Analysis of stability and bifurcation in logistics models with harvesting in the form of the holling type III functional response

Abstract

The logistic model can be applied in the field of biological studies to investigate population growth problems and some important aspects of the ecological situation. This model is a growth model with a limited population growth rate, and ecologists describe this rate as carrying capacity. Carrying capacity can be interpreted as the ideal population size, where individuals in the population can live properly in their environment. The growth rate of a population can be influenced by the harvesting factor, in this case, it is assumed that harvesting is not constant. The effect of the harvest on the growth rate can be analyzed mathematically by using the Holling type III functional response. In this paper, describe the formation of a logistic model taking into account the effects of harvesting, using the Holling type III functional response. Then, perform a nondimensional process in the model, namely simplifying a model that has four parameters to a model that only has two parameters. Next, determine the equilibrium point of the model, perform a stability analysis at that equilibrium point, and investigate the possibility of bifurcation. As result, first obtained a logistic model which has two non-dimensional parameters, where one of the equilibrium points is zero and is unstable. Next, determine another equilibrium point through an implicit equation and investigate its stability through simulation. Finally, obtained two equilibrium points, which are fold bifurcation. Keywords: Logistic Model, Harvesting, Holling Type III, Equilibrium, Stability Analysis, Fol Bifurcation.