GLOBAL DYNAMICS OF A POPULATION MODEL FROM RIVER ECOLOGY

Keyu Li; Fangfang Xu

doi:10.11948/20200081

2020 Volume 10 Issue 4

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Keyu Li, Fangfang Xu. GLOBAL DYNAMICS OF A POPULATION MODEL FROM RIVER ECOLOGY[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1698-1707. doi: 10.11948/20200081

Citation:

Keyu Li, Fangfang Xu. GLOBAL DYNAMICS OF A POPULATION MODEL FROM RIVER ECOLOGY[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1698-1707. doi: 10.11948/20200081

GLOBAL DYNAMICS OF A POPULATION MODEL FROM RIVER ECOLOGY

Keyu Li¹,
Fangfang Xu^2,3, ,

1.
Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
3.
School of Automation, Qingdao University, Qingdao 266071, China

Corresponding author: Email address: happyxufangfang@126.com
Fund Project: The research is supported in part by the Shandong Provincial Natural Science Foundation grant (ZR2019QA017) and NSF of China (11901359 and 11801373)

Abstract

Abstract

In this paper, we investigate the population dynamics of a two-species Lotka-Volterra competition system arising in river ecology. An interesting feature of this modeling system lies in the boundary conditions at the downstream end, where the populations may be exposed to differing magnitudes of loss of individuals. By applying the theory of principal eigenvalue and monotone dynamical systems, we obtain a complete understanding on the global dynamics, which suggests that slower dispersal is selected for. Our results can be seen as a further development of a recent work by Tang and Chen (J. Differential Equations, 2020, 2020(269), 1465–1483).
- Lotka-Volterra competition /
- advection /
- evolution /
- global stability
MSC: 37C65, 35K57, 35P05, 92D25

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References

[1]	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. Google Scholar
[2]	S.-B. Hsu, H. Smith, and P. Waltman, Competitive exclusion and coexistence for competitive systems on ordered Banach spaces, Trans. Amer. Math. Soc., 1996, 348, 4083–4094. doi: 10.1090/S0002-9947-96-01724-2 CrossRef Google Scholar
[3]	M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Mat. Nauk (N. S.), 1948, 3, 3–95. Google Scholar
[4]	K.-Y. Lam and D. Munther, A remark on the global dynamics of competitive systems on ordered Banach spaces, Proc. Amer. Math. Soc., 2016, 144, 1153–1159. Google Scholar
[5]	K.-Y. Lam and N.-W. Ni, Uniqueness and complete dynamics in the heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 2012, 72, 1695–1712. doi: 10.1137/120869481 CrossRef Google Scholar
[6]	Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 2014, 69, 1319–1342. doi: 10.1007/s00285-013-0730-2 CrossRef Google Scholar
[7]	Y. Lou, H. Nie, and Y. E. Wang, Coexistence and bistability of a competition model in open advective evironments, Math. Biosci., 2018, 306, 10–19. doi: 10.1016/j.mbs.2018.09.013 CrossRef Google Scholar
[8]	Y. Lou, D. M. Xiao, and P. Zhou, Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment, Discrete Contin. Dyn. Syst., 2016, 36, 953–969. Google Scholar
[9]	Y. Lou, X.-Q. Zhao, and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. Math. Pures Appl., 2019, 121, 47–82. doi: 10.1016/j.matpur.2018.06.010 CrossRef Google Scholar
[10]	Y. Lou and P. Zhou, Evolution of dispersal in advective homogeneous environment: the effect of boundary conditions, J. Differential Equations, 2015, 259, 141–171. doi: 10.1016/j.jde.2015.02.004 CrossRef Google Scholar
[11]	F. Lutscher, M. A. Lewis, and E. McCauley, Effects of heterogeneity on spread and persistence in rivers, Bull. Math. Biol., 2006, 68, 2129–2160. doi: 10.1007/s11538-006-9100-1 CrossRef Google Scholar
[12]	F. Lutscher, E. McCauley, and M. A. Lewis, Spatial patterns and coexistence mechanisms in rivers, Theor. Pop. Biol., 2007, 71, 267–277. doi: 10.1016/j.tpb.2006.11.006 CrossRef Google Scholar
[13]	D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries, Ecology, 2001, 82, 1219–1237. doi: 10.1890/0012-9658(2001)082[1219:PPIRAE]2.0.CO;2 CrossRef Google Scholar
[14]	D. Tang and Y. M. Chen, Global dynamics of a Lotka-Volterra competition-diffusion system in advective homogeneous environments, J. Differential Equations, 2020, 269, 1465–1483. doi: 10.1016/j.jde.2020.01.011 CrossRef Google Scholar
[15]	D. Tang and P. Zhou, On a Lotka-Volterra competition-diffusion-advection system: Homogeneity vs heterogeneity, J. Differential Equations, 2020, 268, 1570–1599. doi: 10.1016/j.jde.2019.09.003 CrossRef Google Scholar
[16]	F. Xu and W. Gan, On a Lotka-Volterra type competition model from river ecology, Nonlinear Anal. Real World Appl., 2019, 47, 373–384. doi: 10.1016/j.nonrwa.2018.11.011 CrossRef Google Scholar
[17]	P. Zhou, On a Lotka-Volterra competition system: diffusion vs advection, Calc. Var. Partial Differential Equations, 2016, 55, Art. 137, 29 pp. Google Scholar
[18]	P. Zhou and D. M. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 2018, 275, 356–380. doi: 10.1016/j.jfa.2018.03.006 CrossRef Google Scholar
[19]	P. Zhou and X.-Q. Zhao, Evolution of passive movement in advective environments: General boundary condition, J. Differential Equations, 2018, 264, 4176–4198. doi: 10.1016/j.jde.2017.12.005 CrossRef Google Scholar