The differential susceptibility SIR epidemic model with stage structure and pulse vaccination

The differential susceptibility SIR epidemic model with stage structure and pulse vaccination is introduced. By the comparison theorem, some sufficient conditions for the globally attractivity of an infection-free periodic solution and the permanence of this system are presented. Two numerical simulations are also given to illustrate our main results.     

Impulsive control in a stage structure population model with birth pulses

The dynamical behavior of a stage structure population model with birth pulses and impulsive pest management strategy is discussed analytically and numerically. It is assumed that birth pulse and impulsive pest management strategy act with the same period, but not simultaneously. The existence and stability of the positive 2T-period solution are investigated. By using center manifold theorem and bifurcation theorem, the conditions of existence for flip bifurcation are derived. Moreover, some detailed numerical results for phase portraits, periodic solutions, bifurcation diagram, and chaotic attractors, which are illustrated with two examples, are in good agreement with the theoretical analysis.     

Competitive systems with stage structure of distributed-delay type

This paper considers a delay differential equation model for the interaction among n species, the adult members of which are in competition. For each of the n species the model incorporates an infinite distributed time delay which represents the time from birth to maturity of that species. Thus, the time delays appear in the adult recruitment terms. The dynamics of the model are determined, and sharp global stability criteria are established for the interior equilibrium as well as the axial equilibrium.     

Asymptotic behaviors of competitive Lotka-Volterra system with stage structure

In this paper, a two-species competitive model with stage structure is presented and studied. Results on the global extinction and permanence are given, which generalize the well-known three theorems for the two species competitive system and, moreover, they confirm the negative effect of stage structure on the permanence of populations as well as estimate the degree of such effect. Conclusions in this paper suggest that for a competitive community stage structure is also one of the important reasons that cause permanence and extinction.     

Optimal control of harvest and bifurcation of a prey-predator model with stage structure

This paper describes a prey-predator model with stage structure for prey. The adult prey and predator populations are harvested in the proposed system. The dynamic behavior of the model system is discussed. It is observed that singularity induced bifurcation phenomenon is appeared when variation of the economic interest of harvesting is taken into account. State feedback controller is incorporated to stabilize the model system in case of positive economic interest. Harvesting of prey and predator population are used as controls to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent earned from the resource. The Pontryagin’s maximum principle is used to characterize the optimal controls. The optimality system is derived and then solved numerically using an iterative method with Runge-Kutta fourth order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem.     

A predator–prey system with a stage structure for the prey

This paper considers a periodic predator–prey system where the prey has a life history that takes the prey through two stages: immature and mature. We provide a sufficient and necessary condition to guarantee permanence of the system. It is shown that the system is permanent if and only if the growth of the predator by foraging the prey minus its death rate is positive on average during the period.     

Coexistence and stability of predator-prey model with Beddington-DeAngelis functional response and stage structure

We present a predator-prey model of Beddington-DeAngelis type functional response with stage structure on prey. The constant time delay is the time taken from birth to maturity about the prey. By the uniform persistence theories and monotone dynamic theories, sharp threshold conditions which are both necessary and sufficient for the permanence and extinction of the model as well as the sufficient conditions for the global stability of the coexistence equilibria are obtained. Biologically, it is proved that the variation of prey stage structure can affect the permanence of the system and drive the predator into extinction by changing the prey carrying capacity: Our results suggest that the predator coexists with prey permanently if and only if predator's recruitment rate at the peak of prey abundance is larger than its death rate; and that the predator goes extinct if and only if predator's possible highest recruitment rate is less than or equal to its death rate; furthermore, our results also show that a sufficiently large mutual interference by predators can stabilize the system.     

Stability and Hopf bifurcation analysis of an eco-epidemic model with a stage structure

In this paper, an eco-epidemiological model with a stage structure is considered. The asymptotical stability of the five equilibria, the existence of stability switches about positive equilibrium, is investigated. It is found that Hopf bifurcation occurs when the delay τ passes though a critical value. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.     

Traveling waves of a diffusive predator-prey model with nonlocal delay and stage structure

In this paper, a diffusive predator-prey model with nonlocal delay and stage structure is investigated. By using the cross iteration method and Schauder's fixed point theorem, we reduce the existence of traveling wave solutions to the existence of a pair of upper-lower solutions. Numerical simulations are carried out to illustrate the theoretical results.     

A Lotka-Volterra type food chain model with stage structure and time delays

A three-species Lotka-Volterra type food chain model with stage structure and time delays is investigated. It is assumed in the model that the individuals in each species may belong to one of two classes: the immatures and the matures, the age to maturity is presented by a time delay, and that the immature predators (immature top predators) do not have the ability to feed on prey (predator). By using some comparison arguments, we first discuss the permanence of the model. By means of an iterative technique, a set of easily verifiable sufficient conditions are established for the global attractivity of the nonnegative equilibria of the model.     

Permanence and periodicity of a delayed ratio-dependent predator-prey model with stage structure

A periodic ratio-dependent predator-prey model with time delays and stage structure for both prey and predator is investigated. It is assumed that immature individuals and mature individuals of each species are divided by a fixed age, and that immature predators do not have the ability to attack prey. Sufficient conditions are derived for the permanence and existence of positive periodic solutions of the model. Numerical simulations are presented to illustrate the feasibility of our main results.     

Time delayed parabolic system in a two-species competitive model with stage structure

A two-species competitive model with stage structure is discussed. The dynamics of coupled system of semilinear parabolic equations with time delays are investigated. Results on the local and global stabilities of the axial equilibria and positive equilibrium are given. Our results show that the introduction of diffusion does not affect the permanence and extinction of the species though the introduction of stage structure brings negative effect on it.     

Global stability of a stage-structured epidemic model with a nonlinear incidence

In this paper, a stage-structured epidemic model with a nonlinear incidence with a factor S^p is investigated. By using limit theory of differential equations and Theorem of Busenberg and van den Driessche, global dynamics of the model is rigorously established. We prove that if the basic reproduction number R₀ is less than one, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Numerical simulations support our analytical results and illustrate the effect of p on the dynamic behavior of the model.     

Dynamics of a discretized SIR epidemic model with pulse vaccination and time delay

We derive a discretized SIR epidemic model with pulse vaccination and time delay from the original continuous model. The sufficient conditions for global attractivity of an infection-free periodic solution and permanence of our model are obtained. Improving discretization, our results are corresponding to those in the original continuous model.     

Global attractivity and permanence of a delayed SVEIR epidemic model with pulse vaccination and saturation incidence

In this study, we formulate and analyze a new SVEIR epidemic disease model with time delay and saturation incidence, and analyze the dynamic behavior of the model under pulse vaccination. Using the discrete dynamical system determined by the stroboscopic map, we obtain an ‘infection-free’ periodic solution, further, show that the ‘infection-free’ periodic solution is globally attractive for some parameters of the model under appropriate conditions. The permanence of the model is investigated analytically. By computer simulation it is concluded that a large vaccination rate or a short pulse of vaccination or a long latent period are each a sufficient condition for the extinction of the disease.     

Pulse vaccination on SEIR epidemic model with nonlinear incidence rate

In this paper, we consider an SEIR epidemic model with two time delays and nonlinear incidence rate, and study the dynamical behavior of the model with pulse vaccination. By using the Floquet theorem and comparison theorem, we prove that the infection-free periodic solution is globally attractive when R^*<1, and using a new modelling method, we obtain a sufficient condition for the permanence of the epidemic model with pulse vaccination when R_*>1.     

Extinction and permanence in nonautonomous competitive system with stage structure

In this paper, the n-species nonautonomous stage-structured competitive system is constructed and considered. Sufficient conditions for its extinction and permanence are obtained. Results here generalize and unify some previous ones. Moreover, it is concluded that stage structure in this system is one of the important factors that effect the extinction and permanence of species.     

Permanence extinction and global asymptotic stability in a stage structured system with distributed delays

In this paper we consider a nonautonomous stage-structured competitive system of n-species population growth with distributed delays which takes into account the delayed feedback in both interspecific and intraspecific interactions. We obtain, by using the method of repeated replace, sufficient conditions for permanence and extinction of the species. The global attractivity of the unique positive equilibrium is proved in the autonomous case. Our results extend previous ones obtained by Liu et al. in [Nonlinear Anal. 51 (2002) 1347-1361; J. Math. Anal Appl. 274 (2002) 667-684].     

Hopf bifurcation of a predator-prey system with stage structure and harvesting

In this paper, a two-species predator-prey system with stage structure and harvesting is investigated. The existence of Hopf bifurcations of the system is given. And the stability and directions of Hopf bifurcations are determined by applying the normal form theory and the center manifold theorem.     

Permanence of variable coefficients predator–prey system with stage structure

We studied a finite delay predator–prey model with stage structure for predator. By analyzing right hand of function and the standard comparison theorem, some new sufficient conditions are derived for the permanence of population and some biological explanations are made.