Monotone travelling fronts in an age-structured reaction-diffusion model of a single species.

We consider a partially coupled diffusive population model in which the state variables represent the densities of the immature and mature population of a single species. The equation for the mature population can be considered on its own, and is a delay differential equation with a delay-dependent coefficient. For the case when the immatures are immobile, we prove that travelling wavefront solutions exist connecting the zero solution of the equation for the matures with the delay-dependent positive equilibrium state. As a perturbation of this case we then consider the case of low immature diffusivity showing that the travelling front solutions continue to persist. Our findings are contrasted with recent studies of the delayed Fisher equation. Travelling fronts of the latter are known to lose monotonicity for sufficiently large delays. In contrast, travelling fronts of our equation appear to remain monotone for all values of the delay.

Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain.

In this paper we model and analyse nonlocal spatial effects, induced by time delays, in a diffusion model for a single species confined to a finite domain. The nonlocality, a weighted average in space, arises when account is taken of the fact that individuals have been at different points in space at previous times. We show how to correctly derive the spatial averaging kernels for finite domain problems, generalising the ideas of other investigators who restricted attention to the simpler case of an infinite domain. The resulting model is then analysed and results established on linear stability, boundedness, global convergence of solutions and bifurcations.

Travelling front solutions of a nonlocal Fisher equation.

We consider a scalar reaction-diffusion equation containing a nonlocal term (an integral convolution in space) of which Fisher's equation is a particular case. We consider travelling wavefront solutions connecting the two uniform states of the equation. We show that if the nonlocality is sufficiently weak in a certain sense then such travelling fronts exist. We also construct expressions for the front and its evolution from initial data, showing that the main difference between our front and that of Fisher's equation is that for sufficiently strong nonlocality our front is non-monotone and has a very prominent hump.

Linear stability criteria for population models with periodically perturbed delays.

We examine some simple population models that incorporate a time delay which is not a constant but is instead a known periodic function of time. We examine what effect this periodic variation has on the linear stability of the equilibrium states of scalar population models and of a simple predator prey system. The case when the delay differs from a constant by a small amplitude periodic perturbation can be treated analytically by using two-timing methods. Of particular interest is the case when the system is initially marginally stable. The introduction of variation in the delay can then have either a stabilising effect or a destabilizing one, depending on the frequency of the periodic perturbation. The case when the periodic perturbation has large amplitude is studied numerically. If the fluctuation is large enough the effect can be stabilising.

On the problem of asymptotic positivity of solutions for dissipative partial differential equations.

The objective of this paper aims to prove positivity of solutions for the following semilinear partial differential equationu[Formula: see text]. This equation represents a generalised model of the so-called porous medium equation. It arises in a variety of meaningful physical situations including gas flows, diffusion of an electron-ion plasma and the dynamics of biological populations whose mobility is density dependent. In all these situations the solutions of the equation must be positive functions.