Complex dynamics in a tritrophic food chain model with general functional response

Abstract

This paper investigates the rich dynamics in a tritrophic food chain mathematical model, consisting of three species: prey, intermediate predators, and top predators. It is assumed that alternative food are supplied to intermediate predators in addition to feeding on prey. We consider a general Holling type response function and analyze the model. The existence and stability of six possible equilibrium points are established. These equilibrium points describe the various dynamics that could take place in the food chain. Hopf bifurcation, limit cycle, doubling periods, chaotic attractors, boundary crisis are observed in the numerical computations. Our results reveal the rich and complex dynamics of the interactions in the food chain. Recommendations for Resource ManagersOur investigation brings the followings to the attention of management:• Coexistence among predators and prey in the same environment is possible provided a good management of some factors (such as contacts between species, additional food supply, growth rate of species, etc).• The dynamics is complex and highly sensitive to the above factors. Strange or unpredictable behaviors could be observed.• The rate at which species are killed by a single predator (i.e., the functional responses) significantly affects the population sizes of all the species and the overall dynamics in the food chain. K E Y W O R D Scoexistence, food chain model, holling type functions, limit cycle, tritrophic

Coexistence and harvesting optimal policy in three species food chain model with general Holling type functional response

“…The rich dynamics in a tritrophic food chain mathematical model, consisting of three species: source prey (), a generalist intermediate predator (), and top predator () that includes a general Holling type response function is investigated in Dawed et al (2020). Following this approach, we have considered harvesting terms and in the existing model system.…”

“…Transition to chaotic behavior is established (Upadhyay & Raw, 2011) via period‐doubling bifurcation and some sequences of distinctive period‐halving bifurcation leading to limit cycles are observed, furthermore, Holling type IV predator response function is considered. The rich dynamics in a tritrophic food chain mathematical model, consists of three species: prey, intermediate predator, and top predator while both the prey and intermediate predator are being subjected to Holling type I–IV functional response is investigated in Dawed et al (2020). Some research have been proposed to understand the dynamics in a food chain (one predator–two prey).…”

“…This means that the consumption rate of an individual predator depends on the prey density. The detail about each Holling type response function is found in Dawed et al (2020).…”

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 ).…”