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.     

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 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.     

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.     

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.     

Asymptotic behaviour and bifurcation in competitive Lotka-Volterra Systems

A conjecture about global attraction in autonomous competitive Lotka-Volterra systems is clarified by investigating a special system with a circular matrix. Under suitable assumptions, this system meets the condition of the conjecture but Hopf bifurcation occurs in a particular instance. This shows that the conjecture is not true in general and the condition of the conjecture is too weak to guarantee global attraction of an equilibrium. Sufficient conditions for global attraction are also obtained for this system.     

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.     

Permanence and periodicity of a delayed ratio-dependent predator–prey model with Holling type functional response and stage structure

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

一个具有时滞和阶段结构的捕食-被捕食模型

研究一个具有时滞和阶段结构的捕食者-食饵模型.通过构造适当的Lyapunov泛函,讨论了该模型的正平衡点和非负边界平衡点的全局吸引性,从而得到了保证该生态系统永久持续生存与绝灭的充分性条件.     

Average conditions for permanence and extinction in nonautonomous Lotka-Volterra system

An n species nonautonomous competitive Lotka-Volterra system is considered in this paper. The average conditions on the coefficients are given to guarantee that all but one of the species are driven to extinction. The generalization for the result is presented, that is, for each r?n the average conditions on the coefficients are provided to guarantee that r of the species in the system are permanent while the remaining n−r are driven to extinction. It is shown that these average conditions are improvement of those of Ahmad and Montes de Oca [Appl. Math. Comput. 90 (1998) 155-166] and Montes de Oca and Zeeman [Proc. Amer. Math. Soc. 124 (1996) 3677-3687] and [J. Math. Anal. Appl. 192 (1995) 360-370].     

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.     

Extinction in nonautonomous competitive Lotka-Volterra systems with infinite delay

The principle of competitive exclusion is extended to n-species nonautonomous Lotka-Volterra competition systems of differential equations with infinite delay. It is shown that if the coefficients are bounded, continuous and satisfy certain inequalities, then any solution with initial function in an appropriate space will have n−1 of its components tend to zero, while the remaining one will stabilize at a certain solution of a logistic differential equation.     

Stability and travelling waves for a time-delayed population system with stage structure

We study a time-delayed population system with stage structure for the interaction between two species, the adult members of which are in competition. For each of the two species the model incorporates a time delay which represents the time from birth to maturity of that species. The global stability results are established for each equilibrium. The criteria for global convergence to each equilibrium are sharp and involve these delays. By using lower and upper travelling wave solutions, we show that the model has travelling wave solutions that connect the origin and the coexistence equilibrium with speeds greater than the spreading speed of each species in the absence of its rival.     

时滞Lotka-Volterra系统的持久性和周期解

本文研究具有时滞周期捕食系统的持久性和周期解,得到了系统永久持续生存的充要条件。在此条件下系统存在正的ω周期解,改进了一些已知结果。     

Vanishing components in autonomous competitive Lotka-Volterra systems

Sufficient conditions are found for an n-dimensional autonomous competitive Lotka-Volterra system to have a component vanishing in an exponential rate as t→∞. These conditions incorporate a typical known result in the literature as a particular case. Moreover, if the n-dimensional system degenerates asymptotically to an m-dimensional subsystem as t→∞, then, under these conditions on the subsystem, the property that the ith component of every solution of the subsystem vanishes in an exponential rate is also preserved for the n-dimensional system.     

Necessary-Sufficient Conditions for Permanence and Extinction in Lotka-Volterra System with Discrete Delays

In this article, we give the necessary and sufficient conditions for permanence and extinction of a two-species Lotka-Volterra system with different discrete delays. We improve some previous results. Moreover, we show the permanence and extinction for this system is equivalent to that of its corresponding nondelayed system.     

具有时滞的非自治Lotka-Volterra竞争系统的持久与灭绝

本文研究具有纯时滞的一般Ｎ－种群非自治Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ竞争系统的持久性和灭绝性．一些新的判别准则被建立．文献［８－１０］中得到的关于无时滞非自治Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ竞争系统的结果被改进和推广．     

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].     

广义的Lotka—Volterra方程的全局渐近稳定性

通过构造李亚普诺夫函数的方法,研究了广义的Lotka—Volterra时滞模型方程,而且给出了正平衡点的全局渐近稳定性的充分必要条件,同时对前人的结果进行了改进和推广．     

Mathematical models of innovation diffusion with stage structure

Mathematical models with stage structures are proposed to describe the process of awareness, evaluation and decision-making. First, a system of ordinary differential equations is presented that incorporates the awareness stage and the decision-making stage. If the adoption rate is bilinear and imitations are dominant, we find a threshold above which innovation diffusion is successful. Further, if the adoption rate has a higher nonlinearity, it is shown that there exist bistable equilibria and a region such that an innovation diffusion is successful inside and is unsuccessful outside. Secondly, a model with a time delay is proposed that includes an evaluation stage of a product. It is proved that the system exhibits stability switches. The bifurcation direction of equilibria is also discussed.