# American Institute of Mathematical Sciences

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

 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.

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

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##### 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
Nullclines $f(u,v) = 0$ and $g(u,v) = 0$. The red curve represents $f(u,v) = 0$, while the blue curve represents $g(u,v) = 0.$
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)
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 $m_{2}(x)$
Pattern formation in Example 2. The red curve represents receptor; the blue curve represents ligand; light blue is $\mu_{2}(x)$ and brown is $m_{2}(x)$
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 $m_{2}(x)$
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