The conjecture E(k)≤k is proved to be true if and only if k=1, 2, 3, where E(k) is the cyclicity of condimension k generic elementary polycycles. It is also proved that the cyclicity of any codimension 3 ensembles ex...The conjecture E(k)≤k is proved to be true if and only if k=1, 2, 3, where E(k) is the cyclicity of condimension k generic elementary polycycles. It is also proved that the cyclicity of any codimension 3 ensembles except ensembles with "lips" is ≤6. By the way, the methods usually used in the study of cyclicity of polycycles such as derivation division algorithm, Khovanskii procedure and the method of critical point analysis are introduced.展开更多
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 uni...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.展开更多
In this paper, we present a method for constructing a Dulac function for mathematical models in population biology, in the form of systems of ordinary differential equations in the plane.
A SIQS epidemic model with saturated incidence rate is studied. Two equilibrium points exist for the system, disease-free and endemic equilibrium. The stability of the disease-free equilibrium and endemic equilibrium ...A SIQS epidemic model with saturated incidence rate is studied. Two equilibrium points exist for the system, disease-free and endemic equilibrium. The stability of the disease-free equilibrium and endemic equilibrium exists when the basic reproduction number R0, is less or greater than unity respectively. The global stability of the disease-free and endemic equilibrium is proved using Lyapunov functions and Poincare-Bendixson theorem plus Dulac’s criterion respectively.展开更多
Qualitative properties of critical points, integral lines and limit cycles are studied. Interesting relations between quantities characterizing local properties and those characterizing global properties are obtained.
The cyclicity of four classes of codimension 3 plnnar polycycles and ensembles containing a saddle-node and two hyperbdic saddles is dealt with. The exnct cyclicity or cyclicity bound of them is obtained by finitely-s...The cyclicity of four classes of codimension 3 plnnar polycycles and ensembles containing a saddle-node and two hyperbdic saddles is dealt with. The exnct cyclicity or cyclicity bound of them is obtained by finitely-smooth normal form theory.展开更多
or a class of planar polynomial differential systems of degree p+q, arising from biochemical reactions, there are given conditions of parameters under which the systems produce Hopf bifurcation with limit cycles appea...or a class of planar polynomial differential systems of degree p+q, arising from biochemical reactions, there are given conditions of parameters under which the systems produce Hopf bifurcation with limit cycles appearing, or have no closed orbits.展开更多
In this paper, we give complete qualitative analysis on a class of biological system and prove the nonexistence, existence and uniqueness of a limit cycle for the system.
In this paper, the problem of limit cycles for a class of nonpolynomial planar vector felds is investigated. First, based on Liapunov method theory, we obtain some sufcient conditions for determining the origin as the...In this paper, the problem of limit cycles for a class of nonpolynomial planar vector felds is investigated. First, based on Liapunov method theory, we obtain some sufcient conditions for determining the origin as the critical point of such nonpolynomial planar vector felds to be the focus or center. Then, using Dulac criterion, we establish some sufcient conditions for the nonexistence of limit cycles of this nonpolynomial planar vector felds. And then, according to Hopf bifurcation theory, we analyze some sufcient conditions for bifurcating limit cycles from the origin. Finally, by transforming the nonpolynomial planar vector felds into the generalized Li′enard planar vector felds, we discuss the existence, uniqueness and stability of limit cycles for the former and latter planar vector felds. Some examples are also given to illustrate the efectiveness of our theoretical results.展开更多
United Nations Political Declaration 2011 on HIV and AIDS calls to reduce the sexual transmission and the transmission of HIV among people, who inject drugs by 50% by 2015, through different control strategies and pre...United Nations Political Declaration 2011 on HIV and AIDS calls to reduce the sexual transmission and the transmission of HIV among people, who inject drugs by 50% by 2015, through different control strategies and precautionary measures. In this paper, we propose and study a simple SI type model that considers the effect of various precaution- ary measures to control HIV epidemic. We show, unlike conventional epidemic models, that the basic reproduction number which essentially considered as the disease eradica- tion condition is no longer sufficient to eliminate HIV infection. In particular, we show that even when the basic reproduction number is made less than unity, the disease may persist if the initial outbreak is not low. Eradication of disease is however guaranteed if the ensemble control measure exceeds some upper critical value. It is also shown that an epidemic model with mass action incidence may exhibit backward bifurcation and bistability if density-dependent demography is considered. Our theoretical study thus indicates that extra attention should be given in controlling HIV epidemic to achieve the desired result.展开更多
In this paper, I consider two species feeding on limiting substrate in a chemostat taking into account some possible effects of each species on the other one. System of differential equations is proposed as model of t...In this paper, I consider two species feeding on limiting substrate in a chemostat taking into account some possible effects of each species on the other one. System of differential equations is proposed as model of these effects with general inter-specific density-dependent growth rates. Three cases were considered. The first one for a mutual inhibitory relationship where it is proved that at most one species can survive which confirms the competitive exclusion principle. Initial concentrations of species have great importance in determination of which species is the winner. The second one for a food web relationship where it is proved that under general assumptions on the dilution rate, both species persist for any initial conditions. Finally, a third case dealing with an obligate mutualistic relationship was discussed. It is proved that initial condition has a great importance in determination of persistence or extinction of both species.展开更多
文摘The conjecture E(k)≤k is proved to be true if and only if k=1, 2, 3, where E(k) is the cyclicity of condimension k generic elementary polycycles. It is also proved that the cyclicity of any codimension 3 ensembles except ensembles with "lips" is ≤6. By the way, the methods usually used in the study of cyclicity of polycycles such as derivation division algorithm, Khovanskii procedure and the method of critical point analysis are introduced.
基金the National Natural Science Foundation of China(No.10171044)Jiangsu Province(No.BK2001024)the Foundation for University Key Teachers of the Ministry of Education
文摘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.
文摘In this paper, we present a method for constructing a Dulac function for mathematical models in population biology, in the form of systems of ordinary differential equations in the plane.
文摘A SIQS epidemic model with saturated incidence rate is studied. Two equilibrium points exist for the system, disease-free and endemic equilibrium. The stability of the disease-free equilibrium and endemic equilibrium exists when the basic reproduction number R0, is less or greater than unity respectively. The global stability of the disease-free and endemic equilibrium is proved using Lyapunov functions and Poincare-Bendixson theorem plus Dulac’s criterion respectively.
文摘Qualitative properties of critical points, integral lines and limit cycles are studied. Interesting relations between quantities characterizing local properties and those characterizing global properties are obtained.
文摘The cyclicity of four classes of codimension 3 plnnar polycycles and ensembles containing a saddle-node and two hyperbdic saddles is dealt with. The exnct cyclicity or cyclicity bound of them is obtained by finitely-smooth normal form theory.
文摘or a class of planar polynomial differential systems of degree p+q, arising from biochemical reactions, there are given conditions of parameters under which the systems produce Hopf bifurcation with limit cycles appearing, or have no closed orbits.
文摘In this paper, we give complete qualitative analysis on a class of biological system and prove the nonexistence, existence and uniqueness of a limit cycle for the system.
基金Supported by the Natural Science Foundation of Anhui Education Committee(KJ2007A003)the"211 Project"for Academic Innovative Teams of Anhui University(KJTD002B)+3 种基金the Doctoral Scientifc Research Project for Anhui Medical University(XJ201022)the Key Project for Hefei Normal University(2010kj04zd)the Provincial Excellent Young Talents Foundation for Colleges and Universities of Anhui Province(2011SQRL126)the Academic Innovative Scientifc Research Project of Postgraduates for Anhui University(yfc100020,yfc100028)
文摘In this paper, the problem of limit cycles for a class of nonpolynomial planar vector felds is investigated. First, based on Liapunov method theory, we obtain some sufcient conditions for determining the origin as the critical point of such nonpolynomial planar vector felds to be the focus or center. Then, using Dulac criterion, we establish some sufcient conditions for the nonexistence of limit cycles of this nonpolynomial planar vector felds. And then, according to Hopf bifurcation theory, we analyze some sufcient conditions for bifurcating limit cycles from the origin. Finally, by transforming the nonpolynomial planar vector felds into the generalized Li′enard planar vector felds, we discuss the existence, uniqueness and stability of limit cycles for the former and latter planar vector felds. Some examples are also given to illustrate the efectiveness of our theoretical results.
文摘United Nations Political Declaration 2011 on HIV and AIDS calls to reduce the sexual transmission and the transmission of HIV among people, who inject drugs by 50% by 2015, through different control strategies and precautionary measures. In this paper, we propose and study a simple SI type model that considers the effect of various precaution- ary measures to control HIV epidemic. We show, unlike conventional epidemic models, that the basic reproduction number which essentially considered as the disease eradica- tion condition is no longer sufficient to eliminate HIV infection. In particular, we show that even when the basic reproduction number is made less than unity, the disease may persist if the initial outbreak is not low. Eradication of disease is however guaranteed if the ensemble control measure exceeds some upper critical value. It is also shown that an epidemic model with mass action incidence may exhibit backward bifurcation and bistability if density-dependent demography is considered. Our theoretical study thus indicates that extra attention should be given in controlling HIV epidemic to achieve the desired result.
文摘In this paper, I consider two species feeding on limiting substrate in a chemostat taking into account some possible effects of each species on the other one. System of differential equations is proposed as model of these effects with general inter-specific density-dependent growth rates. Three cases were considered. The first one for a mutual inhibitory relationship where it is proved that at most one species can survive which confirms the competitive exclusion principle. Initial concentrations of species have great importance in determination of which species is the winner. The second one for a food web relationship where it is proved that under general assumptions on the dilution rate, both species persist for any initial conditions. Finally, a third case dealing with an obligate mutualistic relationship was discussed. It is proved that initial condition has a great importance in determination of persistence or extinction of both species.
