An existence results on positive solutions for a reaction-diffusion model with logistics growth and indefinite weight

In this paper, using sub-supersolution argument, we prove an existence result on positive solution for an ecological model under certain conditions. It also describes the dynamics of the fish population with natural predation and constant yield harvesting. The assumptions are that the ecosystem is spatially homogeneous and the herbivore density is a constant which are valid assumptions for managed grazing systems. This term saturates to c at high levels of vegetation density as the grazing population is a constant. This model tries to capture the phenomena of bistability and hysteresis and provide qualitative and quantitative information for ecosystem managements. This model has also been applied to describe the dynamics of fish populations. This model describes grazing of a fixed number of grazers on a logistically growing species. The general logistic function is characterized by a declining growth rate per capita function (Equation) Here P is the population, r > 0 is the growth rate and is positive constant[21]. But there are some ecosystems where the growth rate per capita may achieve its peak at a positive density. This is called the Allee effect This effect can be caused by shortage of mates, lack of effective pollinations predator saturation and cooperative behaviors. In this pape, we restrict ourselves to logistic models.