Nonlocal Reaction-diffusion Equation Research Articles

An inverse problem for nonlocal reaction-diffusion equations with time-delay

The reaction-diffusion equations have been studied in various aspects of nature, such as heat transfer and biological dynamic modelling. In this paper, we study a nonlocal reaction-diffusion model with time delay associated with the density and diffusion behaviour of biological populations. Based on our previous conclusions and numerical strategy of the direct problem, we continue to study the inverse problem about the diffusion coefficient as well as the parameters in the birth function, and also perform numerical simulations with error analysis. In addition, we further generalize the constant diffusion coefficient to the position-dependent diffusion rate, which is intended to improve the generalizability to multiple organisms diffusion in nature.

Random dynamics for a stochastic nonlocal reaction-diffusion equation with an energy functional

<abstract><p>In this paper, the asymptotic behavior of solutions to a fractional stochastic nonlocal reaction-diffusion equation with polynomial drift terms of arbitrary order in an unbounded domain was analysed. First, the stochastic equation was transformed into a random one by using a stationary change of variable. Then, we proved the existence and uniqueness of solutions for the random problem based on pathwise uniform estimates as well as the energy method. Finally, the existence of a unique pullback attractor for the random dynamical system generated by the transformed equation is shown.</p></abstract>

Dynamical behavior of a temporally discrete non-local reaction-diffusion equation on bounded domain

This paper focuses on the study of global dynamics of a class of temporally discrete non-local reaction-diffusion equations on bounded domains. Similar to classical reaction diffusion equations and integro-difference equations, temporally discrete reaction-diffusion equations can also be used to describe the dispersal phenomena in population dynamics. In this paper, we first derived a temporally discrete reaction diffusion equation model with time delay and nonlocal effects to model the evolution of a single species population with age-structured located in a bounded domain. By establishing a new maximum principle and applying the monotone iteration method, the global stabilities of the trivial solution and the positive steady state solution are obtained respectively under some appropriate assumptions.

Concentration phenomenon of single phytoplankton species with changing-sign advection term

In this paper, a nonlocal reaction-diffusion equation modeling the growth of phytoplankton species with changing-sign advection in a vertical water column is investigated, where the species depends solely on light for its metabolism. We mainly study the concentration phenomenon of the phytoplankton with large advection amplitude and small diffusion rate. Firstly, we study the threshold-type dynamics of the population by critical death rate d⁎. Secondly, we examine the concentration phenomenon with large advection amplitude and small diffusion in two cases: (i) the advection function h(x) changes sign only once from positive to negative in water column [0,1]. We find that the phytoplankton will concentrate at certain critical point with large advection amplitude and small diffusion; (ii) the advection function h(x)<0 in [0,κ) with ∫0xh(s)a(s)ds<0 for all x∈(0,1], the phytoplankton will concentrate at the surface of water column with large advection amplitude and small diffusion. We also investigate the limiting distribution of phytoplankton as diffusion rate D→+∞: the phytoplankton tends to even distribution in water column.

Conforming and nonconforming virtual element methods for fourth order nonlocal reaction diffusion equation

In this work, we have designed conforming and nonconforming virtual element methods (VEM) to approximate non-stationary nonlocal biharmonic equation on general shaped domain. By employing Faedo–Galerkin technique, we have proved the existence and uniqueness of the continuous weak formulation. Upon applying Brouwer’s fixed point theorem, the well-posedness of the fully discrete scheme is derived. Further, following [J. Huang and Y. Yu, A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations, J. Comput. Appl. Math. 386 (2021) 113229], we have introduced Enrichment operator and derived a priori error estimates for fully discrete schemes on polygonal domains, not necessarily convex. The proposed error estimates are justified with some benchmark examples.

Emergence and competition of virus variants in respiratory viral infections.

The emergence of new variants of concern (VOCs) of the SARS-CoV-2 infection is one of the main factors of epidemic progression. Their development can be characterized by three critical stages: virus mutation leading to the appearance of new viable variants; the competition of different variants leading to the production of a sufficiently large number of copies; and infection transmission between individuals and its spreading in the population. The first two stages take place at the individual level (infected individual), while the third one takes place at the population level with possible competition between different variants. This work is devoted to the mathematical modeling of the first two stages of this process: the emergence of new variants and their progression in the epithelial tissue with a possible competition between them. The emergence of new virus variants is modeled with non-local reaction-diffusion equations describing virus evolution and immune escape in the space of genotypes. The conditions of the emergence of new virus variants are determined by the mutation rate, the cross-reactivity of the immune response, and the rates of virus replication and death. Once different variants emerge, they spread in the infected tissue with a certain speed and viral load that can be determined through the parameters of the model. The competition of different variants for uninfected cells leads to the emergence of a single dominant variant and the elimination of the others due to competitive exclusion. The dominant variant is the one with the maximal individual spreading speed. Thus, the emergence of new variants at the individual level is determined by the immune escape and by the virus spreading speed in the infected tissue.

Dynamics of a stochastic fractional nonlocal reaction-diffusion model driven by additive noise

In this paper, we are concerned with the long-time behavior of stochastic fractional nonlocal reaction-diffusion equations driven by additive noise. We use the techniques of random dynamical systems to transform the stochastic model into a random one. To deal with the new nonlocal term appeared in the transformed equation, we first use a generalization of Peano's theorem to prove the existence of local solutions, and then adopt the Galerkin method to prove existence and uniqueness of weak solutions. Next, the existence of pullback attractors for the equation and its associated Wong-Zakai approximation equation driven by colored noise are shown, respectively. Furthermore, we establish the upper semi-continuity of random attractors of the Wong-Zakai approximation equation as $ \delta\rightarrow0^{+} $.

Long-time behaviour for a nonlocal model from directed polymers

We consider the long time behaviour of solutions to a nonlocal reaction diffusion equation that arises in the study of directed polymers in a random environment. The model is characterized by convolution with a kernel R and an L 2 inner product. In one spatial dimension, we extend a previous result of the authors (arXiv:2002.02799), where only the case was considered; in particular, we show that solutions spread according to a power law consistent with the KPZ scaling conjectured for directed polymers. In the special case when , we find the exact profile of the solution in the rescaled coordinates. We also consider the behaviour in higher dimensions. When the dimension is three or larger, we show that the long-time behaviour is the same as the heat equation in the sense that the solution converges to a standard Gaussian. In contrast, when the dimension is two, we construct a non-Gaussian self-similar solution.

UNIVERSAL ATTRACTOR FOR A NONLOCAL REACTION-DIFFUSION PROBLEM WITH DYNAMICAL BOUNDARY CONDITIONS

A nonlocal reaction-diffusion equation is presented in this article, based on a model proposed by J. Rubinstein and P. Sternberg [6] with a nonlinear strictly monotone operator. A dynamical boundary condition is considered, rather then the usual ones such as Neumann or Dirichlet boundary conditions. The well-posedness and the existence of a universal attractor of this problem, which describes the long time behavior of the solution, are established.

Dynamics of stochastic nonlocal reaction–diffusion equations driven by multiplicative noise

This paper deals with fractional stochastic nonlocal partial differential equations driven by multiplicative noise. We first prove the existence and uniqueness of solution to this kind of equations with white noise by applying the Galerkin method. Then, the existence and uniqueness of tempered pullback random attractor for the equation are ensured in an appropriate Hilbert space. When the fractional nonlocal partial differential equations are driven by colored noise, which indeed are approximations of the previous ones, we show the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as [Formula: see text].

Threshold Solutions for Nonlocal Reaction Diffusion Equations

We study the Cauchy problem for nonlocal reaction diffusion equations with bistable nonlinearity in 1D spatial domain and investigate the asymptotic behaviors of solutions with a one-parameter family of monotonically increasing and compactly supported initial data. We show that for small values of the parameter the corresponding solutions decay to 0, while for large values the related solutions converge to 1 uniformly on compacts. Moreover, we prove that the transition from extinction (converging to 0) to propagation (converging to 1) is sharp and describe the behavior of the corresponding threshold solution: it is bounded, symmetric and has a global maximum at zero. Numerical examples are shown to verify our results.

Global existence and finite time blow-up for a stochastic non-local reaction-diffusion equation

In this paper, we investigate the existence and blow-up of positive solutions for a stochastic non-local reaction-diffusion equation. Firstly, we give the conditions for the existence of global positive solutions and estimate the probability of existence of non-trivial positive global solution. Secondly, by choosing some special initial data, we prove that stochastic non-local reaction-diffusion equation blows up in finite time in the point-wise sense with probability 1. The random upper bounds for blow-up times are obtained. Furthermore, by increasing the amplitude of the initial data, we can get a blow-up in any short time with a positive probability. Finally, we prove that the blow-up set is the whole region.

Weak mean random attractors for non-local random and stochastic reaction–diffusion equations

In this paper, we prove the existence of weak pullback mean random attractors for a non-local stochastic reaction–diffusion equation with a nonlinear multiplicative noise. The existence and uniqueness of solutions and weak pullback mean random attractors are also established for a deterministic non-local reaction–diffusion equations with random initial data.

Development of the reproducing kernel Hilbert space algorithm for numerical pointwise solution of the time-fractional nonlocal reaction-diffusion equation

It is notable that, the nonlocal reaction-diffusion equation carries math and computational physics to the core of extremely dynamic multidisciplinary studies that emerge from a huge assortment of uses. In this investigation, a totally new methodology for building a locally numerical pointwise solution is given by the agent the reproducing kernel algorithm. This is done utilizing a couple of generalized Hilpert spaces and their corresponding Green functions. The proposed calculation algorithm is applied to certain scalar issues problems to figure the arrangement solutions with Dirichlet constraints. By applying the procedures of the Gram–Schmidt process, orthonormalizing the basis, and truncating the optimized series, the approximate solutions are drawn, tabulated, and sketched. Introduced mathematical outcomes not only show the hidden superiority of the algorithm but also show its accurate efficiency. Finally, focused notes and futures planning works are mentioned with the most-used references.

Nonlocal Reaction–Diffusion Equations in Biomedical Applications

Nonlocal reaction-diffusion equations describe various biological and biomedical applications. Their mathematical properties are essentially different in comparison with the local equations, and this difference can lead to important biological implications. This review will present the state of the art in the investigation of nonlocal reaction-diffusion models in biomedical applications. We will consider various models arising in mathematical immunology, neuroscience, cancer modelling, and we will discuss their mathematical properties, nonlinear dynamics, resulting spatiotemporal patterns and biological significance.

Existence of solutions for a nonlocal reaction-diffusion equation in biomedical applications

The paper is devoted to a nonlocal semi-linear elliptic equation in ℝn arising in various biological and biomedical applications. The Fredholm property studied for the corresponding linear elliptic operators with discontinuous coefficients allows the application of the implicit function theorem to prove the persistence of solutions under a small perturbation of the problem. Furthermore, the existence of solutions is established by the Leray—Schauder method based on the topological degree for Fredholm and proper operators and on a priori estimates of solutions in some special weighted spaces.

Traveling Fronts in a Reaction–Diffusion Equation with a Memory Term

Based on a recent work on traveling waves in spatially nonlocal reaction–diffusion equations, we investigate the existence of traveling fronts in reaction–diffusion equations with a memory term. We will explain how such memory terms can arise from reduction of reaction–diffusion systems if the diffusion constants of the other species can be neglected. In particular, we show that two-scale homogenization of spatially periodic systems can induce spatially homogeneous systems with temporal memory. The existence of fronts is proved using comparison principles as well as a reformulation trick involving an auxiliary speed that allows us to transform memory terms into spatially nonlocal terms. Deriving explicit bounds and monotonicity properties of the wave speed of the arising traveling front, we are able to establish the existence of true traveling fronts for the original problem with memory. Our results are supplemented by numerical simulations.

Spreading Speeds of Nonlocal KPP Equations in Heterogeneous Media

This paper is devoted to studying the asymptotic behavior of the solution to nonlocal Fisher-KPP type reaction diffusion equations in heterogeneous media. The kernel K is assumed to depend on the media. First, we give an estimate of the upper and lower spreading speeds by generalized principal eigenvalues. Second, we prove the existence of spreading speeds in the case where the media is periodic or almost periodic by showing that the upper and lower generalized principal eigenvalues are equal.

Space and Genotype-Dependent Virus Distribution during Infection Progression

The paper is devoted to a nonlocal reaction-diffusion equation describing the development of viral infection in tissue, taking into account virus distribution in the space of genotypes, the antiviral immune response, and natural genotype-dependent virus death. It is shown that infection propagates as a reaction-diffusion wave. In some particular cases, the 2D problem can be reduced to a 1D problem by separation of variables, allowing for proof of wave existence and stability. In general, this reduction provides an approximation of the 2D problem by a 1D problem. The analysis of the reduced problem allows us to determine how viral load and virulence depend on genotype distribution, the strength of the immune response, and the level of immunity.

Spatio-temporal modelling of tree-grass dynamics in humid savannas: Interplay between nonlocal competition and nonlocal facilitation

For about twenty years, the question about the essential factors promoting the long-lasting coexistence of trees and grasses in humid savannas is at the center of several mathematical works, by the construction of deterministic and/or stochastic mathematical models. A closely related topic is coexistence of open savanna and forest patches at landscape scales, which raises the challenge of accounting for contrasted spatial patterns under similar climate conditions through fire mediated tree-grass interaction models. In this work, we propose and study a deterministic spatio-temporal fire-mediated tree-grass interactions model. The model is based on two nonlocal reaction-diffusion equations with kernels of intra and inter-specific interactions, corresponding to woody and grassy biomasses. A novelty in this paper is the consideration of a kernel-based nonlocal facilitation of trees by other trees to promote growth of seedlings/shrubs and, indirectly, limit fire propagation and its impact. We also take into account a kernel-based nonlocal competition of trees on grasses for light availability and nutrients. A qualitative analysis of the model is carried out and it reveals several ecological thresholds that shape the overall dynamics of the system. Depending on these thresholds, monostability of the forest, grassland or savanna space-homogeneous stationary state and multistabilities (i.e. involving more that one space-homogeneous stationary state) are proven possible. Thanks to the nonlocal biomasses interactions, our model accounts for the occurrence of space inhomogeneous solutions, including a possibly periodic spatial structuring sometimes observed in the humid savanna zone. Specifically, linear stability analyses, performed in the vicinity of space-homogeneous stationary states, provides conditions for the appearance of space inhomogeneous solutions including spatially periodic or aperiodic ones. Finally, numerical simulations are presented to illustrate our theoretical results. Notably, we verify that the computed spatial wavelengths were in good agreement with the predictions from the theoretical analysis.