Figure 1.
Typical configurations of the zero sets of the kinetic functions
-
Previous Article
Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation
- DCDS Home
- This Issue
-
Next Article
Asymmetric dispersal and evolutional selection in two-patch system
June
2020, 40(6): 3595-3627.
doi: 10.3934/dcds.2020170
Hysteresis-driven pattern formation in reaction-diffusion-ODE systems
| 1. | Institute of Applied Mathematics and Bioquant, Heidelberg University, Heidelberg, 69120, Germany |
| 2. | Institute of Applied Mathematics, Bioquant and Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Heidelberg, 69120, Germany |
| 3. | Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China |
| 4. | Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan |
- Abstract
- Full Text(HTML)
- Figure(8)
- Related Papers
Under a Creative Commons license
The paper is devoted to analysis of far-from-equilibrium pattern formation in a system of a reaction-diffusion equation and an ordinary differential equation (ODE). Such systems arise in modeling of interactions between cellular processes and diffusing growth factors. Pattern formation results from hysteresis in the dependence of the quasi-stationary solution of the ODE on the diffusive component. Bistability alone, without hysteresis, does not result in stable patterns. We provide a systematic description of the hysteresis-driven stationary solutions, which may be monotone, periodic or irregular. We prove existence of infinitely many stationary solutions with jump discontinuity and their asymptotic stability for a certain class of reaction-diffusion-ODE systems. Nonlinear stability is proved using direct estimates of the model nonlinearities and properties of the strongly continuous diffusion semigroup.
Keywords:
Reaction-diffusion-ODE system,
pattern formation,
bistable kinetics,
hysteresis,
stationary solution with jump discontinuity,
asymptotic stability.
Mathematics Subject Classification:
Primary: 35K57, 35B36; Secondary: 35J25, 35K20.
Citation:
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
![]()
References:
| [1] |
D. Angeli, J. E. Ferrell and E. D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, PNAS, 101 (2004), 1822–1827.
doi: 10.1073/pnas.0308265100. |
| [2] |
V. I. Arnold, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2063-1. |
| [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] |
J. E. Ferrell and W. Xiong,
Bistability in cell signaling: How to make continuos processes discontinous, and reversible processes irreversible, Chaos, 11 (2001), 227-236.
doi: 10.1063/1.1349894. |
| [5] |
T. Gregor, E. F. Wieschaus, A. P. McGregor, W. Bialek and D. W. Tank,
Stability and nuclear dynamics of the bicoid morphogen gradient, Cell, 130 (2007), 141-152.
doi: 10.1016/j.cell.2007.05.026. |
| [6] |
S. Härting, A. Marciniak-Czochra and I. Takagi,
Stable patterns with jump discontinuity in systems with Turing instability and hysteresis, Discrete Contin. Dyn. Syst. Ser. A., 37 (2017), 757-800.
doi: 10.3934/dcds.2017032. |
| [7] |
S. Härting and A. Marciniak-Czochra,
Spike patterns in a reaction-diffusion-ode model with Turing instability, Math. Meth. Appl. Sci., 37 (2014), 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 Syst. Biol., 7 (2013), 48.
doi: 10.1186/1752-0509-7-48. |
| [9] |
J. Jaros and T. Kusano,
A picone type identity for second order half-linear differential equations, Acta Math. Univ. Comenian, 68 (1999), 137-151.
|
| [10] |
V. Klika, R. Baker, D. Headon and E. Gaffney,
The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organization, Bulletin of Mathematical Biology, 74 (2012), 935-957.
doi: 10.1007/s11538-011-9699-4. |
| [11] |
S. Kondo and T. Miura,
Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620.
doi: 10.1126/science.1179047. |
| [12] |
K. Korvasová, E. A. Gaffney, P. K. Maini, M. A. Ferreira and V. Klika,
Investigating the turing conditions for diffusion-driven instability in the presence of a binding immobile substrate, J. Theor. Biol., 367 (2015), 286-295.
doi: 10.1016/j.jtbi.2014.11.024. |
| [13] |
Y. Li, A. Marciniak-Czochra, I. Takagi and B. 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. |
| [14] |
W. S. Loud, "Periodic solutions of $x'' +cx' +g(x) = \epsilon f(t)''$, Mem. Amer. Math. Soc., 31 1959, 58 pp. |
| [15] |
A. Marasco and et al.,
Vegetation pattern formation due to interactions between water availability and toxicity in plant-soil feedback, Bull. Math. Biol., 76 (2014), 2866-2883.
doi: 10.1007/s11538-014-0036-6. |
| [16] |
A. Marciniak-Czochra,
Receptor-based models with diffusion-driven instability for pattern formation in hydra, J. Biol. Sys., 11 (2003), 293-324.
doi: 10.1142/S0218339003000889. |
| [17] |
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. |
| [18] |
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. |
| [19] |
A. Marciniak-Czochra, G. Karch and K. Suzuki, Instability of Turing patterns in reaction-diffusion-ODE systems, J. Math. Biol. 74 (2017), 583-618.
doi: 10.1007/s00285-016-1035-z. |
| [20] |
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. |
| [21] |
A. Marciniak-Czochra and M. Kimmel,
Modeling 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. |
| [22] |
A. Marciniak-Czochra, M. Nakayama and I. Takagi,
Pattern formation in a diffusion-ODE model with hysteresis, Differential Integral Equations, 28 (2015), 655-694.
|
| [23] |
A. Marciniak-Czochra and M. Ptashnyk,
Derivation of a macroscopic receptor-based model using homogenization techniques., SIAM J. Math. Anal., 40 (2008), 215-237.
doi: 10.1137/050645269. |
| [24] |
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. |
| [25] |
C. Niehrs,
The Spemann organizer and embryonic head induction, EMBO J., 20 (2001), 631-637.
|
| [26] |
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. |
| [27] |
F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, Springer, 1984.
doi: 10.1007/BFb0099278. |
| [28] |
R. Schaaf, Global Solution Branches of Two Point Boundary Value Problems, Lecture Notes in Mathematics, 1458, Springer, 1990.
doi: 10.1007/BFb0098346. |
| [29] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 258, Springer, New York; Heidelberg; Berlin, 1983. |
| [30] |
A. M. Turing,
The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
| [31] |
D. M. Umulis, M. Serpe, M. B. O'Connor and H. G. Othmer,
Robust, bistable patterning of the dorsal surface of the Drosophila embryo,, Proc. Nat. Ac. Sci., 103 (2006), 11613-11618.
doi: 10.1073/pnas.0510398103. |
show all references
References:
| [1] |
D. Angeli, J. E. Ferrell and E. D. Sontag, Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, PNAS, 101 (2004), 1822–1827.
doi: 10.1073/pnas.0308265100. |
| [2] |
V. I. Arnold, Mathematical Methods of Classical Mechanics, Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2063-1. |
| [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] |
J. E. Ferrell and W. Xiong,
Bistability in cell signaling: How to make continuos processes discontinous, and reversible processes irreversible, Chaos, 11 (2001), 227-236.
doi: 10.1063/1.1349894. |
| [5] |
T. Gregor, E. F. Wieschaus, A. P. McGregor, W. Bialek and D. W. Tank,
Stability and nuclear dynamics of the bicoid morphogen gradient, Cell, 130 (2007), 141-152.
doi: 10.1016/j.cell.2007.05.026. |
| [6] |
S. Härting, A. Marciniak-Czochra and I. Takagi,
Stable patterns with jump discontinuity in systems with Turing instability and hysteresis, Discrete Contin. Dyn. Syst. Ser. A., 37 (2017), 757-800.
doi: 10.3934/dcds.2017032. |
| [7] |
S. Härting and A. Marciniak-Czochra,
Spike patterns in a reaction-diffusion-ode model with Turing instability, Math. Meth. Appl. Sci., 37 (2014), 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 Syst. Biol., 7 (2013), 48.
doi: 10.1186/1752-0509-7-48. |
| [9] |
J. Jaros and T. Kusano,
A picone type identity for second order half-linear differential equations, Acta Math. Univ. Comenian, 68 (1999), 137-151.
|
| [10] |
V. Klika, R. Baker, D. Headon and E. Gaffney,
The influence of receptor-mediated interactions on reaction-diffusion mechanisms of cellular self-organization, Bulletin of Mathematical Biology, 74 (2012), 935-957.
doi: 10.1007/s11538-011-9699-4. |
| [11] |
S. Kondo and T. Miura,
Reaction-diffusion model as a framework for understanding biological pattern formation, Science, 329 (2010), 1616-1620.
doi: 10.1126/science.1179047. |
| [12] |
K. Korvasová, E. A. Gaffney, P. K. Maini, M. A. Ferreira and V. Klika,
Investigating the turing conditions for diffusion-driven instability in the presence of a binding immobile substrate, J. Theor. Biol., 367 (2015), 286-295.
doi: 10.1016/j.jtbi.2014.11.024. |
| [13] |
Y. Li, A. Marciniak-Czochra, I. Takagi and B. 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. |
| [14] |
W. S. Loud, "Periodic solutions of $x'' +cx' +g(x) = \epsilon f(t)''$, Mem. Amer. Math. Soc., 31 1959, 58 pp. |
| [15] |
A. Marasco and et al.,
Vegetation pattern formation due to interactions between water availability and toxicity in plant-soil feedback, Bull. Math. Biol., 76 (2014), 2866-2883.
doi: 10.1007/s11538-014-0036-6. |
| [16] |
A. Marciniak-Czochra,
Receptor-based models with diffusion-driven instability for pattern formation in hydra, J. Biol. Sys., 11 (2003), 293-324.
doi: 10.1142/S0218339003000889. |
| [17] |
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. |
| [18] |
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. |
| [19] |
A. Marciniak-Czochra, G. Karch and K. Suzuki, Instability of Turing patterns in reaction-diffusion-ODE systems, J. Math. Biol. 74 (2017), 583-618.
doi: 10.1007/s00285-016-1035-z. |
| [20] |
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. |
| [21] |
A. Marciniak-Czochra and M. Kimmel,
Modeling 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. |
| [22] |
A. Marciniak-Czochra, M. Nakayama and I. Takagi,
Pattern formation in a diffusion-ODE model with hysteresis, Differential Integral Equations, 28 (2015), 655-694.
|
| [23] |
A. Marciniak-Czochra and M. Ptashnyk,
Derivation of a macroscopic receptor-based model using homogenization techniques., SIAM J. Math. Anal., 40 (2008), 215-237.
doi: 10.1137/050645269. |
| [24] |
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. |
| [25] |
C. Niehrs,
The Spemann organizer and embryonic head induction, EMBO J., 20 (2001), 631-637.
|
| [26] |
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. |
| [27] |
F. Rothe, Global Solutions of Reaction-Diffusion Systems, Lecture Notes in Mathematics, 1072, Springer, 1984.
doi: 10.1007/BFb0099278. |
| [28] |
R. Schaaf, Global Solution Branches of Two Point Boundary Value Problems, Lecture Notes in Mathematics, 1458, Springer, 1990.
doi: 10.1007/BFb0098346. |
| [29] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 258, Springer, New York; Heidelberg; Berlin, 1983. |
| [30] |
A. M. Turing,
The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
| [31] |
D. M. Umulis, M. Serpe, M. B. O'Connor and H. G. Othmer,
Robust, bistable patterning of the dorsal surface of the Drosophila embryo,, Proc. Nat. Ac. Sci., 103 (2006), 11613-11618.
doi: 10.1073/pnas.0510398103. |
Figure 2.
(A) Phase plane of $\gamma^{-1}U_{xx}+q \bar{u}(U)=0$. The blue trajectory $(U(x),U_x(x))$ connects the points $(u_0,0)$ and $(u_e,0)$ and is a solution of the boundary value problem satisfying $U_x(0)=U_x(1)=0$. (B) A monotone increasing stationary solution with jump at $\bar{u}$ and layer position at $\bar{x}$
Figure 3.
The time-maps for the kinetic functions $f(u,v)=1.4v-u$, $g(u,v)=u-(v^3-6.3v^2+10v)$ and the jump $\bar{u}=3.1$. Here, it holds $Q \bar{u}(u_2)<0$ and umin=0.8957. We see that Tu1 is decreasing, whereas $T_{\bar{u}}^{2}$ is increasing, which leads to $\frac{\mathrm{d}}{\mathrm{d} u_{0}} T_{\bar{u}}(u_0)<0$. Furthermore, we observe that $\lim_{u_0\to {\bar{u}}}T_{\bar{u}}{}(u_0)=0$ and $\lim_{u_0\to u_{\rm{min}}}T_{\bar{u}}{}(u_0)=\infty$
Figure 4.
Simulations of the generic model with hysteresis for different types of perturbations of a stationary solution. The plots show initial conditions (dotted lines) and the approached stationary solution (continuous lines) after a sufficiently large time $ t_{end} $
Figure 5.
The layer position $ \bar{x}( \bar{u}) $ and the Interval $ I^0 $ . (A) and (B) Plots for the kinetic functions eq. (60) and for diffusion coefficients $ 1/\gamma $ with $ \gamma=50 $ in A and $ \gamma=200 $ in B. (C) Plots for the kinetic functions eq. (61) and the diffusion coefficient $ 1/200 $
Figure 6.
The phase planes of $\frac{1}{\gamma} U_{x x}+q_{H}(U)=0$ and $\frac{1}{\gamma} U_{x x}+q{T}(U)=0$ are overlapping. In blue we see a periodic solution with jump at u. We cannot determine the mode of a periodic solution in the phase plane. It corresponds to how often the trajectory has been traveled through. In red we see a irregular solution with three different jumps
Figure 7.
An irregular solution $ \big(U(x), V(x)\big) $ with three jumps $ \bar{u}^{{1}}, \bar{u}^{{2}}, \bar{u}^{{3}} $ , which is monotone increasing restricted to $ [0, x^1] $ . We see that for continuity of $ U(x) $ we need to have $ u_{e}^1=u_0^2 $ and $ u_{e}^2=u_0^3 $ fulfilled. Furthermore, we see how the partition of the interval is determined: $ x^1=T(\bar{u}^{{1}}, u_0^1), x^2=x^1+T(\bar{u}^{{2}}, u_0^2) $ and $ 1=x^2+T(\bar{u}^{{3}}, u_0^3) $ . The layer positions are given by $ \bar{x}^{{1}}=T_1(\bar{u}^{{1}}, u_0^1) $ and $ \bar{x}^{{3}}=x^2+T_1(\bar{u}^{{3}}, u_0^3) $ , because $ U(x) $ is increasing on the corresponding subintervals and $ \bar{x}^{{2}}=x^1+T_2(\bar{u}^{{2}}, u_0^2)=x^2-T_1(\bar{u}^{{2}}, u_0^3) $
Figure 8.
Simulations of model (1)-(3) for admissible kinetic functions, diffusion coefficient $ 1/\gamma=1/1000 $ and initial conditions of type (59) having four discontinuities. The $ u $ -component is plotted in blue, whereas the $ v $ -component is red. The initial condition $ \big(u(0, x), v(0, x)\big)=\big(u_0(x), v_0(x)\big) $ is indicated by dotted lines and the stationary solution $ \big(u(t_{end}, x), v(t_{end}, x)\big) $ is indicated by continuous bold lines. Here, $ t_{end} $ is a sufficiently large timepoint, such that the solution $ \big(u(t, x), v(t, x)\big) $ does not change in time anymore. A-C the kinetic functions are given by (60) D the kinetic functions are given by (62)
| [1] |
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 |
| [2] |
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 |
| [3] |
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 |
| [4] |
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 |
| [5] |
Pengchao Lai, Qi Li. Asymptotic behavior for the solutions to a bistable-bistable reaction diffusion equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3313-3323. doi: 10.3934/dcdsb.2021186 |
| [6] |
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 |
| [7] |
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 |
| [8] |
Meng-Xue Chang, Bang-Sheng Han, Xiao-Ming Fan. Global dynamics of the solution for a bistable reaction diffusion equation with nonlocal effect. Electronic Research Archive, 2021, 29 (5) : 3017-3030. doi: 10.3934/era.2021024 |
| [9] |
Gaetana Gambino, Valeria Giunta, Maria Carmela Lombardo, Gianfranco Rubino. Cross-diffusion effects on stationary pattern formation in the FitzHugh-Nagumo model. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022063 |
| [10] |
Qi Wang, Ling Jin, Zengyan Zhang. Global well-posedness, pattern formation and spiky stationary solutions in a Beddington–DeAngelis competition system. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2105-2134. doi: 10.3934/dcds.2020108 |
| [11] |
Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229 |
| [12] |
Philip M. J. Trevelyan. Approximating the large time asymptotic reaction zone solution for fractional order kinetics $A^n B^m$. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 219-234. doi: 10.3934/dcdss.2012.5.219 |
| [13] |
Congming Li, Eric S. Wright. Global existence of solutions to a reaction diffusion system based upon carbonate reaction kinetics. Communications on Pure and Applied Analysis, 2002, 1 (1) : 77-84. doi: 10.3934/cpaa.2002.1.77 |
| [14] |
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 |
| [15] |
Hiroshi Matsuzawa. On a solution with transition layers for a bistable reaction-diffusion equation with spatially heterogeneous environments. Conference Publications, 2009, 2009 (Special) : 516-525. doi: 10.3934/proc.2009.2009.516 |
| [16] |
Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115 |
| [17] |
Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure and Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141 |
| [18] |
Alessandro Audrito. Bistable reaction equations with doubly nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 2977-3015. doi: 10.3934/dcds.2019124 |
| [19] |
Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775 |
| [20] |
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 and Continuous Dynamical Systems - B, 2001, 1 (3) : 339-362. doi: 10.3934/dcdsb.2001.1.339 |
2021 Impact Factor: 1.588
Tools
Article outline
Figures and Tables
[Back to Top]