Population Dynamics with Moderate Tails of the Underlying Random Walk

Symmetric random walks in R^dand Z^dare 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 xand 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...