Geometric criteria for the nonexistence of cycles in Gause-type predator-prey systems

The global stability of a multi-species interacting system has apparently important biological implications. In this paper we study the global stability of Gause-type predator-prey models by providing new criteria for the nonexistence of cycles and limit cycles. Our criteria have clear geometrical interpretations and are easier to apply than other methods employed in recent studies. Using these criteria and related techniques we are able to develop new results on the existence and uniqueness of cycles in Gause-type models with various growth and response functions.

     

广义Liénard方程极限环的惟一性问题

本文证明了广义Lienard方程极限环的一个惟一性定理,并用它证明了具有 稀疏效应的捕食-食饵系统在其正奇点外围至多有一个极限环.     

一类捕食者-食饵系统的全局结构

本文中我们证明了关于一般捕食者-食饵系统不存在闭轨线的定理,即文中定理2.应用这一定理和关于捕食者-食饵系统极限环的存在唯一性定理[1],我们完成了在各种参数条件对一个具体的捕食者-食饵系统模型[2]的研究.     

Uniqueness of Limit Cycles in a Predator-Prey Model with Sigmoid Functional Response

In this paper, we prove that a predator-prey model with sigmoid functional response and logistic growth for the prey has a unique stable limit cycle, if the equilibrium point is locally unstable. This extends the results of the literature where it was proved that the equilibrium point is globally asymptotically stable, if it is locally stable. For the proof, we use a combination of three versions of Zhang Zhifen''s uniqueness theorem for limit cycles in Li$\acute{\rm e}$nard systems to cover all possible limit cycle configurations. This technique can be applied to a wide range of differential equations where at most one limit cycle occurs.     

一类具有Watt型功能性反应的捕食系统的极限环与稳定性

研究一类具有Watt型功能性反应的捕食模型.讨论了该系统正平衡点的存在性以及非负平衡点的性态,应用Poincare-Bendixson定理和张芷芬定理,证明了极限环的存在性和唯一性,并采用构造Dulac函数的方法,获得了正平衡点全局渐近稳定性的一个充分条件.     

ON GENERALIZED LIENARD EQUATION AND THE UNIQUENESS OF LIMIT CYCLES IN GAUSS－TYPE PREDATOR－PREY SYSTEM

ＯＮＧＥＮＥＲＡＬＩＺＥＤＬＩＥＮＡＲＤＥＱＵＡＴＩＯＮＡＮＤＴＨＥＵＮＩＱＵＥＮＥＳＳＯＦＬＩＭＩＴＣＹＣＬＥＳＩＮＧＡＵＳＳ－ＴＹＰＥ　ＰＲＥＤＡＴＯＲ－ＰＲＥＹ　ＳＹＳＴＥＭＳｈａｏＭｉｎｇｈｕａ（邵明华）（ＨａｎｇｚｈｏｕＴｅａｃｈｅｒｓ＇Ｃ...     

Limit cycles of a predator-prey model with intratrophic predation

We present some properties of a differential system that can be used to model intratrophic predation in simple predator-prey models. In particular, for the model we determine the maximum number of limit cycles that can exist around the only fine focus in the first quadrant and show that this critical point cannot be a centre.     

具功能性反应的食饵-捕食者两种群模型的定性分析

本文对具Holling Ⅲ型功能性反应的食饵-捕食者两种群微分方程模型进行定性分析,得到了非平凡平衡点全局稳定性的条件,正平衡点周围存在唯一极限环的条件;同时,对这些条件的生态意义加以讨论。作者们相信这些条件在文献中还是新的。     

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.     

QUALITATIVE ANALYSIS OF A PREDATOR-PREY SYSTEM WITH ALLEE EFFECT FOR PREY

In this paper,we study some predator-prey system with Allee effect for prey. In addition,we discuss the properties of equilibrium points and the existence and uniqueness of limit cycle.     

The uniqueness of limit cycles for Liénard system

In monographs [Theory of Limit Cycles, 1984] and [Qualitative Theory of Differential Equations, 1985], eleven propositions by several mathematicians are listed on the uniqueness of limit cycles for equations of type (I), (II), and (III) of the quadratic ordinary differential systems. In this paper, we first point out that all these propositions were not completely proved since the equations under consideration do not satisfy the conditions of the theorems used to guarantee the uniqueness of limit cycles. Then we give a new set of theorems that guarantee the uniqueness of limit cycles for the Liénard systems, which not only can be applied to complete the proof of the propositions mentioned above but generalize many other uniqueness theorems as well. The conditions in these uniqueness theorems, which are independent and were obtained by different methods, can be combined into one improved general theorem that is easy to apply. Thus many of the most frequently used theorems on the uniqueness of limit cycles are corollaries of the results in this paper.     

Existence of a polycycle in non-Lipschitz Gause-type predator-prey models

We study Gause-type predator-prey models when the interaction between predator and prey is not locally-Lipschitz continuous in the absence of one of them. We shall show that in this case there appears a polycycle, which affects the existence of limit cycles for the system. We apply the results to study the existence of limit cycles for a classical Gause system.     

UNIQUENESS AND DISTRIBUTION OF LIMIT CYCLES FOR BOUNDED QUADRATIC SYSTEM

1 IntroductionAs we know, any given quadratic system which may have limit cycle (LC,fOr abbreviation) can be written in the fOllowing fOrm (see [1] 512)where 6, l, m, n, a, 6 are all real parameters.If all trajectories of a quadratic system remain bounded fOr t 2 0, we saythat the system is bounded, and fOr abbreviation denote by BQS in this paper.The research work for BQS begin with Dickson-Perko [3]. And then, in [4],they made use of the conclusions of [51 to give a detailed classifica…     

一类生化系统的数学模型及定性分析

研究了一类非线性生化系统极限环的存在性与唯一稳定性,利用定性分析的方法研究了生化系统轨线的全局结构,给出了极限环存在与稳定的判别条件,改进和推广了已有的结果.     

Analysis and computation of limit cycles for systems with separable nonlinearities

The variational system obtained by linearizing a dynamical system along a limit cycle is always non-invertible. This follows because the limit cycle is only a unique modulo time translation. It is shown that questions such as uniqueness, robustness, and computation of limit cycles can be addressed using a right inverse of the variational system. Small gain arguments are used in the analysis.     

Analysis and computation of limit cycles for systems with separable nonlinearities

The variational system obtained by linearizing a dynamical system along a limit cycle is always non-invertible. This follows because the limit cycle is only a unique modulo time translation. It is shown that questions such as uniqueness, robustness, and computation of limit cycles can be addressed using a right inverse of the variational system. Small gain arguments are used in the analysis.     

一类异宿环分支极限环的唯一性

本文研究平面上一类两点或三点异宿环附近极限环的分支,在一简洁条件下证明了异宿环分支极限环的唯一性,并给出了极限环唯一存在的充要条件.作为对三维余维2分支的应用,解决了所出现的两点异宿环产生唯一极限环的问题.     

THE LIMIT CYCLES AND HOPF BIFURCATION OF A CLASS OF SIMPLIFIED HOLLING TYPE-IV PREDATOR-PREY SYSTEM WITH LINEAR STATE FEEDBACK

In this paper, a class of simplified Type-IV predator-prey system with linear state feedback is investigated. We prove the boundedness of the positive solutions to this system, and analyze the quality of the equilibria and the existence of limit cycles of the system surrounding the positive equilibra. By Hopf bifurcation theory, the result of having two limit cycles to the system is obtained.     

Some Conditions for the Nonexistence of Limit Cycles in a Predator-Prey System

In this paper, we study the problem of the existence of limit cycles for a predator-prey system with a functional response. It is assumed that the functional response is positive, increasing, concave down, and its third derivative has a unique root. A necessary condition for the nonexistence of limit cycles is presented. Some conditions are given under which the necessary condition is also the sufficient condition for the nonexistence of limit cycles.