International Journal of Biomathematics

In this paper, we investigate the eco-epidemiological model with impulsive perturbation. Using Floquet theory and small amplitude perturbation method, we prove that the boundary periodic solution is asymptotically stable if the impulsive parameter satisfies some conditions. Moreover, the permanence of the system is discussed by analytical method. Finally, the complexity of the system is verified by numerical simulation.

In this work, optimal harvesting policy for an age-dependent and spatial diffusion n-dimensional competing species is discussed. The existence and uniqueness of non-negative solution to the system are investigated by using the fixed point theorem. The existence of optimal control strategy is discussed and optimality conditions are obtained. Our results extend some known criteria.

Two different vaccination and treatment strategies in the SIR epidemic model with saturation infectious force are analyzed. With the continuous vaccination and treatment, it is obtained that the disease free equilibrium and endemic equilibrium are globally asymptotically stable by using Lassall theorem and Pioncare–Bendixon trichotomy. Moreover, with pulse vaccination and treatment at different time, the dynamics of the epidemic model is globally investigated by using Floquet theory and comparison theorem of impulsive differential equation and analytic method. We obtain the conditions of global asymptotical stability of the infection-free periodic solution and permanence of the model. Finally, we compare the two different vaccination and treatment strategies, and obtain that the elimination of disease is independent of treatment in the case of the pulse vaccination.

The effects of time delay on two neurons interaction Morris–Lecar model are investigated. It is assumed the two neurons are coupled via gap junction thus time delay arises naturally since information signals take time transmitting from pre-synaptic to post-synaptic of the neurite. It is found that time delay leads the both neurons to quiescent states at the lower reversal potential value of Ca⁺ channel. However, the bifurcation diagram of inter-spike interval (ISI) uncovers the complex firing behavior as time delay is above a certain critical value. Finally, the complete synchronization and lag synchronization of two neurons are reported under an efficient time delay and reversal potential value of Ca⁺ channel.

Based on the complex network and non-equilibrium statistic theory, the birth and death equation (BDE) is proposed for the virus spreading on biological networks. This equation, in which the Susceptible Infected Susceptible model is extended, can predict the distribution rule of infection density by generating function. It is a better approach to a real infection process.

In this paper, we formulate a robust impulsive Lotka–Volterra n-species competitive system with both discrete delays and continuous delays. Our results in this paper indicate that under the appropriate linear bounded impulsive perturbations, the impulsive delay Lotka–Volterra system remains the original permanence and globally asymptotical stability of the nonimpulsive delay Lotka–Volterra system. We show that the conditions for the permanence and globally asymptotical stability of the system depend on time delays, so, we call it "profitless".

In this work, we consider a delayed stage-structured variable coefficients predator-prey system with impulsive perturbations on predators. By using the discrete dynamical system determined by stroboscopic map and the standard comparison theorem, we obtain the sufficient conditions which guarantee the global attractivity of prey-extinction periodic solution of the investigated system. We also prove that all solutions of the system are uniformly ultimately bounded. Our results provide reliable tactic basis for the practical pest management.

In this paper, a predator-prey model with prey dispersal and time delay is investigated. By analyzing the corresponding characteristic equation of a positive equilibrium, the local stability of the positive equilibrium and the existence of Hopf bifurcation are discussed. By using the normal form theory and center manifold reduction, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. Numerical simulations are given to illustrate the theoretical predictions.

In this paper, an impulsive exploitation of single species modelled by a general nonautonomous equation is considered. The persistence of the population can be obtained under sufficiently weak conditions and also be distinguished from conditions that cause extinction. These attributes resemble corresponding features of the related autonomous population growth model and improve the results of [6–8].

By means of asymptotically stable theory and infection model theory of ordinary differential equation, we do research on SIQS model with nonlinear and isolation. Firstly, we obtain the existence of threshold value R₀ of disease-free equilibration point and local disease equilibration point. Secondly, we prove disease-free equilibration point is locally asymptotically stable when R₀ < 1, and local disease equilibration point is locally asymptotically stable when R₀ > 1. Furthermore, we have disease-free equilibration point and local disease equilibration point are globally asymptotically stable with the help of Liapunov function. Lastly, we explain at the point of biology.

In this paper, using the method of upper and lower solutions and its associated monotone iterations, we establish a new monotone-iterative scheme for finding periodic solutions of an impulsive differential equations. This method leads to the existence of maximal and minimal periodic solutions which can be computed from a linear iteration process in the same fashion as for impulsive differential equations initial value problem. This method is constructive and can be used to develop a computational algorithm for numerical solution of the periodic impulsive system. Our existence result improves a result established in [1]. The result is applied to a model of mutualism of Lotka–Volterra type which involves interactions among a mutualist-competitor, a competitor and a mutualist, the existence of positive periodic solutions of the model is obtained.

This paper is concerned with the estimates of solutions for an impulsive parabolic equations under general boundary condition. We prove that the solutions of impulsive parabolic equations can be controlled and estimated by the solutions of dominating impulsive ordinary differential equations. We also apply the above results to a model problem arising from population biology.