Due to the importance of this topic in population dynamics, our main goal in this paper is to investigate persistence and extinction in the considered model. There are a variety of causes that can contribute directly or indirectly to the extinction of species or group of species, such as lack of food and space or toxic pollution of the entire population habitat, competition for food to better adapted competitors, predation, etc. In ecology, extinction is often used informally to refer to local extinction, in which a species ceases to exist in the chosen area of study, but may still exist elsewhere. Therefore, extinction becomes a certainty when there are no surviving individuals that can reproduce and create a new generation. A species is said to be extinct when the last existing member dies. ![]() Extinction is one of the most important terms in population dynamics. Many authors have investigated this phenomenon (see, e.g., ).Ī key question in population biology is understanding the conditions under which populations coexist or go extinct. Therefore, it is useful to see how changes in environment affect the relationship between predator and prey populations. In reality, natural phenomena counter an environmental noise and usually do not follow deterministic laws strictly but oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time. Ecological systems are often subject to environmental noise (e.g., temperature, precipitation), which is an important factor in ecosystems, to suppress a potential population explosion. However, the natural growth of populations is always affected by environmental stochastic perturbations which should be taken into account in the process of mathematical modeling. Most of these studies utilize deterministic models, which of course supported us with useful results for protecting species. Many studies have explored the effect of predator hunting cooperation on PP systems. The classic prey–predator models are mostly variations of the Lotka–Volterra model, which was proposed by Lotka and Volterra. Prey–predator (PP) interaction is one of the most extensively studied issues in ecological and mathematical literature see.
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