# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021250
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## Reaction-advection-diffusion competition models under lethal boundary conditions

 1 College of General Education, Kookmin University, 77, Jeongneung-Ro, Seoul, 02707, Republic of Korea 2 Department of Mathematics, Korea University, 145, Anam-Ro, Seoul, 02841, Republic of Korea 3 Department of Mathematics, Korea University, 2511, Sejong-Ro, Sejong, 30019, Republic of Korea

* Corresponding author: Inkyung Ahn

Received  January 2021 Revised  June 2021 Early access October 2021

In this study, we consider a Lotka–Volterra reaction–diffusion–advection model for two competing species under homogeneous Dirichlet boundary conditions, describing a hostile environment at the boundary. In particular, we deal with the case in which one species diffuses at a constant rate, whereas the other species has a constant rate diffusion rate with a directed movement toward a better habitat in a heterogeneous environment with a lethal boundary. By analyzing linearized eigenvalue problems from the system, we conclude that the species dispersion in the advection direction is not always beneficial, and survival may be determined by the convexity of the environment. Further, we obtain the coexistence of steady-states to the system under the instability conditions of two semi-trivial solutions and the uniqueness of the coexistence steady states, implying the global asymptotic stability of the positive steady-state.

Citation: Kwangjoong Kim, Wonhyung Choi, Inkyung Ahn. Reaction-advection-diffusion competition models under lethal boundary conditions. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021250
##### References:
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Differential Equations, 154 (1999), 157-190.  doi: 10.1006/jdeq.1998.3559.  Google Scholar [31] K. Ryu and I. Ahn, Positive solutions for ratio-dependent predator-prey interaction systems, J. Differential Equations, 218 (2005), 117-135.  doi: 10.1016/j.jde.2005.06.020.  Google Scholar [32] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [33] P. Zhou, D. Tang and D. Xiao, On Lotka-Volterra competitive parabolic systems: Exclusion, coexistence and bistability, J. Differential Equations, 282 (2021), 596-625.  doi: 10.1016/j.jde.2021.02.031.  Google Scholar [34] P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356-380.  doi: 10.1016/j.jfa.2018.03.006.  Google Scholar

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##### References:
 [1] I. Averill, K.-Y. Lam and Y. Lou, The role of advection in a two-species competition model: A bifurcation approach, Mem. Amer. Math. Soc., 245 (2017), 117 pp. doi: 10.1090/memo/1161.  Google Scholar [2] F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Canad. Appl. Math. Quart., 3 (1995), 379-397.   Google Scholar [3] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, 2003. doi: 10.1002/0470871296.  Google Scholar [4] R. S. Cantrell, C. Cosner and Y. Lou, Movement toward better environments and the evolution of rapid diffusion, Math. Biosci., 204 (2006), 199-214.  doi: 10.1016/j.mbs.2006.09.003.  Google Scholar [5] R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518.  doi: 10.1017/S0308210506000047.  Google Scholar [6] R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Math. Biosci. Eng., 7 (2010), 17-36.  doi: 10.3934/mbe.2010.7.17.  Google Scholar [7] X. Chen, K.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst., 32 (2012), 3841-3859.  doi: 10.3934/dcds.2012.32.3841.  Google Scholar [8] C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete Contin. Dyn. Syst., 34 (2014), 1701-1745.  doi: 10.3934/dcds.2014.34.1701.  Google Scholar [9] C. Cosner and A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion, SIAM J. Appl. Math., 44 (1984), 1112-1132.  doi: 10.1137/0144080.  Google Scholar [10] C. Cosner and Y. Lou, Does movement toward better environments always benefit a population?, J. Math. Anal. Appl., 277 (2003), 489-503.  doi: 10.1016/S0022-247X(02)00575-9.  Google Scholar [11] E. N. Dancer, On positive solutions of some pairs of differential equations, Trans. Amer. Math. Soc., 284 (1984), 729-743.  doi: 10.1090/S0002-9947-1984-0743741-4.  Google Scholar [12] E. N. Dancer, On the existence and uniqueness of positive solutions for competing species models with diffusion, Trans. Amer. Math. Soc., 326 (1991), 829-859.  doi: 10.1090/S0002-9947-1991-1028757-9.  Google Scholar [13] J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.  Google Scholar [14] J. C. Eilbeck, J. E. Furter and J. López-Gómez, Coexistence in the competition model with diffusion, J. Differential Equations, 107 (1994), 96-139.  doi: 10.1006/jdeq.1994.1005.  Google Scholar [15] S. Fernández-Rincón and J. López-Gómez, Spatially heterogeneous Lotka–Volterra competition, Nonlinear Anal., 165 (2017), 33-79.  doi: 10.1016/j.na.2017.09.008.  Google Scholar [16] J. E. Furter and J. López-Gómez, On the existence and uniqueness of coexistence states for the Lotka-Volterra competition model with diffusion and spatially dependent coefficients, Nonlinear Anal., 25 (1995), 363-398.  doi: 10.1016/0362-546X(94)00139-9.  Google Scholar [17] J. E. Furter and J. López-Gómez, Diffusion-mediated permanence problem for a heterogeneous Lotka–Volterra competition model, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 281-336.  doi: 10.1017/S0308210500023659.  Google Scholar [18] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman, New York, 1991.  Google Scholar [19] M. G. Kreǐn and M. A. Rutman, Linear operators leaving invariant a cone in Banach space (in Russian), Usp. Mat. Nauk. 3 (1948), 3–95. English translation in Amer. Math. Soc. Transl., 26 (1950).  Google Scholar [20] K. Kuto and T. Tsujikawa, Limiting structure of steady-states to the Lotka-Volterra competition model with large diffusion and advection, J. Differential Equations, 258 (2015), 1801-1858.  doi: 10.1016/j.jde.2014.11.016.  Google Scholar [21] K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differential Equations, 250 (2011), 161-181.  doi: 10.1016/j.jde.2010.08.028.  Google Scholar [22] K.-Y. Lam, Limiting profiles of semilinear elliptic equations with large advection in population dynamics II, SIAM J. Math. Anal., 44 (2012), 1808-1830.  doi: 10.1137/100819758.  Google Scholar [23] K.-Y. Lam and W.-M. Ni, Advection-mediated competition in general environments, J. Differential Equations, 257 (2014), 3466-3500.  doi: 10.1016/j.jde.2014.06.019.  Google Scholar [24] K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst., 28 (2010), 1051-1067.  doi: 10.3934/dcds.2010.28.1051.  Google Scholar [25] L. Li, Coexistence theorems of steady states for predator-prey interacting systems, Trans. Amer. Math. Soc., 305 (1988), 143-166.  doi: 10.1090/S0002-9947-1988-0920151-1.  Google Scholar [26] L. Li and R. Logan, Positive solutions to general elliptic competition models, Differential Integral Equations, 4 (1991), 817-834.   Google Scholar [27] Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342.  doi: 10.1007/s00285-013-0730-2.  Google Scholar [28] Y. Lou and S. Martínez, Evolution of cross-diffusion and self-diffusion, J. Biol. Dyn., 3 (2009), 410-429.  doi: 10.1080/17513750802491849.  Google Scholar [29] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157.  Google Scholar [30] Y. Lou and W.-M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.  doi: 10.1006/jdeq.1998.3559.  Google Scholar [31] K. Ryu and I. Ahn, Positive solutions for ratio-dependent predator-prey interaction systems, J. Differential Equations, 218 (2005), 117-135.  doi: 10.1016/j.jde.2005.06.020.  Google Scholar [32] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar [33] P. Zhou, D. Tang and D. Xiao, On Lotka-Volterra competitive parabolic systems: Exclusion, coexistence and bistability, J. Differential Equations, 282 (2021), 596-625.  doi: 10.1016/j.jde.2021.02.031.  Google Scholar [34] P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356-380.  doi: 10.1016/j.jfa.2018.03.006.  Google Scholar
Case $\Delta m > 0$: (a) Instabilities of $(0,\theta_\nu)$ when $\alpha = 0$, (b) Stability of $(0,\theta_\nu)$ when $\alpha = 0.1$ ($m(x) = 1-0.4\sin(\pi x), \mu = 0.01, \nu = 0.02$)
Case $\Delta m < 0$ : (a) Stabilities of $(0,\theta_\nu)$ when $\alpha = 0$, (b) Instability of $(0,\theta_\nu)$ when $\alpha = 0.1$ ($m(x) = 0.5+0.4\sin(\pi x), \mu = 0.02, \nu = 0.01$)
$\Delta m$ changes its sign : (a) Stability of $(0,\theta_\nu)$ when $\alpha = 0$, (b) Instability of $(0,\theta_\nu)$ when $\alpha = 0.5$ ($m(x) = 0.8+0.2\cos(4\pi x), \mu = 0.02, \nu = 0.01$)
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