# Spatial population dynamics: beyond the Kirkwood superposition approximation by advancing to the Fisher-Kopeliovich ansatz

## TL;DR

This paper applies the Fisher-Kopeliovich closure to spatial population dynamics, improving the accuracy of population density and distribution predictions over traditional methods by comparing with simulations.

## Contribution

It introduces the Fisher-Kopeliovich ansatz to the hierarchy of master equations, advancing beyond the Kirkwood superposition approximation for spatial population models.

## Key findings

- Fisher-Kopeliovich closure improves model accuracy.
- Enhanced predictions of population distributions.
- Better agreement with individual-based simulations.

## Abstract

The superior Fisher-Kopeliovich closure is applied to the hierarchy of master equations for spatial moments of population dynamics for the first time. As a consequence, the population density, pair and triplet distribution functions are calculated within this closure for a birth-death model with nonlocal dispersal and competition in continuous space. The new results are compared with those obtained by ``exact'' individual-based simulations as well as by the inferior mean-field and Kirkwood superposition approximations. It is shown that the Fisher-Kopeliovich approach significantly improves the quality of the description in a wide range of varying parameters of the model.