symmetry-breaking bifurcations in a delayed reaction–diffusion equation

Hopf bifurcation in a spatial heterogeneous and nonlocal delayed reaction–diffusion equation

A spatial heterogeneous and nonlocal delayed reaction–diffusion equation is investigated. The existence of the nontrivial steady-state solution is shown by steady-state bifurcation at the trivial solution. The form of the positive spatially nonhomogeneous steady state is given under certain conditions. Based on the form, its stability and Hopf bifurcation are analyzed so that the periodic and the quasi-periodic solutions near the spatially nonhomogeneous steady state are demonstrated.

A stable second-order difference scheme for the generalized time-fractional non-Fickian delay reaction-diffusion equations

A stable second-order difference scheme for the generalized time-fractional non-Fickian delay reaction-diffusion equations

Equivalence of MTS and CMR methods associated with the normal form of Hopf bifurcation for delayed reaction–diffusion equations

In this paper, we prove the equivalence of the multiple time scales (MTS) method and the center manifold reduction (CMR) method for deriving the normal form of Hopf bifurcation for delayed reaction–diffusion equations. First, we generalize the MTS method associated with functional differential equations to delayed reaction–diffusion equation for solving the normal form of Hopf bifurcation. This provides a user-friendly approach of analyzing the stability of time-periodic solutions, particularly, for the engineering researchers who are not familiar with CMR method. Second, we outline the CMR method associated with the computation of normal form of Hopf bifurcation for delayed reaction–diffusion equations. Third, we prove that the two methods can derive equivalent up to the third order normal form of Hopf bifurcation. Finally, different types of examples are presented, such as reaction–diffusion equations with Neumann boundary condition and Dirichlet boundary conditions, single delay and two delays, spatially homogeneous and spatially inhomogeneous periodic solutions. We obtain the equivalent Hopf bifurcation normal forms results derived by using the CMR and MTS methods for above examples, respectively.

Mathematical modeling of respiratory viral infection and applications to SARS-CoV-2 progression.

Viral infection in cell culture and tissue is modeled with delay reaction‐diffusion equations. It is shown that progression of viral infection can be characterized by the viral replication number, time‐dependent viral load, and the speed of infection spreading. These three characteristics are determined through the original model parameters including the rates of cell infection and of virus production in the infected cells. The clinical manifestations of viral infection, depending on tissue damage, correlate with the speed of infection spreading, while the infectivity of a respiratory infection depends on the viral load in the upper respiratory tract. Parameter determination from the experiments on Delta and Omicron variants allows the estimation of the infection spreading speed and viral load. Different variants of the SARS‐CoV‐2 infection are compared confirming that Omicron is more infectious and has less severe symptoms than Delta variant. Within the same variant, spreading speed (symptoms) correlates with viral load allowing prognosis of disease progression.

Existence, exponential mixing and convergence of periodic measures of fractional stochastic delay reaction-diffusion equations on [formula omitted

This paper is concerned with periodic measures of fractional stochastic reaction-diffusion equations with variable time delay defined on unbounded domains. We first prove the existence of periodic measures of the Markov process associated with the equation by showing the weak compactness of distribution laws of a family of solutions based on the uniform estimates and the equicontinuity of solutions in probability. We then establish the regularity of periodic measures and prove the tightness of the set of all periodic measures of the equation for small noise intensity. As a result, any sequence of periodic measures possesses at least one limit point, and we further prove every such limit must be a periodic measure of the corresponding limiting system. Finally, under further assumptions on the nonlinear terms, we establish the uniqueness and the exponential mixing property of periodic measures in the sense of Wasserstein metric. All the results of the paper are also valid for invariant measures of the corresponding autonomous version of the stochastic equation with constant time delay. The idea of uniform tail estimates of solutions in probability is employed to overcome the difficulty introduced by the non-compactness of Sobolev embeddings on unbounded domains.

Hopf Bifurcation in a Delayed Equation with Diffusion Driven by Carrying Capacity

In this paper, a delayed reaction–diffusion equation with carrying capacity-driven diffusion is investigated. The stability of the positive equilibrium solutions and the existence of the Hopf bifurcation of the equation are considered by studying the principal eigenvalue of an associated elliptic operator. The properties of the bifurcating periodic solutions are also obtained by using the normal form theory and the center manifold reduction. Furthermore, some representative numerical simulations are provided to illustrate the main theoretical results.

Galerkin finite element method for the generalized delay reaction-diffusion equation

Galerkin finite element method for the generalized delay reaction-diffusion equation

Qualitative properties of traveling wave solutions in delayed reaction-diffusion equations with degenerate monostable nonlinearity

This paper deals with the asymptotic behavior, uniqueness up to translation and Lyapunov stability of traveling wave solutions with non-critical speeds in a class of delayed reaction-diffusion equations with degenerate monostable nonlinearity. With the help of Harnack inequalities and the detailed asymptotic higher order expansion, the precise asymptotic behavior of traveling wave solutions is established. Our results show that these traveling wave solutions decay algebraically, and the corresponding decaying rates of traveling wave solutions are independent of the delay. To illustrate our results on the asymptotic behavior, several degenerate equations are investigated. For a special equation, we present that the delay could affect the asymptotic behavior of traveling wave solutions. Furthermore, based on the special asymptotic behavior, we obtain the uniqueness up to translation and Lyapunov stability of traveling wave solutions.