Yongzhen Pei

TWO DIFFERENT VACCINATION STRATEGIES IN AN SIR EPIDEMIC MODEL WITH SATURATED INFECTIOUS FORCE

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.

Two types of predator–prey models with harvesting: Non-smooth and non-continuous

Abstract This article investigates continuous and impulsive threshold harvesting strategies on the predator which needs to be applied only when the predator population is above or reaches the harvesting threshold. For the continuous threshold model, the system is nonsmooth and has complex dynamics with multiple internal equilibria, limit cycle, homoclinic orbit, saddle–node bifurcation, transcritical bifurcation, subcritical and supercritical Hopf bifurcation, Bogdanov–Takens bifurcation and discontinuous Hopf bifurcation. In order to prevent the predator population being above the threshold, we further extend our model with impulsive threshold harvesting strategies. The model is non-continuous and the existence and stability of positive order-1 and order-2 periodic solutions were obtained by using the Poincare map. It is seen that the impulsive threshold harvesting strategies are more effective than the continuous. Furthermore, some numerical simulations are given to illustrate our results.

Hopf bifurcation and global stability of a diffusive Gause-type predator–prey models

Abstract This paper mainly provides Hopf bifurcation formulas for a general Gause type predator–prey system with diffusion and Neumann boundary condition by using the center manifold theory and normal form method, where the spectral and stability analysis around an equilibrium is addressed, and our results can be applied to the case without diffusion. As an application of these results, we give a complete and rigorous analysis of the global dynamics of a diffusive predator–prey model with herd behavior, especially, the Hopf bifurcation and its direction, and the stability of the bifurcating periodic solutions.

Effect of harvesting, delay and diffusion in a generalist predator-prey model

In compared with specialist predators which feed almost exclusively on a specific species of prey, generalist predators feed on many types of species. Consequently, their dynamics is not coupled to the dynamics of a specific prey population, and the generalist predators has itself growth function which be extended a well-known logistic growth term. We develop a generalist predator-prey model with diffusion and study the effect of harvesting and delay under Neumann conditions. The stability of the equilibria is firstly investigated, and the existence of traveling wave solutions is then established by constructing a pair of upper-lower solutions and using the cross iteration method and Schauders fixed point theorem.

Construction of positivity preserving numerical method for stochastic age-dependent population equations

The aim of this paper is to construct a numerical method preserving positivity for stochastic age-dependent population equations. We use the balanced implicit numerical techniques to maintain the nonnegative path of the exact solution. It is proved that the Balanced Implicit Method (BIM) preserves positivity and converges with strong order 1/2 under given conditions. Finally, two examples are simulated to verify the positivity and efficiency of the proposed method.

EFFECT OF HARVESTING AND PREY REFUGE IN A PREY–PREDATOR SYSTEM

Considering that the ecological system is often deeply perturbed by human exploiting activities, this paper deals with a prey–predator model with prey refuge in which both species are independently harvested. First, some sufficient conditions for global stability of equilibria are obtained, and the existence and uniqueness of limit cycles are established. Our results indicate that over-exploitation would result in the extinction of the population and an appropriate harvesting strategy should ensure the sustainability of the population, which is in line with reality. Furthermore, the existence of bionomic equilibrium is discussed. Finally, the influences of prey refuge and harvesting efforts on equilibrium density values are considered and some numerical simulations are given to illustrate our results.

The impact of predation on the coexistence and competitive exclusion of pathogens in prey.

A two-strain epidemic model with saturating contact rate under a generalist predator is proposed. For a generalist predator which feeds on many types of prey, we assume that the predator can discriminate among susceptible and infected with each strain prey. First, mathematical analysis of the model with regard to invariance of nonnegativity, boundedness of solutions, nature of equilibria, persistence and global stability are analyzed. Second, the two strains will competitively exclude each other in the absence of predation with the strain with the larger reproduction number persisting. If predation is discriminate, then depending on the predation level, a dominant strain may occur. Thus, for some predation levels, the strain one may persist while for other predation levels strain two may persist. Furthermore, coexistence line and coexistent asymptotic-periodic solution are obtained when coexistence occur while heteroclinic is obtained when the two strains competitively exclude each other. Finally, the impact of predation is mentioned along with numerical results to provide some support to the analytical findings.

IMPULSIVE SELECTIVE HARVESTING IN A LOGISTIC FISHERY MODEL WITH TIME DELAY

In this work, we consider a logistic fishery model and discuss the selective impulsive harvesting of fish above a certain age or size by incorporating a time delay in the impulsive harvesting term. It is proved that there exists an asymptotically stable positive periodic solution when the catchability coefficient h is less than some critical value . It is concluded that and are increasing with respect to l. Simulations shows that the delayed harvesting is advantageous to the sustainability of the population.

Convergence of the split-step θ-method for stochastic age-dependent population equations with Poisson jumps

In this paper, a new split-step ? (SS?) method for stochastic age-dependent population equations with Poisson jumps is constructed. The main aim of this paper is to investigate the convergence of the SS? method for stochastic age-dependent population equations with Poisson jumps. It is proved that the proposed method is convergent with strong order 1/2 under given conditions. Finally, an example is simulated to verify the results obtained from theory, the results show that the SS? method has better accuracy compared to the Euler method.

Modeling and analysis of a predator–prey model with state-dependent delay

We propose and study a predator–prey model with state-dependent delay where the prey population is assumed to have an age structure. The state-dependent delay appears due to the mature condition that the prey must spend an amount of time in the immature stage sufficient to accumulate a threshold amount of food. We perform a qualitative analysis of the solutions, which includes studying positivity and boundedness, existence and local stability of equilibria. For the global dynamics of the system, we discuss an attracting region which is determined by solutions, and the region collapses to the interior equilibrium in the constant delay case.