Dynamic behaviors of a delay differential equation model of plankton allelopathy

AbstractIn this paper, we consider a modified delay differential equation model of the growth of n-species of plankton having competitive and allelopathic effects on each other. We first obtain the sufficient conditions which guarantee the permanence of the system. As a corollary, for periodic case, we obtain a set of delay-dependent condition which ensures the existence of at least one positive periodic solution of the system. After that, by means of a suitable Lyapunov functional, sufficient conditions are derived for the global attractivity of the system. For the two-dimensional case, under some suitable assumptions, we prove that one of the components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation. Examples show the feasibility of the main results

Existence of periodic positive solutions to a nonlinear Lotka-Votlerra competition systems

We investigate the existence of positive periodic solutions of a nonlinear Lotka-Volterra competition system with deviating arguments. The main tool we use to obtain our result is the Krasnoselskii fixed point theorem. In particular, this paper improves important and interesting work [X.H. Tang, X. Zhou, On positive periodic solution of Lotka–Volterra competition systems with deviating arguments, Proc. Amer. Math. Soc. 134 (2006), 2967–2974]. Moreover, as an application, we also exhibit some special cases of the system, which have been studied extensively in the literature

Spatial patterns of competing random walkers

We review recent results obtained from simple individual-based models of biological competition in which birth and death rates of an organism depend on the presence of other competing organisms close to it. In addition the individuals perform random walks of different types (Gaussian diffusion and L\'{e}vy flights). We focus on how competition and random motions affect each other, from which spatial instabilities and extinctions arise. Under suitable conditions, competitive interactions lead to clustering of individuals and periodic pattern formation. Random motion has a homogenizing effect and then delays this clustering instability. When individuals from species differing in their random walk characteristics are allowed to compete together, the ones with a tendency to form narrower clusters get a competitive advantage over the others. Mean-field deterministic equations are analyzed and compared with the outcome of the individual-based simulations.Comment: 38 pages, including 6 figure

Global exponential stability of nonautonomous neural network models with unbounded delays

For a nonautonomous class of n-dimensional di erential system with in nite delays, we give su cient conditions for its global exponential stability, without showing the existence of an equilibrium point, or a periodic solution, or an almost periodic solution. We apply our main result to several concrete neural network models, studied in the literature, and a comparison of results is given. Contrary to usual in the literature about neural networks, the assumption of bounded coe cients is not need to obtain the global exponential stability. Finally, we present numerical examples to illustrate the e ectiveness of our results.The paper was supported by the Research Center of Mathematics of University of Minho with the Portuguese Funds from the FCT - “Fundação para a Ciência e a Tecnologia”, through the Project UID/MAT/00013/2013. The author thanks the referees for valuable comments.info:eu-repo/semantics/publishedVersio