In this paper we consider a kind of predator-prey model named Holling-Tanner model. Firstly, we prove all solutions of this model to be bounded from above. Secondly, we find a positive invariant set of the model, and prove the existence of stable limit cycle in this invariant set by Poincaré-Bendixson theorem for the unstable equilibrium. Thirdly, we get the region of parameters in which the corresponding stable equilibrium are also globally asymptotically stable. Lastly, we give a bifurcation diagram and illustration with two limit cycles for special parameters through numerical simulation. By our knowledge, the invariant set constructed in this paper is better than that in the book written by Murray.

In this paper, we study a predator-prey system, the modified Holling-Tanner model with strong Allee effect. The existence and stability of the non-negative equilibria are discussed first. Several kinds of bifurcation phenomena, which the model may undergo, such as saddle-node bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation, are studied second. Bifurcation diagram for Bogdanov-Takens bifurcation of codimension 2 is given. Then, possible dynamical behaviors of this model are illustrated by numerical simulations. This paper appears to be the first study of the modified Holling-Tanner model that includes the influence of a strong Allee effect.

The Leslie–Gower model, a kind of predator–prey model with weak Allee effect, is studied in this paper. The existence and stability of non-negative equilibria are first discussed. Then, we investigate several bifurcation phenomena undergoing positive equilibria, such as saddle-node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, etc. Some possible dynamical behaviors of this model are illustrated by numerical simulation. The bifurcation diagrams for the cases of codimensions 2 and 3 are given respectively. The coexistence of a periodic cycle and a homoclinic cycle, and two limit cycles enclosing an unstable equilibrium are also proved. This appears to be the first study of the Leslie–Gower model including the influence of weak Allee effect on prey.

Fractional linear maps have played a key role in mathematical biology, population dynamics, and other research areas. In this paper, a special kind of Ricatti map is studied in detail in order to determine the asymptotical behaviors of fixed points and periodic solutions. Making use of composition operation of maps and the methods of dynamical systems and qualitative theory, fixed points or periodic orbits are expressed precisely, average value of periodic solution is estimated concretely, and several different bounds are obtained for periodic solutions of the Beverton?Holt map when both intrinsic growth rate and carrying capacity change periodically. In addition, some sufficient conditions are given about the attenuation of periodic solution of the non?autonomous Beverton?Holt equation. Compared with present works in literature, our results about bounds of periodic solutions are more precise, and our proofs about the attenuation of periodic solution are more concise.

In this paper, the famous logistic map is studied in a new point of view. We study the boundedness and the periodicity of non-autonomous logistic map x(n+1) = r(n)x(n)(1 - x(n)), n=0, 1, ..., where {r(n)} is a positive p-periodic sequence. The sufficient conditions are given to support the existence of asymptotically stable and unstable p-periodic orbits. This appears to be the first study of the map with variable parameter r.

In this paper, we consider the dynamics of a generalized threedimensional Henon map. Necessary and sufficient conditions on the existence and stability of the fixed points of this system are established. By applying the center manifold theorem and bifurcation theory, we show that the system has the fold bifurcation, flip bifurcation, and Neimark-Sacker bifurcation under certain conditions. Numerical simulations are presented to not only show the consistence between examples and our theoretical analysis, but also exhibit complexity and interesting dynamical behaviors, including period-10, -13, -14, -16, -17, -20, and -34 orbits, quasi-periodic orbits, chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors. These results demonstrate relatively rich dynamical behaviors of the three-dimensional Henon map.

In this paper, we consider several kinds of maps frequently appearing in the population dynamics, e.g., logistic mapping: $f(x)=rx(1-x)$ $(x\in \mathbb{R}, $ $r>0$ is a parameter), unimodel Allee maps with Allee effect and Sigmoid mapping, etc. We study the persistence of some properties for these maps such as the numbers of fixed point, their positions, stabilities and so on. We prove that they are closed under composition of maps. That means they form a semigroup. The examples are given to illustrate the results.

The dynamics of a discrete-time predator-prey system is investigated in detail in this paper. It is shown that the system undergoes flip bifurcation and Hopf bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto's chaos is proved when some certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-6, 7, 8, 10, 14, 18, 24, 36, 50 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.

We investigate the dynamics of a discrete-time predator-prey system. Firstly, we give necessary and sufficient conditions of the existence and stability of the fixed points. Secondly, we show that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation by using center manifold theorem and bifurcation theory. Furthermore, we present numerical simulations not only to show the consistence with our theoretical analysis, but also to exhibit the complex but interesting dynamical behaviors, such as the period-6, -11, -16, -18, -20, -21, -24, -27, and -37 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors, which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Finally, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.

In this paper, we investigate the dynamics of a nonlinear economic cycle model. The necessary and sufficient conditions are given to guarantee the existence and stability of the fixed point. It is also shown that the system undergoes a Neimark-Sacker bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto's chaos is proved when certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviour, such as the period-10, -16, -20 orbits, attracting invariant cycles, quasi-periodic orbits, 10-coexisting chaotic attractors, and boundary crisis. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.

在传统的食饵-捕食者模型的基础上,生物学家根据化学元素对系统的影响,建立了含有元素循环的食饵-捕食者模型(LKE模型).考虑具有Holling Ⅱ型功能反应的LKE模型,分析了系统内部平衡点存在的条件及其稳定性,对可能出现的几种分支现象,给出了对应的参数值,最后通过数值模拟验证所得理论结果.

An explicit upper bound B(n) is derived for the number of zeros of Abelian integrals I(h)=∮Γhg(x,y)dx-f(x,y)dy on the open interval (0,1/4), where Γh is an oval lying on the algebraic curve H(x,y)=x2+y2-x4+ax2y2+y4 with a>-2,f(x,y) and g(x,y) are polynomials in x and y of degrees not exceeding n. Assume I(h) not vanish identically, then B(n)≤3n-14+12n-34+23.

In this paper, we study the number of limit cycles of some polynomial Lienard systems. Using the methods of the Melnikov functions and Hopf, homoclinic and heteroclinic bifurcation theory, we prove that H(2, 5) ≥ 4, H(6, 5) ≥ 7, H(10, 5) ≥ 11.

In this paper, we study the existence of transversal homoclinic orbits in a planar circular restricted four-body problem, based on the perturbation theory of integrable Hamiltonian systems. We start from a planar circular restricted four-body model and regard it as a perturbation of the two-body model. Then, in order to conveniently study unbounded orbits, we transform the infinite points to finite points by a non-canonical transformation, arriving at a non-Hamiltonian system with degenerate fixed points. According to the extended Melnikov method, we finally prove that there exist transversal homoclinic orbits in this four-body model.

It was proved that the sharp upper bound of the number of zeros of Abelian integrals and the number of limit cycles bifurcated from Poincare bifurcation were B(n).An explicit B(n) is derived for the number of zeros of Abelian integrals I(h) = Γ(h) f (x, y) dy-g(x, y) dx on the open interval (0, ∞), where Γ(h) is an oval lying on the algebraic curve H(x, y) = x 2a /A + y 2b /B = h, f (x, y),g(x, y) are polynomials of x and y, and max{deg f (x, y), deg g(x, y)} = n.Assume I(h) not vanish identically, c = gcd (a, b)2 ] -1 2 (λ -1)(λ -2) for n ≥ 2λ + 1.

In this paper, we study the number of limit cycles of some polynomial Liénard systems. By the Melnikov functions and the methods of Hopf, homoclinic and heteroclinic bifurcation theory, we prove that H(2,5)⩾3, H(4,5)⩾5, H(6,5)⩾10, H(8,5)⩾10.

This paper is concerned with the bifurcation of limit cycles from a quadratic reversible system under polynomial perturbations. It is proved that the cyclicity of the period annulus is two, and also a linear estimate of the number of zeros of the Abelian integral for the system under polynomial perturbations of arbitrary degree n is given.

In this paper,we obtain the finite generators of Abelian integral I(h) =∮_(Γh)(M(x,y)g(x,y))dx -(M(x,y)f(x,y))dy,whereΓ_h is a family of closed ovals defined by H(x,y) = x~k(1/2y~2 + Ax~2 + Bx + C) = h,h∈E,k is a positive integer,E is the open interval on whichΓ_h is defined,f(x,y) and g(x,y) are real polynomials in x and y of degrees,not exceeding n.An upper bound of the number of zeros of Abelian integral I(h),for the above system with one centre,is given by its algebraic structure for a special case.

An explicit upper bound Z(2, n) <= n + m - 1 is derived for the number of zeros of Abelian integrals M-1(h) = closed integral(gamma(h)) P(x, y) dy - Q(x, y) dx on the open interval (0, 1/6), where gamma(h) is an oval lying on the algebraic curve H-lambda = (1/2)x(2)+(1/2)y(2)-(1/3)x(3)-lambda y(3) = h, P(x, y), Q(x, y) are polynomials of x and y, and max{deg P(x, y), degQ(x, y)} = n. The proof exploits the expansion of the first order Melnikov function M-1(h, lambda) near lambda = 0 and assume (partial derivative(m)/partial derivative lambda(m)) M-1(h, lambda)vertical bar(lambda=0) not vanish identically.

Most of predator-prey systems are continuous;but the growth of organisms is not necessarily continuous and the effect of the environmental change is not instantaneous,either.Having considered the interaction between the predator and the prey in continuous system,Beverton-Holt equation and the influence of the stoichiometric factor to the system,we construct predator-prey model.The analysis shows that they retain most of the fundamental features of the continuous model and also reveal the paradox of energy;Also analytical and numerical analyses of the model provide more insight into system dynamics including a set of global attraction,meanwhile we notes the fine distinctions between the two states of organisms—facing to die out and having died out,which indicate that it contains more biologic phenomena than continuous model and discrete one deduced from continuous one by using the continuous solution discretely.