# American Institute of Mathematical Sciences

 College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

* Corresponding author: Qi Wang

Received  November 2018 Revised  May 2019 Published  September 2019

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.

Citation: Qi Wang. On steady state of some Lotka-Volterra competition-diffusion-advection model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019193
##### References:
 [1] I. Averill, The Effect of Intermediate Advection on Two Competing Species, Doctoral Thesis, Ohio State University, 2012.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] R. S. Cantrell, C. Cosner and V. Huston, Permanence in ecological systems with diffusion, Proc. Roy. Soc. Edinburgh A, 123 (1993), 553-559. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] A. Hastings, Spatial heterogeneity and ecological models, Ecology, 71 (1990), 426-428. doi: 10.2307/1940296. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] J. López-Gómez, Coexistence and meta-coexistence for competing species, Houston J. Math., 29 (2003), 483-536. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### References:
 [1] I. Averill, The Effect of Intermediate Advection on Two Competing Species, Doctoral Thesis, Ohio State University, 2012.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] R. S. Cantrell, C. Cosner and V. Huston, Permanence in ecological systems with diffusion, Proc. Roy. Soc. Edinburgh A, 123 (1993), 553-559. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] A. Hastings, Spatial heterogeneity and ecological models, Ecology, 71 (1990), 426-428. doi: 10.2307/1940296. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] J. López-Gómez, Coexistence and meta-coexistence for competing species, Houston J. Math., 29 (2003), 483-536. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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