# American Institute of Mathematical Sciences

February  2017, 37(2): 757-800. doi: 10.3934/dcds.2017032

## Stable patterns with jump discontinuity in systems with Turing instability and hysteresis

 1 Institute of Applied Mathematics and BIOQUANT, Heidelberg University, Im Neuenheimer Feld 205,69120 Heidelberg, Germany 2 Institute of Applied Mathematics, IWR and BIOQUANT, Heidelberg University, Im Neuenheimer Feld 205,69120 Heidelberg, Germany 3 Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

* Corresponding author: takagi@m.tohoku.ac.jp.

Received  May 2015 Revised  November 2015 Published  November 2016

Fund Project: This work was undertaken in the framework of German-Japanese University Partnership Program (HeKKSaGOn Alliance). The first two authors were supported by European Research Council Starting Grant No 210680 'Multiscale mathematical modelling of dynamics of structure formation in cell systems' and Emmy Noether Program of DFG. The second author was supported by 'Heidelberg Graduate School of Mathematical and Computational Methods for the Sciences' and 'Baden-Württemberg Stipendium plus' scholarship of Baden-Württemberg Stiftung. The third author was supported in part by JSPS Grant-in-Aid for Scientific Research (A) #22244010 'Theory of Differential Equations Applied to Biological Pattern Formation|from Analysis to Synthesis' and #26610027 'Control of Patterns by Multi-component Reaction-Diffusion Systems of Degenerate Type'.

Classical models of pattern formation are based on diffusion-driven instability (DDI) of constant stationary solutions of reaction-diffusion equations, which leads to emergence of stable, regular Turing patterns formed around that equilibrium. In this paper we show that coupling reaction-diffusion equations with ordinary differential equations (ODE) may lead to a novel pattern formation phenomenon in which DDI causes destabilization of both constant solutions and Turing patterns. Bistability and hysteresis effects in the null sets of model nonlinearities yield formation of far from the equilibrium patterns with jump discontinuity. We derive conditions for stability of stationary solutions with jump discontinuity in a suitable topology which allows us to include the discontinuity points and leads to the definition of $(\varepsilon_0, A)$-stability. Additionally, we provide conditions on stability of patterns in a quasi-stationary model reduction. The analysis is illustrated on the example of three-component model of receptor-ligand binding. The proposed model provides an example of a mechanism of de novo formation of far from the equilibrium patterns in reaction-diffusion-ODE models involving co-existence of DDI and hysteresis.

Citation: Steffen Härting, Anna Marciniak-Czochra, Izumi Takagi. Stable patterns with jump discontinuity in systems with Turing instability and hysteresis. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 757-800. doi: 10.3934/dcds.2017032
##### References:
 [1] M. Akam, Making stripes inelegantly, Nature, 341 (1989), 282-283.  doi: 10.1038/341282a0. [2] A. Anma, K. Sakamoto and T. Yoneda, Unstable subsystems cause Turing instability, Kodai Math. J., 35 (2012), 215-247.  doi: 10.2996/kmj/1341401049. [3] 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. [4] W. Bangerth, R. Hartmann and G. Kanschat, deal. Ⅱ -a general purpose object oriented finite element library, ACM Trans. Math. Softw. , 33 (2007), Art. 24, 27 pp. [5] R. Casten and C. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations, 27 (1978), 266-273.  doi: 10.1016/0022-0396(78)90033-5. [6] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin), 12 (1972), 30-39.  doi: 10.1007/BF00289234. [7] S. Härting and A. Marciniak-Czochra, Spike patterns in a reaction-diffusion-ode model with Turing instability, Math. Methods Appl. Sci., 37 (2013), 1377-1391. [8] S. Hock, Y. Ng, J. Hasenauer, D. Wittmann, D. Lutter, D. Trümbach, W. Wurst, N. Prakash and F. J. Theis, Sharpening of expression domains induced by transcription and microRNA regulation within a spatio-temporal model of mid-hindbrain boundary formation, BMC Systems Biol., 7 (2013), p48.  doi: 10.1186/1752-0509-7-48. [9] V. Klika, R. Baker, D. Headon and E. Gaffney, The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organisation, Bull. Math. Biol., 74 (2012), 935-957.  doi: 10.1007/s11538-011-9699-4. [10] A. Köthe and A. Marciniak-Czochra, Multistability and hysteresis-based mechanism of pattern formation in biology, in Pattern Formation in Morphogenesis-problems and their Mathematical Formalization (eds. V. Capasso, M. Gromov and N. Morozova), Springer Proceedings in Mathematics, 15 (2012), 153-173. [11] D. A. Lauffenburger and J. J. Linderman, Receptors. Models for Binding, Trafficking, and Signaling, Oxford University Press, 1993. [12] 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. [13] 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. [14] A. Marciniak-Czochra, Strong two-scale convergence and corrector result for the receptor-based model of the intercellular communication, IMA J. Appl. Math., 77 (2012), 855-868.  doi: 10.1093/imamat/hxs052. [15] A. Marciniak-Czochra, G. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl., 99 (2013), 509-543.  doi: 10.1016/j.matpur.2012.09.011. [16] A. Marciniak-Czochra, G. Karch and K. Suzuki, Instability of turing patterns in reaction-diffusion-ODE systems, J. Math. Biol., (2016).  doi: 10.1007/s00285-016-1035-z. [17] A. Marciniak-Czochra and M. Kimmel, Modelling of early lung cancer progression: Influence of growth factor production and cooperation between partially transformed cells, Math. Models Methods Appl. Sci., 17 (2007), 1693-1719.  doi: 10.1142/S0218202507002443. [18] A. Marciniak-Czochra, M. Nakayama and I. Takagi, Pattern formation in a diffusion-ODE model with hysteresis, Differential Integral Equations, 28 (2015), 655-694. [19] A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenisation techniques, SIAM J. Math. Anal., 40 (2008), 215-237.  doi: 10.1137/050645269. [20] 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. [21] J. D. Murray, Mathematical Biology. Ⅱ. Spatial Models and Biomedical Applications 3rd edition, Interdisciplinary Applied Mathematics, 18, 2003, Springer-Verlag, New York. [22] W. A. Müller, Developmental Biology, Springer-Verlag, New York, 1997. [23] K. Pham, A. Chauviere, H. Hatzikirou, X. Li, H.M. Byrne, V. Cristini and J. Lowengrub, Density-dependent quiescence in glioma invasion: Instability in a simple reaction-diffusion model for the migration/proliferation dichotomy, J. Biol. Dyn., 6 (2012), 54-71.  doi: 10.1080/17513758.2011.590610. [24] F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, 1984, Springer-Verlag, Berlin. [25] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2 edition, Springer-Verlag, New York, 1994.  doi: 10.1007/978-1-4612-0873-0. [26] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012. [27] D. M. Umulis, M. Serpe, M. B. O'Connor and H. G. Othmer, Robust, bistable patterning of the dorsal surface of the Drosophila embryo, PNAS, 103 (2006), 11613-11618.  doi: 10.1073/pnas.0510398103. [28] 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. [29] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Heidelberg/Dordrecht/London/New York, 2010.  doi: 10.1007/978-3-642-04631-5.

show all references

##### References:
 [1] M. Akam, Making stripes inelegantly, Nature, 341 (1989), 282-283.  doi: 10.1038/341282a0. [2] A. Anma, K. Sakamoto and T. Yoneda, Unstable subsystems cause Turing instability, Kodai Math. J., 35 (2012), 215-247.  doi: 10.2996/kmj/1341401049. [3] 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. [4] W. Bangerth, R. Hartmann and G. Kanschat, deal. Ⅱ -a general purpose object oriented finite element library, ACM Trans. Math. Softw. , 33 (2007), Art. 24, 27 pp. [5] R. Casten and C. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Differential Equations, 27 (1978), 266-273.  doi: 10.1016/0022-0396(78)90033-5. [6] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin), 12 (1972), 30-39.  doi: 10.1007/BF00289234. [7] S. Härting and A. Marciniak-Czochra, Spike patterns in a reaction-diffusion-ode model with Turing instability, Math. Methods Appl. Sci., 37 (2013), 1377-1391. [8] S. Hock, Y. Ng, J. Hasenauer, D. Wittmann, D. Lutter, D. Trümbach, W. Wurst, N. Prakash and F. J. Theis, Sharpening of expression domains induced by transcription and microRNA regulation within a spatio-temporal model of mid-hindbrain boundary formation, BMC Systems Biol., 7 (2013), p48.  doi: 10.1186/1752-0509-7-48. [9] V. Klika, R. Baker, D. Headon and E. Gaffney, The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organisation, Bull. Math. Biol., 74 (2012), 935-957.  doi: 10.1007/s11538-011-9699-4. [10] A. Köthe and A. Marciniak-Czochra, Multistability and hysteresis-based mechanism of pattern formation in biology, in Pattern Formation in Morphogenesis-problems and their Mathematical Formalization (eds. V. Capasso, M. Gromov and N. Morozova), Springer Proceedings in Mathematics, 15 (2012), 153-173. [11] D. A. Lauffenburger and J. J. Linderman, Receptors. Models for Binding, Trafficking, and Signaling, Oxford University Press, 1993. [12] 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. [13] 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. [14] A. Marciniak-Czochra, Strong two-scale convergence and corrector result for the receptor-based model of the intercellular communication, IMA J. Appl. Math., 77 (2012), 855-868.  doi: 10.1093/imamat/hxs052. [15] A. Marciniak-Czochra, G. Karch and K. Suzuki, Unstable patterns in reaction-diffusion model of early carcinogenesis, J. Math. Pures Appl., 99 (2013), 509-543.  doi: 10.1016/j.matpur.2012.09.011. [16] A. Marciniak-Czochra, G. Karch and K. Suzuki, Instability of turing patterns in reaction-diffusion-ODE systems, J. Math. Biol., (2016).  doi: 10.1007/s00285-016-1035-z. [17] A. Marciniak-Czochra and M. Kimmel, Modelling of early lung cancer progression: Influence of growth factor production and cooperation between partially transformed cells, Math. Models Methods Appl. Sci., 17 (2007), 1693-1719.  doi: 10.1142/S0218202507002443. [18] A. Marciniak-Czochra, M. Nakayama and I. Takagi, Pattern formation in a diffusion-ODE model with hysteresis, Differential Integral Equations, 28 (2015), 655-694. [19] A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenisation techniques, SIAM J. Math. Anal., 40 (2008), 215-237.  doi: 10.1137/050645269. [20] 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. [21] J. D. Murray, Mathematical Biology. Ⅱ. Spatial Models and Biomedical Applications 3rd edition, Interdisciplinary Applied Mathematics, 18, 2003, Springer-Verlag, New York. [22] W. A. Müller, Developmental Biology, Springer-Verlag, New York, 1997. [23] K. Pham, A. Chauviere, H. Hatzikirou, X. Li, H.M. Byrne, V. Cristini and J. Lowengrub, Density-dependent quiescence in glioma invasion: Instability in a simple reaction-diffusion model for the migration/proliferation dichotomy, J. Biol. Dyn., 6 (2012), 54-71.  doi: 10.1080/17513758.2011.590610. [24] F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, 1984, Springer-Verlag, Berlin. [25] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2 edition, Springer-Verlag, New York, 1994.  doi: 10.1007/978-1-4612-0873-0. [26] A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012. [27] D. M. Umulis, M. Serpe, M. B. O'Connor and H. G. Othmer, Robust, bistable patterning of the dorsal surface of the Drosophila embryo, PNAS, 103 (2006), 11613-11618.  doi: 10.1073/pnas.0510398103. [28] 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. [29] A. Yagi, Abstract Parabolic Evolution Equations and their Applications, Springer-Verlag, Heidelberg/Dordrecht/London/New York, 2010.  doi: 10.1007/978-3-642-04631-5.
Illustration of the topology applied to problem (1.1) for scalar $u$. Model (1.1) exhibits steady states with jump discontinuity and global existence of classical solutions. $\tilde{u}$ represents a steady state while $u(t,x)$ represents a solution for some $t$
Plot of the nullclines of $f_r(u,v)=-(1+v)u+m_1(u^2/(1+ku^2))$ for $u\neq 0$ and $g_r(u,v)=-(\mu_3+u)v+m_2 (u^2/(1+ku^2))$.
Numerically obtained solution to model (3.3)-(3.5) for parameters $m_1 =1.44, m_2 = 2,\mu_3 \approx 4.1, k=0.01, D=1$. We observe convergence towards a steady state with jump discontinuity. Left: Non-diffusive component $u$. Right: Diffusive component $v$.
Numerically obtained solution component $u$ of model (3.3)-(3.5) for parameters $m_1 =1.44, m_2 = 2,\mu_3 \approx 4.1, k=0.01$ with varying diffusion coefficient: upper left: $D=5$, upper right: $D=1$, lower left: $D=0.5$, lower right: $D=0.1$. We observe emergence of more jump-type discontinuities for smaller diffusion coefficient. Initial conditions are $u_0(x)=1.725-0.1\cos(2 \pi x^2), v_0(x)=2.48615$
Numerically obtained solution component $u$ of model (3.3)-(3.5) for parameters $m_1 =1.44, m_2 = 2,\mu_3 \approx 4.1, k=0.01, D=5$. We observe emergence of jump-type discontinuities around local maxima of the initial conditions. Initial conditions are $u_0(x)=1.725-0.1x^4\cos(8 \pi x^2), v_0(x)=2.48615$
Nullclines of $f_r$ and $g_r$ and $v_{f_r,g_r}$ for parameters $m_1 = 1.44, m_2 = 2, \mu_3 = 4.2, k=0.1$
Illustration of the right-hand side of $-\partial^2 v/\partial x^2 = g_r(u,v)$ for different branches of the solution $u(v)$ of $u_t=f_r(u,v)=0$. The parameters for illustration are $D=1,m_1=1.44,m_2=2,\mu_3=4.2$. We can observe that all nontrivial homogeneous steady states are of type $u_-$
Illustration of the construction of weak steady states in the proof of Lemma 3.6
 [1] Alexandra Köthe, Anna Marciniak-Czochra, Izumi Takagi. Hysteresis-driven pattern formation in reaction-diffusion-ODE systems. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3595-3627. doi: 10.3934/dcds.2020170 [2] Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3109-3140. doi: 10.3934/dcds.2020400 [3] Weihua Jiang, Xun Cao, Chuncheng Wang. Turing instability and pattern formations for reaction-diffusion systems on 2D bounded domain. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1163-1178. doi: 10.3934/dcdsb.2021085 [4] Lena Noethen, Sebastian Walcher. Tikhonov's theorem and quasi-steady state. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 945-961. doi: 10.3934/dcdsb.2011.16.945 [5] 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 and Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031 [6] Yansu Ji, Jianwei Shen, Xiaochen Mao. Pattern formation of Brusselator in the reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022103 [7] Theodore Kolokolnikov, Michael J. Ward, Juncheng Wei. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1373-1410. doi: 10.3934/dcdsb.2014.19.1373 [8] Joseph G. Yan, Dong-Ming Hwang. Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 255-270. doi: 10.3934/dcds.1996.2.255 [9] Shin-Ichiro Ei, Kota Ikeda, Eiji Yanagida. Instability of multi-spot patterns in shadow systems of reaction-diffusion equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 717-736. doi: 10.3934/cpaa.2015.14.717 [10] Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi, Gregory I. Sivashinsky. A fully nonlinear equation for the flame front in a quasi-steady combustion model. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1415-1446. doi: 10.3934/dcds.2010.27.1415 [11] 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 [12] Samira Boussaïd, Danielle Hilhorst, Thanh Nam Nguyen. Convergence to steady state for the solutions of a nonlocal reaction-diffusion equation. Evolution Equations and Control Theory, 2015, 4 (1) : 39-59. doi: 10.3934/eect.2015.4.39 [13] 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 and Continuous Dynamical Systems - S, 2021, 14 (10) : 3785-3801. doi: 10.3934/dcdss.2020433 [14] Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21 [15] Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589 [16] 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 [17] Yoshihisa Morita, Kunimochi Sakamoto. Turing type instability in a diffusion model with mass transport on the boundary. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3813-3836. doi: 10.3934/dcds.2020160 [18] Georg Hetzer, Anotida Madzvamuse, Wenxian Shen. Characterization of turing diffusion-driven instability on evolving domains. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3975-4000. doi: 10.3934/dcds.2012.32.3975 [19] Tian Ma, Shouhong Wang. Dynamic transition and pattern formation for chemotactic systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2809-2835. doi: 10.3934/dcdsb.2014.19.2809 [20] 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 and Applied Analysis, 2019, 18 (5) : 2325-2347. doi: 10.3934/cpaa.2019105

2020 Impact Factor: 1.392

## Metrics

• HTML views (68)
• Cited by (8)

• on AIMS