On steady state of some Lotka-Volterra competition-diffusion-advection model

Qi Wang

doi:10.3934/dcdsb.2019193

Article Contents

2020, Volume 25, Issue 3: 859-875. Doi: 10.3934/dcdsb.2019193

This issue Previous Article Input-to-state stability of continuous-time systems via finite-time Lyapunov functions Next Article Global dynamics of a reaction-diffusion system with intraguild predation and internal storage

On steady state of some Lotka-Volterra competition-diffusion-advection model

Qi Wang^,

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

^* Corresponding author: Qi Wang
^* Corresponding author: Qi Wang

Received: November 2018

Revised: May 2019

Early access: September 2019

Published: March 2020

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 92B05, 35B35, 35B40; Secondary: 35B30, 35J47.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.