Multiparametric Bifurcation Analysis of a Basic Two-Stage Population Model

Baer, Steven; Kooi, Bob W.; Kuznetsov, Yu. А.; Thieme, Horst R.

doi:10.1137/050627757

Cited by 52 publications

(42 citation statements)

References 27 publications

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“…qualitative behavior not involving structural changes in the sliding segments [21], or determined by sliding dynamics of Filippov system (7), i.e. qualitative behavior involving some structural changes in the sliding segments.…”

Section: Preliminaries and Lemmasmentioning

confidence: 99%

Global qualitative analysis of a non-smooth Gause predator–prey model with a refuge

Tang

Liang

2013

Nonlinear Analysis: Theory, Methods & Applications

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Section: Preliminaries and Lemmasmentioning

confidence: 99%

Global qualitative analysis of a non-smooth Gause predator–prey model with a refuge

Tang

Liang

2013

Nonlinear Analysis: Theory, Methods & Applications

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“…However, recent work (Alexander and Moghadas [1,2], Liu, Hethcote, and Levin [17], Moghadas and Alexander [22], Ruan and Wang [25], Wang [26]) indicates that some epidemic models can have two limit cycles. One may expect that the appearance of two limit cycles is due to the fact that degenerate Hopf and degenerate Bogdanov-Takens bifurcations [4] may occur in such epidemic models as well. However, to the best of our knowledge, so far there is no such study on the degenerate Hopf bifurcation and degenerate Bogdanov-Takens bifurcation on epidemic models.…”

Section: Degenerate Bogdanov-takens Bifurcation By Lemma 23 Whenmentioning

confidence: 99%

Coexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate

Tang¹,

Huang²,

Ruan³

et al. 2008

SIAM J. Appl. Math.

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Abstract. 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. In most epidemic models (see Anderson and May [3]), the incidence rate (the number of new cases per unit time) takes the mass-action form with bilinear interactions, namely, κS(t)I(t), where S(t) and I(t) are the numbers of susceptible and infectious individuals at time t, respectively, and the constant κ is the probability of transmission per contact. Epidemic models with such bilinear incidence rates usually have at most one endemic equilibrium and do not exhibit periodicity; the disease will be eradicated if the basic reproduction number is less than one and will persist otherwise (Anderson and May [3], Hethcote [11]). There are many reasons for using nonlinear incidence rates, and various forms of nonlinear incidence rates have

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“…We systematically explored the four possible unfoldings of this planar singularity: namely the focus, elliptic, saddle and cusp case [22–24]. For each of them we checked the time forward and time reversed behavior and located paths for slow-wave and hysteresis-loop bursters.…”

Section: Discussionmentioning

confidence: 99%

Fast–Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultra-slow Transitions of Classes

et al. 2017

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Bursting is a phenomenon found in a variety of physical and biological systems. For example, in neuroscience, bursting is believed to play a key role in the way information is transferred in the nervous system. In this work, we propose a model that, appropriately tuned, can display several types of bursting behaviors. The model contains two subsystems acting at different time scales. For the fast subsystem we use the planar unfolding of a high codimension singularity. In its bifurcation diagram, we locate paths that underlie the right sequence of bifurcations necessary for bursting. The slow subsystem steers the fast one back and forth along these paths leading to bursting behavior. The model is able to produce almost all the classes of bursting predicted for systems with a planar fast subsystem. Transitions between classes can be obtained through an ultra-slow modulation of the model’s parameters. A detailed exploration of the parameter space allows predicting possible transitions. This provides a single framework to understand the coexistence of diverse bursting patterns in physical and biological systems or in models.

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Multiparametric Bifurcation Analysis of a Basic Two-Stage Population Model

Cited by 52 publications

References 27 publications

Global qualitative analysis of a non-smooth Gause predator–prey model with a refuge

Global qualitative analysis of a non-smooth Gause predator–prey model with a refuge

Coexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate

Fast–Slow Bursters in the Unfolding of a High Codimension Singularity and the Ultra-slow Transitions of Classes

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