Stability Of Positive Periodic Solution Research Articles

Dynamics of impulsive reaction–diffusion predator–prey system with Holling type III functional response

An impulsive reaction–diffusion periodic predator–prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the predator–prey system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given at the end.

THE DYNAMICS OF A CHEMOSTAT MODEL WITH STATE DEPENDENT IMPULSIVE EFFECTS

We consider the dynamic behaviors of a mathematical chemostat model with state dependent impulsive perturbations. By using the Poincaré map and analogue of Poincaré's criterion, some conditions for the existence and stability of positive periodic solution are obtained. Moreover, we show that there is no periodic solution with order larger than or equal to three. Numerical simulation are carried out to illustrate the feasibility of our main results, thus implying that the presence of pulses makes the dynamic behavior more complex.

Global Attractivity and Periodic Solution of a Discrete Multispecies Cooperation and Competition Predator‐Prey System

We propose a discrete multispecies cooperation and competition predator‐prey systems. For general nonautonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained.

Permanence and periodic solutions for an impulsive reaction-diffusion food-chain system with holling type III functional response

An impulsive reaction-diffusion periodic food-chain system with Holling type III functional response is presented and studied in this paper. Sufficient conditions for the ultimate boundedness and permanence of the food-chain system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing appropriate auxiliary function , the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are shown to illustrate our results. A discussion is given in the end of the paper.

Positive periodic solution for a multispecies competition-predator system with Holling III functional response and time delays

In this paper, a delayed multispecies ecological competition-predator system with Holling-III functional response is studied. By using the continuation theorem of coincidence degree theory and constructing suitable Lyapunov functionals, some sufficient conditions are derived for the existence, uniqueness and global asymptotic stability of positive periodic solutions to the system.

Global behaviors of a periodic budworm population model with impulsive perturbations

In this paper, a periodic budworm population model with impulsive perturbations is investigated. The impulse is realized at fixed moments of time. A good understanding of the existence and global asymptotic stability of positive periodic solutions is gained. It turns out that the impulsive perturbations play an important role and have effects on the above dynamics of the system. Numerical simulations are presented to verify the validity of the proposed criteria. Copyright © 2010 John Wiley & Sons, Ltd.

Periodicity and asymptotic stability of a predator–prey system with infinite delays

By using a fixed point theorem and Lyapunov functional, an especially easily checked criterion is obtained for the global existence and global asymptotic stability of positive periodic solutions of a periodic predator–prey system with infinite delays. Moreover, the global existence theorem is also sufficient and necessary. This result improves and generalizes noticeably some known results.

Modeling and analysis of a periodic delayed two-species model of facultative mutualism

In this paper, we propose a periodic delayed two-species system modeling facultative mutualism. By using the method of coincidence degree and Lyapunov functional, easily verifiable sufficient conditions for the existence and globally asymptotic stability of positive periodic solutions of the above system.

Existence and globally asymptotical stability of periodic solutions for two-species non-autonomous diffusion competition n-patch system with time delay and impulses

By using Mawhin continuation theorem and constructing suitable Lyapunov functional, a set of easily verifiable sufficient conditions ensuring the existence and global asymptotical stability of positive periodic solutions for two-species non-autonomous diffusion competition n-patch system with time delay and impulses are established. Finally, applications and an illustrative example are given. The main results generalize and improve the previously known results in Zhang and Wang (J. Math. Anal. Appl. 265:38–48, 2002), Dong et al. (Chaos Solitons Fractals 32:1916–1926, 2007).

The study of a predator–prey system with group defense and impulsive control strategy

A predator–prey system with group defense and impulsive control strategy is established. By using Floquet theorem and small amplitude perturbation skills, a locally asymptotically stable prey-eradication periodic solution is obtained when the impulsive period is less than some critical value. Otherwise, if the impulsive period is larger than the critical value, the system is permanent. By using bifurcation theory, we show the existence and stability of positive periodic solution when the pest-eradication lost its stability. Further, numerical examples show that the system considered has more complicated dynamics, such as: (1) quasi-periodic oscillating, (2) period-doubling bifurcation, (3) period-halving bifurcation, (4) non-unique dynamics (meaning that several attractors coexist), (5) attractor crisis, etc. Finally, the biological implications of the results and the impulsive control strategy are discussed.

Dynamics of a three-species ratio-dependent diffusive model

Abstract In this paper, we study a nonautonomous ratio-dependent diffusive model of three species. By using the fixed point theorem of Brouwer and the theory of differential inequality and constructing a suitable Lyapunov function, sufficient conditions are obtained which guarantee the existence, uniqueness and stability of positive periodic solution. At last, some results are proved with the technology of numerical simulation.

The dynamics of a Lotka–Volterra predator–prey model with state dependent impulsive harvest for predator

According to the economic and biological aspects of renewable resources management, we propose a Lotka–Volterra predator–prey model with state dependent impulsive harvest. By using the Poincaré map, some conditions for the existence and stability of positive periodic solution are obtained. Moreover, we show that there is no periodic solution with order larger than or equal to three under some conditions. Numerical results are carried out to illustrate the feasibility of our main results. The bifurcation diagrams of periodic solutions are obtained by using the numerical simulations, and it is shown that a chaotic solution is generated via a cascade of period-doubling bifurcations, which implies that the presence of pulses makes the dynamic behavior more complex.

Permanence and global attractivity of a discrete semi-ratio dependent predator-prey system with Holling II type functional response

In this paper, we propose a discrete semi-ratio dependent predator-prey system with Holling II type functional response. For general nonautonomous case, sufficient conditions which ensure the permanence and the global stability of the system are obtained; for periodic case, sufficient conditions which ensure the existence of a globally stable positive periodic solution of the system are obtained.

Permanence and Global Stability of Positive Periodic Solutions of a Discrete Competitive System

We consider the dynamic behaviors of a discrete competitive system. A good understanding of the permanence, existence, and global stability of positive periodic solutions is gained. Numerical simulations are also presented to substantiate the analytical results.

A nonautonomous predator–prey system with stage structure and double time delays

In the present paper we study a nonautonomous predator–prey model with stage structure and double time delays due to maturation time for both prey and predator. We assume that the immature and mature individuals of each species are divided by a fixed age, and the mature predator only attacks the immature prey. Based on some comparison arguments we discuss the permanence of the species. By virtue of the continuation theorem of coincidence degree theory, we prove the existence of positive periodic solution. By means of constructing an appropriate Lyapunov functional, we obtain sufficient conditions for the uniqueness and the global stability of positive periodic solution. Two examples are given to illustrate the feasibility of our main results.

Existence and global asymptotic stability of positive periodic solution for a Predator–Prey system with mutual interference

In this paper, a Predator–Prey system with mutual interference is studied. Some sufficient conditions are obtained for the existence and global asymptotic stability of positive periodic solution for the system by utilizing Mawhin’s coincidence degree theorem and constructing a suitable Lyapunov functional. It is interesting that our results depend on the mutual interference constant. Furthermore, an example is illustrated to verify the results by simulating.

A FOOD CHAIN SYSTEM WITH HOLLING IV FUNCTIONAL RESPONSES AND IMPULSIVE EFFECT

In the paper, according to biological and chemical control strategy for pest control, our main purpose is to construct a three trophic level food chain system with Holling IV functional responses and periodic constant impulsive effect concerning integrated pest management (IPM), and investigate the dynamic behaviors of this system. By using the Floquet theory and comparison theorem of impulsive differential equation and analytic method, we prove that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Further, condition for permanence of the system is established. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level.

Dynamic behavior of an eco-epidemic system with impulsive birth

In the paper, we investigate an eco-epidemic system with impulsive birth. The conditions for the stability of infection-free periodic solution are given by applying Floquet theory of linear periodic impulsive equation. And we give the conditions of persistence by constructing a consequence of some abstract monotone iterative schemes. By using the method of coincidence degree, a set of sufficient conditions are derived for the existence of at least one strictly positive periodic solution. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels.

Effect of periodic environmental fluctuations on the Pearl–Verhulst model

We address the effect of periodic environmental fluctuations on the Pearl–Verhulst model in population dynamics and clarify several important issues very actively discussed in the recent papers by Lakshmi [Lakshmi BS. Oscillating population models. Chaos Solitons & Fractals 2003;16:183–6; Lakshmi BS. Population models with time dependent parameters. Chaos Solitons & Fractals 2005;26:719–21], Leach and Andriopoulos [Leach PGL, Andriopoulos K. An oscillatory population model. Chaos Solitons & Fractals 2004;22:1183–8], Swart and Murrell [Swart JH, Murrell HC. An oscillatory model revisited. Chaos Solitons & Fractals 2007;32:1325–7]. Firstly, we review general results regarding existence and properties of periodic solutions and examine existence of a unique positive asymptotically stable periodic solution of a non-autonomous logistic differential equation when r ( t ) > 0 . Proceeding to the case where r ( t ) is allowed to take on negative values, we consider a modified Pearl–Verhulst equation because, as emphasized by Hallam and Clark [Hallam TG, Clark CE. Non-autonomous logistic equations as models of populations in deteriorating environment. J Theor Biol 1981;93:303–11], use of the classic one leads to paradoxical biological conclusions. For a modified logistic equation with ω -periodic coefficients, we establish existence of a unique asymptotically stable positive periodic solution with the same period. Special attention is paid to important cases where time average of the intrinsic growth rate is non-positive. Results of computer simulation demonstrating advantages of a modified equation for modeling periodic environmental fluctuations are presented.

Globally asymptotic stability in two periodic delayed competitive systems

Based on two types of well known periodic single-species population growth models with time delay, we propose two corresponding periodic competitive systems with multiple delays. By using some new analysis techniques, we derive the same criteria for the existence and globally asymptotic stability of positive periodic solutions of the above two competitive systems. The easily verifiable criteria show that the existence and globally asymptotic stability of positive periodic solutions of two delayed nonautonomous competitive systems correspond to the existence and global stability of positive equilibrium of corresponding undelayed autonomous system. As an application, some special cases of the above systems are examined and some earlier results are extended and improved. Biological interpretations on the main results are also given.