Multiple limit cycles and global stability in predator-prey model

1.IntroductionWeconsideranextensivesystemofthefamiliarLotka-Volterrasysteminwhichthepopulationshaveself-inhibit(i.e.,withtheadditionofdampingterm)thatcanbemodeledbytheequationswherexandyrepresentthepreydensityandpredatordensityrespectively.Thespecifi...     

A polycycle and limit cycles in a non-differentiable predator-prey model

For a non-differentiable predator-prey model, we establish conditions for the existence of a heteroclinic orbit which is part of one contractive polycycle and for some values of the parameters, we prove that the heteroclinic orbit is broken and generates a stable limit cycle. In addition, in the parameter space, we prove that there exists a curve such that the unique singularity in the realistic quadrant of the predator-prey model is a weak focus of order two and by Hopf bifurcations we can have at most two small amplitude limit cycles.     

Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response

The goal of this paper is to establish the uniqueness of limit cycles of the predator-prey systems with Beddington-DeAngelis functional response. Through a change of variables, the predator-prey system can be transformed into a better studied Gause-type predator-prey system. As a result, the uniqueness of limit cycles can be solved.     

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.     

Prime number selection of cycles in a predator-prey model

The fact that some species of cicadas appear every 7, 13, or 17 years and that these periods are prime numbers has been regarded as a coincidence. We found a simple evolutionary predator-prey model that yields prime-periodic preys having cycles predominantly around the observed values. An evolutionary game on a spatial array leads to travelling waves reminiscent of those observed in excitable systems. The model marks an encounter of two seemingly unrelated disciplines: biology and number theory. A restriction to the latter, provides an evolutionary generator of arbitrarily large prime numbers. © 2001 John Wiley & Sons, Inc.     

Uniqueness of limit cycles in predator-prey system: the role of weight functions

We consider a Gause type model of interactions between predator and prey populations. Using the ideas of Cheng and Liou we give a sufficient condition for uniqueness of the limit cycle, which is more general than their condition. That is, we include a kind of weight function in the condition. It was motivated by a result due to Hwang, where the prey isocline plays a role of weight function. Moreover, we show that the interval where the condition from Hwang's result is to be fulfilled can be narrowed.     

Bifurcation of limit cycles from a heteroclinic loop with two cusps

In this paper, we study the expansion of the first Melnikov function for general near-Hamiltonian systems near a heteroclinic loop with two cusps of order 1 or 2, obtain the formulas for the first coefficients appearing in the expansion, and establish some bifurcation theorems on the number of limit cycles. We also give some application examples.     

Uniqueness of limit cycles bounded by two invariant parabolas

In this paper we consider a class of cubic polynomial systems with two invariant parabolas and prove in the parameter space the existence of neighborhoods such that in one the system has a unique limit cycle and in the other the system has at most three limit cycles, bounded by the invariant parabolas.     

Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays

In this paper, the Leslie-Gower predator-prey system with two delays is investigated. By choosing the delay as a bifurcation parameter, we show that Hopf bifurcations can occur as the delay crosses some critical values. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using a global Hopf bifurcation theorem for functional differential equations, we show the global existence of periodic solutions.     

Simultaneous bifurcation of limit cycles from two nests of periodic orbits

Let be an holomorphic differential equation having a center at p, and consider the following perturbation . We give an integral expression, similar to an Abelian integral, whose zeroes control the limit cycles that bifurcate from the periodic orbits of the period annulus of p. This expression is given in terms of the linearizing map of at p. The result is applied to control the simultaneous bifurcation of limit cycles from the two period annuli of , after a polynomial perturbation.     

A cubic system with thirteen limit cycles

We construct a planar cubic system and demonstrate that it has at least 13 limit cycles. The construction is essentially based on counting the number of zeros of some Abelian integrals.     

Piecewise-linearized methods for oscillators with limit cycles

A piecewise linearization method based on the linearization of nonlinear ordinary differential equations in small intervals, that provides piecewise analytical solutions in each interval and smooth solutions everywhere, is developed for the study of the limit cycles of smooth and non-smooth, conservative and non-conservative, nonlinear oscillators. It is shown that this method provides nonlinear maps for the displacement and velocity which depend on the previous values through the nonlinearity and its partial derivatives with respect to time, displacement and velocity, and yields non-standard finite difference formulae. It is also shown by means of five examples that the piecewise linearization method presented here is more robust and yields more accurate (in terms of displacement, energy and frequency) solutions than the harmonic balance procedure, the method of slowly varying amplitude and phase, and other non-standard finite difference equations.     

Foliation without limit cycles

The following theorem is proved for a closed manifold M with an oriented foliated structure of codimension 1 without limit cycles, supplemented by a foliation of one-dimensional normals: if every normal in M intersects every leaf, the same is true of the induced foliation on M (a universal covering of M).Translated from Matematicheskie Zametki, Vol. 9, No. 2, pp. 181–191, February, 1971     

The number of limit cycles of cubic Hamiltonian system with perturbation

In this paper, by using qualitative analysis, we investigate the number of limit cycles of perturbed cubic Hamiltonian system with perturbation in the form of (2n+2m) or (2n+2m+1)th degree polynomials . We show that the perturbed systems has at most (n+m) limit cycles, and has at most n limit cycles if m=1. If m=1, n=1 and m=1, n=2, the general conditions for the number of existing limit cycles and the stability of the limit cycles will be established, respectively. Such conditions depend on the coefficients of the perturbed terms. In order to illustrate our results, two numerical examples on the location and stability of the limit cycles are given.     

Bifurcation of limit cycles near equivariant compound cycles

In this paper we study some equivariant systems on the plane. We first give some criteria for the outer or inner stability of compound cycles of these systems. Then we investigate the number of limit cycles which appear near a compound cycle of a Hamiltonian equivariant system under equivariant perturbations. In the last part of the paper we present an application of our general theory to show that a Z3 equivariant system can have 13 limit cycles.     

Stability of limit cycles in a pluripotent stem cell dynamics model

This paper is devoted to the study of the stability of limit cycles of a nonlinear delay differential equation with a distributed delay. The equation arises from a model of population dynamics describing the evolution of a pluripotent stem cells population. We study the local asymptotic stability of the unique nontrivial equilibrium of the delay equation and we show that its stability can be lost through a Hopf bifurcation. We then investigate the stability of the limit cycles yielded by the bifurcation using the normal form theory and the center manifold theorem. We illustrate our results with some numerics.     

A cubic differential system with nine limit cycles

Advances in Computer Algebra software have made calculations possible that were previously intractable. Our particular interest is in the investigation of limit cycles of nonlinear differential equations. We describe some recent developments in handling very large computations involving resultants and present an example of a nonlinear differential system of degree three with nine small amplitude limit cycles surrounding a focus. We know of no examples of cubic systems with more than this number bifurcating from a fine focus, as opposed to a centre. Our example appears to be the first to have been obtained without recourse to some numerical calculation.     

Some minimal bimolecular mass-action systems with limit cycles

We discuss three examples of bimolecular mass-action systems with three species, due to Feinberg, Berner, Heinrich, and Wilhelm. Each system has a unique positive equilibrium which is unstable for certain rate constants and then exhibits stable limit cycles, but no chaotic behaviour. For some rate constants in the Feinberg–Berner system, a stable equilibrium, an unstabe limit cycle, and a stable limit cycle coexist. All three networks are minimal in some sense.By way of homogenising these three examples, we construct bimolecular mass-conserving mass-action systems with four species that admit a stable limit cycle. The homogenised Feinberg–Berner system and the homogenised Wilhelm–Heinrich system admit the coexistence of a stable equilibrium, an unstable limit cycle, and a stable limit cycle.     

Quadratic systems with maximum number of limit cycles

We suggest a method for obtaining quadratic systems with a given distribution of limit cycles. We use it to obtain a set of quadratic systems with the distributions (3, 1), (3, 0), and 3 of limit cycles and with different configurations of singular points. The distributions are justified with the use of a modified Dulac function in a natural domain of existence of limit cycles.