Analysis of logistic growth models.

*(English)*Zbl 0993.92028Summary: A variety of growth curves have been developed to model both unpredated, intraspecific population dynamics and more general biological growth. Most predictive models are shown to be based on variations of the classical Verhulst logistic growth equation. We review and compare several such models and analyse properties of interest for these. We also identify and detail several associated limitations and restrictions.

A generalized form of the logistic growth curve is introduced which incorporates these models as special cases. Several properties of the generalized growth are also presented. We furthermore prove that the new growth form incorporates additional growth models which are markedly different from the logistic growth and its variants, at least in their mathematical representation. Finally, we give a brief outline of how the new curve could be used for curve-fitting.

A generalized form of the logistic growth curve is introduced which incorporates these models as special cases. Several properties of the generalized growth are also presented. We furthermore prove that the new growth form incorporates additional growth models which are markedly different from the logistic growth and its variants, at least in their mathematical representation. Finally, we give a brief outline of how the new curve could be used for curve-fitting.

##### MSC:

92D25 | Population dynamics (general) |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

92B05 | General biology and biomathematics |

##### Keywords:

biological growth dynamics; logistic growth; generalized logistic growth; inflection point; incomplete betafunction; beta function; gamma function; mimimax; saddle curve; finite difference method##### Software:

GenStat
PDF
BibTeX
XML
Cite

\textit{A. Tsoularis} and \textit{J. Wallace}, Math. Biosci. 179, No. 1, 21--55 (2002; Zbl 0993.92028)

Full Text:
DOI

##### References:

[1] | Verhulst, P.F., Notice sur la loi que la population suit dans son accroissement, Curr. math. phys., 10, 113, (1838) |

[2] | Carlson, T., Über geschwindigkeit und grösse der hefevermehrung in würze, Biochem. Z, 57, 313, (1913) |

[3] | Pearl, R., The growth of populations, Quart. rev. biol., 2, 532, (1927) |

[4] | Pearl, R., Introduction of medical biometry and statistics, (1930), Saunders Philadelphia, PA |

[5] | Morgan, B.J.T., Stochastic models of groupings changes, Adv. appl. probability, 8, 30, (1976) |

[6] | Krebs, C.J., The experimental analysis of distribution and abundance, (1985), Harper and Row New York |

[7] | Fisher, T.C.; Fry, R.H., Tech. forecasting soc. changes, 3, 75, (1971) |

[8] | C. Marchetti, N. Nakicenovic, The Dynamics of Energy Systems and the Logistic Substitution Model, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1980 |

[9] | Herman, R.; Montroll, E.W., Proc. nat. acad. sci. USA, 69, 3019, (1972) |

[10] | Nelder, J.A., The Fitting of a generalization of the logistic curve, Biometrics, 17, 89, (1961) · Zbl 0099.14303 |

[11] | Abramowitz, M.; Stegun, I.M., Handbook of mathematical functions, (1965), Dover New York |

[12] | Pearl, R.; Reed, L.J., On the rate of growth of the population of united states Since 1790 and its mathematical representation, Proc. nat. acad. sci. USA G, 275, (1920) |

[13] | Richards, F.J., A flexible growth function for empirical use, J. exp. botany, 10, 29, 290, (1959) |

[14] | Von Bertalanffy, L., A quantitative theory of organic growth, Human biol., 10, 2, 181, (1938) |

[15] | Blumberg, A.A., Logistic growth rate functions, J. theor. biol., 21, 42, (1968) |

[16] | Turner, M.E.; Blumenstein, B.A.; Sebaugh, J.L., A generalization of the logistic law of growth, Biometrics, 25, 577, (1969) |

[17] | Turner, M.E.; Bradley, E.; Kirk, K.; Pruitt, K., A theory of growth, Math. biosci., 29, 367, (1976) · Zbl 0328.92014 |

[18] | Buis, R., On the generalization of the logistic law of growth, Acta biotheoretica, 39, 185, (1991) |

[19] | Smith, F.E., Population dynamics in daphnia magna and a new model for population growth, Ecology, 44, 4, 651, (1963) |

[20] | Lotka, A.J., Elements of mathematical biology, (1956), Dover New York · Zbl 0074.14404 |

[21] | Spencer, R.P.; Coulombe, M.J., Quantitation of hepatic growth and regeneration, Growth, 30, 277, (1966) |

[22] | Schnute, J., A versatile growth model with statistically stable parameters, Can. J. fish. aquat. sci., 38, 1128, (1981) |

[23] | Zeide, B., Analysis of growth equations, Forestscience, 39, 3, 594, (1993) |

[24] | Birch, C.P.D., A new generalized logistic Sigmoid growth equation compared with the Richards growth equation, Ann. botany, 83, 713, (1999) |

[25] | Heinen, M., Analytical growth equations and their genstat 5 equivalents, Netherlands J. agricult. sci., 47, 67, (1999) |

[26] | Savageau, M.A., Growth of complex systems can be related to the properties of their underlying determinants, Proc. nat. acad. sci. USA, 76, 11, 5413, (1979) · Zbl 0411.92005 |

[27] | Savageau, M.A., Growth equations: a general equation and a survey of special cases, Math. biosci., 48, 267, (1980) · Zbl 0419.92001 |

[28] | Ratkowsky, D., Non-linear regression modeling: a unified practical approach, (1983), Marcel Dekker New York · Zbl 0572.62054 |

[29] | Pearson, K., Tables of the incomplete beta function, (1968), Library Binding, Lubrecht and Cramer · Zbl 0157.24103 |

[30] | Banks, R.B., Growth and diffusion phenomena, (1994), Springer Berlin · Zbl 0788.92001 |

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.