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January
2021, 26(1): 173-190.
doi: 10.3934/dcdsb.2020329
The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect
Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan |
In this paper, we analyze the interaction of localized patterns such as traveling wave solutions for reaction-diffusion systems with nonlocal effect in one space dimension. We consider the case that a nonlocal effect is given by the convolution with a suitable integral kernel. At first, we deduce the equation describing the movement of interacting localized patterns in a mathematically rigorous way, assuming that there exists a linearly stable localized solution for general reaction-diffusion systems with nonlocal effect. When the distances between localized patterns are sufficiently large, the motion of localized patterns can be reduced to the equation for the distances between them. Finally, using this equation, we analyze the interaction of front solutions to some nonlocal scalar equation. Under some assumptions, we can show that the front solutions are interacting attractively for a large class of integral kernels.
Keywords:
Interaction of pulse solutions,
interaction of front solutions,
nonlocal effect,
convolution,
integral kernel.
Mathematics Subject Classification:
Primary: 35B36, 37L10; Secondary: 35K57.
Citation:
Shin-Ichiro Ei, Hiroshi Ishii. The motion of weakly interacting localized patterns for reaction-diffusion systems with nonlocal effect. Discrete and Continuous Dynamical Systems - B,
2021, 26
(1)
: 173-190.
doi: 10.3934/dcdsb.2020329
![]()
References:
| [1] |
F. Andreu-Vaillo, J. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI, Real Sociedad Matemática Española, Madrid, 2010.
doi: 10.1090/surv/165. |
| [2] |
P. W. Bates,
On some nonlocal evolution equations arising in materials science, Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 13-52.
|
| [3] |
P. W. Bates and F. Chen,
Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation, J. Math. Anal. Appl., 273 (2002), 45-57.
doi: 10.1016/S0022-247X(02)00205-6. |
| [4] |
P. W. Bates and F. Chen,
Spectral analysis of traveling waves for nonlocal evolution equations, SIAM J. Math. Anal., 38 (2006), 116-126.
doi: 10.1137/S0036141004443968. |
| [5] |
P. W. Bates, X. Chen and A. J. J. Chmaj,
Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions, Calc. Var., 24 (2005), 261-281.
doi: 10.1007/s00526-005-0308-y. |
| [6] |
P. W. Bates, P. C. Fife, X. Ren and X. Wang,
Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136.
doi: 10.1007/s002050050037. |
| [7] |
J. A. Carrillo, H. Murakawa, M. Sato, H. Togashi and O. Trush,
A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation, J. Theor. Biology, 474 (2019), 14-24.
doi: 10.1016/j.jtbi.2019.04.023. |
| [8] |
F. Chen,
Almost periodic traveling waves of nonlocal evolution equations, Nonlinear Anal., 50 (2002), 807-838.
doi: 10.1016/S0362-546X(01)00787-8. |
| [9] |
X. Chen,
Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
|
| [10] |
A. Chmaj and X. Ren,
Homoclinic solutions of an integral equation: Existence and stability, J. Differential Equations, 155 (1999), 17-43.
doi: 10.1006/jdeq.1998.3571. |
| [11] |
J. Coville and L. Dupaigne,
On a non-local equation arising in population dynamics, Proc. R. Soc. Edinb. A, 137 (2007), 727-755.
doi: 10.1017/S0308210504000721. |
| [12] |
A. Doleman, R. A. Gardner and T. J. Kaper,
Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotics approach, Physica D, 122 (1998), 1-36.
doi: 10.1016/S0167-2789(98)00180-8. |
| [13] |
S.-I. Ei,
The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynam. Diff. Eqns., 14 (2002), 85-137.
doi: 10.1023/A:1012980128575. |
| [14] |
S.-I. Ei, J.-S. Guo, H. Ishii and C.-C. Wu, Existence of traveling waves solutions to a nonlocal scalar equation with sign-changing kernel, Journal of Mathematical Analysis and Applications, 487 (2020), 124007, 14 pp.
doi: 10.1016/j.jmaa.2020.124007. |
| [15] |
S.-I. Ei, H. Ishii, S. Kondo, T. Miura and Y. Tanaka, Effective nonlocal kernels on reaction-diffusion networks, Journal of Theoretical Biology, 509 (2021), 110496.
doi: 10.1016/j.jtbi.2020.110496. |
| [16] |
S.-I. Ei and H. Matsuzawa,
The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment, Discrete Contin. Dyn. Syst., 26 (2010), 901-921.
doi: 10.3934/dcds.2010.26.901. |
| [17] |
P. C. Fife and J. B. Mcleod,
The approach of solutions of nonlinear diffusion equations to traveling wave front solutions, Arch. Rat. Mech. Anal., 65 (1977), 335-361.
doi: 10.1007/BF00250432. |
| [18] |
A. Gierer and H. Meinhardt,
A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
| [19] |
V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers,
The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.
doi: 10.1007/s00285-003-0210-1. |
| [20] |
S. Ishihara, M. Otsuji and A. Mochizuki, Transient and steady state of mass conserved reaction-diffusion systems, Phys. Rev. E, 75 (2007), 015203.
doi: 10.1103/PhysRevE.75.015203. |
| [21] |
C. K. R. T. Jones,
Stability of the traveling wave solution of the FitzHugh-Nagumo system, Trans. A. M. S., 286 (1984), 431-469.
doi: 10.1090/S0002-9947-1984-0760971-6. |
| [22] |
S. Kondo,
An updated kernel-based Turing model for studying the mechanisms of biological pattern formation, J. Theoretical Biology, 414 (2017), 120-127.
doi: 10.1016/j.jtbi.2016.11.003. |
| [23] |
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. |
| [24] |
M. Mimura and M. Nagayama,
Nonannihilation dynamics in an exothermic reaction-diffusion system with mono-stable excitability, Chaos, 7 (1997), 817-826.
doi: 10.1063/1.166282. |
| [25] |
J. D. Murray, Mathematical Biology. I. An Introduction, Third edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.
doi: 10.1007/b98869. |
| [26] |
Y. Nagashima, S. Tsugawa, A. Mochizuki, T. Sasaki, H. Fukuda and Y. Oda, A Rho-based reaction-diffusion system governs cell wall patterning in metaxylem vessels, Sci. Rep., 8 (2018), 11542.
doi: 10.1038/s41598-018-29543-y. |
| [27] |
A. Nakamasu, G. Takahashi, A. Kanbe and S. Kondo,
Interactions between zebrafish pigment cells responsible for the generation of Turing patterns, PNAS, 106 (2009), 8429-8434.
doi: 10.1073/pnas.0808622106. |
| [28] |
H. Ninomiya, Y. Tanaka and H. Yamamoto,
Reaction, diffusion and non-local interaction, J. Math. Biol., 75 (2017), 1203-1233.
doi: 10.1007/s00285-017-1113-x. |
| [29] |
K. J. Painter, J. M. Bloomfield, J. A. Sherratt and A. Gerisch,
A nonlocal model for contact attraction and repulsion in heterogeneous cell populations, Bulletin of Mathematical Biology, 77 (2015), 1132-1165.
doi: 10.1007/s11538-015-0080-x. |
| [30] |
J. Siebert and E. Schöll, Front and turing patterns induced by mexican-hat-like nonlocal feedback, Europhys. Lett., 109 (2015), 40014.
doi: 10.1209/0295-5075/109/40014. |
| [31] |
T. Sushida, S. Kondo, K. Sugihara and M. Mimura,
A differential equation model of retinal processing for understanding lightness optical illusions, Japan Journal of Industrial and Applied Mathematics, 35 (2018), 117-156.
doi: 10.1007/s13160-017-0272-x. |
| [32] |
A. M. Turing,
The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. Ser. B, 237 (1953), 37-72.
doi: 10.1098/rstb.1952.0012. |
| [33] |
E. Yanagida,
Stability of fast traveling pulse solutions of the FitzHugh-Nagumo equations, J. Math. Biol., 22 (1985), 81-104.
doi: 10.1007/BF00276548. |
| [34] |
G. Zhao and S. Ruan,
The decay rates of traveling waves and spectral analysis for a class of nonlocal evolution equations, Math. Model. Nat. Phenom., 10 (2015), 142-162.
doi: 10.1051/mmnp/20150610. |
show all references
References:
| [1] |
F. Andreu-Vaillo, J. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, 165. American Mathematical Society, Providence, RI, Real Sociedad Matemática Española, Madrid, 2010.
doi: 10.1090/surv/165. |
| [2] |
P. W. Bates,
On some nonlocal evolution equations arising in materials science, Nonlinear Dynamics and Evolution Equations, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 48 (2006), 13-52.
|
| [3] |
P. W. Bates and F. Chen,
Spectral analysis and multidimensional stability of traveling waves for nonlocal Allen-Cahn equation, J. Math. Anal. Appl., 273 (2002), 45-57.
doi: 10.1016/S0022-247X(02)00205-6. |
| [4] |
P. W. Bates and F. Chen,
Spectral analysis of traveling waves for nonlocal evolution equations, SIAM J. Math. Anal., 38 (2006), 116-126.
doi: 10.1137/S0036141004443968. |
| [5] |
P. W. Bates, X. Chen and A. J. J. Chmaj,
Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions, Calc. Var., 24 (2005), 261-281.
doi: 10.1007/s00526-005-0308-y. |
| [6] |
P. W. Bates, P. C. Fife, X. Ren and X. Wang,
Traveling waves in a convolution model for phase transitions, Arch. Ration. Mech. Anal., 138 (1997), 105-136.
doi: 10.1007/s002050050037. |
| [7] |
J. A. Carrillo, H. Murakawa, M. Sato, H. Togashi and O. Trush,
A population dynamics model of cell-cell adhesion incorporating population pressure and density saturation, J. Theor. Biology, 474 (2019), 14-24.
doi: 10.1016/j.jtbi.2019.04.023. |
| [8] |
F. Chen,
Almost periodic traveling waves of nonlocal evolution equations, Nonlinear Anal., 50 (2002), 807-838.
doi: 10.1016/S0362-546X(01)00787-8. |
| [9] |
X. Chen,
Existence, uniqueness and asymptotic stability of traveling waves in nonlocal evolution equations, Adv. Differential Equations, 2 (1997), 125-160.
|
| [10] |
A. Chmaj and X. Ren,
Homoclinic solutions of an integral equation: Existence and stability, J. Differential Equations, 155 (1999), 17-43.
doi: 10.1006/jdeq.1998.3571. |
| [11] |
J. Coville and L. Dupaigne,
On a non-local equation arising in population dynamics, Proc. R. Soc. Edinb. A, 137 (2007), 727-755.
doi: 10.1017/S0308210504000721. |
| [12] |
A. Doleman, R. A. Gardner and T. J. Kaper,
Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotics approach, Physica D, 122 (1998), 1-36.
doi: 10.1016/S0167-2789(98)00180-8. |
| [13] |
S.-I. Ei,
The motion of weakly interacting pulses in reaction-diffusion systems, J. Dynam. Diff. Eqns., 14 (2002), 85-137.
doi: 10.1023/A:1012980128575. |
| [14] |
S.-I. Ei, J.-S. Guo, H. Ishii and C.-C. Wu, Existence of traveling waves solutions to a nonlocal scalar equation with sign-changing kernel, Journal of Mathematical Analysis and Applications, 487 (2020), 124007, 14 pp.
doi: 10.1016/j.jmaa.2020.124007. |
| [15] |
S.-I. Ei, H. Ishii, S. Kondo, T. Miura and Y. Tanaka, Effective nonlocal kernels on reaction-diffusion networks, Journal of Theoretical Biology, 509 (2021), 110496.
doi: 10.1016/j.jtbi.2020.110496. |
| [16] |
S.-I. Ei and H. Matsuzawa,
The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment, Discrete Contin. Dyn. Syst., 26 (2010), 901-921.
doi: 10.3934/dcds.2010.26.901. |
| [17] |
P. C. Fife and J. B. Mcleod,
The approach of solutions of nonlinear diffusion equations to traveling wave front solutions, Arch. Rat. Mech. Anal., 65 (1977), 335-361.
doi: 10.1007/BF00250432. |
| [18] |
A. Gierer and H. Meinhardt,
A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
| [19] |
V. Hutson, S. Martinez, K. Mischaikow and G. T. Vickers,
The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.
doi: 10.1007/s00285-003-0210-1. |
| [20] |
S. Ishihara, M. Otsuji and A. Mochizuki, Transient and steady state of mass conserved reaction-diffusion systems, Phys. Rev. E, 75 (2007), 015203.
doi: 10.1103/PhysRevE.75.015203. |
| [21] |
C. K. R. T. Jones,
Stability of the traveling wave solution of the FitzHugh-Nagumo system, Trans. A. M. S., 286 (1984), 431-469.
doi: 10.1090/S0002-9947-1984-0760971-6. |
| [22] |
S. Kondo,
An updated kernel-based Turing model for studying the mechanisms of biological pattern formation, J. Theoretical Biology, 414 (2017), 120-127.
doi: 10.1016/j.jtbi.2016.11.003. |
| [23] |
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. |
| [24] |
M. Mimura and M. Nagayama,
Nonannihilation dynamics in an exothermic reaction-diffusion system with mono-stable excitability, Chaos, 7 (1997), 817-826.
doi: 10.1063/1.166282. |
| [25] |
J. D. Murray, Mathematical Biology. I. An Introduction, Third edition, Interdisciplinary Applied Mathematics, 17. Springer-Verlag, New York, 2002.
doi: 10.1007/b98869. |
| [26] |
Y. Nagashima, S. Tsugawa, A. Mochizuki, T. Sasaki, H. Fukuda and Y. Oda, A Rho-based reaction-diffusion system governs cell wall patterning in metaxylem vessels, Sci. Rep., 8 (2018), 11542.
doi: 10.1038/s41598-018-29543-y. |
| [27] |
A. Nakamasu, G. Takahashi, A. Kanbe and S. Kondo,
Interactions between zebrafish pigment cells responsible for the generation of Turing patterns, PNAS, 106 (2009), 8429-8434.
doi: 10.1073/pnas.0808622106. |
| [28] |
H. Ninomiya, Y. Tanaka and H. Yamamoto,
Reaction, diffusion and non-local interaction, J. Math. Biol., 75 (2017), 1203-1233.
doi: 10.1007/s00285-017-1113-x. |
| [29] |
K. J. Painter, J. M. Bloomfield, J. A. Sherratt and A. Gerisch,
A nonlocal model for contact attraction and repulsion in heterogeneous cell populations, Bulletin of Mathematical Biology, 77 (2015), 1132-1165.
doi: 10.1007/s11538-015-0080-x. |
| [30] |
J. Siebert and E. Schöll, Front and turing patterns induced by mexican-hat-like nonlocal feedback, Europhys. Lett., 109 (2015), 40014.
doi: 10.1209/0295-5075/109/40014. |
| [31] |
T. Sushida, S. Kondo, K. Sugihara and M. Mimura,
A differential equation model of retinal processing for understanding lightness optical illusions, Japan Journal of Industrial and Applied Mathematics, 35 (2018), 117-156.
doi: 10.1007/s13160-017-0272-x. |
| [32] |
A. M. Turing,
The chemical basis of morphogenesis, Philos. Trans. R. Soc. Lond. Ser. B, 237 (1953), 37-72.
doi: 10.1098/rstb.1952.0012. |
| [33] |
E. Yanagida,
Stability of fast traveling pulse solutions of the FitzHugh-Nagumo equations, J. Math. Biol., 22 (1985), 81-104.
doi: 10.1007/BF00276548. |
| [34] |
G. Zhao and S. Ruan,
The decay rates of traveling waves and spectral analysis for a class of nonlocal evolution equations, Math. Model. Nat. Phenom., 10 (2015), 142-162.
doi: 10.1051/mmnp/20150610. |
Figure 2.
The images of localized patterns. (a) Pulse solution when $ P^{+} = P^{-} = \mathit{\boldsymbol{0}} $ . (b) Front solution when $ n = 1 $ and $ P^{+}>P^{-} $
Figure 4.
The image of $ P(z;\mathit{\boldsymbol{h}}) $ . (a) $ (N^+, N^-) = (1,1) $ . (b) $ (N^+, N^-) = (2,1) $
Figure 5.
The image of the interaction of two standing fronts, when the kernel is a non-negative function
Figure 6.
(a) The graph of (30) when $ \varepsilon = 0.01,\ q_1 = 1.0,\ q_2 = 2.0 $ . (b) The graph of $ G(\lambda) $ when $ d = 1.0 $ , $ f'(1) = -1 $ and the integral kernel is same as Figure 6 (a)
Figure 7.
(a) The numerical solution of $ (2) $ on the interval $ (0,40) $ when $ t = 100.0 $ , where $ f(u) = \frac{1}{2}u(1-u^2) $ and the other parameters are same as that in >Figure 6. (b) The graph of $ \log|u(t,x)-1| $ on the interval $ (20,35) $ when $ t = 100.0 $ , where $ u(t,x) $ is the numerical solution of $ (2) $
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