Figure 1

(a)–(d) Numerical solutions of system (32) using forward Euler scheme. Here, \({u}_{0} =1+\sin ({x})\) and \({v}_{0} =2+ \cos ({x})\), with \({l}=5\), \(m = 1\), \(\kappa_{1} = 0.01\), and \(\kappa_{2} =0.01\). The concentration of \(u(\tau )\) is at the top (2D and 3D plots) while the concentration of \(v(\tau )\) is at the bottom (2D and 3D plots). The solutions \(v(x,\tau )\) and \(u(x,\tau )\) tend to the constant steady state. (e)–(h) Numerical solutions of system (32) using forward Euler 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 \(v(x,\tau )\) and \(u(x,\tau )\)