Evolution of infinite populations of immigrants: Micro- and mesoscopic description

2021

J. Evol. Equ.

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An individual-based model of stochastic branching is proposed and studied, in which point particles drift in $$\bar{\mathbb {R}}_{+}:=[0,+\infty )$$ R ¯ + : = [ 0 , + ∞ ) toward the origin (edge) with unit speed, where each of them splits into two particles that instantly appear in $$\bar{\mathbb {R}}_{+}$$ R ¯ + at random positions. During their drift, the particles are subject to a random disappearance (death). The model is intended to capture the main features of the proliferation of tumor cells, in which trait $$x\in \bar{\mathbb {R}}_{+}$$ x ∈ R ¯ + of a given cell is time to its division and the death is caused by therapeutic factors. The main result of the paper is proving the existence of an honest evolution of this kind and finding a condition that involves the death rate and cell cycle distribution parameters, under which the mean size of the population remains bounded in time.

“…where 7). As all such k u are continuous, one may pick t ε so that I 3 (t) < ε/2 for t ≤ t ε which by (2.19) yields I 2 (t) < ε for such t. This completes the proof.…”

Section: )mentioning

confidence: 64%

Section: Further Developmentmentioning

confidence: 99%

Section: Further Developmentmentioning

confidence: 99%

See 1 more Smart Citation

Stochastic branching at the edge: individual-based modeling of tumor cell proliferation

2021

J. Evol. Equ.

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“…Without mathematical justification-i.e., without an explicit passage to the mesoscopic scaling limit-kinetic equations are derived by the so called moment closure (decoupling) procedure [31]. Basing on the results of [17,26], we can state here that the kinetic equation corresponding to our model is obtained from the first equation (3.15) of the chain encrypted in (3.14) by 'decoupling' k…”

Section: The Kinetic Equation and Beyondmentioning

confidence: 97%

“…The Poisson measures from P exp are then characterized by densities q ∈ L ∞ (X ), see (2.14). At the mesoscopic level obtained by a scaling procedure, see [17,26], the corpuscular structure of the system is lost, i.e., it appears now as a continuous medium characterized solely by its density, the evolution of which is now the evolution of the whole system. The inter-particle interactions give rise to a state-dependent modulation of the environment, which corresponds to the mean-field approach widely used in statistical physics [38].…”

Section: The Kinetic Equation and Beyondmentioning

confidence: 99%

An interplay between attraction and repulsion in infinite populations

2021

Anal.Math.Phys.

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We propose and study a model describing an infinite population of point entities arriving in and departing from $$X=\mathbb {R}^d$$ X = R d , $$d\ge 1$$ d ≥ 1 . The already existing entities force each other to leave the population (repulsion) and attract the newcomers. The evolution of the population states is obtained by solving the corresponding Fokker-Planck equation. Without interactions, the evolution preserves states in which the probability $$p(n,\Lambda )$$ p ( n , Λ ) of finding n points in a compact vessel $$\Lambda \subset X$$ Λ ⊂ X obeys the Poisson law. As we show, for pure attraction the decay of $$p(n,\Lambda )$$ p ( n , Λ ) with $$n\rightarrow +\infty $$ n → + ∞ may be essentially slower. The main result is the statement that in the presence of repulsion—even of an arbitrary short range—the evolution preserves states in which the decay of $$p(n,\Lambda )$$ p ( n , Λ ) is at most Poissonian. We also derive the corresponding kinetic equation, the numerical solutions of which can provide more detailed information on the interplay between attraction and repulsion. Further possibilities in studying the proposed model are also discussed.

Dynamics of an infinite age‐structured particle system

Jasinska

Math Methods in App Sciences

2021

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The Markov evolution is studied of an infinite age‐structured population of migrants arriving in and departing from a continuous habitat X⊆ℝd—at random and independently of each other. Each population member is characterized by its age a ≥ 0 (time of presence in the population) and location x ∈ X. The population states are probability measures on the space of the corresponding marked configurations. The result of the paper is an explicit construction of the evolution μ0 → μt of such states by solving a standard Fokker–Planck equation for this models. We also found a stationary state μ existing if the emigration rate is separated away from zero. Under additional conditions imposed on μ0, it is shown that μt weakly converges to μ as t → +∞.