Narrow Results

In this paper, we focus on a network system which describes spatiotemporal dynamics of single species population at different patches since species can have different features in various life stages and different behaviors in various spatial environments. With the effect of time delay and spatial dispersion, homogenous, periodic and spatiotemporally nonhomogeneous distributions are identified. The stability analysis is carried out for the discrete-space and continuous-time network on single species with time delay and the Hopf bifurcation of the single species population model in a network is explored. Formulas for determining the direction of Hopf bifurcation are derived by using the center manifold method and the normal form theorem. It is found that the network can generate spatial patterns only when time delay is present. Finally, numerical simulations are performed which agree well with our theoretical result, i.e. this discrete-space and continuous-time model admits regular temporal patterns since the delay induces Hopf bifurcations with network structure.

This article concentrates on the study of delay effect on a model of single species in a polluted closed environment. The cases of constant emission and instantaneous spill of a toxicant are considered in the model study. Sufficient criteria on persistence of the population are derived. Global attractivity condition for the positive equilibrium is obtained.

The effect of toxicants on ecological systems is an important and urgent issue from mathematical and experimental point of view. In this paper, we have studied the dynamical behavior of a single species population affected by environmental toxicant and population toxicant. Boundedness, local and global stabilities, bifurcation are addressed. We have also studied the effect of discrete delay of the environmental toxicant on the instantaneous growth rates of the population biomass and population toxicant due to incubation period. The length of delay preserving the stability is also estimated. Computer simulations are carried out to illustrate our analytical findings.