Stability and bifurcation analysis of predator-prey model with Allee effect using conformable derivatives
M. B. Almatrafi, Messaoud Berkal
Abstract
Some organisms coexist on the expense of others. This coexistence is called predation which has been successfully investigated using differential equations. In this work, we aim to analyse a fractional order predator-prey dynamical system with Allee effect using bifurcation theory. The Allee effect is a density-dependent phenomenon where the population growth and individual fitness increase as population density increases. Several mechanisms, such as cooperative feeding, mate limitation, and predator satiation, can cause Allee effects. {The piecewise-constant approximation method and the conformable derivatives are utilized to discretise the propose model.} We explore equilibrium points, the local stability, the Neimark-Sacker bifurcation, periodic-doubling bifurcation, chaos control, and numerical simulations of the proposed model. The linear theory of stability is used to examine the local attractivity of the fixed points. Our findings include that the coexistence equilibrium point is locally stable, source, unstable under certain constraints. We also prove that the considered discrete model goes through Neimark-Sacker and periodic-doubling bifurcations according to specific conditions. The used techniques can be applied for other nonlinear discrete systems.