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Note on the permanence of stochastic population models

Abstract The concept of permanence of any system is an important technical issue. This concept is very significant to all kind of systems, e.g., social, medical, biological, population, mechanical, or electrical. It is desirable by scientists and investigators that any system under consideration must be long time survival. For example, if we consider any ecosystem, it is always pre-requisite that this system is permanent. In general language, permanence is just the persistent and bounded system in a particular surface time frame. But the meaning may vary with the type of systems. For example, deterministic and stochastic biological systems have different concepts of permanence in an abstract mathematical platform. The reason is simple: it is due to the mathematical nature of parameters, methods of derivations of the model, biological assumptions, details of the study, etc. In this short note, we consider the stochastic models for their permanence. To address stochastic permanence of biological systems, many different approaches have been proposed in the literature. In this note, we propose a new definition of permanence for stochastic population models (SPM). The proposed definition is applied to the well-known Lotka–Volterra two species stochastic population model. The note is closed with the open ended discussion on the topic.