Stochastic hybrid delay population dynamics: well-posed models and extinction

Nonlinear differential equations have been used for decades for studying fluctuations in the populations of species, interactions of species with the environment, and competition and symbiosis between species. Over the years, the original non-linear models have been embellished with delay terms, stochastic terms and more recently discrete dynamics. In this paper, we investigate stochastic hybrid delay population dynamics (SHDPD), a very general class of population dynamics that comprises all of these phenomena. For this class of systems, we provide sufficient conditions to ensure that SHDPD have global positive, ultimately bounded solutions, a minimum requirement for a realistic, well-posed model. We then study the question of extinction and establish conditions under which an ecosystem modelled by SHDPD is doomed.