
STABILITY ANALYSIS OF A TRITROPHIC FOOD CHAIN MODEL WITH AN ADAPTIVE PARAMETER FOR THE PREDATOR
Author(s) -
TCHUENCHE JEAN M.,
CHIYAKA CHRISTINAH
Publication year - 2009
Publication title -
natural resource modeling
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.28
H-Index - 32
eISSN - 1939-7445
pISSN - 0890-8575
DOI - 10.1111/j.1939-7445.2008.00035.x
Subject(s) - equilibrium point , predator , apex predator , stability (learning theory) , food chain , predation , food web , mathematics , saddle point , ecology , statistical physics , biology , control theory (sociology) , economics , computer science , physics , mathematical analysis , control (management) , machine learning , geometry , management , differential equation
The study of three‐species communities have become the focus of considerable attention, and because the studies of ecological communities start with their food web, we consider a tritrophic food chain model comprised of the prey, the predator, and the super‐predator. The classical assumption of the domino effect is supplemented with an adaptive parameter for the predator (in the absence of prey). Thus, the model exhibits an equilibrium with the predator‐top‐predator steady state, which is a saddle point. Dynamical behaviors such as boundedness, existence of periodic orbits, persistence, as well as stability are analyzed. The long‐term coexistence of the three interacting species is addressed, and the stability analysis of the model shows that the biologically most relevant equilibrium point is globally asymptotically stable whenever it satisfies a certain criterion. Practical implications are explored and related to real populations.