Joseph W.-H. So

Numerical steady state and Hopf bifurcation analysis on the diffusive Nicholson's blowflies equation1Research partially supported by Natural Sciences and Engineering Research Council of Canada.1

For the Dirichlet boundary value problem of the diffusive Nicholsons blowflies equation, it was shown in Ref. [17] that in a certain range of the parameter space, there is a unique positive steady state solution. In this paper, we propose a scheme to compute this steady state numerically. In addition, we describe an iterative procedure to locate the critical values of the delay where a Hopf bifurcation of time periodic solutions takes place near the steady state. Some numerical simulations of both schemes are given.

A reaction-diffusion model for a single species with age structure. I Travelling wavefronts on unbounded domains

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 Nicholsons blowflies equation and we consider the dependence of the minimal wave speed on the mobility of the immature population.

Global stability and persistence of simple food chains

Abstract The main purpose of this paper is to develop criteria for which a simple food-chain model of intermediate type and of arbitrary length has a globally stable positive equilibrium and to develop criteria under which such a food chain exhibits uniform persistence. The same techniques are used to obtain conditions for a model of a predator-prey system with mutual interference of the predator to possess a globally stable positive equilibrium.

Uniform persistence and repellors for maps

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.

Traveling waves for the diffusive Nicholson's blowflies equation

We consider traveling wave front solutions for the diffusive Nicholsons blowflies equation on the real line. The existence of such solutions is proved using the technique developed by J. Wu and X. Zou (J. Dyn. Differ. Equations 13 (3) (2001)). Some numerical simulation using the iteration formula of Wu and Zou [7] is also provided.

Coexistence in a model of competition in the Chemostat incorporating discrete delays

A model of two species competing for a single nutrient in the chemostat is proposed, incorporating general monotone uptake functions with discrete time delays. After transforming the model, an anal...

Asymptotic stability of travelling waves for Nicholson's blowflies equation with diffusion

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

Diagonal dominance and harmless off-diagonal delays

Systems of linear differential equations with constant coefficients, as well as Lotka–Volterra equations, with delays in the off–diagonal terms are considered. Such systems are shown to be asymptotically stable for any choice of delays if and only if the matrix has a negative weakly dominant diagonal.

Centre manifolds for partial differential equations with delays

A centre manifold theory for reaction-diffusion equations with temporal delays is developed. Besides an existence proof, we also show that the equation on the centre manifold is a coupled system of scalar ordinary differential equations of higher order. As an illustration, this reduction procedure is applied to the Hutchinson equation with diffusion.

Multiple Limit Cycles for Predator-Prey Models

We construct a Gause-type predator-prey model with concave prey isocline and (at least) two limit cycles. This serves as a counter-example to the global stability criterion of Hsu [Math. Biosci. 39:1-10 (1978)].