Global stability analysis of a general Leslie type prey-predator system involving harvesting and refuge effect

We consider a general Leslie type prey-predator system involving harvesting and refuge effects. In the model, we use a general function to define how the population of prey grows. In this paper, we explore dynamical behavior of this system by taking three different growth forms for prey, namely, (a) exponential growth, (b) logistic growth and (c) logistic growth including Allee effect, respectively. For each model, we present global stability analysis by applying linear stability theory, utilizing Bendixson-Dulac criteria and constructing an appropriate Lyapunov function, respectively. From a mathematical point of view, we have concluded that harvesting and refuge have no effects on global stability of each model's consistent condition. Furthermore, we investigate the impacts of an external harvesting and a refuge for prey population on densities of prey and predator species. Mathematical analysis yields the result that prey population density decreases under the harvesting effect and increases under the refuge effect, while predator population density decreases as the harvesting and refuge effects increase. At last, in order to support analytical results and also to explore further impacts of prey refuge and harvesting on dynamics, we perform numerical simulations. We numerically discuss their impacts on the densities in early time. The results obtained underline that these effects are important for the continuation of the balanced functioning between prey and predator species, so they should be accounted into prey-predator models together.