Equilibria of a cyclin structured cell population model

Borges, Ricardo; Calsina, Àngel; Cuadrado, Śılvia

doi:10.3934/dcdsb.2009.11.613

Cited by 11 publications

(15 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…If we find P * ∈ R N for which the operator A P * has zero eigenvalue, then letting φ * ∈ D(A P * ) be a corresponding eigenvector, a nonzero stationary solution φ is constructed by putting its ith component as φ i = P i * φ i * /(P φ * ) i . Such an approach was used recently for similar structured single-species population models by Borges et al [3] and Farkas et al [6].…”

Section: Stationary Solutionsmentioning

confidence: 99%

“…Stability properties of stationary solutions have been investigated by semigroup theory by Farkas and Hagen [5]. Recently, Borges et al [3] and Farkas et al [6] have studied the existence problem of nonzero stationary solutions for certain structured population models by using zero eigenvalue problems. Also, Walker [10] has studied nonzero stationary solutions for age-and spatially structured population models by using a fixed-point problem.…”

Section: ⎤ ⎦ U I (X T)mentioning

confidence: 99%

See 1 more Smart Citation

Stationary solutions to a system of size-structured populations with nonlinear growth rate

Kato

2012

Journal of Biological Dynamics

View full text Add to dashboard Cite

To cite this article: Nobuyuki Kato (2012) Stationary solutions to a system of size-structured populations with nonlinear growth rate, Journal of Biological Dynamics, 6:sup1, 42-53, DOI: 10.1080/17513758.2011 We study stationary solutions to a system of size-structured population models with nonlinear growth rate. Several characterizations of stationary solutions are provided. It is shown that the steady-state problem can be converted into different problems such as two types of eigenvalue problems and a fixed-point problem.In the two-species case, we give an existence result of nonzero stationary solutions by using the fixed-point problem.

show abstract

Section: Stationary Solutionsmentioning

confidence: 99%

Section: ⎤ ⎦ U I (X T)mentioning

confidence: 99%

Stationary solutions to a system of size-structured populations with nonlinear growth rate

Kato

2012

Journal of Biological Dynamics

View full text Add to dashboard Cite

show abstract

“…[15]) that there exists a zero eigenvalue. A similar strategy was employed in [2] to establish the existence and uniqueness of an equilibrium solution of a cyclin structured cell population model.…”

Section: The General Casementioning

confidence: 99%

Semigroup Analysis of Structured Parasite Populations

Farkas

Green

Hinow

2010

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

Abstract. Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a population is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral theory of semigroups to establish conditions for the existence of a positive equilibrium solution of our model. Then, we formulate conditions that guarantee that the linearised system is governed by a positive quasicontraction semigroup on the biologically relevant state space. We also show that the governing linear semigroup is eventually compact, hence growth properties of the semigroup are determined by the spectrum of its generator. In the case of a separable fertility function, we deduce a characteristic equation, and investigate the stability of equilibrium solutions in the general case using positive perturbation arguments.

show abstract

“…A similar operator theoretic framework was previously utilised for simpler problems (in particular with finite dimensional nonlinearities and classical boundary conditions), see e.g. [5,15]. The key idea to treat the steady state problem is to define a linear operator for a fixed environment (nonlinearity) and to study spectral properties of that operator.…”

Section: Introduction Of the Modelmentioning

confidence: 99%

“…Our main goal in this paper is to establish sufficient conditions for the existence of positive steady state solutions of model (1.1)-(1.3). We shall refer the interested reader to [5,8,15,17] where different size-structured models with distributed recruitment processes were investigated. The boundary condition (1.2) is the so called generalized Wentzell-Robin or dynamic boundary condition.…”

Section: Introduction Of the Modelmentioning

confidence: 99%

Steady states in a structured epidemic model with Wentzell boundary condition

Calsina

Farkas

2012

J. Evol. Equ.

Self Cite

View full text Add to dashboard Cite

Abstract. We introduce a nonlinear structured population model with diffusion in the state space. Individuals are structured with respect to a continuous variable which represents a pathogen load. The class of uninfected individuals constitutes a special compartment that carries mass, hence the model is equipped with generalized Wentzell (or dynamic) boundary conditions. Our model is intended to describe the spread of infection of a vertically transmitted disease, for example Wolbachia in a mosquito population. Therefore the (infinite dimensional) nonlinearity arises in the recruitment term. First we establish global existence of solutions and the Principle of Linearised Stability for our model. Then, in our main result, we formulate simple conditions, which guarantee the existence of non-trivial steady states of the model. Our method utilizes an operator theoretic framework combined with a fixed point approach. Finally, in the last section we establish a sufficient condition for the local asymptotic stability of the positive steady state.

show abstract

Equilibria of a cyclin structured cell population model

Cited by 11 publications

References 0 publications

Stationary solutions to a system of size-structured populations with nonlinear growth rate

Stationary solutions to a system of size-structured populations with nonlinear growth rate

Semigroup Analysis of Structured Parasite Populations

Steady states in a structured epidemic model with Wentzell boundary condition

Contact Info

Product

Resources

About