Hostname: page-component-cb9f654ff-65tv2 Total loading time: 0 Render date: 2025-09-02T00:26:30.085Z Has data issue: false hasContentIssue false
  • English
  • Français

Global well-posedness of advective Lotka–Volterra competition systems with nonlinear diffusion

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  03 April 2019

Qi Wang ,
Jingyue Yang  and
Feng Yu
Qi Wang*
Affiliation:
Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan611130, China (qwang@swufe.edu.cn)
Jingyue Yang
Affiliation:
Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan611130, China (yjy@2011.swufe.edu.cn)
Feng Yu
Affiliation:
Department of Mathematics, University of Central Florida, Orlando, FL32816, USA (yfeng@knights.ucf.edu)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper investigates the global well-posedness of a class of reaction–advection–diffusion models with nonlinear diffusion and Lotka–Volterra dynamics. We prove the existence and uniform boundedness of the global-in-time solutions to the fully parabolic systems under certain growth conditions on the diffusion and sensitivity functions. Global existence and uniform boundedness of the corresponding parabolic–elliptic system are also obtained. Our results suggest that attraction (positive taxis) inhibits blowups in Lotka–Volterra competition systems.

Keywords

Lotka–Volterra competition systemnonlinear diffusionglobal existenceboundedness

MSC classification

Primary: 35K51: Initial-boundary value problems for second-order parabolic systems
Secondary: 35K55: Nonlinear parabolic equations

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2322 - 2348
Check for updates
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Alikakos, N. D.. L p bounds of solutions of reaction-diffusion equations. Comm. Partial Diff. Equ. 4 (1979), 827–868.CrossRefGoogle Scholar
2Amann, H.. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. Funct. Spaces Diff. Operators Nonlinear Anal. Teubner, Stuttgart, Leipzig 133 (1993), 9–126.Google Scholar
3Burger, M., Di Francesco, M., Dolak-Struss, Y.. The Keller–Segel model for chemotaxis with prevention of overcrowding: linear vs. nonlinear diffusion. SIAM J. Math. Anal. 38 (2006), 1288–1315.CrossRefGoogle Scholar
4Calvez, V., Carrillo, J. A.. Volume effects in the Keller–Segel model: energy estimates preventing blow-up. J. Math. Pures Appl. 86 (2006), 155–175.CrossRefGoogle Scholar
5Cantrell, R., Cosner, C. and Lou, Y.. Approximating the ideal free distribution via reaction–diffusion–advection equations. J. Diff. Equ. 245 (2008), 3687–3703.CrossRefGoogle Scholar
6Cantrell, R., Cosner, C., Lou, Y. and Xie, C.. Random dispersal versus fitness-dependent dispersal. J. Diff. Equ. 254 (2013), 2905–2941.CrossRefGoogle Scholar
7Chen, L. and Jüngel, A.. Analysis of a parabolic cross-diffusion population model without self-diffusion. J. Diff. Equ. 224 (2006), 39–59.CrossRefGoogle Scholar
8Choi, Y. S., Lui, R. and Yamada, Y.. Existence of global solutions for the Shigesada–Kawasaki–Teramoto model with weak cross-diffusion. Discrete Contin. Dyn. Syst. 9 (2003), 1193–1200.CrossRefGoogle Scholar
9Choi, Y. S., Lui, R. and Yamada, Y.. Existence of global solutions for the Shigesada–Kawasaki–Teramoto model with strongly coupled cross-diffusion. Discrete Contin. Dyn. Syst. 10 (2004), 719–730.CrossRefGoogle Scholar
10Conway, E. and Smoller, J.. A comparison technique for systems of reaction–diffusion equations. Comm. Partial Diff. Equ. 2 (1977), 679–697.CrossRefGoogle Scholar
11Conway, E., Hoff, D. and Smoller, J.. Large time behavior of solutions of systems of nonlinear reaction–diffusion equations. SIAM J. Appl. Math. 35 (1978), 1–16.CrossRefGoogle Scholar
12Cosner, C.. Reaction–diffusion–advection models for the effects and evolution of dispersal. Discrete Contin. Dyn. Syst. 34 (2014), 1701–1745.CrossRefGoogle Scholar
13De Mottoni, P. and Rothe, F.. Convergence to homogeneous equilibrium state for generalized Volterra–Lotka systems with diffusion. SIAM J. Appl. Math. 37 (1979), 648–663.CrossRefGoogle Scholar
14Desvillettes, L., Lepoutre, T., Moussa, A. and Trescases, A.. On the entropic structure of reaction–cross diffusion systems. Comm. Partial Diff. Equ. 40 (2015), 1705–1747.CrossRefGoogle Scholar
15Haroske, D. and Triebel, H.. Distributions, Sobolev Spaces, Elliptic equations (Zurich: European Mathematical Society, 2008).Google Scholar
16Hoang, L., Nguyen, T. and Phan, T.. Gradient estimates and global existence of smooth solutions to a cross-diffusion system. SIAM J. Math. Anal. 47 (2015), 2122–2177.CrossRefGoogle Scholar
17Ishida, S., Seki, K. and Yokota, T.. Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains. J. Diff. Equ. 256 (2014), 2993–3010.CrossRefGoogle Scholar
18Jüngel, A.. The boundedness-by-entropy principle for cross-diffusion systems. Nonlinearity 28 (2015), 1963–2001.CrossRefGoogle Scholar
19Kishimoto, K. and Weinberger, H.. The spatial homogeneity of stable equilibria of some reaction-diffusion systems in convex domains. J. Diff. Equ. 58 (1985), 15–21.CrossRefGoogle Scholar
20Kolokolnikov, T. and Wei, J.. Stability of spiky solutions in a competition model with cross-diffusion. SIAM J. Appl. Math. 71 (2011), 1428–1457.CrossRefGoogle Scholar
21Kuto, K.. Limiting structure of shrinking solutions to the stationary Shigesada–Kawasaki–Teramoto model with large cross-diffusion. SIAM J. Math. Anal. 47 (2015), 3993–4024.CrossRefGoogle Scholar
22Kuto, K. and Tsujikawa, T.. Limiting structure of steady states to the Lotka–Volterra competition model with large diffusion and advection. J. Diff. Equ. 258 (2015), 1801–1858.CrossRefGoogle Scholar
23Lankeit, J.. Chemotaxis can prevent thresholds on population density. Discrete Contin. Dyn. Syst., Ser. B 20 (2015), 1499–1527.CrossRefGoogle Scholar
24Le, D.. Cross diffusion systems on n spatial dimensional domains. Indiana Univ. Math. J. 51 (2002), 625–643.CrossRefGoogle Scholar
25Le, D. and Nguyen, V.. Global solutions to cross diffusion parabolic systems on 2D domains. Proc. Amer. Math. Soc. 143 (2015), 2999–3010.CrossRefGoogle Scholar
26Le, D. and Nguyen, V.. Global and blow up solutions to cross diffusion systems on 3D domains. Proc. Amer. Math. Soc. (2016), 144 (2016), 4845–4859.CrossRefGoogle Scholar
27Le, D., Nguyen, L. and Nguyen, T.. Coexistence in cross diffusion systems. Indiana Univ. J. Math. 56 (2007), 1749–1791.CrossRefGoogle Scholar
28Lou, Y. and Ni, W.-M.. Diffusion, self-diffusion and cross-diffusion. J. Diff. Equ. 131 (1996), 79–131.CrossRefGoogle Scholar
29Lou, Y., Ni, W.-M.. Diffusion vs cross-diffusion: an elliptic approach. J. Diff. Equ. 154 (1999), 157–190.CrossRefGoogle Scholar
30Lou, Y., Ni, W.-M. and Wu, Y.. On the global existence of a cross-diffusion system. Discrete Contin. Dynam. Systems 4 (1998), 193–203.CrossRefGoogle Scholar
31Lou, Y., Ni, W.-M. and Yotsutani, S.. On a limiting system in the Lotka–Volterra competition with cross-diffusion. Discrete Contin. Dyn. Syst. 10 (2004), 435–458.Google Scholar
32Lou, Y., Winkler, M. and Tao, Y.. Approaching the ideal free distribution in two-species competition models with fitness-dependent dispersal. SIAM J. Math. Anal. 46 (2014), 1228–1262.CrossRefGoogle Scholar
33Lou, Y., Ni, W.-M. and Yotsutani, S.. Pattern formation in a cross-diffusion system. Discrete Contin. Dyn. Syst. 35 (2015), 1589–1607.CrossRefGoogle Scholar
34Matano, H. and Mimura, M.. Pattern formation in competition-diffusion systems in nonconvex domains. Publ. Res. Inst. Math. Sci. 19 (1983), 1049–1079.CrossRefGoogle Scholar
35Mimura, M. and Kawasaki, K.. Spatial segregation in competitive interaction–diffusion equations. J. Math. Biol. 9 (1980), 49–64.CrossRefGoogle Scholar
36Mimura, M., Nishiura, Y., Tesei, A. and Tsujikawa, T.. Coexistence problem for two competing species models with density-dependent diffusion. Hiroshima Math. J. 14 (1984), 425–449.CrossRefGoogle Scholar
37Mimura, M., Ei, S.-I. and Fang, Q.. Effect of domain-shape on coexistence problems in a competition-diffusion system. J. Math. Biol. 29 (1991), 219–237.CrossRefGoogle Scholar
38Ni, W.-M., Wu, Y. and Xu, Q.. The existence and stability of nontrivial steady states for S–K–T competition model with cross diffusion. Discrete Contin. Dyn. Syst. 34 (2014), 5271–5298.CrossRefGoogle Scholar
39Wang, Q., Song, Y. and Shao, L.. Boundedness and persistence of populations in advective Lotka–Volterra competition system. Discrete Contin. Dyn. Syst., Ser. B 23 (2018), 2245–2263.Google Scholar
40Shigesada, N., Kawasaki, K. and Teramoto, E.. Spatial segregation of interacting species. J. Theoret. Biol. 79 (1979), 83–99.CrossRefGoogle ScholarPubMed
41Shim, S.-A. Uniform boundedness and convergence of solutions to cross-diffusion systems. J. Diff. Equ. 185 (2002), 281–305.CrossRefGoogle Scholar
42Sugiyama, Y. and Kunii, H.. Global existence and decay properties for a degenerate Keller–Segel model with a power factor in drift term. J. Diff. Equ. 227 (2006), 333–364.CrossRefGoogle Scholar
43Tao, Y. and Winkler, M.. Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with subcritical sensitivity. J. Diff. Equ. 252 (2012), 692–715.CrossRefGoogle Scholar
44Tuoc, P.. Global existence of solutions to Shigesada–Kawasaki–Teramoto cross–diffusion systems on domains of arbitrary dimensions. Proc. Amer. Math. Soc. 135 (2007), 3933–3941.CrossRefGoogle Scholar
45Tuoc, P. and Phan, V.. On global existence of solutions to a cross-diffusion system. J. Math. Anal. Appl. 343 (2008), 826–834.CrossRefGoogle Scholar
46Wang, Y.. Boundedness in the higher-dimensional chemotaxis-haptotaxis model with nonlinear diffusion. J. Diff. Equ. 260 (2016), 1975–1989.CrossRefGoogle Scholar
47Wang, Q., Gai, C. and Yan, J.. Qualitative analysis of a Lotka–Volterra competition system with advection. Discrete Contin. Dyn. Syst. 35 (2015), 1239–1284.CrossRefGoogle Scholar
48Wang, Q.. On the steady state of a shadow system to the SKT competition model. Discrete Contin. Dyn. Syst. Ser. B 19 (2014), 2941–2961.Google Scholar
49Wang, L., Mu, C. and Zhou, S.. Boundedness in a parabolic–parabolic chemotaxis system with nonlinear diffusion. Z. Angew. Math. Phys. 65 (2014), 1137–1152.CrossRefGoogle Scholar
50Wang, Q. and Zhang, L.. On the multi-dimensional advective Lotka–Volterra competition systems. Nonlinear Anal., Real World Appl. 37 (2017), 329–349.CrossRefGoogle Scholar
51Winkler, M.. Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model. J. Diff. Equ. 248 (2010), 2889–2905.CrossRefGoogle Scholar
52Winkler, M.. Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source. Comm. Partial Diff. Equ. 35 (2010), 1516–1537.CrossRefGoogle Scholar
53Winkler, M.. Blow-up in a higher-dimensional chemotaxis system despite logistic growth restriction. J. Math. Anal. Appl. 384 (2011), 261–272.CrossRefGoogle Scholar
54Winkler, M.. How far can chemotactic cross-diffusion enforce exceeding carrying capacities?. J. Nonlinear Sci. 24 (2014), 809–855.CrossRefGoogle Scholar
55Wu, Y. and Xu, Q.. The existence and structure of large spiky steady states for SKT competition systems with cross-diffusion. Discrete Contin. Dyn. Syst. 29 (2011), 367–385.CrossRefGoogle Scholar
56Yamada, Y.. Global solutions for the Shigesada–Kawasaki–Teramoto model with cross-diffusion. In Recent progress on reaction-diffusion systems and viscosity solutions, (Edited by Du, Yihong, Ishii, Hitoshi and Lin, Wei-Yueh), pp. 282–299 (NJ: World Scientific River Edge, 2009)CrossRefGoogle Scholar
57Zhang, Q. and Li, Y.. Boundedness in a quasilinear fully parabolic Keller–Segel system with logistic source. Z. Angew. Math. Phys. 66 (2015), 2473–2484.CrossRefGoogle Scholar
58Zheng, P., Mu, C. and Hu, X.. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete Contin. Dyn. Syst. 35 (2015), 2299–2323.CrossRefGoogle Scholar