Permanence and stability of an Ivlev-type predator–prey system with impulsive control strategies

In this paper, we study a predator–prey system with an Ivlev-type functional response and impulsive control strategies containing a biological control (periodic impulsive immigration of the predator) and a chemical control (periodic pesticide spraying) with the same period, but not simultaneously. We find conditions for the local stability of the prey-free periodic solution by applying the Floquet theory of an impulsive differential equation and small amplitude perturbation techniques to the system. In addition, it is shown that the system is permanent under some conditions by using comparison results of impulsive differential inequalities. Moreover, we add a forcing term into the prey population’s intrinsic growth rate and find the conditions for the stability and for the permanence of this system.     

A predator–prey model with disease in the prey and two impulses for integrated pest management

In this paper, a predator–prey model with disease in the prey is constructed and investigated for the purpose of integrated pest management. In the first part of the main results, the sufficient condition for the global stability of the susceptible pest-eradication periodic solution is obtained, which means if the release amount of infective prey and predator satisfy the condition, then the pest will be controlled. The sufficient condition for the permanence of the system is also obtained subsequently, which means if the release amount of infective prey and predator satisfy the condition, then the prey and the predator will coexist. At last, we interpret our mathematical results.     

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.     

Periodicity in a generalized semi-ratio-dependent predator–prey system with time delays and impulses

With the help of a continuation theorem based on coincidence degree theory, we establish necessary and sufficient conditions for the existence of positive periodic solutions in a generalized semi-ratio-dependent predator–prey system with time delays and impulses, which covers many models appeared in the literature. When the results reduce to the semi-ratio-dependent predator–prey system without impulses, they generalize and improve some known ones.     

Multiple periodic solutions in delayed Gause-type ratio-dependent predator–prey systems with non-monotonic numerical responses

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of multiple periodic solutions in delayed Gause-type ratio-dependent predator–prey systems with numerical responses. As corollaries, some applications are listed.     

Existence of periodic solutions in predator–prey and competition dynamic systems

In this paper, we systematically explore the periodicity of some dynamic equations on time scales, which incorporate as special cases many population models (e.g., predator–prey systems and competition systems) in mathematical biology governed by differential equations and difference equations. Easily verifiable sufficient criteria are established for the existence of periodic solutions of such dynamic equations, which generalize many known results for continuous and discrete population models when the time scale is chosen as or , respectively. The main approach is based on a continuation theorem in coincidence degree theory, which has been extensively applied in studying existence problems in differential equations and difference equations but rarely applied in dynamic equations on time scales. This study shows that it is unnecessary to explore the existence of periodic solutions of continuous and discrete population models in separate ways. One can unify such studies in the sense of dynamic equations on general time scales.     

An impulsive ratio-dependent predator–prey system with diffusion

We investigate the predator–prey system with diffusion, when biological and environmental parameters are assumed to change in periodical manner over time. The system is affected by impulses which can be considered as a control. Conditions for the permanence of the predator–prey system and for the existence of a unique globally stable periodic solutions are obtained.     

Dynamics of a predator–prey model with disease in the predator

The present investigation deals with a predator–prey model with disease that spreads among the predator species only. The predator species is split out into two groups—the susceptible predator and the infected predator both of which feeds on prey species. The stability and bifurcation analyses are carried out and discussed at length. On the basis of the normal form theory and center manifold reduction, the explicit formulae are derived to determine stability and direction of Hopf bifurcating periodic solution. An extensive quantitative analysis has been performed in order to validate the applicability of our model under consideration. Copyright © 2013 John Wiley & Sons, Ltd.     

Periodic solutions for discrete predator–prey systems with the Beddington–DeAngelis functional response

Sufficient criteria are established for the existence of positive periodic solutions of discrete nonautonomous predator–prey systems with the Beddington–DeAngelis functional response using a continuation theorem.     

Delay induced oscillation in predator–prey system with Beddington–DeAngelis functional response

The Beddington–DeAngelis predator–prey system with distributed delay is studied in this paper. At first, the positive equilibrium and its local stability are investigated. Then, with the mean delay as a bifurcation parameter, the system is found to undergo a Hopf bifurcation. The bifurcating periodic solutions are analyzed by means of the normal form and center manifold theorems. Finally, numerical simulations are also given to illustrate the results.     

Analysis of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with time delay

Two-dimensional delayed continuous time dynamical system modeling a predator–prey food chain, and based on a modified version of Holling type-II scheme is investigated. By constructing a Liapunov function, we obtain a sufficient condition for global stability of the positive equilibrium. We also present some related qualitative results for this system.     

Positive periodic solution of two-species ratio-dependent predator–prey system with time delay in two-patch environment

In this paper, we are concerned with the existence of positive periodic solution to a class of two-species ratio-dependent predator–prey diffusion model with time delay. By using the continuation theorem of coincidence degree theory, we transform this problem into a problem of calculating the topological degree of a continuous mapping, and then some sufficient conditions of the existence of positive periodic solution is established for the system.     

Extinction and permanence in delayed stage-structure predator–prey system with impulsive effects

Based on the classical stage-structured model and Lotka–Volterra predator–prey model, an impulsive delayed differential equation to model the process of periodically releasing natural enemies at fixed times for pest control is proposed and investigated. We show that the conditions for global attractivity of the ‘pest-extinction’ (‘prey-eradication’) periodic solution and permanence of the population of the model depend on time delay. We also show that constant maturation time delay and impulsive releasing for the predator can bring great effects on the dynamics of system by numerical analysis. As a result, the pest maturation time delay is considered to establish a procedure to maintain the pests at an acceptably low level in the long term. In this paper, the main feature is that we introduce time delay and pulse into the predator–prey (natural enemy-pest) model with age structure, exhibit a new modelling method which is applied to investigate impulsive delay differential equations, and give some reasonable suggestions for pest management.     

Limit cycles in a general Kolmogorov model

The problem of limit cycles is interesting and significant both in theory and applications. In mathematical ecology, finding models that display a stable limit cycle—an attracting stable self-sustained oscillation, is a primary work.In this paper, a general Kolmogorov system, which includes the Gause-type model (Math. Biosci. 88 (1988) 67), the general predator-prey model (J. Phys. A: Math. Gen. 21 (1988) L685; Math. Biosci. 96 (1989) 47), and many other models (J. Biomath. 15(3) (2001) 266; J. Biomath. 16(2) (2001) 156; J. Math. 21(22) (2001) 145), is studied. The conditions for the existence and uniqueness of limit cycles in this model are proved. Some known results are easily derived as an illustration of our work.     

Mathematical model for the control of a pest population with impulsive perturbations on diseased pest

In this paper, a stage-structured pest management SI model with impulsive perturbations on infected pest is introduced. Sufficient conditions of the global attractivity of pest-extinction periodic solution and permanence of the system are obtained. We also prove that all solutions of system are uniformly ultimately bounded.     

Survival of three species in a nonautonomous Lotka–Volterra system

In Ahmad and Stamova (2004) [1], the author considers a competitive Lotka–Volterra system of three species with constant interaction coefficients. In this paper, we study a nonautonomous Lotka–Volterra model with one predator and two preys. The explorations involve the persistence, extinction and global asymptotic stability of a positive solution.     

Lyapunov–Razumikhin method for impulsive functional differential equations and applications to the population dynamics

The present paper deals with the investigation of the stability of the zero solution of impulsive functional differential equations. By means of piecewise continuous functions coupled with the Razumikhin technique sufficient conditions for stability, uniform stability and asymptotic stability of the zero solution of such equations are found.     

An impulsive controlled eco-epidemic model with disease in the prey

In this paper, we investigate the dynamic behavior of an eco-epidemic model with impulsive control strategy. By using Floquet theorem of impulsive differential equation, we show there is a globally stable prey eradication periodic solution when the impulsive period is less than some critical value. We study the permanence of the system. Numerical simulations show that the complex dynamics of the system depends on the values of impulsive period and impulsive perturbation, for example double period, triple period solutions.     

Uniform persistence of asymptotically periodic multispecies competition predator–prey systems with Holling III type functional response

In this paper, we studied the persistence of the asymptotically periodic multispecies competition predator–prey system with Holling III type functional response. Further, by use of the Standard Comparison Theorem, we improved the results of paper [C. Chen, F. Chen, Conditions for global attractivity of multispecies ecological competition-predator system with Holling III type functional response, Journal of Biomathematics 19(2) (2004) 136–140].     

Periodicity and stability in a single-species model governed by impulsive differential equation

A periodic single-species model with periodic impulsive perturbations was investigated. By using Brouwer’s fixed point theorem and the Lyapunov function, sufficient conditions for the existence and global asymptotic stability of positive periodic solutions of the system were derived. Numerical simulations were presented to verify the feasibilities of our main results.