Population dynamics with moderate tails of the underlying random walk

TL;DR

This paper analyzes the asymptotic behavior of symmetric random walks with moderate tail distributions in continuous and discrete spaces, and explores ecological wave propagation in related population models.

Contribution

It provides the global asymptotic behavior of transition probabilities for random walks with finite moments, including the second, and describes ecological wave front propagation.

Findings

Asymptotic behavior of transition probabilities is characterized.

Ecological wave front propagation is described.

Results apply to random walks with finite moments, including the second.

Abstract

Symmetric random walks in $R^d$ and $Z^d$ are considered. It is assumed that the jump distribution density has moderate tails, i.e., several density moments are finite, including the second one. The global (for all $x$ and $t$) asymptotic behavior at infinity of the transition probability (fundamental solution of the corresponding parabolic convolution operator) is found. Front propagation of ecological waves in the corresponding population dynamics models is described.