On stability and periodicity in phosphorus nutrient dynamics
Author:
E. M. Arnold
Journal:
Quart. Appl. Math. 38 (1980), 139-141
MSC:
Primary 92A15; Secondary 58F10
DOI:
https://doi.org/10.1090/qam/575838
MathSciNet review:
575838
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Abstract: The stability of an equilibrium and the existence of limit cycles in a three-dimensional dynamical system arising in predator-prey-nutrient dynamics are demonstrated, using center manifold theory. Some implications of this result for limnological applications are discussed.
D. M. DiToro, D. J. O’Connor and R. V. Thomann, A dynamic model of phytoplankton population in the Sacramento-San Joaquin Delta, Adv. Chem. Series 106, Am. Chem. Soc., pp. 131–180 (1971)
- Morris W. Hirsch and Stephen Smale, Differential equations, dynamical systems, and linear algebra, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Pure and Applied Mathematics, Vol. 60. MR 0486784
E. M. Arnold, Aspects of a zooplankton, phytoplankton, phosphorus system, Ecological Modeling 5, 293–300 (1978)
- H. I. Freedman and P. Waltman, Perturbation of two-dimensional predator-prey equations, SIAM J. Appl. Math. 28 (1975), 1–10. MR 357961, DOI https://doi.org/10.1137/0128001
- Al Kelley, Stability of the center-stable manifold, J. Math. Anal. Appl. 18 (1967), 336–344. MR 210998, DOI https://doi.org/10.1016/0022-247X%2867%2990061-3
- Juan Lin and Peter B. Kahn, Averaging methods in predator-prey systems and related biological models, J. Theoret. Biol. 57 (1976), no. 1, 73–102. MR 496849, DOI https://doi.org/10.1016/S0022-5193%2876%2980006-9
D. M. DiToro, D. J. O’Connor and R. V. Thomann, A dynamic model of phytoplankton population in the Sacramento-San Joaquin Delta, Adv. Chem. Series 106, Am. Chem. Soc., pp. 131–180 (1971)
M. Hirsch and S. Smale, Differential equations, dynamical systems and linear algebra, Academic Press, New York, pp. 261–262, 1974
E. M. Arnold, Aspects of a zooplankton, phytoplankton, phosphorus system, Ecological Modeling 5, 293–300 (1978)
H. I. Freedman and P. Waltman, Perturbation of two-dimensional predator-prey equations, SIAM J. Appl. Math. 28, 1–11 (1975)
A. Kelley, Stability of the center-stable manifold, J. Math. Anal. Appl. 18, 336–344 (1967)
J. Lin and P. B. Kahn, Averaging methods in predator-prey systems and related biological models, J. Theor. Biol. 57, 73–102 (1976)
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Article copyright:
© Copyright 1980
American Mathematical Society
