How many species can two essential resources support?

SIAM Journal on Applied Mathematics

Volumn 62, Issue 1, 2001, Pages 336-366

  Li, Bingtuan ^a,b   Smith, Hal ^c  

^a UNIVERSITY OF UTAH   (United States)

^b UNIVERSITY OF LOUISVILLE   (United States)

^c UNIVERSITY OF MINNESOTA   (United States)

Author keywords

Chemostat; Coexistence; Competition for two resources; Competitive exclusion principle; Competitive system; Global asymptotic behavior

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; GROWTH KINETICS; MICROORGANISMS; PROBLEM SOLVING;

CHEMOSTAT MODEL; COMPETITION FOR TWO RESOURCES; COMPETITIVE EXCLUSION PRINCIPLE; GROWTH LIMITING NUTRIENTS;

COMPUTER SIMULATION;

EID: 0036228880     PISSN: 00361399     EISSN: None     Source Type: Journal    
DOI: 10.1137/S003613999935319X     Document Type: Article

References (42)

1. R. A. ARMSTRONG AND R. MCGEHEE, Competitive exclusion, Amer. Natur., 115 (1980), pp. 151-170.
2. F. G. BADER, Analysis of double-substrate limited growth, Biotechnol. Bioengrg., 20 (1978), pp. 183-202.
3. M. M. BALLYK AND G. S. K. WOLKOWICZ, An examination of the thresholds of enrichment: A resource-based growth model, J. Math. Biol., 33 (1995), pp. 435-457.
4. M. M. BALLYK AND G. S. K. WOLKOWICZ, Exploitative competition in the chemostat for two perfectly substitutable resources, Math. Biosci., 118 (1993), pp. 127-180.
5. G. J. BUTLER AND G. S. K. WOLKOWICZ, A mathematical model of the chemostat with a general class of functions describing nutrient uptake, SIAM J. Appl. Math., 45 (1985), pp. 138-151.
6. G. J. BUTLER AND G. S. K. WOLKOWICZ, Exploitative competition in a chemostat for two complementary, and possible inhibitory, resources, Math. Biosci., 83 (1987), pp. 1-48.
7. J. C. GOTTSCHAL AND T. F. THINGSTAD, Mathematical description of competition between two and three bacterial species under dual substrate limitation in a chemostat: A comparison and experimental data, Biotechnol. Bioengrg., 24 (1982), pp. 1403-1418.
8. J. P. GROVER, Resource Competition, Population and Community Biology Series 19, Chapman and Hall, New York, 1997.
9. S. R. HANSEN AND S. P. HUBBELL, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 213 (1981), pp. 927-979.
10. W. M. HIRSCH, H. HANISCH, AND J. P. GABRIEL, Differential equation models for some parasitic infections: Methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), pp. 733-753.
11. M. W. HIRSCH, Systems of differential equations which are competitive or cooperative. I. Limit sets, SIAM J. Appl. Math., 13 (1982), pp. 167-179.
12. C. S. HOLLING, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Soc. Canada, 45 (1965), pp. 3-60.
13. J. HOFBAUER AND K. SIGMUND, The Theory of Evolution and Dynamical Systems, London Math. Soc. Stud. Texts 7, Cambridge University Press, Cambridge, UK, 1988.
14. S. B. HSU, Limiting behavior for competing species, SIAM J. Appl. Math., 34 (1978), pp. 760-763.
15. S. B. HSU, K. S. CHENG, AND S. P. HUBBELL, Exploitative competition of microorganisms for two complementary nutrients in continuous cultures, SIAM J. Appl. Math., 41 (1981), pp. 422-444.
16. S. B. HSU, S. HUBBELL, AND P. WALTMAN, A mathematical theory of single-nutrient competition in continuous cultures of micro-organisms, SIAM J. Appl. Math., 32 (1977), pp. 366-383.
17. J. HUISMAN AND F. J. WEISSING, Biodiversity of plankton by species oscillations and chaos, Nature, 402 (1999), pp. 407-410.
18. J. A. LEON AND D. B. TUMPSON, Competition between two species for two complementary or substitutable resources, J. Theor. Biol., 50 (1975), pp. 185-201.
19. A. J. LOTKA, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925.
20. B. LI, Global asymptotic behavior of the chemostat: General response functions and different removal rates, SIAM J. Appl. Math., 59 (1998), pp. 411-422.
21. B. LI, Analysis of Chemostat-Related Models with Distinct Removal Rates, Ph.D thesis, Arizona State University, Tempe, AZ, 1998.
22. B. LI, G. S. K. WOLKOWICZ, AND Y. KUANG, Global asymptotic behavior of a chemostat model with two perfectly complementary resources and distributed delay, SIAM J. Appl. Math., 60 (2000), pp. 2058-2086.
23. R. M. MAY AND W. J. LEONARD, Nonlinear aspects of competition between three species, SIAM J. Appl. Math., 29 (1975), pp. 243-253.
24. A. NOVICK AND L. SZILIARD, Description of the chemostat, Science, 112 (1950), pp. 715-716.
25. O. M. PHILLIPS, The equilibrium and stability of simple marine biological systems I. Primary nutrient consumers, Amer. Natur., 107 (1973), pp. 73-93.
26. D. RAPPORT, An optimization model of food selection, Amer. Natur. 105 (1971), pp. 575-587.
27. L. A. REAL, The kinetics of functional response, Amer. Natur., 111 (1977), pp. 287-300.
28. D. N. RYDER AND C. G. SINCLAIR, Model for the growth of aerobic microorganisms under oxygen limiting conditions, Biotechnol. Bioengrg., 14 (1972), pp. 787-798.
29. U. SOMMER, Competition and coexistence, Nature, 402 (1999), pp. 366-367.
30. R. N. SKYES, Identification of the limiting nutrient and specific growth rate, J. Water Pollut. Control Fed., 45 (1973), pp. 888-895.
31. H. L. SMITH AND P. WALTMAN, The Theory of the Chemostat, Cambridge University Press, Cambridge, UK, 1995.
32. H. L. SMITH, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, AMS, Providence, RI, 1995.
33. H. L. SMITH, Periodic orbits of competitive and cooperative systems, J. Differential Equations, 65 (1986), pp. 361-373.
34. D. TILMAN AND S. S. KOLHAM, Phosphate and silicate growth and uptake kinetics of the diatoms, aterionella formosa and cyclotella meneghiniana, J. Phycol., 12 (1976), pp. 375-383.
35. D. TILMAN, Resource competition between planktonic algae: An experimental and theoretical approach, Ecology, 58 (1977), pp. 338-348.
36. D. TILMAN, Resource Competition and Community Structure, Princeton University Press, Princeton, NJ, 1982.
37. H. R. THIEME, Persistence under relaxed point-dissipativity (with an application to an endemic model), SIAM J. Math. Anal, 24 (1993), pp. 407-435.
38. V. VOLTERRA, Variations and fluctuations of the number of individuals in animal species living together, J. Conserv. (Conserv. Int. Explor. Mer.), 3 (1928), pp. 3-51.
39. P. WALTMAN, S. P. HUBBELL, AND S. B. HSU, Theoretical and experimental investigation of microbial competition in continuous culture, in Modeling and Differential Equations, T. Burton, ed., Marcel Dekker, New York, 1980, pp. 105-172.
40. G. S. K. WOLKOWICZ AND Z. LU, Global dynamics of a mathematical model of competition in the chemostat: General response function and differential death rates, SIAM J. Appl. Math., 52 (1992), pp. 222-233.
41. G. S. K. WOLKOWICZ, H. XIA, AND S. RUAN, Competition in the chemostat: A distributed delay model and its global asymptotic behavior, SIAM J. Appl. Math., 57 (1997), pp. 1281-1310.
42. X.-Q. ZHAO, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications, Canad. Appl. Math. Quart., 3 (1995), pp. 473-495.