Xinzhu Meng

The dynamics of a new SIR epidemic model concerning pulse vaccination strategy

Abstract A new SIR epidemic model with vertical and horizontal transmission is proposed, and the dynamics of this disease model under constant and pulse vaccination are analyzed. Firstly, global stability of the endemic equilibrium states of the model with constant vaccination is thereby established. Further, we show that there exists a stable ‘infection-free’ periodic solution when the period of impulsive effect is less than some critical value. The condition for the permanence of the system with pulse vaccination is also given, which implies the periodic bursts of epidemic occurs. Numerical simulation shows system with pulse vaccination has more complex dynamic behavior for positive periodic oscillation, ‘infection free’ quasi-periodic oscillation than system with constant vaccination. Finally, we compare validity of the strategy of pulse vaccination with no vaccination and constant vaccination, and conclude that pulse vaccination strategy is more effective than no vaccination and continuous vaccination.

Periodic solution of a prey-predator model with nonlinear state feedback control

Assume that when the number of pests reaches the certain threshold, pest management strategy will be taken to control pests. Based on this assumption, in this paper, we propose a pest management model with nonlinear state feedback control. We then analyze the dynamic behavior of the model. More precisely, we first investigate the singularity of the model by using method of qualitative analysis; secondly the existence of periodic solution of the model is studied by using successor functions and Poincare-Bendixson theorem; and then it is followed by the study of the stability of periodic solution; finally, an example with numerical simulations is given to illustrate our conclusions.

A stage-structured Holling mass defence predator–prey model with impulsive perturbations on predators

Abstract In this work, we consider a stage-structured Holling mass defence predator–prey model with time delay and impulsive transmitting on predators. Sufficient conditions which guarantee the global attractivity of pest-extinction periodic solution and permanence of the system are obtained. We also prove that all solutions of the system are uniformly ultimately bounded. Our results provide reliable tactic basis for the practical pest management.

Stochastic inequalities and applications to dynamics analysis of a novel SIVS epidemic model with jumps

This paper proposes a new nonlinear stochastic SIVS epidemic model with double epidemic hypothesis and Lévy jumps. The main purpose of this paper is to investigate the threshold dynamics of the stochastic SIVS epidemic model. By using the technique of a series of stochastic inequalities, we obtain sufficient conditions for the persistence in mean and extinction of the stochastic system and the threshold which governs the extinction and the spread of the epidemic diseases. Finally, this paper describes the results of numerical simulations investigating the dynamical effects of stochastic disturbance. Our results significantly improve and generalize the corresponding results in recent literatures. The developed theoretical methods and stochastic inequalities technique can be used to investigate the high-dimensional nonlinear stochastic differential systems.

Stability of a novel stochastic epidemic model with double epidemic hypothesis

Abstract In this paper, a novel SIR disease transmission model is formulated under double epidemic hypothesis and stochastic perturbation. We present the stability conditions of the disease-free equilibrium of the SIR model without stochastic perturbation and with stochastic perturbation. We obtain the deterministic stability threshold β 1 ∗ ∗ and β 2 ∗ ∗ of the disease-free equilibrium, under which the disease-free equilibrium is stochastically stable such that the disease will disappear finally leaving all the population susceptible. The results show that double epidemic hypothesis and stochastic perturbation have significant effects on the dynamics behaviors of the model.

Stability analysis of a chemostat model with maintenance energy

Abstract In this paper, we dedicate ourselves to the study of a diffusive model for unstirred membrane reactors with maintenance energy and subject to a homogeneous Neumann boundary condition. It shows that the unique constant steady state is globally asymptotically stable when it exists. This result further implies the non-existence of any spatial patterns.

A new stage structured predator–prey Gomportz model with time delay and impulsive perturbations on the prey

Abstract In this paper, we formulate a robust prey-dependent consumption predator–prey Gomportz model with periodic harvesting (catching or poisoning ) for the prey and stage structure for the predator with constant maturation time delay (through-stage time delay) and perform a systematic mathematical and ecological study. By use of the discrete dynamical system determined by the stroboscopic map, we obtain a ‘predator-extinction’ periodic solution. Further, we show it is globally attractive when some parameters of the system are under appropriate conditions. By using the theory on delay functional and impulsive differential equation, we obtain sufficient condition with time delay for the permanence of the system. In this paper, the main feature is that we introduce time delay and pulse into the predator–prey (natural enemy–pest) model with stage structure, exhibit a new modeling method which is applied to investigate delay impulsive differential equations, and give some reasonable suggestions for pest management.

A Stage-Structured Predator-Prey SI Model with Disease in the Prey and Impulsive Effects

This paper aims to develop a high-dimensional SI model with stage structure for both the prey (pest) and the predator, and then to investigate the dynamics of it. The model can be used for the study of Integrated Pest Management (IPM) which is a combination of constant pulse releasing of animal enemies and diseased pests at two different fixed moments. Firstly, we use analytical techniques for impulsive delay differential equations to obtain the conditions for global attractivity of the ‘pest-free’ periodic solution and permanence of the population model. It shows that the conditions strongly depend on time delay, impulsive release of animal enemies and infective pests. Secondly, we present a pest management strategy in which the pest population is kept under the economic threshold level (ETL) when the pest population is permanent. Finally, numerical analysis is presented to illustrate our main conclusion.

Adaptive dynamics analysis of a predator-prey model with selective disturbance

Based on predators capture rate functions, we construct an invasion fitness function.We use a size-selective disturbance function to study evolutionary dynamics.Evolutionarily stable coexistence and branching are explored by numerical simulations.Harvesting may drive evolution towards smaller value of phenotype trait.Large disturbance can go against evolutionary branching and promote evolutionary stability. Evolution problem is always a hot topic in the mathematical biology field. In this paper, we investigate the evolutionary effects of selective disturbance on an evolving trait (e.g. body size and maturation age) of the predator individuals in one-predator two-prey community. By using methods of adaptive dynamics and population dynamics we construct an invasion fitness function and obtain the conditions for evolutionary branching and evolutionary stability under selective disturbance in both monomorphic and dimorphic populations. We further conduct a size-selective disturbance function founded on chi-square distribution to study evolutionary stable coexistence, and considering the evolutionary branching and evolutionary stability by using theoretic analysis and numerical simulations. The evolutionary results from a biological point of view show that (1) two strategies could gradually evolve to form a single ancestral strategy, moreover, higher levels of polymorphism cannot build up during evolution, that is, following first evolutionary branching two species will eventually evolve into two generalist species and reach an evolutionary stable coexistence; (2) smaller disturbance could touch off higher levels of dimorphism during evolution, while large disturbance can go against evolutionary branching and advance evolutionary stability.

The dynamics of an impulsive delay predator–prey model with variable coefficients☆

Abstract In this paper, we formulate a new robust two-species nonautonomous predator–prey model with multi-delays and impulsive effects and perform a systematic mathematical and ecological study. Our results in this paper indicate that under the appropriate linear periodic impulsive perturbations, the system is permanent and has a unique positive globally attractive semi-trivial periodic solution. By using the Brouwer fixed point theorem, we prove that if the periodic system is permanent, then there is at least one positive periodic solution of the system. We show that the conditions for global attractivity of the positive semi-trivial periodic solution and permanence of the population of the model depend on time delay, so, we call it “profitless”. In this paper, the main feature is that we introduce multi-delays and impulses into the predator–prey model, exhibit a new modeling method which is applied to investigate multi-species impulsive multi-delays differential equations.