International Journal of Biomathematics

This paper deals with an impulsive predator-prey model with Beddington–DeAngelis functional response and time delay, in which the evolution of the predators takes them through two stages, juvenile and mature. It is assumed that only mature predators are able to hunt for prey and reproduce and the time delay is understood as being the time spent by the juvenile predators from birth to maturity. It is first seen that the dynamics of the model can be completely determined through the use of a reduced system consisting of the equations for prey and mature predators, respectively. Using the discrete dynamical system determined by the stroboscopic map, one first determines the mature predator-free periodic solution of the reduced system. By means of comparison techniques, one then deduces sufficient criteria for the global stability of the mature predator-free periodic solution and for the permanence of the reduced system, which yield similar properties for the initial system. As a result, it is observed that time delay and pulses have a crucial effect upon the dynamics of our model.

Brusselator model is a very typical autocatalytic reaction diffusion system. The bifurcation of steady-states of Brusselator model can be used to explain spot patterns of certain animals such as leopard and jaguar. Periodic patterns can be found throughout whole natural world, so it is very interesting to study patterns generated by the bifurcation of periodic solutions in extended Brusselator (EB) model, which extends Brusselator to T-periodic coefficients. In this paper, we study extended simplified Brusselator (ESB) model, which is EB model without diffusion terms. We find a unique T-periodic solution x₀(t) in the strictly positively invariant region and prove its stability. This result establishes a foundation to study the bifurcation of EB model from x₀(t). We also develop techniques of using degree theory and Floquet theory to analyze existence, uniqueness and stability of a periodic solution.

In this paper, a class of n-species nonautonomous Lotka–Volterra competitive system with infinite delays and feedback controls is studied. A new criterion on the permanence of species is established. This result shows that the feedback controls do not influence the permanence of species.

A non-autonomous epidemic dynamical system, in which we include variable susceptibility, is proposed. Some threshold conditions are derived which determine whether or not the disease will go to extinction. Some new threshold values, , and , are deduced for this general time-dependent system such that when is greater than 0, the disease is endemic in the sense of permanence and when one of the threshold values and is less than 0, the disease will die out. As an application of these results, the basic reproductive number ℛ₀ will be given if all the coefficients are periodic with common period. In addition, ℛ₀ < 1 implies the global stability of the disease-free periodic solution. Some corollaries are given for periodic and almost-periodic cases. The theoretical results are confirmed by a special example and numerical simulations.

In this paper, we give a compartment model to discuss the influence of media coverage to the spreading and controlling of infectious disease in a given region. The model exhibits two equilibria: a disease-free and a unique endemic equilibrium. Stability analysis of the models shows that the disease-free equilibrium is globally asymptotically stable if the reproduction number (ℝ₀), which depends on parameters, is less than unity. But if ℝ₀ > 1, it is shown that a unique endemic equilibrium appears, which is asymptotically stable. On a special case, the endemic equilibrium is globally stable. We discuss the role of media coverage on the spreading based on the theory results.

We present three different simplest realistic models of the circadian pacemaker that can be interpreted in molecular terms: Goodwin model, piece-wise Goodwin model and delay model. The main purpose is to investigate how the common parameters in these three model affect the dynamical behavior. In particular, the existence and stability of periodic solutions of three models are studied, and the parameter space where the steady state becomes unstable is given which can be used to compare the effect of parameters on the oscillations of three models. Furthermore, the effects of parameters on the period of periodic solutions and phase response curves are investigated and discussed. Our results confirm that all three models can be used to simulate the realistic circadian rhythms. However, in practice, which model is chosen depends on the real data.

The present paper gives two methods to obtain the existence and uniqueness of almost periodic solution of the following discrete Lasota–Wazewska model

In this paper, by using the theory of abstract continuous theorem of k-set contractive operator and some analysis techniques, we obtain some sufficient conditions for the existence of positive periodic solutions for a neutral population model.