December 2018, 23(10): 4255-4266. doi: 10.3934/dcdsb.2018136

Invasion and coexistence of competition-diffusion-advection system with heterogeneous vs homogeneous resources

1. 

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

2. 

Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China

* Corresponding author

Received  September 2017 Revised  December 2017 Published  April 2018

Fund Project: The first author is supported by Shanghai Peak Subject Funding

This paper mainly study the dynamics of a Lotka-Volterra reaction-diffusion-advection model for two competing species which disperse by both random diffusion and advection along environmental gradient. In this model, the species are assumed to be identical except spatial resource distribution: heterogeneity vs homogeneity. It is shown that the species with heterogeneous resources distribution is always in a better position, that is, it can always invade when rare. The ratio of advection strength and diffusion rate of the species with heterogeneous distribution plays a crucial role in the dynamics behavior of the system. Some conditions of invasion, driving extinction, and coexistence are given in term of this ratio and the diffusion rate of its competitor.

Citation: Benlong Xu, Hongyan Jiang. Invasion and coexistence of competition-diffusion-advection system with heterogeneous vs homogeneous resources. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4255-4266. doi: 10.3934/dcdsb.2018136
References:
[1]

I. Averill, K. -Y. Lam and Y. Lou, The role of advection in a two-species competition model: A bifurcation approach, Mem. Am. Math. Soc., 245 (2017), v+117 pp.

[2]

F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Canadian Appl. Math. Quarterly, 3 (1995), 379-397.

[3]

K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120. doi: 10.1016/0022-247X(80)90309-1.

[4]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK, 2003.

[5]

R. S. CantrellC. Cosner and Y. Lou, Movement towards better enviromentsand the evolution of rapid diffusion, Math. Biosciences, 204 (2006), 199-214. doi: 10.1016/j.mbs.2006.09.003.

[6]

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

[7]

X. F. ChenR. Hambrock and Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model, J. Math. Biol., 57 (2008), 361-386. doi: 10.1007/s00285-008-0166-2.

[8]

X. F. ChenK.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrite Contin. Syst., 32 (2012), 3841-3859. doi: 10.3934/dcds.2012.32.3841.

[9]

X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204.

[10]

X. F. Chen and Y. Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operator and their applications, Indiana Univ. Math. J., 61 (2012), 45-80. doi: 10.1512/iumj.2012.61.4518.

[11]

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.

[12]

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.

[13]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.

[14]

R. Hambrock and Y. Lou, The evolution of conditional dispersal strategy in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 1793-1817. doi: 10.1007/s11538-009-9425-7.

[15]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 24 (1983), 244-251. doi: 10.1016/0040-5809(83)90027-8.

[16]

X. Q. He and W.-M. Ni, The effects of diffusion and spatial variation in Lokta-Volterra competition-diffusion system, Ⅰ: Heterogeneity vs. homogeneity, J. Diff. Eqs., 254 (2013), 528-546. doi: 10.1016/j.jde.2012.08.032.

[17]

X. Q. He and W.-M. Ni, The effects of diffusion and spatial variation in Lokta-Volterra competition-diffusion system, Ⅱ: The general case, J. Diff. Eqs., 254 (2013), 4088-4108. doi: 10.1016/j.jde.2013.02.009.

[18]

X. Q. He and W.-M. Ni, Global dynamics of the Lokta-Volterra competition-diffusion system: Diffusion and spatial heterogeneity, Ⅰ, Comm. Pure Appl. Math., 69 (2016), 981-1014. doi: 10.1002/cpa.21596.

[19]

X. Q. He and W. -M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅱ, Calc. Var. Partial Diff. Equ., 55 (2016), Art. 25, 20 pp. doi: 10.1007/s00526-016-0964-0.

[20]

X. Q. He and W. -M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅲ, Calc. Var. Partial Diff. Equ., 56 (2017), Art. 132, 26 pp. doi: 10.1007/s00526-017-1234-5.

[21]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics, Vol. 247, Longman, Harlow, UK, 1991.

[22]

M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-51.

[23]

S. HsuH. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Bnanch spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094. doi: 10.1090/S0002-9947-96-01724-2.

[24]

V. HutsonY. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Diff. Eqs., 185 (2002), 97-136. doi: 10.1006/jdeq.2001.4157.

[25]

K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differ. Equ., 250 (2011), 161-181. doi: 10.1016/j.jde.2010.08.028.

[26]

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.

[27]

K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst. A, 28 (2010), 1051-1067. doi: 10.3934/dcds.2010.28.1051.

[28]

K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in the heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712. doi: 10.1137/120869481.

[29]

K.-Y. Lam and W.-M. Ni, Advection-mediated competition in general environments, J. Differ. Equ., 257 (2014), 3466-3500. doi: 10.1016/j.jde.2014.06.019.

[30]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Diff. Eqs., 223 (2006), 400-426. doi: 10.1016/j.jde.2005.05.010.

[31]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutorials in Mathematical Biosciences Ⅳ, Lecture Notes in Math., 1922, Springer, Berlin, 2008,171–205.

[32]

Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Diff. Eqns, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157.

[33]

W. -M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Sef. in Appl. Math., Vol. 82, SIAM, Philadelphia, 2011.

[34]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, 2nd Ed. Springer, Berlin, 1984.

[35]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1982), 459-470.

[36]

H. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, Math. Survey Monogr. 41, American Mathematical Society, Providence, RI, 1995.

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. Am. Math. Soc., 245 (2017), v+117 pp.

[2]

F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Canadian Appl. Math. Quarterly, 3 (1995), 379-397.

[3]

K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120. doi: 10.1016/0022-247X(80)90309-1.

[4]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK, 2003.

[5]

R. S. CantrellC. Cosner and Y. Lou, Movement towards better enviromentsand the evolution of rapid diffusion, Math. Biosciences, 204 (2006), 199-214. doi: 10.1016/j.mbs.2006.09.003.

[6]

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

[7]

X. F. ChenR. Hambrock and Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model, J. Math. Biol., 57 (2008), 361-386. doi: 10.1007/s00285-008-0166-2.

[8]

X. F. ChenK.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrite Contin. Syst., 32 (2012), 3841-3859. doi: 10.3934/dcds.2012.32.3841.

[9]

X. F. Chen and Y. Lou, Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model, Indiana Univ. Math. J., 57 (2008), 627-658. doi: 10.1512/iumj.2008.57.3204.

[10]

X. F. Chen and Y. Lou, Effects of diffusion and advection on the smallest eigenvalue of an elliptic operator and their applications, Indiana Univ. Math. J., 61 (2012), 45-80. doi: 10.1512/iumj.2012.61.4518.

[11]

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.

[12]

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.

[13]

D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.

[14]

R. Hambrock and Y. Lou, The evolution of conditional dispersal strategy in spatially heterogeneous habitats, Bull. Math. Biol., 71 (2009), 1793-1817. doi: 10.1007/s11538-009-9425-7.

[15]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theor. Pop. Biol., 24 (1983), 244-251. doi: 10.1016/0040-5809(83)90027-8.

[16]

X. Q. He and W.-M. Ni, The effects of diffusion and spatial variation in Lokta-Volterra competition-diffusion system, Ⅰ: Heterogeneity vs. homogeneity, J. Diff. Eqs., 254 (2013), 528-546. doi: 10.1016/j.jde.2012.08.032.

[17]

X. Q. He and W.-M. Ni, The effects of diffusion and spatial variation in Lokta-Volterra competition-diffusion system, Ⅱ: The general case, J. Diff. Eqs., 254 (2013), 4088-4108. doi: 10.1016/j.jde.2013.02.009.

[18]

X. Q. He and W.-M. Ni, Global dynamics of the Lokta-Volterra competition-diffusion system: Diffusion and spatial heterogeneity, Ⅰ, Comm. Pure Appl. Math., 69 (2016), 981-1014. doi: 10.1002/cpa.21596.

[19]

X. Q. He and W. -M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅱ, Calc. Var. Partial Diff. Equ., 55 (2016), Art. 25, 20 pp. doi: 10.1007/s00526-016-0964-0.

[20]

X. Q. He and W. -M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅲ, Calc. Var. Partial Diff. Equ., 56 (2017), Art. 132, 26 pp. doi: 10.1007/s00526-017-1234-5.

[21]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics, Vol. 247, Longman, Harlow, UK, 1991.

[22]

M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math., 383 (1988), 1-51.

[23]

S. HsuH. Smith and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Bnanch spaces, Trans. Amer. Math. Soc., 348 (1996), 4083-4094. doi: 10.1090/S0002-9947-96-01724-2.

[24]

V. HutsonY. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Diff. Eqs., 185 (2002), 97-136. doi: 10.1006/jdeq.2001.4157.

[25]

K.-Y. Lam, Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model, J. Differ. Equ., 250 (2011), 161-181. doi: 10.1016/j.jde.2010.08.028.

[26]

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.

[27]

K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst. A, 28 (2010), 1051-1067. doi: 10.3934/dcds.2010.28.1051.

[28]

K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in the heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712. doi: 10.1137/120869481.

[29]

K.-Y. Lam and W.-M. Ni, Advection-mediated competition in general environments, J. Differ. Equ., 257 (2014), 3466-3500. doi: 10.1016/j.jde.2014.06.019.

[30]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Diff. Eqs., 223 (2006), 400-426. doi: 10.1016/j.jde.2005.05.010.

[31]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutorials in Mathematical Biosciences Ⅳ, Lecture Notes in Math., 1922, Springer, Berlin, 2008,171–205.

[32]

Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Diff. Eqns, 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157.

[33]

W. -M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conf. Sef. in Appl. Math., Vol. 82, SIAM, Philadelphia, 2011.

[34]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, 2nd Ed. Springer, Berlin, 1984.

[35]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1982), 459-470.

[36]

H. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, Math. Survey Monogr. 41, American Mathematical Society, Providence, RI, 1995.

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