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April  2022, 27(4): 2275-2312. doi: 10.3934/dcdsb.2021132

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

1. 

School of Mathematical Sciences, Anhui University, Hefei 230601, China

2. 

Department of Mathematics, Hangzhou Normal University, Hangzhou 311121, China

* Corresponding author: Ranchao Wu

Received  September 2020 Revised  March 2021 Published  April 2022 Early access  April 2021

Fund Project: The second author is supported by NSF grants 11971032 and 62073114. The third author is supported by NSF grant 11671114 and NSF of Zhejiang LY20A010002

A depletion-type reaction-diffusion Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme and the homogeneous Neumann boundary conditions is introduced and investigated in this paper. Firstly, the boundedness of positive solution of the parabolic system is given, and the constant steady state solutions of the model are exhibited by the Shengjin formulas. Through rigorous theoretical analysis, the stability of the corresponding positive constant steady state solution is explored. Next, a priori estimates, the properties of the nonconstant steady states, non-existence and existence of the nonconstant steady state solution for the corresponding elliptic system are investigated by some estimates and the Leray-Schauder degree theory, respectively. Then, some existence conditions are established and some properties of the Hopf bifurcation and the steady state bifurcation are presented, respectively. It is showed that the temporal and spatial bifurcation structures will appear in the reaction-diffusion model. Theoretical results are confirmed and complemented by numerical simulations.

Citation: Mengxin Chen, Ranchao Wu, Yancong Xu. Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2275-2312. doi: 10.3934/dcdsb.2021132
References:
[1]

S. AbdelmalekS. Bendoukha and B. Rebiai, On the stability and nonexistence of Turing patterns for the generalized Lengyel-Epstein model, Math. Meth. Appl. Sci., 40 (2017), 6295-6305.  doi: 10.1002/mma.4457.

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M. D. Angelis and P. Renno, On the Fitzhugh-Nagumo model, Waves and Stability in Continuous Media, (2008), 193–198. doi: 10.1142/9789812772350_0029.

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J. Buceta and K. Lindenberg, Switching-induced Turing instability, Phys. Rev. E, 66 (2002), 046202. doi: 10.1103/PhysRevE.66.046202.

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Q. Cao and J. Wu, Patterns and dynamics in the diffusive model of a nutrient-microorganism system in the sediment, Nonlnera Anal.: RWA, 49 (2019), 331-354.  doi: 10.1016/j.nonrwa.2019.03.008.

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S. Chen, J. Shi and J. Wei, Bifurcation analysis of the Gierer-Meinhardt system with a saturation in the activator production, Appl. Anal., 93 (2014) 1115–1134. doi: 10.1080/00036811.2013.817559.

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S. J. Fan, A new extracting formula and a new distinguishing means on the one variable cubic equation, J. Hainan Teachers College, 2 (1989), 91-98. 

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A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.

[8]

D. GomezM. J. Ward and J. Wei, The linear stability of symmetric spike patterns for a bulk-membrane coupled Gierer-Meinhardt model, SIAM J. Appl. Dyn. Syst., 18 (2019), 729-768.  doi: 10.1137/18M1222338.

[9]

Y. JiaY. Li and J. Wu, Coexistence of activator and inhibitor for Brusselator diffusion system in chemical or biochemical reactions, Appl. Math. Lett., 53 (2016), 33-38.  doi: 10.1016/j.aml.2015.09.018.

[10]

K. Kurata and K. Morimoto, Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation, Commun. Pur. Appl. Anal., 7 (2008), 1443-1482.  doi: 10.3934/cpaa.2008.7.1443.

[11]

S. LiJ. Wu and Y. Dong, Turing patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differential Equations, 259 (2015), 1990-2029.  doi: 10.1016/j.jde.2015.03.017.

[12]

Y. LiJ. Wang and X. Hou, Stripe and spot patterns for the Gierer-Meinhardt model with saturated activator production, J. Math. Anal. Appl., 449 (2017), 1863-1879.  doi: 10.1016/j.jmaa.2017.01.019.

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C.-S. LinW.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.

[14]

J. LiuF. Yi and J. Wei, Multiple bifurcation analysis and spatiotemporal patterns in a 1-D Gierer-Meinhardt model of morphogenesis, Int. J. Bifurcat. Chaos, 20 (2010), 1007-1025.  doi: 10.1142/S0218127410026289.

[15]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.

[16]

H. Meinhardt and M. Klingler, A model for pattern formation on the shells of molluscs, J. Theor. Biol., 126 (1987), 63-89.  doi: 10.1016/S0022-5193(87)80101-7.

[17]

H. Merdan and Ş. Kayan, Hopf bifurcation in Lengyel-Epstein reaction-diffusion model with discrete time delay, Nonlinear Dyn., 79 (2015), 1757-1770.  doi: 10.1007/s11071-014-1772-8.

[18]

K. Morimoto, On positive solutions generated by semi-strong saturation effect for the Gierer-Meinhardt system, J. Math. Soc. Jpn., 65 (2013), 887-929.  doi: 10.2969/jmsj/06530887.

[19]

I. R. Moyles and M. J. Ward, Existence, stability, and dynamics of ring and near-ring solutions to the saturated Gierer-Meinhardt model in the semistrong regime, SIAM J. Appl. Dyn. Syst., 16 (2017), 597-639.  doi: 10.1137/16M1060327.

[20]

Y. Nishiura, Global structure of bifurcating solutions of some reaction diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593.  doi: 10.1137/0513037.

[21]

P. Y. H. Pang and M. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273.  doi: 10.1016/j.jde.2004.01.004.

[22]

P. Y. H. Pang and M. Wang, Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion, Proc. Lond. Math. Soc., 88 (2004), 135-157.  doi: 10.1112/S0024611503014321.

[23]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Springer, New York, 1992.

[24]

I. RozadaS. J. Ruuth and M. J. Ward, The stability of localized spot patterns for the Brusselator on the sphere, SIAM J. Appl. Dyn. Syst., 13 (2014), 564-627.  doi: 10.1137/130934696.

[25]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290.  doi: 10.1016/0022-0396(81)90077-2.

[26]

Y. SongR. Yang and G. Sun, Pattern dynamics in a Gierer-Meinhardt model with a saturating term, Appl. Math. Model., 46 (2017), 476-491.  doi: 10.1016/j.apm.2017.01.081.

[27]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[28]

J. WangJ. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.

[29]

R. Wu, Y. Shao, Y. Zhou and L. Chen, Turing and Hopf bifurcation of Gierer-Meinhardt activator-substrate model, Electron. J. Differential Equations, 173 (2017), No. 173, 19 pp.

[30]

X.-P. YanJ.-Y. Chen and C.-H. Zhang, Dynamics analysis of a chemical reaction-diffusion model subject to Degn-Harrison reaction scheme, Nonlinear Anal.: RWA, 48 (2019), 161-181.  doi: 10.1016/j.nonrwa.2019.01.005.

[31]

F. YiS. Liu and N. Tuncer, Spatiotemporal patterns of a reaction-diffusion substrate-inhibition Seelig model, J. Dyn. Differential Equations, 29 (2017), 219-241.  doi: 10.1007/s10884-015-9444-z.

[32]

F. YiJ. Wei and J. Shi, Bifurcation and spatialtemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.

[33]

Z. ZhengX. ChenY. Qi and S. Zhou, Existence of traveling waves of general Gray-Scott models, J. Dyn. Differential Equations, 30 (2018), 1469-1487.  doi: 10.1007/s10884-017-9603-5.

show all references

References:
[1]

S. AbdelmalekS. Bendoukha and B. Rebiai, On the stability and nonexistence of Turing patterns for the generalized Lengyel-Epstein model, Math. Meth. Appl. Sci., 40 (2017), 6295-6305.  doi: 10.1002/mma.4457.

[2]

M. D. Angelis and P. Renno, On the Fitzhugh-Nagumo model, Waves and Stability in Continuous Media, (2008), 193–198. doi: 10.1142/9789812772350_0029.

[3]

J. Buceta and K. Lindenberg, Switching-induced Turing instability, Phys. Rev. E, 66 (2002), 046202. doi: 10.1103/PhysRevE.66.046202.

[4]

Q. Cao and J. Wu, Patterns and dynamics in the diffusive model of a nutrient-microorganism system in the sediment, Nonlnera Anal.: RWA, 49 (2019), 331-354.  doi: 10.1016/j.nonrwa.2019.03.008.

[5]

S. Chen, J. Shi and J. Wei, Bifurcation analysis of the Gierer-Meinhardt system with a saturation in the activator production, Appl. Anal., 93 (2014) 1115–1134. doi: 10.1080/00036811.2013.817559.

[6]

S. J. Fan, A new extracting formula and a new distinguishing means on the one variable cubic equation, J. Hainan Teachers College, 2 (1989), 91-98. 

[7]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.

[8]

D. GomezM. J. Ward and J. Wei, The linear stability of symmetric spike patterns for a bulk-membrane coupled Gierer-Meinhardt model, SIAM J. Appl. Dyn. Syst., 18 (2019), 729-768.  doi: 10.1137/18M1222338.

[9]

Y. JiaY. Li and J. Wu, Coexistence of activator and inhibitor for Brusselator diffusion system in chemical or biochemical reactions, Appl. Math. Lett., 53 (2016), 33-38.  doi: 10.1016/j.aml.2015.09.018.

[10]

K. Kurata and K. Morimoto, Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation, Commun. Pur. Appl. Anal., 7 (2008), 1443-1482.  doi: 10.3934/cpaa.2008.7.1443.

[11]

S. LiJ. Wu and Y. Dong, Turing patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differential Equations, 259 (2015), 1990-2029.  doi: 10.1016/j.jde.2015.03.017.

[12]

Y. LiJ. Wang and X. Hou, Stripe and spot patterns for the Gierer-Meinhardt model with saturated activator production, J. Math. Anal. Appl., 449 (2017), 1863-1879.  doi: 10.1016/j.jmaa.2017.01.019.

[13]

C.-S. LinW.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.

[14]

J. LiuF. Yi and J. Wei, Multiple bifurcation analysis and spatiotemporal patterns in a 1-D Gierer-Meinhardt model of morphogenesis, Int. J. Bifurcat. Chaos, 20 (2010), 1007-1025.  doi: 10.1142/S0218127410026289.

[15]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.

[16]

H. Meinhardt and M. Klingler, A model for pattern formation on the shells of molluscs, J. Theor. Biol., 126 (1987), 63-89.  doi: 10.1016/S0022-5193(87)80101-7.

[17]

H. Merdan and Ş. Kayan, Hopf bifurcation in Lengyel-Epstein reaction-diffusion model with discrete time delay, Nonlinear Dyn., 79 (2015), 1757-1770.  doi: 10.1007/s11071-014-1772-8.

[18]

K. Morimoto, On positive solutions generated by semi-strong saturation effect for the Gierer-Meinhardt system, J. Math. Soc. Jpn., 65 (2013), 887-929.  doi: 10.2969/jmsj/06530887.

[19]

I. R. Moyles and M. J. Ward, Existence, stability, and dynamics of ring and near-ring solutions to the saturated Gierer-Meinhardt model in the semistrong regime, SIAM J. Appl. Dyn. Syst., 16 (2017), 597-639.  doi: 10.1137/16M1060327.

[20]

Y. Nishiura, Global structure of bifurcating solutions of some reaction diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593.  doi: 10.1137/0513037.

[21]

P. Y. H. Pang and M. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273.  doi: 10.1016/j.jde.2004.01.004.

[22]

P. Y. H. Pang and M. Wang, Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion, Proc. Lond. Math. Soc., 88 (2004), 135-157.  doi: 10.1112/S0024611503014321.

[23]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Springer, New York, 1992.

[24]

I. RozadaS. J. Ruuth and M. J. Ward, The stability of localized spot patterns for the Brusselator on the sphere, SIAM J. Appl. Dyn. Syst., 13 (2014), 564-627.  doi: 10.1137/130934696.

[25]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290.  doi: 10.1016/0022-0396(81)90077-2.

[26]

Y. SongR. Yang and G. Sun, Pattern dynamics in a Gierer-Meinhardt model with a saturating term, Appl. Math. Model., 46 (2017), 476-491.  doi: 10.1016/j.apm.2017.01.081.

[27]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[28]

J. WangJ. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004.

[29]

R. Wu, Y. Shao, Y. Zhou and L. Chen, Turing and Hopf bifurcation of Gierer-Meinhardt activator-substrate model, Electron. J. Differential Equations, 173 (2017), No. 173, 19 pp.

[30]

X.-P. YanJ.-Y. Chen and C.-H. Zhang, Dynamics analysis of a chemical reaction-diffusion model subject to Degn-Harrison reaction scheme, Nonlinear Anal.: RWA, 48 (2019), 161-181.  doi: 10.1016/j.nonrwa.2019.01.005.

[31]

F. YiS. Liu and N. Tuncer, Spatiotemporal patterns of a reaction-diffusion substrate-inhibition Seelig model, J. Dyn. Differential Equations, 29 (2017), 219-241.  doi: 10.1007/s10884-015-9444-z.

[32]

F. YiJ. Wei and J. Shi, Bifurcation and spatialtemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.

[33]

Z. ZhengX. ChenY. Qi and S. Zhou, Existence of traveling waves of general Gray-Scott models, J. Dyn. Differential Equations, 30 (2018), 1469-1487.  doi: 10.1007/s10884-017-9603-5.

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 $
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