Comp. Appl. Math.
The Hopf bifurcation and stability of delayed
· Imane Agmour
· Youssef El Foutayeni
Received: 14 January 2018 / Revised: 5 May 2018 / Accepted: 29 May 2018
© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract In this paper, a mathematical model consisting of three populations with discrete
time delays is considered. By analyzing the corresponding characteristic equations, the local
stability of each of the feasible equilibria of the system is addressed and the existence of
Hopf bifurcations at the coexistence equilibrium is established. The direction of the Hopf
bifurcations and the stability of the bifurcating periodic solutions are analyzed using the
theory of normal form and center manifold. Discussion with some numerical simulation
examples is given to support the theoretical results.
Keywords Predator–prey · Stability analysis · Hopf bifurcation · Discrete delay
Mathematics Subject Classiﬁcation 91B05 · 91A06 · 91B02 · 91B50
Over the last decade, the dynamic behavior of predator–prey systems has received much
attention from many applied mathematicians and ecologists. Many theoreticians and experi-
Communicated by Maria do Rosário de Pinho.
Youssef El Foutayeni
Analysis, Modeling and Simulation Laboratory, Hassan II University, Casablanca, Morocco
Unit for Mathematical and Computer Modeling of Complex Systems, IRD, Paris, France