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On steady state of some Lotka-Volterra competition-diffusion-advection model
College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China |
In this paper, we study a shadow system of a two species Lotka-Volterra competition-diffusion-advection system, where the ratio of diffusion and advection rates are supposed to be a positive constant. We show that for any given migration, if the product of interspecific competition coefficients of competitors is small, then the shadow system has coexistence state; otherwise we can always find some migration such that it has no coexistence state. Moreover, these findings can be applied to steady state of the two-species Lotka-Volterra competition-diffusion-advection model. Particularly, we show that if the interspecific competition coefficient of the invader is sufficiently small, then rapid diffusion of the invader can drive to coexistence state.
References:
[1] |
I. Averill, The Effect of Intermediate Advection on Two Competing Species, Doctoral Thesis, Ohio State University, 2012. |
[2] |
F. Belgacem and C. Cosner,
The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Can. Appl. Math. Q., 3 (1995), 379-397.
|
[3] |
R. S. Cantrell and C. Cosner,
The effect of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338.
doi: 10.1007/BF00167155. |
[4] |
R. S. Cantrell and C. Cosner,
Should a park be an island?, SIAM J. Appl. Math., 53 (1993), 219-252.
doi: 10.1137/0153014. |
[5] |
R. S. Cantrell and C. Cosner,
On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145.
doi: 10.1007/s002850050122. |
[6] |
R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology, Wiley, Chichester, UK, 2003.
doi: 10.1002/0470871296. |
[7] |
R. S. Cantrell, C. Cosner and V. Huston,
Permanence in ecological systems with diffusion, Proc. Roy. Soc. Edinburgh A, 123 (1993), 553-559.
|
[8] |
R. S. Cantrell, C. Cosner and V. Huston,
Ecological models, permanence and spatial heterogeneity, Rocky Mount. J. Math., 26 (1996), 1-35.
doi: 10.1216/rmjm/1181072101. |
[9] |
R. S. Cantrell, C. Cosner and Y. Lou,
Multiple reversals of competitive dominance in ecological reserves via external habitat degradation, J. Dynam. Differential Equations, 16 (2004), 973-1010.
doi: 10.1007/s10884-004-7831-y. |
[10] |
C. Cosner and Y. Lou,
When does movement toward better environment benefit a population?, J. Math. Analysis Applic., 277 (2003), 489-503.
doi: 10.1016/S0022-247X(02)00575-9. |
[11] |
J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski,
The evolution of slow dispersal rates: A reaction-diffusion equations, J. Math. Biol., 37 (1998), 61-83.
doi: 10.1007/s002850050120. |
[12] |
Y. Du,
Effects of a degeneracy in the competition model, Part I. Classical and generalized steady-state solutions, J. Diff. Eqs., 181 (2002), 92-132.
doi: 10.1006/jdeq.2001.4074. |
[13] |
Y. Du,
Effects of a degeneracy in the competition model, Part II. Perturbation and dynamical behavior, J. Diff. Eqs., 181 (2002), 133-164.
doi: 10.1006/jdeq.2001.4075. |
[14] |
Y. Du,
Realization of prescribed patterns in the competition model, J. Diff. Eqs., 193 (2003), 147-179.
doi: 10.1016/S0022-0396(03)00056-1. |
[15] |
J. E. Furter and J. López-Gómez,
Diffusion-mediated permanence problem for a heterogeneous Lotka-Volterra competition model, Proc. Roy. Soc. Edinburgh A, 127 (1997), 281-336.
doi: 10.1017/S0308210500023659. |
[16] |
A. Hastings,
Spatial heterogeneity and ecological models, Ecology, 71 (1990), 426-428.
doi: 10.2307/1940296. |
[17] |
X. He and W.-M. Ni,
The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system I: heterogeneity vs. homogeneity, J. Diff. Eqs., 254 (2013), 528-546.
doi: 10.1016/j.jde.2012.08.032. |
[18] |
X. He and W.-M. Ni,
The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system II: the general case, J. Diff. Eqs., 254 (2013), 4088-4108.
doi: 10.1016/j.jde.2013.02.009. |
[19] |
X. He and W.-M. Ni,
Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity I, Comm. Pure Appl. Math., 69 (2016), 981-1014.
doi: 10.1002/cpa.21596. |
[20] |
X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, II, Calc. Var. Partial Differential Equations, 55 (2016), Art. 25, 20 pp.
doi: 10.1007/s00526-016-0964-0. |
[21] |
X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, III, Calc. Var. Partial Differential Equations 56 (2017), Art. 132, 26 pp.
doi: 10.1007/s00526-017-1234-5. |
[22] |
E. E. Holmes, M. A. Lewis, J. E. Banks and R. R. Veit,
Partial differential equations in ecology: Spatial interactions and population dynamics, Ecology, 75 (1994), 17-29.
doi: 10.2307/1939378. |
[23] |
V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behavior for a competing species problem with diffusion, in Dynamical Systems and applications, World Scientiffc Series Applicable Analysis, 4, World Scientiffc, River Edge, NJ, (1995), 343–358.
doi: 10.1142/9789812796417_0022. |
[24] |
V. Hutson, Y. 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] |
V. Hutson, Y. Lou and K. Mischaikow,
Convergence in competition models with small diffusion coeffcients, J. Diff. Eqs., 211 (2005), 135-161.
doi: 10.1016/j.jde.2004.06.003. |
[26] |
V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik,
Competing species near the degenerate limit, SIAM J. Math. Anal., 35 (2003), 453-491.
doi: 10.1137/S0036141002402189. |
[27] |
V. Hutson, S. Martinez, K. Mischaikow and G. T. Vicker,
The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.
doi: 10.1007/s00285-003-0210-1. |
[28] |
V. Hutson, K. Mischaikow and P. Polá$\breve{c}$ik,
The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.
doi: 10.1007/s002850100106. |
[29] |
M. G. Krein and M. A. Rutman,
Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk, 3 (1948), 3-95.
|
[30] |
F. Li, L. Wang and Y. Wang,
On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 669-686.
doi: 10.3934/dcdsb.2011.15.669. |
[31] |
J. López-Gómez,
Coexistence and meta-coexistence for competing species, Houston J. Math., 29 (2003), 483-536.
|
[32] |
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. |
[33] |
Y. Lou, S. Martinez and P. Polá$\breve{c}$ik,
Loops and branches of coexistence states in a Lotka-Volterra competition model, J. Diff. Eqs., 230 (2006), 720-742.
doi: 10.1016/j.jde.2006.04.005. |
[34] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
show all references
References:
[1] |
I. Averill, The Effect of Intermediate Advection on Two Competing Species, Doctoral Thesis, Ohio State University, 2012. |
[2] |
F. Belgacem and C. Cosner,
The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Can. Appl. Math. Q., 3 (1995), 379-397.
|
[3] |
R. S. Cantrell and C. Cosner,
The effect of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338.
doi: 10.1007/BF00167155. |
[4] |
R. S. Cantrell and C. Cosner,
Should a park be an island?, SIAM J. Appl. Math., 53 (1993), 219-252.
doi: 10.1137/0153014. |
[5] |
R. S. Cantrell and C. Cosner,
On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145.
doi: 10.1007/s002850050122. |
[6] |
R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology, Wiley, Chichester, UK, 2003.
doi: 10.1002/0470871296. |
[7] |
R. S. Cantrell, C. Cosner and V. Huston,
Permanence in ecological systems with diffusion, Proc. Roy. Soc. Edinburgh A, 123 (1993), 553-559.
|
[8] |
R. S. Cantrell, C. Cosner and V. Huston,
Ecological models, permanence and spatial heterogeneity, Rocky Mount. J. Math., 26 (1996), 1-35.
doi: 10.1216/rmjm/1181072101. |
[9] |
R. S. Cantrell, C. Cosner and Y. Lou,
Multiple reversals of competitive dominance in ecological reserves via external habitat degradation, J. Dynam. Differential Equations, 16 (2004), 973-1010.
doi: 10.1007/s10884-004-7831-y. |
[10] |
C. Cosner and Y. Lou,
When does movement toward better environment benefit a population?, J. Math. Analysis Applic., 277 (2003), 489-503.
doi: 10.1016/S0022-247X(02)00575-9. |
[11] |
J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski,
The evolution of slow dispersal rates: A reaction-diffusion equations, J. Math. Biol., 37 (1998), 61-83.
doi: 10.1007/s002850050120. |
[12] |
Y. Du,
Effects of a degeneracy in the competition model, Part I. Classical and generalized steady-state solutions, J. Diff. Eqs., 181 (2002), 92-132.
doi: 10.1006/jdeq.2001.4074. |
[13] |
Y. Du,
Effects of a degeneracy in the competition model, Part II. Perturbation and dynamical behavior, J. Diff. Eqs., 181 (2002), 133-164.
doi: 10.1006/jdeq.2001.4075. |
[14] |
Y. Du,
Realization of prescribed patterns in the competition model, J. Diff. Eqs., 193 (2003), 147-179.
doi: 10.1016/S0022-0396(03)00056-1. |
[15] |
J. E. Furter and J. López-Gómez,
Diffusion-mediated permanence problem for a heterogeneous Lotka-Volterra competition model, Proc. Roy. Soc. Edinburgh A, 127 (1997), 281-336.
doi: 10.1017/S0308210500023659. |
[16] |
A. Hastings,
Spatial heterogeneity and ecological models, Ecology, 71 (1990), 426-428.
doi: 10.2307/1940296. |
[17] |
X. He and W.-M. Ni,
The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system I: heterogeneity vs. homogeneity, J. Diff. Eqs., 254 (2013), 528-546.
doi: 10.1016/j.jde.2012.08.032. |
[18] |
X. He and W.-M. Ni,
The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system II: the general case, J. Diff. Eqs., 254 (2013), 4088-4108.
doi: 10.1016/j.jde.2013.02.009. |
[19] |
X. He and W.-M. Ni,
Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity I, Comm. Pure Appl. Math., 69 (2016), 981-1014.
doi: 10.1002/cpa.21596. |
[20] |
X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, II, Calc. Var. Partial Differential Equations, 55 (2016), Art. 25, 20 pp.
doi: 10.1007/s00526-016-0964-0. |
[21] |
X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, III, Calc. Var. Partial Differential Equations 56 (2017), Art. 132, 26 pp.
doi: 10.1007/s00526-017-1234-5. |
[22] |
E. E. Holmes, M. A. Lewis, J. E. Banks and R. R. Veit,
Partial differential equations in ecology: Spatial interactions and population dynamics, Ecology, 75 (1994), 17-29.
doi: 10.2307/1939378. |
[23] |
V. Hutson, J. López-Gómez, K. Mischaikow and G. Vickers, Limit behavior for a competing species problem with diffusion, in Dynamical Systems and applications, World Scientiffc Series Applicable Analysis, 4, World Scientiffc, River Edge, NJ, (1995), 343–358.
doi: 10.1142/9789812796417_0022. |
[24] |
V. Hutson, Y. 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] |
V. Hutson, Y. Lou and K. Mischaikow,
Convergence in competition models with small diffusion coeffcients, J. Diff. Eqs., 211 (2005), 135-161.
doi: 10.1016/j.jde.2004.06.003. |
[26] |
V. Hutson, Y. Lou, K. Mischaikow and P. Poláčik,
Competing species near the degenerate limit, SIAM J. Math. Anal., 35 (2003), 453-491.
doi: 10.1137/S0036141002402189. |
[27] |
V. Hutson, S. Martinez, K. Mischaikow and G. T. Vicker,
The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.
doi: 10.1007/s00285-003-0210-1. |
[28] |
V. Hutson, K. Mischaikow and P. Polá$\breve{c}$ik,
The evolution of dispersal rates in a heterogeneous time-periodic environment, J. Math. Biol., 43 (2001), 501-533.
doi: 10.1007/s002850100106. |
[29] |
M. G. Krein and M. A. Rutman,
Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk, 3 (1948), 3-95.
|
[30] |
F. Li, L. Wang and Y. Wang,
On the effects of migration and inter-specific competitions in steady state of some Lotka-Volterra model, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 669-686.
doi: 10.3934/dcdsb.2011.15.669. |
[31] |
J. López-Gómez,
Coexistence and meta-coexistence for competing species, Houston J. Math., 29 (2003), 483-536.
|
[32] |
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. |
[33] |
Y. Lou, S. Martinez and P. Polá$\breve{c}$ik,
Loops and branches of coexistence states in a Lotka-Volterra competition model, J. Diff. Eqs., 230 (2006), 720-742.
doi: 10.1016/j.jde.2006.04.005. |
[34] |
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984.
doi: 10.1007/978-1-4612-5282-5. |
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