Authors: Bootan Rahman; Muhammad A. Yau; Yuliya N. Kyrychko; Konstantin B. Blyuss
Addresses: Mathematics Unit, School of Science and Engineering, University of Kurdistan Hewlêr (UKH), Erbil, 44001, Kurdistan Region, Iraq ' Department of Mathematical Sciences, Nasarawa State University Keffi, 234, Nigeria ' Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, UK ' Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, UK
Abstract: This paper considers a predator-prey model with discrete time delay representing prey handling time and assumed equal to the predator maturation period, and a distributed time delay describing intra-species interactions. We show that due to the delayed logistic growth of the prey, it is impossible for the species to become extinct through predation. Conditions for existence and local stability of the co-existence equilibrium are derived in terms of system parameters. Using techniques of centre manifold reduction and the normal form theory, we establish the direction of Hopf bifurcation of the co-existence equilibrium, as well as the stability of the bifurcating period solution. Numerical bifurcation analysis and simulations are performed to illustrate regions of stability of the co-existence equilibrium, to investigate how the amplitude and the period of bifurcating periodic solutions depend on parameters, and to demonstrate different types of dynamics of the system.
Keywords: stability; discrete and distributed delay; predator-prey model; Hopf bifurcation; periodic solutions.
International Journal of Dynamical Systems and Differential Equations, 2020 Vol.10 No.5, pp.427 - 449
Received: 23 Sep 2018
Accepted: 22 Feb 2019
Published online: 15 Nov 2020 *