Abstract. For a class of nonlocal time-delayed reaction-diffusion equations, we prove that all noncritical wavefronts are globally exponentially stable, and critical wavefronts are globally algebraically stable when the initial perturbations around the wavefront decay to zero exponentially near the negative infinity regardless of the magnitude of time delay. This work also improves and develops the existing stability results for local and nonlocal reaction-diffusion equations with delays. Our approach is based on the combination of the weighted energy method and the Green function technique.

We develop a perturbation argument based on existing results on asymptotic autonomous systems and the Fredholm alternative theory that yields the persistence of traveling wavefronts for reaction-diffusion equations with nonlocal and delayed nonlinearities, when the time lag is relatively small. This persistence result holds when the nonlinearity of the corresponding ordinary reaction-diffusion system is either monostable or bistable. We then illustrate this general result using five different models from population biology, epidemiology and bio-reactors.

Abstract. In this paper, we derive the conditions for the existence of stationary solutions (i.e., nonconstant steady states) of a volume-filling chemotaxis model with logistic growth over a bounded domain subject to homogeneous Neumann boundary conditions. At the same time, we show that the same system without the chemotaxis term does not admit pattern formations. Moreover, based on an explicit formula for the stationary solutions, which is derived by asymptotic bifurcation analysis, we establish the stability criteria and find a selection mechanism of the principal wave modes for the stable stationary solution by estimating the leading term of the principal eigenvalue. We show that all bifurcations except the one at the first location of the bifurcation parameter are unstable, and if the pattern is stable, then its principal wave mode must be a positive integer which minimizes the bifurcation parameter. For a special case where the carrying capacity is one half, we find a necessary and sufficient condition for the stability of pattern solutions. Numerical simulations are presented, on the one hand, to illustrate and fit our analytical results and, on the other hand, to demonstrate a variety of interesting spatio-temporal patterns, such as chaotic dynamics and the merging process, which motivate an interesting direction to pursue in the future. 1. Introduction. The process of generation of spontaneous patterns (i.e., selforganization) involves many instances where symmetry is broken or a more symmetric state develops into a less symmetric one. It is striking that many of these breaks in symmetry have no external trigger, but are instigated internally. Hence a natural question arises: how can this process be done? The significant progress toward this question was made in 1952 by Alan Turing, whose paper [29] was considered one of the most influential works in theoretical biology. Turing thought about a simple model with two morphogens (chemical species): a short-range activator and a long-range inhibitor. Both morphogens diffuse in space, but at different rates, and the inhibitor diffuses faster than the activator. The combination of local strong activation and longrange inhibition is able to instigate a spatial patterning process. Turing proposed a mathematical model based on a couple of partial differential equations encapsulating these elements, known as a reaction-diffusion model (see, e.g., [18]). Turing's revolutionary contribution was that passive diffusion could interact with chemical reaction

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In this paper a new approach based on a shooting method in a half line coupled with the technique of upper-lower solution pair is used to study the existence and nonexistence of monotone waves for one form of the delayed Fisher equation that does not have the quasimonotonicity property. A necessary and sufficient condition is provided. This new method can be extended to investigate many other nonlocal and non-monotone delayed reaction-diffusion equations.

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Abstract. In this paper we develop a new method to establish the existence of traveling wavefronts for a food-limited population model with nonmonotone delayed nonlocal effects. Our approach is based on a combination of perturbation methods, the Fredholm theory, and the Banach fixed point theorem. We also develop and theoretically justify Canosa's asymptotic method for the wavefronts with large wave speeds. Numerical simulations are provided to show that there exists a prominent hump when the delay is large.

We consider a simple phytoplankton model introduced by Shigesada and Okubo which incorporates the sinking and self-shading effect of the phytoplankton. The amount of light the phytoplankton receives is assumed to be controlled by the density of the phytoplankton population above the given depth. We show the existence of non-homogeneous solutions for any water depth and study their profiles and stability. Depending on the sinking rate of the phytoplankton, light intensity and water depth, the plankton can concentrate either near the surface, at the bottom of the water column, or both, resulting in a "double-peak" profile. As the buoyancy passes a certain critical threshold, a sudden change in the phytoplankton profile occurs. We quantify this transition using asymptotic techniques. In all cases we show that the profile is locally stable. This generalizes the results of Shigesada and Okubo where infinite depth was considered.