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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. [ABSTRACT FROM AUTHOR]

We continue the study on a reaction–diffusion system with non-local terms, which arises as a mean field limit of a master equation using a reaction radius. This work was initiated in Kavallaris & Suzuki (2012, Non-local reaction–diffusion system involved by reaction radius I. IMA J. Appl. Math., 1–19), and here, we investigate the rate of convergence of the solution towards the unique stationary solution. [ABSTRACT FROM PUBLISHER]

This paper is to study the existence of travelling wave solutions in a non-local and time-delayed reaction–diffusion malaria model proposed by Lou and Zhao (2011, A reaction–diffusion malaria model with incubation period in the vector population. J. Math. Biol., 62, 543–568). We first analyse the positivity and invariance of solutions for the corresponding Cauchy problem in an unbounded domain. Then, according to the basic reproduction ratio R0 which serves as a threshold that predicts whether epidemics will spread, we show that there exist travelling wave solutions connecting the two steady states: the disease-free steady state and the endemic steady state if R0>1, and there do not exist travelling wave solutions connecting the disease-free steady state itself if R0<1. This explores how a malaria infected state invades into the previously uninfected state in the spatial domain. Numerical simulation is provided to show that the travelling wave solutions can be non-monotone. [ABSTRACT FROM AUTHOR]

A reaction–diffusion system with non-local term arises as a mean field limit of a master equation using reaction radius. We show that a two-phase Stefan problem is derived in the limit of infinite chemical reaction rate. The convergence of the global-in-time solutions of the preceding system to a stationary solution is also discussed. [ABSTRACT FROM AUTHOR]