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In this paper we study bifurcations in predator-prey systems with seasonal prey harvesting. First, when the seasonal harvesting reduces to constant yield, it is shown that various kinds of bifurcat...

Recently, Ruan and Wang [J. Differential Equations, 188 (2003), pp. 135–163] studied the global dynamics of a SIRS epidemic model with vital dynamics and a nonlinear saturated incidence rate. Under certain conditions they showed that the model undergoes a Bogdanov–Takens bifurcation; i.e., it exhibits saddle-node, Hopf, and homoclinic bifurcations. They also considered the existence of none, one, or two limit cycles. In this paper, we investigate the coexistence of a limit cycle and a homoclinic loop in this model. One of the difficulties is to determine the multiplicity of the weak focus. We first prove that the maximal multiplicity of the weak focus is 2. Then feasible conditions are given for the uniqueness of limit cycles. The coexistence of a limit cycle and a homoclinic loop is obtained by reducing the model to a universal unfolding for a cusp of codimension 3 and studying degenerate Hopf bifurcations and degenerate Bogdanov–Takens bifurcations of limit cycles and homoclinic loops of order 2.

The existence of a heteroclinic bifurcation for the Michaelis–Menten-type ratio-dependent predator-prey system is rigorously established. Limit cycles related to the heteroclinic bifurcation are also discussed. It is shown that the heteroclinic bifurcation is characterized by the collision of a stable limit cycle with the origin, and the bifurcation triggers a catastrophic shift from the state of large oscillations of predator and prey populations to the state of extinction of both populations. It is also shown that the limit cycles related to the heteroclinic bifurcation originally bifurcate from the Hopf bifurcation.

We consider a predator-prey system with nonmonotonic functional response: $p(x)=\frac{mx}{ax^2+bx+1}$. By allowing b to be negative ($b > -2\sqrt a$), p(x) is concave up for small values of x > 0 a...

The Holling--Tanner model for predator-prey systems has two Hopf bifurcation points for certain parameters. The dependence of the environmental parameters on the underlying bifurcation structure is uncovered using two-timing. Emphasis is on how the bifurcation diagram changes as the Hopf bifurcation points separate. Two degenerate cases require a modification of conventional two-timing. When the two Hopf bifurcation points nearly coalesce, the two stable periodic solution branches are shown to be connected. As a ratio of linear growth rates varies, the Hopf bifurcation points separate further and one limit cycle becomes unstable. This situation can correspond to an outbreak in populations. The modified two-timing analysis analytically captures the unstable and stable limit cycles of the new branch.

A predator-prey system with nonmonotonic functional response is considered. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. The bifurcation a...

Hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system. The problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks. Most of these instabilities are starting from a single Hopf bifurcation. Because of the large delay, a delayed amplitude appears in the slow time bifurcation equation which generates new bifurcations to periodic and quasi-periodic states. We determine analytical expressions for all branches of periodic solutions and show the emergence of secondary bifurcation points from double Hopf bifurcation points. We study numerically different cases of bistability between steady, periodic, and quasi-periodic regimes. Finally, the validity of the Hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the original laser equations and the slow time amplitude equation.

Hamiltonian systems of one degree of freedom depending on a parameter are considered. It is assumed that the equilibrium states have a known and simple bifurcation structure with respect to the parameter. When the parametric dependence is replaced by a slow time dependence the question arises whether solutions of the fully time-dependent system follow branches of equilibrium solutions that are stable on the short timescale, and, when they do, whether they follow them promptly. This is the problem of dynamic bifurcation. We consider paradigms for supercritical and transcritical bifurcation, and find that for supercritical pitchfork bifurcation, most orbits follow the stable branches; for transcritical bifurcation most orbits do not. Asymptotic and numerical estimates of the probability of the rare “hovering” orbits, which fail to follow stable branches in the supercritical case, are given.

Perturbation and asymptotic methods are used to obtain the secondary bifurcation of quasi-periodic solutions from periodic solutions for a model problem. It is a two-cell model consisting of a coupled system of van der Pol-Duffing oscillators. The qualitative features of the secondary bifurcation states depends on the detuning of the oscillators. For special values of the detuning the secondary states are periodic. Then it is possible for period multiplication of solutions to occur at the secondary bifurcation point.

The existence of a stable periodic solution in the positive octant of a two-predator–one-prey competition model is shown to result from the interaction of a steady state bifurcation with a Hopf bifurcation. This result confirms the conjectures, based on numerical simulations of several authors, that coexistence in the form of periodic oscillations in all three population sizes takes place for some parameter values of the model.

Branches of periodic solutions which exhibit the alternatives of the global Hopf bifurcation theorem are calculated for two general systems of differential equations. In the first, a branch of solu...

Secondary bifurcation of general nonlinear operator equations is studied in the neighborhood of multiple eigenvalues of the linearized problem. Necessary conditions for the existence of secondary bifurcation are derived and the location of points of secondary bifurcation are found. The geometrical significance of these conditions is discussed and conditions on primary bifurcation at a multiple eigenvalue are found which guarantee the existence of points of secondary bifurcation.

Relaxation oscillations characterized by two quite different time scales are described by mathematical models of the form $x_t = f(x,y,\lambda ,\varepsilon )$ and $y_t = \varepsilon g(x,y,\lambda ,\varepsilon )$ where $\varepsilon \ll 1$ and $\lambda $ is the control parameter. In this paper, we study the singular Hopf bifurcation from a basic steady state to these relaxation oscillations. Our bifurcation analysis shows how the harmonic oscillations near the bifurcation point progressively change to become pulsed, triangular oscillations.In the second part of the paper, we present a numerical study of the FitzHugh–Nagumo equations for nerve conduction. We first observe that the numerical results are in good agreement with the analytical predictions. We then consider the switching from a stable steady state to a stable periodic solution, or the reverse transition. Our purpose is to explain the annihilation experiments described in the nerve conduction literature.