Hopf Bifurcation for a Maturity Structured Population Dynamic Model

Chu, Jianhua; Magal, Pierre; Yuan, Rong

doi:10.1007/s00332-010-9091-9

Cited by 7 publications

(5 citation statements)

References 43 publications

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“…Based on the center manifold theorem proved in Magal and Ruan [33], a Hopf bifurcation theorem has been presented for abstract non-densely defined Cauchy problem in Liu et al [28]. These theorems have been successfully applied to study the existence of Hopf bifurcation for some age-/size-structured models (see [3,[9][10][11]34,35,39,48]). Recently, much attention has been focused on high-codimensional bifurcations, since they may reveal some complex dynamical behaviors.…”

Section: (T)u (T)mentioning

confidence: 99%

Bogdanov–Takens bifurcation in a predator–prey model

Liu

Magal

Xiao

2016

Z. Angew. Math. Phys.

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Abstract. In this paper, we investigate a class of predator-prey model with age structure and discuss whether the model can undergo Bogdanov-Takens bifurcation. The analysis is based on the normal form theory and the center manifold theory for semilinear equations with non-dense domain combined with integrated semigroup theory. Qualitative analysis indicates that there exist some parameter values such that this predator-prey model has an unique positive equilibrium which is Bogdanov-Takens singularity. Moreover, it is shown that under suitable small perturbation, the system undergoes the Bogdanov-Takens bifurcation in a small neighborhood of this positive equilibrium.

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Section: (T)u (T)mentioning

confidence: 99%

Bogdanov–Takens bifurcation in a predator–prey model

Liu

Magal

Xiao

2016

Z. Angew. Math. Phys.

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“…These theorems have been successfully applied to study the existence of Hopf bifurcation for some age/size-structured models, see [2,4,5,6,19,21,26,31]. More results can be found in the context of cell population dynamics in Doumic et al [13], and in the context of structured neuron population in Pakdaman et al [22].…”

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confidence: 92%

Hopf bifurcation for a spatially and age structured population dynamics model

Liu¹,

Tang²,

Magal³

2015

DCDS-B

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This paper is devoted to the study of a spatially and age structured population dynamics model. We study the stability and Hopf bifurcation of the positive equilibrium of the model by using a bifurcation theory in the context of integrated semigroups. This problem is a first example for Hopf bifurcation for a spatially and age/size structured population dynamics model. Bifurcation analysis indicates that Hopf bifurcation occurs at a positive age/size dependent steady state of the model. The results are confirmed by some numerical simulations.2010 Mathematics Subject Classification. 37L10, 37G15, 35B32, 35K55, 92D25.

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“…The system formed with equations (2.3), (2.4) and (2.7) is such that the tools developed in [51] can be used to obtain a system of differential equations with a fixed delay on which the tools developed in [6] can be used. However, subsequent versions of the model might not be suitable for such a transformation, and it might be necessary to perform the stability analysis on the state-dependent delay formulation following works like [3,31] or even on the age-and maturity-structured PDEs formulation using results on Hopf bifurcation such as [13,36]. This is the object of our future work.…”

Section: Application To a System With A Third-degree Transcendental Ementioning

confidence: 99%

A Model for Megakaryopoiesis with State-Dependent Delay

Boullu¹,

Praly²,

Wu³

2019

SIAM J. Appl. Math.

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We analyze the stability of a system of differential equations with a threshold defined delay arising from a model for platelet production. We consider a maturity-structured population of megakaryocyte progenitors and an age-structured population of platelets, where the cytokine thrombopoietin (TPO) increases the maturation rate of progenitors. Using the quasi-steady-state approximation for TPO dynamics and the method of characteristics, partial differential equations are reduced to a system of two differential equations with a state-dependent delay accounting for the variable maturation rate. We start by introducing the model and proving the positivity and boundedness of the solutions. Then we use a change of variables to obtain an equivalent system of two differential equations with a constant delay, from which we prove existence and uniqueness of the solution. As linearization around the unique positive steady state yields a transcendental characteristic equation of third degree, we introduce the main result, a new framework for stability analysis on models with fixed delays. This framework is then used to describe the stability of the megakaryopoiesis with respect to its parameters. Finally, with parameters being obtained and estimated from data, we give an example in which oscillations appear when the death rate of progenitors is increased 10-fold.

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Hopf Bifurcation for a Maturity Structured Population Dynamic Model

Cited by 7 publications

References 43 publications

Bogdanov–Takens bifurcation in a predator–prey model

Bogdanov–Takens bifurcation in a predator–prey model

Hopf bifurcation for a spatially and age structured population dynamics model

A Model for Megakaryopoiesis with State-Dependent Delay

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