Hopf bifurcation analysis in a multiple delayed innovation diffusion model with Holling II functional response

Abstract

A nonlinear mathematical model with Holling II functional response describing the dynamics of nonadopter and adopters population in a stage structured innovation diffusion model, which incorporates the evaluation stage (multiple delays), is proposed. Firstly, we study the stability and the existence of periodic solutions via Hopf bifurcation with respect to both delays at the positive equilibrium by analyzing the distribution of the roots of the corresponding exponential characteristic equation obtained through the variational matrix. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined with the help of normal form theory and center manifold theorem. Meanwhile, various cases are discussed to examine the effect of different delays on the stability of delayed innovation diffusion system and are also established numerically. It is also observed that the cumulative density of external influences has a significant role in developing maturity stage (adoption stage) in the system. Finally, numerical simulations are carried out to support and supplement the analytical findings. KEYWORDS center manifold theorem, Hopf bifurcation, innovation diffusion model, multiple delays, normal form theory, stability analysis MSC CLASSIFICATION 34C11; 34C23; 34D20; 92D25 2056

Dynamic Analysis and Hopf Bifurcation of a Lengyel–Epstein System with Two Delays

“…In this section, we use the method in [21] to discuss the system's positivity and uniform boundedness. en, the solutions u(t) and v(t) of system (6) are nonnegative, for all t > 0.…”

Stability and Hopf Bifurcation Analysis of a Vector-Borne Disease Model with Two Delays and Reinfection

“…That is, reinfection likely occurs [6,7]. It is shown that time delay is of great significance in many biological modelling, and its change may affect the dynamic behavior of the system [4][5][6][9][10][11][12][13][14][15][16][17][18][19], such as stability, uniqueness, and oscillation of solution. So, it is instructive to consider a mathematical model with time delays to research the influence of immunity on disease control of malaria transmission.…”