Spatial population dynamics: Beyond the Kirkwood superposition approximation by advancing to the Fisher–Kopeliovich ansatz

Omelyan, I. P.

doi:10.1016/j.physa.2019.123546

Cited by 6 publications

(4 citation statements)

References 60 publications

(204 reference statements)

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“…The numerical simulations can be extended to systems of higher dimensions. The Poisson approximation used for the derivation of the kinetic equation can be improved by advancing to the Kirkwood [22] or Fisher-Kopeliovich [23] ansatz like for birth-death models. Mass and size of particles can also be taken into account.…”

Section: Discussionmentioning

confidence: 99%

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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Section: Discussionmentioning

confidence: 99%

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

2020

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“…A formal definition of spatial moments is provided in section 3 of the electronic supplementary material. It is possible to define higher-order moments similarly, but for the present study, we restrict our attention to the first three spatial moments [43,51].…”

Section: Spatial Moment Dynamicsmentioning

confidence: 99%

Population dynamics with spatial structure and an Allee effect

2020

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Population dynamics including a strong Allee effect describe the situation where long-term population survival or extinction depends on the initial population density. A simple mathematical model of an Allee effect is one where initial densities below the threshold lead to extinction, whereas initial densities above the threshold lead to survival. Mean-field models of population dynamics neglect spatial structure that can arise through short-range interactions, such as competition and dispersal. The influence of non-mean-field effects has not been studied in the presence of an Allee effect. To address this, we develop an individual-based model that incorporates both short-range interactions and an Allee effect. To explore the role of spatial structure we derive a mathematically tractable continuum approximation of the IBM in terms of the dynamics of spatial moments. In the limit of long-range interactions where the mean-field approximation holds, our modelling framework recovers the mean-field Allee threshold. We show that the Allee threshold is sensitive to spatial structure neglected by mean-field models. For example, there are cases where the mean-field model predicts extinction but the population actually survives. Through simulations we show that our new spatial moment dynamics model accurately captures the modified Allee threshold in the presence of spatial structure.

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“…Various moment closure schemes have been proposed [21]. We use the well-known Kirkwood [22] superposition approximation, which is the most analytically tractable, as well as having comparable accuracy with other closure schemes for the third moment in a variety of scenarios [23],…”

Section: Moment Closure Approximationmentioning

confidence: 99%

Asymptotic expansion approximation for spatial structure arising from directionally biased movement

Plank¹

2020

Physica A: Statistical Mechanics and its Applications

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Spatial structure can arise in spatial point process models via a range of mechanisms, including neighbour-dependent directionally biased movement. This spatial structure is neglected by mean-field models, but can have important effects on population dynamics. Spatial moment dynamics are one way to obtain a deterministic approximation of a dynamic spatial point process that retains some information about spatial structure. However, the applicability of this approach is limited by the computational cost of numerically solving spatial moment dynamic equations at a sufficient resolution. We present an asymptotic expansion for the equilibrium solution to the spatial moment dynamics equations in the presence of neighbour-dependent directional bias. We show that the asymptotic expansion provides a highly efficient scheme for obtaining approximate equilibrium solutions to the spatial moment dynamics equations when bias is weak. This scheme will be particularly useful for performing parameter inference on spatial moment models.

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Spatial population dynamics: Beyond the Kirkwood superposition approximation by advancing to the Fisher–Kopeliovich ansatz

Cited by 6 publications

References 60 publications

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

Population dynamics with spatial structure and an Allee effect

Asymptotic expansion approximation for spatial structure arising from directionally biased movement

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