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We study the Fisher-KPP equation with a fractional laplacian of order {\alpha} \in (0, 1). We know that the stable state invades the unstable one at constant speed for {\alpha} = 1, and at an exponential in time velocity for {\alpha} \in…
We consider a nonlocal semi-linear parabolic equation on a connected exterior domain of the form $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a compact "obstacle". The model we study is motivated by applications in biology and…
The Fisher-KPP model, and generalisations thereof, is a simple reaction-diffusion models of biological invasion that assumes individuals in the population undergo linear diffusion with diffusivity $D$, and logistic proliferation with rate…
How should dispersal strategies be chosen to increase the likelihood of survival of a species? We obtain the answer for the spatially extended versions of three well-known models of two competing species with unequal diffusivities. Though…
This paper is concerned with the spreading speeds of nonlocal dispersal predator-prey systems in shifting habitats under general initial conditions. By employing geometric optics techniques and theory of viscosity solutions, we reformulate…
This paper is concerned with non-cooperative parabolic reaction--diffusion systems which share structural similarities with the scalar Fisher--KPP equation. These similarities make it possible to prove, among other results, an extinction…
The problem of front propagation in flowing media is addressed for laminar velocity fields in two dimensions. Three representative cases are discussed: stationary cellular flow, stationary shear flow, and percolating flow. Production terms…
A general reaction-diffusion equation with spatiotemporal delay and homogeneous Dirichlet boundary condition is considered. The existence and stability of positive steady state solutions are proved via studying an equivalent…
The diffusive Holling-Tanner predator-prey model with no-flux boundary conditions and nonlocal prey competition is considered in this paper. We show the existence of spatial nonhomogeneous periodic solutions, which is induced by nonlocal…
This is part two of our study on the spreading properties of the Lotka-Volterra competition-diffusion systems with a stable coexistence state. We focus on the case when the initial data are exponential decaying. By establishing a comparison…
We provide an asymptotic analysis of a fractional Fisher-KPP type equation in periodic non-connected 1-dimensional media with Dirichlet conditions outside the domain. After demonstrating the existence and uniqueness of a non-trivial bounded…
We develop general heterogeneous nonlocal diffusion models and investigate their connection to local diffusion models by taking a singular limit of focusing kernels. We reveal the link between the two groups of diffusion equations which…
We study the velocity of travelling waves of a reaction-diffusion system coupling a standard reaction-diffusion equation in a strip with a one-dimensional diffusion equation on a line. We show that it grows like the square root of the…
We study the existence and uniqueness of mild and strong solutions of nonlocal nonlinear diffusion problems of $p$-Laplacian type with nonlinear boundary conditions posed in metric random walk spaces. These spaces include, among others,…
This paper is devoted to the study of propagation dynamics for a large class of non-monotone evolution systems. In two directions of the spatial variable, such a system has two limiting systems admitting the spatial translation invariance.…
Diffusion of information in networks is at the core of many problems in AI. Common examples include the spread of ideas and rumors as well as marketing campaigns. Typically, information diffuses at a non-linear rate, for example, if markets…
We study a variant of the Fisher-KPP equation with nonlocal dispersal. Using the theory of large deviations, we show the emergence of a "Bramson-like" logarithmic delay for the linearised equation with step-like initial data. We conclude…
We focus on the spreading properties of solutions of monostable equations with non-linear diffusion. We consider both the porous medium diffusion and the fast diffusion regimes. Initial data may have heavy tails, which tends to accelerate…
The dynamics of a population expanding into unoccupied habitat has been primarily studied for situations in which growth and dispersal parameters are uniform in space or vary in one dimension. Here we study the influence of finite-sized…
We perform an asymptotic analysis of models of population dynamics with a fractional Laplacian and local or nonlocal reaction terms. The first part of the paper is devoted to the long time/long range rescaling of the fractional Fisher-KPP…