Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays

For a Nicholson’s blowflies model with patch structure and multiple discrete delays, we study some aspects of its global dynamics. Conditions for the absolute global asymptotic stability of both the trivial equilibrium and a positive equilibrium (when it exists) are given. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. We further consider a diffusive Nicholson-type model with patch structure, and establish a criterion for the existence of positive travelling wave solutions, for large wave speeds. Several applications illustrate the results, improving some criteria in the recent literature.     

Spreading speeds of a Lotka-Volterra predator-prey system: The role of the predator

This paper is concerned with the spreading speeds of a modified Lotka-Volterra predator-prey system; the intention is to describe the propagation mode whereby the two species coinvade a new habitat. Under some assumptions, it is proved that the spreading speed of the prey is slower than in the case where the predator vanishes, and the density of the prey on the coexistence domain is also smaller compared with the case where interspecies actions disappear. Therefore, the predator has a negative effect on the evolution of the prey by slowing the spreading speed and decreasing the population density of the prey.     

Periodicity and stability for a Lotka-Volterra type competition system with feedback controls and deviating arguments

This paper deals with the general periodic Lotka-Volterra type competition systems with feedback controls and deviating arguments. By employing fixed point index theory on cone, an explicit necessary and sufficient condition for the global existence of the positive periodic solution of the systems is proved. By constructing a suitable Lyapunov functional, a set of easily verifiable sufficient conditions for the global asymptotic stability of the positive periodic solution of the systems is given.     

Dynamics of multi-species competition-predator system with impulsive perturbations and Holling type III functional responses

We investigate the dynamics of a class of multi-species predator-prey interaction models with Holling type III functional responses based on systems of nonautonomous differential equations with impulsive perturbations. Sufficient conditions for existence of a positive periodic solution are investigated by using a continuation theorem in coincidence degree theory, which have been extensively applied in studying existence problems in differential equations and difference equations. In addition, sufficient criteria are established for the global stability and the globally exponential stability of the system by using the comparison principle and the Lyapunov method.     

Some generalizations of locally and weakly locally uniformly convex space

In this paper, we prove that strongly convex space and almost locally uniformly rotund space, very convex space and weakly almost locally uniformly rotund space are respectively equivalent. We also investigate a few properties of k-strongly convex space and k-very convex space, and discuss the applications of strongly convex space and very convex space in approximation theory.     

Stochastically forced cardiac bidomain model

The bidomain system of degenerate reaction–diffusion equations is a well-established spatial model of electrical activity in cardiac tissue, with “reaction” linked to the cellular action potential and “diffusion” representing current flow between cells. The purpose of this paper is to introduce a “stochastically forced” version of the bidomain model that accounts for various random effects. We establish the existence of martingale (probabilistic weak) solutions to the stochastic bidomain model. The result is proved by means of an auxiliary nondegenerate system and the Faedo–Galerkin method. To prove convergence of the approximate solutions, we use the stochastic compactness method and Skorokhod–Jakubowski a.s. representations. Finally, via a pathwise uniqueness result, we conclude that the martingale solutions are pathwise (i.e., probabilistic strong) solutions.     

Minimal-speed selection of traveling waves to the Lotka–Volterra competition model

In this paper the minimal-speed determinacy of traveling wave fronts of a two-species competition model of diffusive Lotka–Volterra type is investigated. First, a cooperative system is obtained from the classical Lotka–Volterra competition model. Then, we apply the upper-lower solution technique on the cooperative system to study the traveling waves as well as its minimal-speed selection mechanisms: linear or nonlinear. New types of upper and lower solutions are established. Previous results for the linear speed selection are extended, and novel results on both linear and nonlinear selections are derived.     

Evolution of a semilinear parabolic system for migration and selection without dominance

The semilinear parabolic system that describes the evolution of the gene frequencies in the diffusion approximation for migration and selection at a multiallelic locus without dominance is investigated. The population occupies a finite habitat of arbitrary dimensionality and shape (i.e., a bounded, open domain in Rd). The selection coefficients depend on position; the drift and diffusion coefficients may depend on position. The primary focus of this paper is the dependence of the evolution of the gene frequencies on λ, the strength of selection relative to that of migration. It is proved that if migration is sufficiently strong (i.e., λ is sufficiently small) and the migration operator is in divergence form, then the allele with the greatest spatially averaged selection coefficient is ultimately fixed. The stability of each vertex (i.e., an equilibrium with exactly one allele present) is completely specified. The stability of each edge equilibrium (i.e., one with exactly two alleles present) is fully described when either (i) migration is sufficiently weak (i.e., λ is sufficiently large) or (ii) the equilibrium has just appeared as λ increases. The existence of unexpected, complex phenomena is established: even if there are only three alleles and migration is homogeneous and isotropic (corresponding to the Laplacian), (i) as λ increases, arbitrarily many changes of stability of the edge equilibria and corresponding appearance of an internal equilibrium can occur and (ii) the conditions for protection or loss of an allele can both depend nonmonotonically on λ. Neither of these phenomena can occur in the diallelic case.