×

zbMATH — the first resource for mathematics

Transient behavior of biological loop models with application to the Droop model. (English) Zbl 0822.92001
Summary: We study the transient behavior of a class of nonlinear differential systems verifying sign conditions through the succession of extrema of the state variables. This analysis does not depend, for the main part, on the analytical formulation of the model. The possible scenarios of sequences for the extrema are represented on a graph and can be compared with the experimental data to validate the model. An application to the Droop model illustrates this method; we obtain as a result the global stability of the equilibrium and the possible successions of the extrema.

MSC:
92B05 General biology and biomathematics
05C90 Applications of graph theory
93A30 Mathematical modelling of systems (MSC2010)
92D40 Ecology
93A99 General systems theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Burmaster, D.E., The unsteady continuous culture of phosphate-limited monochrysis lutheri droop: experimental and theoretical analysis, J. exp. mar. biol. ecol., 39, 2, 167-186, (1979)
[2] Clarke, B.L., Stability of complex chemical reaction networks, (), 1-215
[3] Droop, M.R., Vitamin B12 and marine ecology. IV. the kinetics of uptake growth and inhibition in monochrysis lutheri, J. mar. biol. assoc., 48, 3, 689-733, (1968)
[4] Edelstein, L., Mathematical models in biology, (1988), Random House New York · Zbl 0674.92001
[5] Feinberg, M., Chemical reaction network structure and the stability of complex isothermal reactors I, Chem. eng. sci., 42, 2229-2268, (1987)
[6] Glass, L.; Pasternack, J.S., Stable oscillations in mathematical models of biological control systems, J. math. biol., 6, 207-223, (1978) · Zbl 0391.92001
[7] Goodwin, B.C., Oscillatory behaviour in enzymatic control processes, (), 425-438
[8] Jacquez, J.J.; Simon, C.P., Qualitative theory of compartmental systems, SIAM rev., 35, 1, 43-79, (1993) · Zbl 0776.92001
[9] Jeffries, C., Qualitative stability of certain nonlinear systems, Linear algebra appl., 75, 133-144, (1986) · Zbl 0694.93084
[10] Lange, K.; Oyarzun, F.J., The attractiveness of the droop equations, Math. biosci., 111, 261-278, (1992) · Zbl 0762.92021
[11] Levine, D.S., Qualitative theory of a third order nonlinear system with examples in population dynamics and chemical kinetics, Math. biosci., 77, 17-33, (1985) · Zbl 0579.92020
[12] Monod, J., Recherches sur la croissance des cultures bactériennes, (1942), Hermann Paris
[13] Segel, L.A., Modeling dynamic phenomena in molecular and cellular biology, (1984), Cambridge University Press Cambridge · Zbl 0639.92001
[14] Segel, L.A., On the validity of the steady state assumption of enzyme kinetics, Bull. math. biol., 50, 6, 579-593, (1988) · Zbl 0653.92006
[15] Snoussi, E.H.; Thomas, R., Logical identification of all steady states: the concept of feedback loop characteristic states, Bull. math. biol., 55, 5, 973-991, (1993) · Zbl 0784.92002
[16] (), Lecture Notes in Biomathematics
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.