Jumps and coalescence in the continuum: A numerical study

Kozitsky, Yuri; Omelyan, I. P.; Pilorz, Krzysztof

doi:10.1016/j.amc.2020.125610

Cited by 3 publications

(7 citation statements)

References 41 publications

(102 reference statements)

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“…Despite the mentioned loss of information, such equations can provide much more detailed description of the system's evolution. They are pretty well accessible to numerical methods, especially tailored to study such models [27,35]. We plan to realize this in a forthcoming paper.…”

Section: The Kinetic Equation and Beyondmentioning

confidence: 99%

An interplay between attraction and repulsion in infinite populations

Kozitsky

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.

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Section: The Kinetic Equation and Beyondmentioning

confidence: 99%

An interplay between attraction and repulsion in infinite populations

Kozitsky

2021

Anal.Math.Phys.

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“…Recently, considerable attention is directed to multiscale modeling based upon a consistent description at different scales. The kinetic equation studied in this work is derived by a certain procedure from the corresponding chain of evolution equations obtained at a microscopic level (see [8,18] for more detail).…”

Section: A Kinetic Equation Of a Population Modelmentioning

confidence: 99%

“…Time evolution of the population will be described in terms of the particle density n(x, t). Performing the passage from the microscopic individual-based dynamics to the mesoscopic description by means of a Vlasov-type scaling, one derives the following kinetic equation for the density in the Poisson approximation with the jump-coalescence model [7,8,18]:…”

Section: A Kinetic Equation Of a Population Modelmentioning

confidence: 99%

“…Quite recently [18], first numerical results on the jump-coalescence kinetic equation were reported, where the role of repulsion in the appearance of a spatial heterogeneity was elucidated. There it was shown also how the kinetic equation can rigorously be obtained from the corresponding microscopic evolution equations obtained and studied in [8].…”

Section: Introductionmentioning

confidence: 99%

“…There it was shown also how the kinetic equation can rigorously be obtained from the corresponding microscopic evolution equations obtained and studied in [8]. However, in [18], the numerical consideration was restricted to a few particular examples of the model parameters. Moreover, very little was said about the algorithm used to obtain the numerical solutions.…”

Section: Introductionmentioning

confidence: 99%

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Algorithm for numerical solutions to the kinetic equation of a spatial population dynamics model with coalescence and repulsive jumps

2020

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An algorithm is proposed for finding numerical solutions of a kinetic equation that describes an infinite system of point particles placed in $\mathbb {R}^{d} (d \geq 1)$ ℝ d ( d ≥ 1 ) . The particles perform random jumps with pair-wise repulsion in the course of which they can also merge. The kinetic equation is an essentially nonlinear and nonlocal integro-differential equation, which can hardly be solved analytically. The numerical algorithm which we use to solve it is based on a space-time discretization, boundary conditions, composite Simpson and trapezoidal rules, Runge-Kutta methods, and adjustable system-size schemes. We show that, for special choices of the model parameters, the solutions manifest unusual time behavior. A numerical error analysis of the obtained results is also carried out.

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Algorithm for numerical solutions to the kinetic equation of a spatial population dynamics model with coalescence and repulsive jumps

Omelyan¹,

Kozitsky²,

Pilorz³

2020

Preprint

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An algorithm is proposed for finding numerical solutions of a kinetic equation that describes an infinite system of point articles placed in R d (d ≥ 1). The particles perform random jumps with pair wise repulsion, in the course of which they can also merge. The kinetic equation is an essentially nonlinear and nonlocal integro-differential equation, which can hardly be solved analytically. The derivation of the algorithm is based on the use of space-time discretization, boundary conditions, composite Simpson and trapezoidal rules, Runge-Kutta methods, adjustable system-size schemes, etc. The algorithm is then applied to the one-dimensional version of the equation with various initial conditions. It is shown that for special choices of the model parameters, the solutions may have unexpectable time behaviour. A numerical error analysis of the obtained results is also carried out.

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Jumps and coalescence in the continuum: A numerical study

Cited by 3 publications

References 41 publications

An interplay between attraction and repulsion in infinite populations

An interplay between attraction and repulsion in infinite populations

Algorithm for numerical solutions to the kinetic equation of a spatial population dynamics model with coalescence and repulsive jumps

Algorithm for numerical solutions to the kinetic equation of a spatial population dynamics model with coalescence and repulsive jumps

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