A Comparative Study of Allee Effects and Fear-Induced Responses: Exploring Hyperbolic and Ratio-Dependent Models

The present article deals with the employment of parameter-dependent center manifold reduction in order to comprehend the structure of codimension one and two bifurcations from two different stand points of hyperbolic and ratio-dependent functional response in the proposed system. At the onset one may explore the number of equilibria of the system including their characteristics by means of the equations of equilibria and the distribution of eigenvalues. The system appears to exhibit a multi-stable structure with refinement of the bifurcation parameters, the stable-unstable manifolds of saddles and limit cycles resulting from the respective Hopf and Bogdanov-Takens bifurcations so as to constitute the attractive domains of various attractors. The complex algebraic treatment generates the Bogdanov-Takens normal form and three different types of bifurcation curves. The structure of codimension one and two bifurcations outlines several phase portraits corresponding to the parameters in proximity to different local bifurcating points. The theoretical outcomes of the present system are finally corroborated by the analytical findings together with its numerical simulations counterpart.