Evolution of states in a continuum migration model

The Markov evolution of states of a continuum migration model is studied. The model describes an infinite system of entities placed in $\mathds{R}^d$ in which the constituents appear (immigrate) with rate $b(x)$ and disappear, also due to competition. For this model, we prove the existence of the evolution of states $μ_0 \mapsto μ_t$ such that the moments $μ_t(N_Λ^n)$, $n\in \mathds{N}$, of the number of entities in compact $Λ\subset \mathds{R}^d$ remain bounded for all $t>0$. Under an additional condition, we prove that the density of entities and the second correlation function remain bounded globally in time.

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