Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme

Figure 1.  Comparison of $ f(h) = h $ and $ f(h) = \frac{h}{1+mh} $ in the plane of $ h-f(h) $

Figure 2.  Upper: there exists a unique $ \tilde{\lambda}\in(\pi_u, \pi_v) $ such that $ h_1(\tilde{\lambda}) = 0 $, we have $ h_1(\lambda)>0 $ when $ \lambda\in(\pi_u, \tilde{\lambda}) $, and $ h_1(\lambda)<0 $ when $ \lambda\in(\tilde{\lambda}, \pi_v) $; Lower left: there exists a unique $ \widetilde{\lambda} $ satisfying $ \lambda_\sharp<\widetilde{\lambda}<\pi_v $ such that $ h_3(\widetilde{\lambda}) = 0 $, and one has $ h_3(\lambda)>0 $ when $ \lambda\in(\lambda_\sharp, \widetilde{\lambda}) $, and $ h_3(\lambda)<0 $ when $ \lambda\in(\widetilde{\lambda}, \pi_v) $; Lower right: there exists a unique $ \lambda^\ddagger\in(\lambda^\dagger, \pi_v) $ such that $ f_u(\lambda^\ddagger) = 0 $, and $ f_u(\lambda)>0 $ when $ \lambda\in(\lambda^\dagger, \lambda^\ddagger) $, and $ f_u(\lambda)<0 $ when $ \lambda\in(\lambda^\ddagger, \pi_v) $. Parameters are chosen as follows. Upper: $ r = 0.5, m = 1.3, \alpha = 0.08, \beta = 0.25 $; Lower left: $ r = 1.15, m = 0.02, \alpha = 0.15, \beta = 3.8 $; Lower right: $ r = 1.15, m = 0.02, \alpha = 0.15, \beta = 2.8 $

Figure 3.  Positive constant steady state $ E_* = (0.9022568, $ $ 0.0462763) $ is locally asymptotically stable. Here $ d_1 = 1.5, d_2 = 0.5, r = 1.15, m = 0.02, \alpha = 0.1, \beta = 0.05 $

Figure 4.  Graph of the curve $ \phi(\lambda) $. There exists a unique $ \underline{\lambda}<\overline{\lambda}<\pi_v $ such that $ \phi(\overline{\lambda}) = 0 $. Thus, $ \phi(\lambda)>0 $ when $ \lambda\in(\underline{\lambda}, \overline{\lambda}) $ and $ \phi(\lambda)<0 $ when $ \lambda\in(\overline{\lambda}, \pi_v) $, and $ \phi(\lambda) $ achieves its maximum $ \phi(\underline{\lambda}) = \phi_* $ at $ \lambda = \underline{\lambda} $ for $ \lambda\in(\pi_u, \pi_v) $. Here $ r = 0.25, m = 2.7, \alpha = 0.098, \beta = 0.5 $

Figure 5.  Left: there exist two positive constants $ \lambda_* $ and $ \lambda^* $ satisfying $ \pi_u<\lambda_*<\lambda^*<\pi_v $, such that $ \Delta(\lambda_*) = \Delta(\lambda^*) = 0 $ and $ \Delta(\lambda)>0 $ when $ \lambda\in(\pi_u, \lambda_*)\cup(\lambda^*, \pi_v) $; Right: the existence of $ p_+(\lambda) $ and $ p_-(\lambda) $ in $ \mathcal{C}_1 $, it is found that $ p_+(0)<0, p_-(0)<0 $ and $ \lim_{\lambda\rightarrow \lambda_{min}}p_+(\lambda) = \lim_{\lambda\rightarrow \lambda_{min}}p_-(\lambda)>0 $. Here $ d_1 = 0.4, d_2 = 1, r = 0.6, m = 0.5, \alpha = 0.2, \beta = 1.25 $

Figure 6.  There exist spatially homogeneous periodic solutions of system (4)

Figure 7.  There exist steady state solutions of system (4). Here $ d_1 = 8, d_2 = 0.5, r = 0.058, m = 0.15, \alpha = 4.0, \beta = 1.35 $