Harvesting a population model with Allee effect in a periodically varying environment

A nonautonomous logistic population model with a feature of an Allee threshold has been investigated in a periodically fluctuating environment. A slow periodicity of the harvesting effort was considered and may arise in response to relatively slow fluctuations of the environment. This assumption permits obtaining the analytical approximate solutions of such model using the perturbation approach based on the slow variation. Thus, the analytical expressions of the population evolution in the situation of subcritical and the supercritical harvesting were obtained and discussed in the framework of the Allee effect. Since the exact solution was not available due to the nonlinearity of the system, the numerical computation was considered to validate our analytical approximation. The comparison between the two methods showed a remarkable agreement as the time progressed, while such agreement fell off when the time was close to the initial density. Moreover, in the absence of the periodicity of the harvesting term, the expressions of the population evolution reduced to the exact solutions but in implicit forms. The finding results were appropriate for a wide range of parameter values, which lead to avoiding extensive recalculations while displaying the population behavior.