Relationships between limit cycles and algebraic invariant curves for quadratic systems

Algebraic limit cycles for quadratic systems started to be studied in 1958. Up to now we know 7 families of quadratic systems having algebraic limit cycles of degree 2, 4, 5 and 6. Here we present some new results on the limit cycles and algebraic limit cycles of quadratic systems. These results provide sometimes necessary conditions and other times sufficient conditions on the cofactor of the invariant algebraic curve for the existence or nonexistence of limit cycles or algebraic limit cycles. In particular, it follows from them that for all known examples of algebraic limit cycles for quadratic systems those cycles are unique limit cycles of the system.     

Exact number of limit cycles for a family of rigid systems

For a given family of planar differential equations it is a very difficult problem to determine an upper bound for the number of its limit cycles. Even when this upper bound is one it is not always an easy problem to distinguish between the case of zero and one limit cycle. This note mainly deals with this second problem for a family of systems with a homogeneous nonlinear part. While the condition that allows us to separate the existence and the nonexistence of limit cycles can be described, it is very intricate.

     

一类具功能反应的食饵—捕食系统模型的定性分析

研究一类具功能反应的食饵—捕食系统:x=xg(x)-yφ(x),y=y(-d+eφ(x))在g(x)=a-bx~m,φ(x)=cx~θ及m=θ=1/n,n>2为正整数情形下,分析了该系统的平衡点性态,并得到了系统在正平衡点外围的极限环的不存在性,存在性与唯一性的相关条件.     

一类Kolmogorov捕食系统的极限环

研究一类Kolmogorov捕食系统dx/dt=x(a_0-a_1x+a_2x~(n-1)-a_3x~n+a_4xy~m),dy/dt=y(b_1x~n-b2),得到了存在唯一极限环和不存在极限环的充要条件,从而推广了前人相关的结果.     

一类具功能反应的食饵——捕食者系统定性分析

研究一类具功能反应的食饵-捕食者系统:x=xg(x)-y(?)(x),y=y(-d+e(?)(x)．在g(x)=α-bxm,(?)(x)=cxθ及m+θ=1,m=1/n,n>2为正整数情形下,分析了该系统的平衡点性态,并得到了系统在正平衡点外围的极限环的不存在性、存在性与唯一性的相关条件．     

On the Predator-Prey System with Holling-（n ＋ 1） Functional Response

The qualitative properties of a predatorprey system with Holling-（n ＋ 1） functional response and a fairly general growth rate are completely investigated. The necessary and sufficient condition to guarantee the uniqueness of limit cycles is given. Our work extends the previous relevant results in the reference.     

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.

     

ON THE LIMIT CYCLES OF PLANAR AUTONOMOUS SYSTEMS

91. IntroductionThe theory of ~ cycles is a very aCtive research field of qualitative theory of ordinary~nilal equationS. There have been many mathemsticians studying the nonetistence,~ence and piqueness of ~ cycles for plane systems, and most atteattiope were paidto some special formS (see [2-4, 610] and the references cited therein)' As we know, for thegeneral system on the Planei = p(z,g), b = Q(z,g), (1.1)where P, Q: RZ - R are continuouSly dmerentiable, there are some well-knoWn res…     

一类Leslie模型的定性分析

对一类Leslie模型进行定性分析，研究了其极限环的存在性，不存在性和唯一性．证明了该系统在细焦点外围至多有一个极限环，以及如果系统有奇数个极限环，则它恰有一个极限环．     

一类二次系统极限环的拓扑结构

利用微分方程定性理论的分析方法,讨论了一类二次系统极限环的存在与结构.     

THE LIMIT CYCLES OF A KIND OF EXPLOITED PREDATOR-PREY SYSTEM

In this paper, a kind of exploited predator-prey system is studied. By using the qualitative theory, we obtain some sufficient conditions for the existence and nonexistence of limit cycles of the system.     

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.     

一类生物捕食系统的定性分析

对一类两种群均有收获率的具较为特殊的功能反应函数的食饵-捕食系统作定性分析,利用常微分方程定性,稳定性及分支理论,此类生物捕食系统平衡点的性态和极限环的存在,不存在及唯一的条件,获得了一类食饵-捕食系统的定性分析结果.     

Limit cycles in a generalized Gause-type predator–prey model

We study the problem of the existence of limit cycles for a generalized Gause-type predator–prey model with functional and numerical responses that satisfy some general assumptions. These assumptions describe the effect of prey density on the consumption and reproduction rates of predator. The model is analyzed for the situation in which the conversion efficiency of prey into new predators increases as prey abundance increases. A necessary and sufficient condition for the existence of limit cycles is given. It is shown that the existence of a limit cycle is equivalent to the instability of the unique positive critical point of the model. The results can be applied to the analysis of many models appearing in the ecological literature for predator–prey systems. Some ecological models are given to illustrate the results.     

ON THE NONEXISTENCE OF THE LIMIT CYCLEFOR QUADRATIC SYSTEM OF TYPE III

1ResultsC()nsid'3rtlloq,,.d,.ti<1sVstemofTypeIll[1]withoutlossofgenerality,wemayassumen>0.TheresultsontilenonexiStenceoflimitcycleforquadraticsystemofTypeillcanbefoundin[2],[3]and[4].Inthispaper,weestablishagainseveralnewcriteriaofdecidingthenonoxistenceoflimitcycle.FOrsimplicity,letN(0)andN(0)denotethenumberoflimitcyclesof(1)aroundthesingularpointsO(0,0)andO(0,1),resl)ccl\ively,weobtail]tilefollowingresults:Theorem1Supposetlsatjn b<0,axldthattilefollowingconditionsaresatisfied,tlsenN(O)…     

ON LIMIT CYCLES FOR A CLASS OF DIFFERENTIAL SYSTEMS WITH POSITIVE DEFINITE POLYNOMIAL

In this paper, the nonexistence, existence and the number of limit cycles for a class of differential systems with positive definite polynomial are considered, and the results obtained generalize and supplement those of [1].     

一类非线性生化系统的定性分析

研究了一类非线性生化系统的定性行为，给出了该系统极限环的不存在性、存在性、惟一性和稳定性的充分条件，补充和完善了戴林勋(1999)的结论。     

Proof of Artés–Llibre–Valls's conjectures for the Higgins–Selkov and the Selkov systems

The aim of this paper is to prove Artés–Llibre–Valls's conjectures on the uniqueness of limit cycles for the Higgins–Selkov system and the Selkov system. In order to apply the limit cycle theory for Liénard systems, we change the Higgins–Selkov and the Selkov systems into Liénard systems first. Then, we present two theorems on the nonexistence of limit cycles of Liénard systems. At last, the conjectures can be proven by these theorems and some techniques applied for Liénard systems.     

一类多分子可逆饱和生化反应模型的定性分析

研究一类多分子可逆饱和生化反应的数学模型：｛ｄｘｄｔ＝ａ－ｂｘ＋ｘ＾ｎｙ－ｃｘ＾ｎ＋１－ｅｘ／（ｘ＋ｋ）,ｄｙｄｔ＝ｂｘ－ｘ＾ｎｙ＋ｃｘ＾ｎ＋１。应用微分方程定性理论,完整地解决了该系统极限环的存在性、惟一性和不存在性等问题。     

非线性系统有理首次积分的不存在性和部分存在性

该文的目的是给出一般非线性系统存在和部分存在有理首次积分的判别准则. 作者给出系统存在有理首次积分的必要条件, 在此基础上进一步给出系统不存在其它有理首次积分(在函数独立的意义下)的判定准则.