September  2022, 27(9): 4749-4767. doi: 10.3934/dcdsb.2021250

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 Published  September 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (9) : 4749-4767. doi: 10.3934/dcdsb.2021250
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.

[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. 

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, 2003. doi: 10.1002/0470871296.

[4]

R. S. CantrellC. 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.

[5]

R. S. CantrellC. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518.  doi: 10.1017/S0308210506000047.

[6]

R. S. CantrellC. 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.

[7]

X. ChenK.-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.

[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.

[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.

[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.

[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.

[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.

[13]

J. DockeryV. HutsonK. 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.

[14]

J. C. EilbeckJ. 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.

[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.

[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.

[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.

[18]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman, New York, 1991.

[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).

[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.

[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.

[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.

[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.

[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.

[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.

[26]

L. Li and R. Logan, Positive solutions to general elliptic competition models, Differential Integral Equations, 4 (1991), 817-834. 

[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.

[28]

Y. Lou and S. Martínez, Evolution of cross-diffusion and self-diffusion, J. Biol. Dyn., 3 (2009), 410-429.  doi: 10.1080/17513750802491849.

[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.

[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.

[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.

[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.

[33]

P. ZhouD. 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.

[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.

show all references

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.

[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. 

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, 2003. doi: 10.1002/0470871296.

[4]

R. S. CantrellC. 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.

[5]

R. S. CantrellC. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518.  doi: 10.1017/S0308210506000047.

[6]

R. S. CantrellC. 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.

[7]

X. ChenK.-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.

[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.

[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.

[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.

[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.

[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.

[13]

J. DockeryV. HutsonK. 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.

[14]

J. C. EilbeckJ. 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.

[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.

[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.

[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.

[18]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman, New York, 1991.

[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).

[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.

[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.

[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.

[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.

[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.

[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.

[26]

L. Li and R. Logan, Positive solutions to general elliptic competition models, Differential Integral Equations, 4 (1991), 817-834. 

[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.

[28]

Y. Lou and S. Martínez, Evolution of cross-diffusion and self-diffusion, J. Biol. Dyn., 3 (2009), 410-429.  doi: 10.1080/17513750802491849.

[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.

[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.

[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.

[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.

[33]

P. ZhouD. 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.

[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.

Figure 1.  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 $)
Figure 2.  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 $)
Figure 3.  $ \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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