Figure 2

(a)–(d) Numerical solutions of system (32) using Crank–Nicolson scheme. Here, \({u}_{0} =1+\sin({x})\) and \({v}_{0} =2+ \cos({x})\), with \(l = 5\), \(m = 1\), \(\kappa_{1} =0.01\), and \(\kappa_{2} =0.01\). (Top) The concentration of \(u(\tau )\) is on the left while the concentration of \(v(\tau )\) is on the right. (Bottom) The solutions \(v(x,\tau )\) and \(u(x,\tau )\) tend to the constant steady state. (e)–(h) Numerical solutions of system (32) using Crank–Nicolson scheme. Here, \(u_{0} = 1 + \sin (x)\) and \(v_{0} = 2 + \cos (x)\), with \(l = 10\), \(m = 5\), \(\kappa_{1} =0.01\), and \(\kappa_{2} =0.01\). (Top) The solutions \(v(x,\tau )\) and \(u(x,\tau )\) tend to the spatially homogeneous periodic orbit. (Bottom) The projected views onto the xτ-plane at \(\tau = 20\) for \(v(x,\tau )\) and \(u(x,\tau )\). The stripe structure invades the homogeneous periodic orbit for \(u(x,\tau )\) and \(v(x,\tau )\)