American Institute of Mathematical Sciences

Advanced
July  2021, 41(7): 3109-3140. doi: 10.3934/dcds.2020400

Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media

Izumi Takagi 1,2, and  Conghui Zhang 1,,

1. 

Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, China
2. 

Institute of Liberal Arts and Sciences, Tohoku University, Sendai, 980-8576, Japan

* Corresponding author: Conghui Zhang

† Currently, Mathematical Institute, Tohoku University, Sendai, 980-8578.
Dedicated to the memory of the late Professor Yuzo Hosono

Received  March 2020 Revised  September 2020 Published  December 2020

Fund Project: This work is supported in part by JSPS Kakenhi, Grant Numbers 16KT0128 and 19K03557; CHZ is sponsored by the China Scholarship Council

This paper is concerned with the existence and stability of steady states of a reaction-diffusion-ODE system arising from the theory of biological pattern formation. We are interested in spontaneous emergence of patterns from spatially heterogeneous environments, hence assume that all coefficients in the equations can depend on the spatial variable. We give some sufficient conditions on the coefficients which guarantee the existence of far-from-the-equilibrium patterns with jump discontinuity and then verify their stability in a weak sense. Our conditions cover the case where the number of equilibria of the kinetic system (i.e., without diffusion) changes from one to three in the spatial interval, which is not obtained by a small perturbation of constant coefficients. Moreover, we consider the asymptotic behavior of steady states as the diffusion coefficient tends to infinity. Some examples and numerical simulations are given to illustrate the theoretical results.

Keywords: Reaction-diffusion-ODE system, pattern formation, hysteresis, steady states with jump discontinuity, asymptotic behavior, stability.
Mathematics Subject Classification: Primary: 35B36, 35K57; Secondary: 35B35, 35J25.
Citation: Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400
References:
[1]

D. G. Aronson, A. Tesei and H. Weinberger, A density-dependent diffusion system with stable discontinuous stationary solutions, Ann. Mat. Pura Appl., 152 (1988), 259-280.  doi: 10.1007/BF01766153.  Google Scholar

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[3]

S. Härting, A. Marciniak-Czochra and I. Takagi, Stable patterns with jump discontinuity in systems with Turing instability and hysteresis, Discrete Contin. Dyn. Syst., 37 (2017), 757-800.  doi: 10.3934/dcds.2017032.  Google Scholar

[4]

Y. Li, A. Marciniak-Czochra, I. Takagi and B. Y. Wu, Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis, Hiroshima Math. J., 47 (2017), 217-247.  doi: 10.32917/hmj/1499392826.  Google Scholar

[5]

A. Marciniak-Czochra, Receptor-based models with diffusion-driven instability for pattern formation in hydra, J. Biol. Systems, 11 (2003), 293-324.  doi: 10.1142/S0218339003000889.  Google Scholar

[6]

A. Marciniak-Czochra, Receptor-based models with hysteresis for pattern formation in Hydra, Math. Biosci., 199 (2006), 97-119.  doi: 10.1016/j.mbs.2005.10.004.  Google Scholar

[7]

A. Marciniak-Czochra, M. Nakayama and I. Takagi, Pattern formation in a diffusion-ODE model with hysteresis, Differential Integral Equations, 28 (2015), 655-694.   Google Scholar

[8]

M. Mimura, M. Tabata and Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal., 11 (1980), 613-631.  doi: 10.1137/0511057.  Google Scholar

[9]

J. D. Murray, Mathematical Biology. II: Spatial Models and Biomedical Applications, Third edition, Springer, 2003.  Google Scholar

[10]

J. A. Sherrat, P. K. Maini, W. Jäger and W. M$\ddot{\mathrm{u}}$ller, A receptor-based model for pattern formation in hydra, Forma, 10 (1995), 77-95.   Google Scholar

[11]

I. Takagi and H. Yamamoto, Locator function for concentration points in a spatially heterogeneous semilinear Neumann problem, Indiana Univ. Math. J., 68 (2019), 63-103.  doi: 10.1512/iumj.2019.68.7560.  Google Scholar

[12]

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

[13]

J. C. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied Mathematical Sciences, Springer, London, 2014. doi: 10.1007/978-1-4471-5526-3.  Google Scholar

[14]

H. F. Weinberger, A simple system with a continuum of stable inhomogeneous steady states, Nonlinear Partial Differential Equations in Applied Science; Proceedings of the U.S.-Japan Seminar, North-Holland Math. Stud., 81 (1983), 345–359.  Google Scholar

show all references

References:
[1]

D. G. Aronson, A. Tesei and H. Weinberger, A density-dependent diffusion system with stable discontinuous stationary solutions, Ann. Mat. Pura Appl., 152 (1988), 259-280.  doi: 10.1007/BF01766153.  Google Scholar

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley and Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[3]

S. Härting, A. Marciniak-Czochra and I. Takagi, Stable patterns with jump discontinuity in systems with Turing instability and hysteresis, Discrete Contin. Dyn. Syst., 37 (2017), 757-800.  doi: 10.3934/dcds.2017032.  Google Scholar

[4]

Y. Li, A. Marciniak-Czochra, I. Takagi and B. Y. Wu, Bifurcation analysis of a diffusion-ODE model with Turing instability and hysteresis, Hiroshima Math. J., 47 (2017), 217-247.  doi: 10.32917/hmj/1499392826.  Google Scholar

[5]

A. Marciniak-Czochra, Receptor-based models with diffusion-driven instability for pattern formation in hydra, J. Biol. Systems, 11 (2003), 293-324.  doi: 10.1142/S0218339003000889.  Google Scholar

[6]

A. Marciniak-Czochra, Receptor-based models with hysteresis for pattern formation in Hydra, Math. Biosci., 199 (2006), 97-119.  doi: 10.1016/j.mbs.2005.10.004.  Google Scholar

[7]

A. Marciniak-Czochra, M. Nakayama and I. Takagi, Pattern formation in a diffusion-ODE model with hysteresis, Differential Integral Equations, 28 (2015), 655-694.   Google Scholar

[8]

M. Mimura, M. Tabata and Y. Hosono, Multiple solutions of two-point boundary value problems of Neumann type with a small parameter, SIAM J. Math. Anal., 11 (1980), 613-631.  doi: 10.1137/0511057.  Google Scholar

[9]

J. D. Murray, Mathematical Biology. II: Spatial Models and Biomedical Applications, Third edition, Springer, 2003.  Google Scholar

[10]

J. A. Sherrat, P. K. Maini, W. Jäger and W. M$\ddot{\mathrm{u}}$ller, A receptor-based model for pattern formation in hydra, Forma, 10 (1995), 77-95.   Google Scholar

[11]

I. Takagi and H. Yamamoto, Locator function for concentration points in a spatially heterogeneous semilinear Neumann problem, Indiana Univ. Math. J., 68 (2019), 63-103.  doi: 10.1512/iumj.2019.68.7560.  Google Scholar

[12]

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

[13]

J. C. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied Mathematical Sciences, Springer, London, 2014. doi: 10.1007/978-1-4471-5526-3.  Google Scholar

[14]

H. F. Weinberger, A simple system with a continuum of stable inhomogeneous steady states, Nonlinear Partial Differential Equations in Applied Science; Proceedings of the U.S.-Japan Seminar, North-Holland Math. Stud., 81 (1983), 345–359.  Google Scholar

Figure 2.  Nullclines $ f(u,v) = 0 $ and $ g(u,v) = 0 $. The red curve represents $ f(u,v) = 0 $, while the blue curve represents $ g(u,v) = 0. $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 1.  The relationship between $ X_{i,\beta} $ and $ Y_{i,\beta} $. Cases (a) and (b) are exclusive each other; and cases (c) and (d) are exclusive each other. Theorem 3.2 treats cases (a) and (c); Theorem 3.1 (i) deals with case (a); Theorem 3.1 (ii) deals with case (c) and Theorem 3.3 treats cases (b) and (d)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Pattern formation in Example 1. The red curve represents receptor; the blue curve represents ligand; light blue is $ \mu_{2} $; green is $ m_{1}(x) $ and brown is $ m_{2}(x) $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Pattern formation in Example 2. The red curve represents receptor; the blue curve represents ligand; light blue is $ \mu_{2}(x) $ and brown is $ m_{2}(x) $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Pattern formation in Example 3. The red curve represents receptor; the blue curve represents ligand; light blue is $ \mu_{2}(x) $; green is $ m_{1}(x) $ and brown is $ m_{2}(x) $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170
[2]

Linda J. S. Allen, B. M. Bolker, Yuan Lou, A. L. Nevai. Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 1-20. doi: 10.3934/dcds.2008.21.1
[3]

Bo Li, Xiaoyan Zhang. Steady states of a Sel'kov-Schnakenberg reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1009-1023. doi: 10.3934/dcdss.2017053
[4]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[5]

Steffen Härting, Anna Marciniak-Czochra, Izumi Takagi. Stable patterns with jump discontinuity in systems with Turing instability and hysteresis. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 757-800. doi: 10.3934/dcds.2017032
[6]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031
[7]

Junping Shi, Jimin Zhang, Xiaoyan Zhang. Stability and asymptotic profile of steady state solutions to a reaction-diffusion pelagic-benthic algae growth model. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105
[8]

Dagny Butler, Eunkyung Ko, R. Shivaji. Alternate steady states for classes of reaction diffusion models on exterior domains. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1181-1191. doi: 10.3934/dcdss.2014.7.1181
[9]

Joseph G. Yan, Dong-Ming Hwang. Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics. Discrete & Continuous Dynamical Systems, 1996, 2 (2) : 255-270. doi: 10.3934/dcds.1996.2.255
[10]

Maxime Breden, Christian Kuehn, Cinzia Soresina. On the influence of cross-diffusion in pattern formation. Journal of Computational Dynamics, 2021, 8 (2) : 213-240. doi: 10.3934/jcd.2021010
[11]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433
[12]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129
[13]

Wenrui Hao, Jonathan D. Hauenstein, Bei Hu, Yuan Liu, Andrew J. Sommese, Yong-Tao Zhang. Multiple stable steady states of a reaction-diffusion model on zebrafish dorsal-ventral patterning. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1413-1428. doi: 10.3934/dcdss.2011.4.1413
[14]

Keng Deng. Asymptotic behavior of an SIR reaction-diffusion model with a linear source. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5945-5957. doi: 10.3934/dcdsb.2019114
[15]

Keng Deng, Yixiang Wu. Asymptotic behavior for a reaction-diffusion population model with delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 385-395. doi: 10.3934/dcdsb.2015.20.385
[16]

Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure & Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141
[17]

Wei-Ming Ni, Yaping Wu, Qian Xu. The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5271-5298. doi: 10.3934/dcds.2014.34.5271
[18]

R.A. Satnoianu, Philip K. Maini, F.S. Garduno, J.P. Armitage. Travelling waves in a nonlinear degenerate diffusion model for bacterial pattern formation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339
[19]

Martin Baurmann, Wolfgang Ebenhöh, Ulrike Feudel. Turing instabilities and pattern formation in a benthic nutrient-microorganism system. Mathematical Biosciences & Engineering, 2004, 1 (1) : 111-130. doi: 10.3934/mbe.2004.1.111
[20]

Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (82)
  • HTML views (179)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences