Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-22T05:16:26.041Z Has data issue: false hasContentIssue false
  • English
  • Français

Steady-state solutions of one-dimensional competition models in an unstirred chemostat via the fixed point index theory

Part of: Nonlinear integral equations Boundary value problems Mathematical biology in general Nonlinear operators and their properties

Published online by Cambridge University Press:  11 March 2020

Kunquan Lan  and
Wei Lin
Kunquan Lan
Affiliation:
Department of Mathematics, Ryerson University, Toronto, Ontario, CanadaM5B 2K3 (klan@ryerson.ca)
Wei Lin
Affiliation:
School of Mathematical Sciences, SCMS, and Centre for Computational Systems Biology, Fudan University, Shanghai200433, P. R. China (wlin@fudan.edu.cn)
Get access Rights & Permissions [Opens in a new window]

Abstract

The existence and nonexistence of semi-trivial or coexistence steady-state solutions of one-dimensional competition models in an unstirred chemostat are studied by establishing new results on systems of Hammerstein integral equations via the classical fixed point index theory. We provide three ranges for the two parameters involved in the competition models under which the models have no semi-trivial and coexistence steady-state solutions or have semi-trivial steady-state solutions but no coexistence steady-state solutions or have semi-trivial or coexistence steady-state solutions. It remains open to find the largest range for the two parameters under which the models have only coexistence steady-state solutions. We apply the new results on systems of Hammerstein integral equations to obtain results on steady-state solutions of systems of reaction-diffusion equations with general separated boundary conditions. Such type of results have not been studied in the literature. However, these results are very useful for studying the competition models in an unstirred chemostat. Our results on Hammerstein integral equations and differential equations generalize and improve some previous results.

Keywords

System of Hammerstein integral equationstwo point boundary value problemsfixed point indexone-dimensional competition modelchemostat

MSC classification

Primary: 45G15: Systems of nonlinear integral equations
Secondary: 34B18: Positive solutions of nonlinear boundary value problems 47H10: Fixed-point theorems 47H30: Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 92B05: General biology and biomathematics
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 1 , February 2021 , pp. 240 - 264
Check for updates
Copyright
Copyright © Royal Society of Edinburgh 2020

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.)

References

1Amann, H.. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM. Rev. 18 (1976), 620–709.CrossRefGoogle Scholar
2Crandall, M. G. and Rabinowitz, P. H.. Nonlinear Sturm-Liouville eigenvalue problem and topological degree. J. Math. Mech. 19 (1970), 1083–1102.Google Scholar
3Fredrickson, A. G. and Stephanopoulos, G.. Microbial competition. Science 213 (1981), 972–979.CrossRefGoogle ScholarPubMed
4Guo, D. and Lakshmikantham, V.. Nonlinear problems in abstract cones (San Diego: Academic Press, 1988).Google Scholar
5Hsu, S. B. and Waltman, P.. On a system of reaction-diffusion equations arising from competition in an unstored chemostat. SIAM J. Appl. Math. 53 (1993), 1026–1044.Google Scholar
6Lan, K. Q.. Multiple positive solutions of Hammerstein integral equations with singularities. Differ. Equ. Dyn. Syst. 8 (2000), 175–192.Google Scholar
7Lan, K. Q.. Multiple positive solutions of semilinear differential equations with singularities. J. London Math. Soc. 63 (2001), 690–704.CrossRefGoogle Scholar
8Lan, K. Q.. Positive solutions of semi-positone Hammerstein integral equations and applications. Commun. Pure Appl. Anal. 6 (2007), 441–451.CrossRefGoogle Scholar
9Lan, K. Q.. Eigenvalues of semi-positone Hammerstein integral equations and applications to boundary value problems. Nonlinear Anal. 71 (2009), 5979–5993.CrossRefGoogle Scholar
10Lan, K. Q.. Existence of nonzero positive solutions of systems of second order elliptic boundary value problems. J. Appl. Anal. Comput. 1 (2011), 21–31.Google Scholar
11Lan, K. Q.. Nonzero positive solutions of systems of elliptic boundary value problems. Proc. Amer. Math. Soc. 139 (2011), 4343–4349.CrossRefGoogle Scholar
12Lan, K. Q. and Lin, W.. Multiple positive solutions of systems of Hammerstein integral equations with applications to fractional differential equations. J. London Math. Soc. 83 (2011), 449–469.CrossRefGoogle Scholar
13Lan, K. Q. and Lin, W.. Positive solutions of systems of singular Hammerstein integral equations with applications to semilinear elliptic equations in annuli. Nonlinear Anal. 74 (2011), 7184–7197.CrossRefGoogle Scholar
14Lan, K. Q. and Zhang, Z. T.. Nonzero positive weak solutions of systems of p-Laplace equations. J. Math. Anal. Appl. 394 (2012), 581–591.CrossRefGoogle Scholar
15Lee, Y. H.. Multiplicity of positive radial solutions for multiparameter semilinear elliptic systems on an annulus. J. Differ. Equ. 174 (2001), 420–441.CrossRefGoogle Scholar
16Nussbaum, R. D.. Periodic solutions of some nonlinear integral equations. Dynamical systems. In Proc. Internat. Sympos., Univ. Florida, Gainesville, Fla., pp. 221–249 (New York: Academic Press, 1977, 1976).Google Scholar
17Smith, H. L. and Waltman, P.. The theory of the chemostat (Cambridge:Cambridge University Press, 1995).CrossRefGoogle Scholar
18Smoller, J.. Shock waves and reaction diffusion equations (New York: Springer-Verlag, 1983).CrossRefGoogle Scholar
19So, J. H. and Waltman, P.. A nonlinear boundary value problem arising from competition in the chemostat. Appl. Math. Comput. 32 (1989), 169–183.Google Scholar
20Waltman, P., Hubbell, S. P. and Hsu, S. B.. Theoretical and experimental investigation of microbial competition in continuous culture. In Modelling and differential equations (ed. Burton, T.) (New York: Marcel Dekker, 1980.Google Scholar
21Webb, J. R. L. and Lan, K. Q.. Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type. Topol. Methods Nonlinear Anal. 27 (2006), 91–116.Google Scholar
22Wu, J. H.. Global bifurcation of coexistence state for the competition model in the chemostat. Nonlinear Anal. 39 (2000), 817–835.CrossRefGoogle Scholar
23Wu, J. H., Nie, H. and Wolkowicz, G. S. K.. The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat. SIAM J. Math. Anal. 38 (2007), 1860–1885.CrossRefGoogle Scholar
24Yang, X. J.. Existence of positive solutions for 2m-order nonlinear differential systems. Nonlinear Anal. 61 (2005), 77–95.Google Scholar
25Yuan, H., Zhang, C. and Li, Y.. Existence and stability of coexistence states in a competition unstirred chemostat. Nonlinear Anal. Real World Appl. 35 (2017), 441–456.CrossRefGoogle Scholar