Negative spectrum of non-local operators with periodic potential

TL;DR

This paper analyzes how negative periodic perturbations of non-local convolution operators, modeling population dynamics, shift the spectrum to induce population extinction across dimensions.

Contribution

It proves that negative periodic perturbations of these operators cause spectral shifts leading to population extinction, even with non-symmetric, spatially heterogeneous kernels.

Findings

Negative perturbations shift spectrum to the left half-plane.

Such shifts lead to population extinction in all dimensions.

Results apply to non-symmetric, spatially heterogeneous kernels.

Abstract

The paper deals with spectral analysis of non-local operators arising in population dynamics models. We consider negative periodic perturbations of non-local operators of the convolution type. Such operators describe evolutions of the first correlation function in the stochastic birth and death dynamcis in the presence of suppression forces that increase mortality. We consider the case when the birth kernel can be non-symmetric and spatially heterogeneous. It has been proven that any negative periodic perturbation of the equilibrium dynamics generator shifts the spectrum to the left half-plane and, consequently, such a perturbation of mortality leads to the population extinction in any dimension.