Hopf bifurcation for a spatially and age structured population dynamics model

Liu, Zhihua; Tang, Hui; Magal, Pierre

doi:10.3934/dcdsb.2015.20.1735

Cited by 12 publications

(8 citation statements)

References 28 publications

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“…Recently, as the significance of age structure in populations has become increasingly prevalent, there has been explosively growing literature dealing with all kinds of aspects of interacting populations with age structure [6,[12][13][14][15][16][17][18][19]. In particular, [14][15][16][17][18][19] considered the age-structured model as a non-densely defined Cauchy problem and discussed the existence of Hopf bifurcation. [3] considered the proliferation of uninfected T cells by a logistic function and formulated the following model    dT dt with the boundary and initial conditions…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Analysis of an Age Structured HIV Infection Model with Both Virus-to-Cell and Cell-to-Cell Transmissions

Zhang

Liu

2018

Int. J. Bifurcation Chaos

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We make a mathematical analysis of an age structured HIV infection model with both virus-to-cell and cell-to-cell transmissions to understand the dynamical behavior of HIV infection in vivo. In the model, we consider the proliferation of uninfected CD4 + T cells by a logistic function and the infected CD4 + T cells are assumed to have an infection-age structure. Our main results concern the Hopf bifurcation of the model by using the theory of integrated semigroup and the Hopf bifurcation theory for semilinear equations with nondense domain. Bifurcation analysis indicates that there exist some parameter values such that this HIV infection model has a non-trivial periodic solution which bifurcates from the positive equilibrium. The numerical simulations are also carried out. .cn (Z. Liu).where T , I and V denote the population of uninfected CD4 + T cells, productively infected CD4 + T cells and virus, respectively. They treated the logistic proliferation term rT 1 − T +I K in which r is the maximum proliferation rate and K is the uninfected CD4 + T cells population density at which proliferation shuts off. Since the proportion of productively infected CD4 + T cells I is very small and thus it is reasonable to ignore this correction. They eventually represented the proliferation of uninfected CD4 + T cells by a logistic function rT 1 − T K . However, the authors only consider the classical virus-to-cell infection and neglect the direct cell-to-cell transmission.[12] incorporated the two modes (virus-to-cell infection and cell-to-cell transmission) of transmission into a classic model and considered the following model system+∞ 0 q(a)i(t, a)da, ∂i(t,a) ∂t + ∂i(t,a) ∂a = −σ(a)i(t, a), dV (t) dt = ∞ 0 p(a)i(t, a)da − cV (t),

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Section: Introductionmentioning

confidence: 99%

Bifurcation Analysis of an Age Structured HIV Infection Model with Both Virus-to-Cell and Cell-to-Cell Transmissions

Zhang

Liu

2018

Int. J. Bifurcation Chaos

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“…On the other hand, in Zhang and Wang's paper ( [14]) the time delay is induced by the maturation process of the prey. In [9], Liu et al describe the growth of trees, producing the first example of Hopf bifurcation in a spatially and age/stage structured population dynamics model. Spatial spread itself is a long standing focus in biological models which has garnered much attention in recent years.…”

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confidence: 99%

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Cantrell¹,

Lenhart²,

Lou³

et al. 2015

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“…[14,15,16,24]. Recently, Liu et al [13] have studied a spatially and size structured-population model as following…”

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confidence: 99%

“…is the number of individuals with a size s ∈ [s 1 , s 2 ] and located in the region [x 1 , x 2 ] at time t. In [13], the authors considered the structured model (3) as the growth of species of trees. In addition, the Laplacian operator ∆ x : W 2,1 (0, 1) −→ L 1 (0, 1) is the diffusion operator with Neumann boundary conditions.…”

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A spatially and size-structured population model with unbounded birth process

Boulouz

2022

DCDS-B

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<p style='text-indent:20px;'>In this paper, we consider a spatially and size structured population model with unbounded birth process. Firstly, the model is transformed into a closed-loop system, and hence the well-posedness is established by using the feedback theory of regular linear systems. Moreover, the solution to the resulting closed-loop system is given by a perturbed semigroup. Secondly, we give a condition on birth and death rates in such a way that the solution decays exponentially. To do this, we show that the semigroup solution is positive and hence we derive a characterization of exponential stability due to the technique tools of positive semigroups. We mention that our results extend a previous work in [D. Yan and X. Fu, Comm. Pure Appl. Anal. 15 (2016), 637–655] to the unbounded situation.</p>

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Hopf bifurcation for a spatially and age structured population dynamics model

Cited by 12 publications

References 28 publications

Bifurcation Analysis of an Age Structured HIV Infection Model with Both Virus-to-Cell and Cell-to-Cell Transmissions

Bifurcation Analysis of an Age Structured HIV Infection Model with Both Virus-to-Cell and Cell-to-Cell Transmissions

Preface

A spatially and size-structured population model with unbounded birth process

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