Asymptotic behavior of age-structured and delayed Lotka-Volterra models

In this work we investigate some asymptotic properties of an age-structured Lotka-Volterra model, where a specific choice of the functional parameters allows us to formulate it as a delayed problem, for which we prove the existence of a unique coexistence equilibrium and characterize the existence of a periodic solution. We also exhibit a Lyapunov functional that enables us to reduce the attractive set to either the nontrivial equilibrium or to a periodic solution. We then prove the asymptotic stability of the nontrivial equilibrium where, depending on the existence of the periodic trajectory, we make explicit the basin of attraction of the equilibrium. Finally, we prove that these results can be extended to the initial PDE problem.Comment: 29 page

On a predator prey model with nonlinear harvesting and distributed delay

A predator prey model with nonlinear harvesting (Holling type-II) with both constant and distributed delay is considered. The boundeness of solutions is proved and some sufficient conditions ensuring the persistence of the two populations are established. Also, a detailed study of the bifurcation of positive equilibria is provided. All the results are illustrated by some numerical simulations.Ministerio de Economía y CompetitividadFondo Europeo de Desarrollo RegionalConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

Harmless delays and global attractivity for nonautonomous predator-prey system with dispersion

AbstractIn this paper, we consider a nonautonomous predator-prey model with dispersion and a finite number of discrete delays. The system consists of two Lotka-Volterra patches and has two species: one can disperse between two patches, but the other is confined to one patch and cannot disperse. Our purpose is to demonstrate that the time delays are harmless for uniform persistence of the solutions of the system. Furthermore, we establish conditions under which the system admits a positive periodic solution which attracts all solutions

Existence and global attractivity of positive almost periodic solutions for a kind of fishing model with pure-delay

summary:By using some analytical techniques, modified inequalities and Mawhin's continuation theorem of coincidence degree theory, some sufficient conditions for the existence of at least one positive almost periodic solution of a kind of fishing model with delay are obtained. Further, the global attractivity of the positive almost periodic solution of this model is also considered. Finally, three examples are given to illustrate the main results of this paper

Advanced Nonlinear Dynamics of Population Biology and Epidemiology

abstract: Modern biology and epidemiology have become more and more driven by the need of mathematical models and theory to elucidate general phenomena arising from the complexity of interactions on the numerous spatial, temporal, and hierarchical scales at which biological systems operate and diseases spread. Epidemic modeling and study of disease spread such as gonorrhea, HIV/AIDS, BSE, foot and mouth disease, measles, and rubella have had an impact on public health policy around the world which includes the United Kingdom, The Netherlands, Canada, and the United States. A wide variety of modeling approaches are involved in building up suitable models. Ordinary differential equation models, partial differential equation models, delay differential equation models, stochastic differential equation models, difference equation models, and nonautonomous models are examples of modeling approaches that are useful and capable of providing applicable strategies for the coexistence and conservation of endangered species, to prevent the overexploitation of natural resources, to control disease’s outbreak, and to make optimal dosing polices for the drug administration, and so forth.View the article as published at https://www.hindawi.com/journals/aaa/2014/214514