In this paper, we derive the equation for a single species population with two age classes and a fixed maturation period living in a spatially unbounded environment. We show that if the mature death and diffusion rates are age independent, then the total mature population is governed by a reaction-diffusion equation with time delay and non-local effect. We also consider the existence, uniqueness and positivity of solution to the initial-value problem for this type of equation. Moreover, we establish the existence of a travelling-wave front for the special case when the birth function is the one which appears in the well-known Nicholson's blowflies equation and we consider the dependence of the minimal wave speed on the mobility of the immature population.

We present a short survey on the biological modeling, dynamics analysis, and numerical simulation of nonlocal spatial effects, induced by time delays, in diffusion models for a single species confined to either a finite or an infinite 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 discuss and compare two existing approaches to correctly derive the spatial averaging kernels, and we summarize some of the recent developments in both qualitative and numerical analysis of the nonlinear dynamics, including the existence, uniqueness (up to a translation), and stability of traveling wave fronts and periodic spatiotemporal patterns of the model equations in unbounded domains and the linear stability, boundedness, global convergence of solutions and bifurcations of the model equations in finite domains.

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This paper considers the nonlinear stability of travelling wavefronts of a time-delayed diffusive Nicholson blowflies equation. We prove that, under a weighted L 2 norm, if a solution is sufficiently close to a travelling wave front initially, it converges exponentially to the wavefront as t → ∞. The rate of convergence is also estimated.

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We establish conditions for an isolated invariant set M of a map to be a repellor. The conditions are first formulated in terms of the stable set of M. They are then refined in two ways by considering (i) a Morse decomposition for M , and (ii) the invariantly connected components of the chain recurrent set of M. These results generalize and unify earlier persistence results.

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