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doi: 10.3934/dcdsb.2022136
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## Front propagation and blocking for the competition-diffusion system in a domain of half-lines with a junction

 1 Department of Applied Mathematics and Informatics, Ryukoku University, Seta Otsu 520-2194, Japan 2 Faculty of Mathematics and Physics, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan 3 Department of Mathematics, Josai University, Keyakidai Sakado 350-0295, Japan

*Corresponding author: Yoshihisa Morita

In Memory of Professor Masayasu Mimura

Received  November 2021 Revised  April 2022 Early access July 2022

Fund Project: The first author was partially supported by JSPS KAKENHI Grant Number JP18H01139. The second author was supported by JSPS KAKENHI Grant Number JP18K03412 and JP21K03368

The two-component Lotka-Volterra competition-diffusion system is well accepted as a model describing the invasion of a superior species into a new habitat. Under a bistable condition, we deal with the system in a domain of half-lines with a single junction and investigate the condition for the invasion from some of the half-lines beyond the junction or blocking the propagation of the superior species. We first give a sufficient condition for the invasion in the whole domain by a subsolution. Then, making use of sub- and supersolutions, we construct a standing front solution blocking the propagation if the number of half-lines occupied by the inferior species is sufficiently larger than that occupied by the superior species.

Citation: Yoshihisa Morita, Ken-Ichi Nakamura, Toshiko Ogiwara. Front propagation and blocking for the competition-diffusion system in a domain of half-lines with a junction. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022136
##### References:
 [1] C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.  doi: 10.1512/iumj.1984.33.33018. [2] R. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), 343-364.  doi: 10.1016/0022-0396(82)90001-8. [3] L. Girardin, The effect of random dispersal on competitive exclusion - A review, Math. Biosci., 318 (2019), 108271.  doi: 10.1016/j.mbs.2019.108271. [4] J.-S. Guo and Y.-C. Lin, The sign of the wave speed for the Lotka-Volterra competition-diffusion system, Commun. Pure Appl. Anal., 12 (2013), 2083-2090. [5] Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competition models, Numerical and Applied Mathematics Part Ⅱ, Baltzer, Montréal, (1989), 687–692. [6] S. Jimbo and Y. Morita, Entire solutions to reaction-diffusion equations in multiple half-lines with a junction, J. Differential Equations, 267 (2019), 1247-1276.  doi: 10.1016/j.jde.2019.02.008. [7] S. Jimbo and Y. Morita, Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph, Discrete Contin. Dyn. Syst., 41 (2021), 4013-4039.  doi: 10.3934/dcds.2021026. [8] S. Jimbo and Y. Takazawa, Y-shaped graph and time entire solutions of a semilinear parabolic equation, preprint, (2022). [9] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556. [10] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I. [11] Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221. [12] K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21.  doi: 10.1016/0022-0396(85)90020-8. [13] M. Ma, Z. Huang and C. Ou, Speed of the traveling wave for the bistable Lotka-Volterra competition model, Nonlinearity, 32 (2019), 3143-3162.  doi: 10.1088/1361-6544/ab231c. [14] M. Ma, Q. Zhang, J. Yue and C. Ou, Bistable wave speed of the Lotka-Volterra competition model, J. Biol. Dynam., 14 (2020), 608-620.  doi: 10.1080/17513758.2020.1795284. [15] H. Matano, $L^\infty$ stability of an exponentially decreasing solution of the problem $\Delta u+f(x, u) = 0$ in $\mathbb{R}^n$, Japan J. Appl. Math., 2 (1985), 85-110.  doi: 10.1007/BF03167040. [16] H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci., 19 (1983), 1049-1079.  doi: 10.2977/prims/1195182020. [17] M. Mimura and P. Fife, A 3-component system of competition and diffusion, Hiroshima Math. J., 16 (1986), 189-207. [18] Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715. [19] A. Okubo, P. Maini, M. Williamson and J. Murray, On the spatial spread of the grey squirrel in Britain, Proc. R. Soc. Lond. Ser. B Biol. Sci., 238 (1989), 113-125. [20] M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270. [21] A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994. doi: 10.1090/mmono/140.

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##### References:
 [1] C. Conley and R. Gardner, An application of the generalized Morse index to travelling wave solutions of a competitive reaction-diffusion model, Indiana Univ. Math. J., 33 (1984), 319-343.  doi: 10.1512/iumj.1984.33.33018. [2] R. Gardner, Existence and stability of travelling wave solutions of competition models: A degree theoretic approach, J. Differential Equations, 44 (1982), 343-364.  doi: 10.1016/0022-0396(82)90001-8. [3] L. Girardin, The effect of random dispersal on competitive exclusion - A review, Math. Biosci., 318 (2019), 108271.  doi: 10.1016/j.mbs.2019.108271. [4] J.-S. Guo and Y.-C. Lin, The sign of the wave speed for the Lotka-Volterra competition-diffusion system, Commun. Pure Appl. Anal., 12 (2013), 2083-2090. [5] Y. Hosono, Singular perturbation analysis of travelling waves for diffusive Lotka-Volterra competition models, Numerical and Applied Mathematics Part Ⅱ, Baltzer, Montréal, (1989), 687–692. [6] S. Jimbo and Y. Morita, Entire solutions to reaction-diffusion equations in multiple half-lines with a junction, J. Differential Equations, 267 (2019), 1247-1276.  doi: 10.1016/j.jde.2019.02.008. [7] S. Jimbo and Y. Morita, Asymptotic behavior of entire solutions to reaction-diffusion equations in an infinite star graph, Discrete Contin. Dyn. Syst., 41 (2021), 4013-4039.  doi: 10.3934/dcds.2021026. [8] S. Jimbo and Y. Takazawa, Y-shaped graph and time entire solutions of a semilinear parabolic equation, preprint, (2022). [9] Y. Kan-on, Parameter dependence of propagation speed of travelling waves for competition-diffusion equations, SIAM J. Math. Anal., 26 (1995), 340-363.  doi: 10.1137/S0036141093244556. [10] Y. Kan-on, Fisher wave fronts for the Lotka-Volterra competition model with diffusion, Nonlinear Anal., 28 (1997), 145-164.  doi: 10.1016/0362-546X(95)00142-I. [11] Y. Kan-on and E. Yanagida, Existence of non-constant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221. [12] K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21.  doi: 10.1016/0022-0396(85)90020-8. [13] M. Ma, Z. Huang and C. Ou, Speed of the traveling wave for the bistable Lotka-Volterra competition model, Nonlinearity, 32 (2019), 3143-3162.  doi: 10.1088/1361-6544/ab231c. [14] M. Ma, Q. Zhang, J. Yue and C. Ou, Bistable wave speed of the Lotka-Volterra competition model, J. Biol. Dynam., 14 (2020), 608-620.  doi: 10.1080/17513758.2020.1795284. [15] H. Matano, $L^\infty$ stability of an exponentially decreasing solution of the problem $\Delta u+f(x, u) = 0$ in $\mathbb{R}^n$, Japan J. Appl. Math., 2 (1985), 85-110.  doi: 10.1007/BF03167040. [16] H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci., 19 (1983), 1049-1079.  doi: 10.2977/prims/1195182020. [17] M. Mimura and P. Fife, A 3-component system of competition and diffusion, Hiroshima Math. J., 16 (1986), 189-207. [18] Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715. [19] A. Okubo, P. Maini, M. Williamson and J. Murray, On the spatial spread of the grey squirrel in Britain, Proc. R. Soc. Lond. Ser. B Biol. Sci., 238 (1989), 113-125. [20] M. Rodrigo and M. Mimura, Exact solutions of a competition-diffusion system, Hiroshima Math. J., 30 (2000), 257-270. [21] A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994. doi: 10.1090/mmono/140.
Occupation of the species $U$ and $V$ in the domain $\Omega$ with $m = 4, \ell = 3$. The vertical line indicates the values of $U$ and $V$ in the domain
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