Hopf and Bautin Bifurcation in a Tritrophic Food Chain Model with Holling Functional Response Types III and IV

Abstract

In this paper, we analyze the Hopf and Bautin bifurcation of a given system of differential equations, corresponding to a tritrophic food chain model with Holling functional response types III and IV for the predator and superpredator, respectively. We distinguish two cases, when the prey has linear or logistic growth. In both cases we guarantee the existence of a limit cycle bifurcating from an equilibrium point in the positive octant of [Formula: see text]. In order to do so, for the Hopf bifurcation we compute explicitly the first Lyapunov coefficient, the transversality Hopf condition, and for the Bautin bifurcation we also compute the second Lyapunov coefficient and verify the regularity conditions.

Influence of impreciseness in designing tritrophic level complex food chain modeling in interval environment

“…Banerjee and Das [24] have shown the impulsive effect on a tritrophic food chain model with mixed functional responses under seasonal perturbations. Castellonas et al [25] have shown both Hopf and Bautin bifurcations in a tritrophic food chain model. Huang et al [26] discussed the dynamical behavior of a food chain model with stage structure and time delays.…”

Stability analysis of a tritrophic model with stage structure in the prey population

“…Proof. If the hypothesis are valid, using Proposition 2.2 for the transversality condition and the Kuznetsov formulae (see [2,7,8]) for the first Lyapunov coefficient, the Mathematica software allows to get by a direct calculation:…”

Bistability and Hopf bifurcation of a tritrophic system with Holling functional responses

“…Furthermore they analyzed and determined conditions in order to obtain a stable limit cycle in tritrophic models. These are systems in which three differential equations are involved, as they analyze the behavior of three species, prey, predator and superpredator ( Francoise and Llibre, 2011 , Castellanos et al, 2018 , Dawed et al, 2020 ) and the references therein.…”

“…As a result of the analysis of the tritrophic models, the authors presented conditions in the parameters of the analyzed system, under which there exists a stable limit cycle for a system with a functional response of type Lotka Volterra or Holling. This was proved by showing the existence of a Hopf or a Bautin bifurcation ( Castellanos et al, 2018 , Bentounsi et al, 2018 , Blé et al, 2018 , Wang and Yu, 2019 , Dawed et al, 2020 ).…”

“…A tritrophic food chain is modeled by the following system of three differential equations: where x represents the density of a prey that gets eaten by a predator of density y , and the species y feeds the top predator z (superpredator). The function represents the growth rate of the prey population in absence of the predators, and the functions and are the functional responses for the mesopredator and the superpredator, respectively ( Castellanos et al, 2018 ). The parameters and are positive and for ecological considerations we focus in finding stable solutions in the positive octant …”