Steady states of lattice population models with immigration

TL;DR

This paper studies the long-term behavior of a lattice population model with immigration, proving the existence of steady states and deriving explicit formulas for limiting distributions in specific cases.

Contribution

It introduces Carleman type estimates for cumulants and establishes the existence of steady states in lattice Galton-Watson models with immigration.

Findings

Proved existence of steady states in the model

Derived formulas for limiting distributions in special cases

Established Carleman estimates for cumulants

Abstract

We consider the time evolution of the lattice subcritical Galton-Watson model with immigration. We prove Carleman type estimation for the cumulants in the simple case (binary splitting) and show the existence of a steady state. We also present the formula of the limiting distribution in a particular solvable case.