Related papers: Travelling waves, blow-up and extinction in the Fi…
Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade into vacant regions, are routinely studied using partial differential equation (PDE) models based upon the classical…
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…
The Fisher-Stefan model involves solving the Fisher-KPP equation on a domain whose boundary evolves according to a Stefan-like condition. The Fisher-Stefan model alleviates two practical limitations of the standard Fisher-KPP model when…
This review provides open-access computational tools that support a range of mathematical approaches to analyse three related scalar reaction-diffusion models used to study biological invasion. Starting with the classic Fisher-Kolmogorov…
We investigate pattern formation in a two-dimensional (2D) Fisher--Stefan model, which involves solving the Fisher--KPP equation on a compactly-supported region with a moving boundary. By combining the Fisher--KPP and classical Stefan…
Reaction-diffusion models are often used to describe biological invasion, where populations of individuals that undergo random motility and proliferation lead to moving fronts. Many models of biological invasion are extensions of the…
We consider a moving boundary mathematical model of biological invasion. The model describes the spatiotemporal evolution of two adjacent populations: each population undergoes linear diffusion and logistic growth, and the boundary between…
In this paper, we propose a novel free boundary problem to model the movement of single species with a range boundary. The spatial movement and birth/death processes of the species found within the range boundary are assumed to be governed…
Single-species reaction-diffusion equations, such as the Fisher-KPP and Porous-Fisher equations, support travelling wave solutions that are often interpreted as simple mathematical models of biological invasion. Such travelling wave…
A family of travelling wave solutions to the Fisher-KPP equation with speeds $c=\pm 5/\sqrt{6}$ can be expressed exactly using Weierstrass elliptic functions. The well-known solution for $c=5/\sqrt{6}$, which decays to zero in the…
We consider a minimal go-or-grow model of cell invasion, whereby cells can either proliferate, following logistic growth, or move, via linear diffusion, and phenotypic switching between these two states is density-dependent. Formal analysis…
Invasion waves are a fundamental building block of theoretical ecology. In this study we aim to take the first steps to link propagation failure and fast acceleration of traveling waves to critical transitions (or tipping points). The…
The FKPP equation with a variable growth rate and advection by an incompressible velocity field is considered as a model for plankton dispersed by ocean currents. If the average growth rate is negative then the model has a…
Stationary solutions of the Fisher-KPP equation with general nonlinear diffusion and arbitrary reactional kinetic orders terms are characterized. Such stationary (separatrix-like) solutions disjoint the blow-up solutions from those showing…
In this paper, we treat the Fisher-KPP equation with a Caputo-type time fractional derivative and discuss the propagation speed of the solution. The equation is a mathematical model that describes the processes of sub-diffusion,…
The famous Fisher-KPP reaction-diffusion model combines linear diffusion with the typical KPP reaction term, and appears in a number of relevant applications in biology and chemistry. It is remarkable as a mathematical model since it…
In this paper we study the existence of traveling wave solutions for a free-boundary problem modeling the phase transition of a material where the heat is transported by both conduction and radiation. Specifically, we consider a…
We analyse a novel mathematical model of malignant invasion which takes the form of a two-phase moving boundary problem describing the invasion of a population of malignant cells into a population of background tissue, such as skin. Cells…
Non-cooperative Fisher-KPP systems with space-time periodic coefficients are motivated for instance by models for structured populations evolving in periodic environments. This paper is concerned with entire solutions describing the…
We consider Fisher-KPP equation with advection: $u_t=u_{xx}-\beta u_x+f(u)$ for $x\in (g(t),h(t))$, where $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions. This equation is used to describe the population dynamics in…