Analysis of a generalized Fujikawa’s growth model

Alejandro Rincón; Fabiola Angulo; Fredy E. Hoyos; Alejandro Rincón; Fabiola Angulo; Fredy E. Hoyos

doi:10.3934/mbe.2020112

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 3: 2103-2137. doi: 10.3934/mbe.2020112

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Research article

Analysis of a generalized Fujikawa’s growth model

1.
Instituto de Investigación en Microbiología y Biotecnología Agroindustrial, Universidad Católica de Manizales, Grupo de Investigaciones Biológicas -GIBI, Carrera 23 N. 60–63, 170002, Manizales, Colombia
2.
Facultad de Ingeniería y Arquitectura Universidad Nacional de Colombia Sede Manizales, Departamento de Ingeniería Eléctrica, Electrónica y Computación-Percepción y Control Inteligente-Bloque Q, Campus La Nubia, 170003, Manizales, Colombia
3.
Universidad Nacional de Colombia, Sede Medellín, Facultad de Ciencias, Escuela de Física, Carrera 65 No. 59A, 110, Medellín 050034, Antioquia, Colombia

Received: 09 October 2019 Accepted: 26 December 2019 Published: 06 January 2020

We analyze a generalized form of the Fujikawas growth model which involves an adaptation function that enhances the representation of the lag phase. This model is autonomous, and combines a power law term, a saturation term and an adaptation function that suppresses the growth rate during initial period corresponding to the lag phase. The properties of the adaptation function are determined, and the proposed model is examined separately for the regular measure and the logarithmic measure, including: Convergence and boundedness properties; population at the inflection point; conditions for the existence of the inflection point and lag phase; effect of model parameters on the existence of the inflection point and lag phase; population size of the inflection point under limiting values of the model parameters; and parameter values that lead to inflection point located at the mean value of the curve. Different combinations of model parameters lead to different possibilities for the existence of the inflection point and the lag phase. It was noticed that the power law term has a strong effect on the representation of the exponential growth phase, whereas the adaptation function has a strong effect on the representation of the lag phase. The lag phase duration depends on the exponent parameter of the adaptation function, and its dependence with respect to the power law parameter is low. Also, an approach is proposed for the analytical determination of the lag time, based on the application of the classical approach to a simplified model. Ascertained lag time values were obtained, what confirms the assumptions. At last, the model is applied to experimental data.
- growth model,
- lag time,
- lag phase,
- inflection point,
- Fujikawa’s model
Citation: Alejandro Rincón, Fabiola Angulo, Fredy E. Hoyos. Analysis of a generalized Fujikawa’s growth model[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2103-2137. doi: 10.3934/mbe.2020112

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Abstract

We analyze a generalized form of the Fujikawas growth model which involves an adaptation function that enhances the representation of the lag phase. This model is autonomous, and combines a power law term, a saturation term and an adaptation function that suppresses the growth rate during initial period corresponding to the lag phase. The properties of the adaptation function are determined, and the proposed model is examined separately for the regular measure and the logarithmic measure, including: Convergence and boundedness properties; population at the inflection point; conditions for the existence of the inflection point and lag phase; effect of model parameters on the existence of the inflection point and lag phase; population size of the inflection point under limiting values of the model parameters; and parameter values that lead to inflection point located at the mean value of the curve. Different combinations of model parameters lead to different possibilities for the existence of the inflection point and the lag phase. It was noticed that the power law term has a strong effect on the representation of the exponential growth phase, whereas the adaptation function has a strong effect on the representation of the lag phase. The lag phase duration depends on the exponent parameter of the adaptation function, and its dependence with respect to the power law parameter is low. Also, an approach is proposed for the analytical determination of the lag time, based on the application of the classical approach to a simplified model. Ascertained lag time values were obtained, what confirms the assumptions. At last, the model is applied to experimental data.

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