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In this paper, we consider the diffusive competition problem with a free boundary and sign-changing intrinsic growth rate in heterogeneous time-periodic environment, consisting of an invasive species with density $u$ and a native species…
Similarity solutions for a one-dimensional mathematical model for thawing in a saturated semi-infinite porous media is considered when change of phase induces a density jump and a convective boundary condition is imposed at the fixed face…
Spatially resolved genetic data is increasingly used to reconstruct the migrational history of species. To assist such inference, we study, by means of simulations and analytical methods, the dynamics of neutral gene frequencies in a…
This paper is concerned with the asymptotic behavior of the solution to the Euler equations with time-depending damping on quadrant $(x,t)\in \mathbb{R}^+\times\mathbb{R}^+$, \begin{equation}\notag \partial_t v - \partial_x u=0, \qquad…
We develop a continuous mathematical model of population dynamics that describes the sequential emergence of new genotypes under limited resources. The framework models genotype density as a nonlinear flow in mutation space, combining…
In this series of papers, we investigate the spreading and vanishing dynamics of time almost periodic diffusive KPP equations with free boundaries. Such equations are used to characterize the spreading of a new species in time almost…
We consider a parabolic non-local free boundary problem that has been derived as a limit of a bulk-surface reaction-diffusion system which models cell polarization. In previous papers, we have established well-posedness of this problem and…
There is now experimental evidence that nearest-neighbour interactions in flocks of birds are metric free, i.e. they have no characteristic interaction length scale. However, models that involve interactions between neighbours that are…
We consider a class of tumor growth models under the combined effects of density-dependent pressure and cell multiplication, with a free boundary model as its singular limit when the pressure-density relationship becomes highly nonlinear.…
Recent results have shown that there is an acceleration in the spread of the size distribution of droplet populations in the region bordering the cloud and undersaturated ambient. We have analyzed the supersaturation balance in this region,…
We consider the Cauchy problem of the porous medium type reaction-diffusion equation \begin{equation*} \partial_t\rho=\Delta\rho^m+\rho g(\rho),\quad (x,t)\in \mathbb{R}^n\times \mathbb{R}_+,\quad n\geq2,\quad m>1, \end{equation*} where $g$…
We investigate the competition between barrier slowing down and proliferation induced superdiffusion in a model of population dynamics in a random force field. Numerical results in $d=1$ suggest that a new intermediate diffusion behaviour…
We expand on a previous study of fronts in finite particle number reaction-diffusion systems in the presence of a reaction rate gradient in the direction of the front motion. We study the system via reaction-diffusion equations, using the…
We examine travelling wave solutions of the Porous-Fisher model, $\partial_t u(x,t)= u(x,t)\left[1-u(x,t)\right] + \partial_x \left[u(x,t) \partial_x u(x,t)\right]$, with a Stefan-like condition at the moving front, $x=L(t)$. Travelling…
For the gang territoriality model \begin{align*} \begin{cases} u_t = D_u \Delta u + \chi_u \nabla \cdot (u \nabla w), \\ v_t = D_v \Delta v + \chi_v \nabla \cdot (v \nabla z), \\ w_t = -w + \frac{v}{1+v}, \\ z_t = -z + \frac{u}{1+u},…
In this article, we analyse the non-local model : $\partial$ t U (t, x) = J $\star$ U (t, x) -- U (t, x) + f (x -- ct, U (t, x)) for t > 0, and x $\in$ R, where J is a positive continuous dispersal kernel and f (x, s) is a heterogeneous KPP…
We investigate general semilinear (obstacle-like) problems of the form $\Delta u = f(u)$, where $f(u)$ has a singularity/jump at $\{u=0\}$ giving rise to a free boundary. Unlike many works on such equations where $f$ is approximately…
A forward diffusion equation describing the evolution of the allele frequency spectrum is presented. The influx of mutations is accounted for by imposing a suitable boundary condition. For a Wright-Fisher diffusion with or without selection…
Invasion fronts in ecology are well studied but very few mathematical results concern the case with variable motility (possibly due to mutations). Based on an apparently simple reaction-diffusion equation, we explain the observed phenomena…
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…