On the Balance between Emigration and Immigration as Random Walks on Non-Negative Integers

Life is on a razor’s edge resulting from the random competitive forces of birth and death. 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 is either determined by external demands or by aging, leading to different conditions of stability.