On the balance between Emigration and Immigration as Random Walks on the non-negative integers

Life is on the razor’s edge as resulting from competitive birth and death random forces. We illustrate this aphorism in the context of three Markov chain population models where systematic random immigration events promoting growth are simultaneously balanced with random emigration ones provoking thinning. The origin of mass removals are either determined by external demands or by aging leading to different conditions of stability.