Equilibrium solutions for microscopic stochastic systems in population dynamics

Lachowicz, Mirosław; Ryabukha, Tatiana V.

doi:10.3934/mbe.2013.10.777

Cited by 6 publications

(3 citation statements)

References 14 publications

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“…The existence of non-homogeneous equilibrium solutions and an analytical proof of the stability of equilibrium solutions are challenging problems, that are, in general, open problems. The study of equilibrium solutions of the conservative system (2) has been addressed in different papers, see, for example, [1,2,6,11,12].…”

Section: 2mentioning

confidence: 99%

“…Paper [12] studies the existence of equilibrium solutions when periodic boundary conditions are considered and interaction rates are expressed in terms of convolution functions. Paper [11] investigates the existence of equilibrium solutions of factorized type and discusses the uniqueness under the assumption of periodic structures.…”

Section: 2mentioning

confidence: 99%

“…In what follows, we are interested in determining the equilibrium solutions of the kinetic system derived in [13], say f i eq , investigating how they are related with the equilibrium solutions of the macroscopic equations, and studying their stability properties. This is a challenging problem, mainly due to the stochastic nature of the kinetic approach in population dynamics, when dealing with interacting cellular systems, in contrast to the physical nature of the kinetic approach in non-equilibrium statistical mechanics, when dealing with colliding particles, see [1,2,6,11,12]. We start by studying, in Section 2, the case of a conservative system in which only conservative interactions are included in the dynamics.…”

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Equilibrium of a kinetic model of oscillating patterns for chronic autoimmune diseases

Miguel Oliveira,

Machado Ramos,

Ribeiro

et al. 2024

KRM

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We study the equilibrium states of a non-linear kinetic system of integro-partial differential equations describing the well known relapsingremitting stages observed in many chronic autoimmune diseases. The kinetic system describes the immune system at the cellular level in terms of conservative, proliferative and destructive interactions between population of cells, with the addition of natural death rates and a constant input source for one of the cell populations. The macroscopic equations describing the global behaviour of the populations have been obtained from the kinetic system by averaging with respect to the activity variable. They constitute an autonomous dynamical system possessing very interesting dynamical properties in terms of equilibrium states, their stability and existence of bifurcations. We study the equilibrium solutions of the kinetic system and relate these states to the equilibrium solutions of the corresponding macroscopic equations. We also analyse the stability of the equilibrium solutions of the kinetic system by establishing an H-theorem for certain variants of the model. First, we deal with the case in which only conservative interactions are involved in the dynamics, and then we study the case in which also non conservative interactions are involved under the assumption that proliferative interactions are counterbalanced by destructive interactions. Finally, we determine numerically equilibrium solutions of the conservative system exhibiting a stable behaviour, and then we obtain both unstable and stable solutions for the non conservative kinetic system.

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Section: 2mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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confidence: 99%

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Equilibrium of a kinetic model of oscillating patterns for chronic autoimmune diseases

Miguel Oliveira,

Machado Ramos,

Ribeiro

et al. 2024

KRM

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A class of microscopic individual models corresponding to the macroscopic logistic growth

Lachowicz

2017

Math Methods in App Sciences

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Communicated by: J. Banasiak MOS Classification: 60J75; 92D25; 35Q92; 35R09; 37N25; 45K05We study a class of microscopic models corresponding to the standard macroscopic logistic equation. The models at microscale refer to a number of interacting individuals and are in terms of linear evolution equations related to Markov jump processes. The asymptotic large time behavior for the microscopic models is obtained. Moreover, it is shown that any, even nonfactorized, initial probability density tends in the evolution to a factorized equilibrium probability density. KEYWORDS asymptotic behavior, individual-based models, logistic equation, microscopic 8446

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