Abstract
The growth of a species feeding on a limiting nutrient supplied at a constant rate is modelled by chemostat-type equations with a general nutrient uptake function and delayed nutrient recycling. Conditions for boundedness of the solutions and the existence of non-negative equilibria are given for the integrodifferential equations with distributed time lags. When the time lags are neglected conditions for the global stability of the positive equilibrium and for the extinction of the species are provided. The positive equilibrium continues to be locally stable when the time lag in recycling is considered and this is proved for a wide class of memory functions. Computer simulations suggest that even in this case the region of stability is very large, but the solutions tend to the equilibrium through oscillations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Caperon, J.: Population growth response of Isochrysis galbana to nitrate variation at limiting concentration. Ecology 49, 866–872 (1968)
Cushing, J. M.: Integrodifferential equations and delay models in population dynamics. (Lect. Notes Biomath., vol. 20) Berlin Heidelberg New York: Springer 1977
Gopalsamy, K.: Transit delays and dispersive populations in patchy habitats. Int. J. System Sci. 14, 239–268 (1983)
Hale, J. K., Somolinos, A. S.: Competition for fluctuating nutrient. J. Math. Biol. 18, 255–280 (1983)
Hsu, S. B., Hubbel, S., Waltman, P.: A mathematical theory for single-nutrient competition in continuous cultures of micro-organisms. SIAM J. Appl. Math. 32, 366–383 (1977)
La Salle, J., Lefschetz, S.: Stability by Liapunov's direct method. New York: Academic Press 1961
MacDonald, N.: Time lags in biological models. (Lect. Notes Biomath., vol. 27) Berlin Heidelberg New York: Springer 1978
Nisbet, R. M., Gurney, W. S. C.: Model of material cycling in a closed ecosystem. Nature 264, 633–635 (1976).
Nisbet, R. M., McKinstry, J., Gurney, W. S. C.: A strategic model of material cycling in a closed ecosystem. Math. Biosci. 64, 99–113 (1983)
Powell, T., Richerson, P. J.: Temporal variation, spatial heterogeneity and competition for resources in plankton system: a theoretical model. Am. Nat. 125, 431–464 (1985)
Smith, H. L.: Competitive coexistence in an oscillating chemostat. SIAM J. Appl. Math. 40, 498–522 (1981)
Svirezhev, Y. M., Logofet, D. O.: Stability of biological communities. Moscow: Mir 1983
Ulanowicz, R. E.: Mass and energy flow in closed ecosystems. J. Theor. Biol. 34, 239–253 (1972)
Waltman, P., Hubbel, S. P., Hsu, S. B.: Theoretical and experimental investigations of microbial competition in continuous culture. In: Burton, T. (ed.) Modeling and differential equations in biology. New York: Dekker 1980
Whittaker, R. H.: Communities and ecosystems. New York: Macmillan 1975
Yoshizawa, T.: Stability theory by Liapunov's second method. The Mathematical Society of Japan 1966
Author information
Authors and Affiliations
Additional information
Work performed within the activity of the research group “Equazioni di evoluzione e applicazioni Fisico-Matematiche”, MPI (Italy), and under the auspices of GNFM, CNR (Italy)
Rights and permissions
About this article
Cite this article
Beretta, E., Bischi, G.I. & Solimano, F. Stability in chemostat equations with delayed nutrient recycling. J. Math. Biol. 28, 99–111 (1990). https://doi.org/10.1007/BF00171521
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00171521