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

Abstract

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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“…where c N is a positive (> 0) constant (that depends on N and a + ). Because f N = 1 we obtain (17) e…”

“…It would lead to the full scale description of the process in question. One may observe that except some particular cases -see [17,25] and references thereinthat generally could be quite complex. The main question is what we are going to consider as a basic scale.…”

“…Then with the presence of experimental data on microscopic or mesoscopic scale we may choose a proper one among the classes. The mathematical framework of such approach was proposed in series of works -see [6,17,25]. An alternative but much difficult approach could be the modeling on microscopic (or mesoscopic) scale and finding a corresponding macroscopic limit (a kind of "hydrodynamic limit").…”

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