Nonlinear Dyn DOI 10.1007/s11071-014-1772-8 ORIGINAL PAPER Hopf bifurcations in Lengyel–Epstein reaction–diffusion model with discrete time delay H. Merdan · ¸ S. Kayan Received: 26 April 2014 / Accepted: 22 October 2014 © Springer Science+Business Media Dordrecht 2014 Abstract We investigate bifurcations of the Lengyel– Epstein reaction–diffusion model involving time delay under the Neumann boundary conditions. Choosing the delay parameter as a bifurcation parameter, we show that Hopf bifurcation occurs. We also determine two properties of the Hopf bifurcation, namely direction and stability, by applying the normal form theory and the center manifold reduction for partial functional dif- ferential equations. Keywords Lengyel–Epstein reaction–diffusion model · Hopf bifurcation · Stability · Time delay · Periodic solutions 1 Introduction In this paper, we consider the following reaction– diffusion model with time delay under the Neumann boundary conditions H. Merdan (B) · ¸ S. Kayan Department of Mathematics, Faculty of Science and Letters, TOBB University of Economics and Technology, Sö˘ gütözü Cad. No 43., 06560 Ankara, Turkey e-mail: merdan@etu.edu.tr ¸ S. Kayan Department of Mathematics and Computer Sciences, Faculty of Science and Letters, Çankaya University, Yukarıyurtçu Mahallesi Mimar Sinan Caddesi No 4., 06790 Etimesgut-Ankara, Turkey e-mail: sbilazeroglu@etu.edu.tr; sbilazeroglu@cankaya.edu.tr ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∂ u(x ,t ) ∂ t = d 1 ∂ 2 u(x ,t ) ∂ x 2 + a − u (x , t ) −4 u(x ,t )v(x ,t −τ) 1+u 2 (x ,t ) for x ∈ Ω, t > 0, ∂v(x ,t ) ∂ t = d 2 ∂ 2 v(x ,t ) ∂ x 2 + σ b  u (x , t ) − u(x ,t )v(x ,t −τ) 1+u 2 (x ,t )  for x ∈ Ω, t > 0, ∂ u ∂ −→ n = ∂v ∂ −→ n = 0 for x ∈ ∂Ω, t >10, u (x , 0) = u 0 (x ), v(x , 0) = v 0 (x ) for x ∈ Ω, (1) where Ω is a bounded domain in R m , m ≥ 1, with smooth boundaries ∂Ω, −→ n is the unit outer normal to ∂Ω and u 0 ,v 0 ∈ C 2 (Ω) ∩ C ( Ω). When there is no time delay, system (1) reduces to the Lengyel–Epstein reaction–diffusion model based on the chlorite–iodide– malonic acid chemical (CIMA) reaction (see Turing [1], Lengyel et al. [2, 3], De Kepper et al. [4], Epstein et al. [5] and the references therein). In the model, u (x , t ) and v(x , t ) denote chemical concentrations of the activator iodide and the inhibitor chlorite, respec- tively. a > 0 and b > 0 are parameters related to the feed concentrations, and σ> 0 is a rescaling parameter depending on the concentration of the starch. Here, the positive constants d 1 and d 2 are diffusion coefficients of the activator and the inhibitor, respectively, and τ is delay parameter. Studies on stability analysis of reaction–diffusion systems have attracted very much interest in mathe- matical biology, medicine, ecology, economics and so on (see, e.g., [5–10] and [11]). In the last two decades, some mathematical investigations were conducted for 123