### Mathematical Biosciences and Engineering

2020, Issue 3: 2103-2137. doi: 10.3934/mbe.2020112
Research article

# Analysis of a generalized Fujikawa’s growth model

• 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.

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

### Related Papers:

• 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.

 [1] J. Baranyi, T.A. Roberts, P. McClure, A non-autonomous differential equation to model bacterial growth, Int. J. Food Microbiol., 10 (1993), 43-59. [2] M.H. Zwietering, I. Jongenburger, F.M Rombouts, K. Van't Riet, Modeling of the bacterial growth curve, Appl. Environ. Microb., 56 (1990), 1875-1881. [3] S. Perni, P.W. Andrew, G. Shama, Estimating the maximum growth rate from microbial growth curves: Definition is everything, Int. J. Food Microbiol., 22 (2005), 491-495. [4] S. Basak, P. Guha, Modelling the effect of essential oil of betel leaf (Piper betle L.) on germination, growth, and apparent lag time of Penicillium expansum on semi-synthetic media, Int. J. Food Microbiol., 215 (2015), 171-178. [5] P.R. Santos, I.C. Tessaro, L.D. Ferreira, Integrating a kinetic microbial model with a heat transfer model to predict Byssochlamys fulva growth in refrigerated papaya pulp, J. Food Eng., 118 (2013), 279-288. [6] P.R. Santos, I.C. Tessaro, L.D. Ferreira, Modeling and simulation of Byssochlamys fulva growth on papaya pulp subjected to evaporative cooling, Chem. Eng. Sci., 114 (2014), 134-143. [7] R.M. de Castro, J.R. de Souza, A.F da Eira, Digital monitoring of mycelium growth kinetics and vigor of shiitake (Lentinula edodes (Berk.) Pegler) on agar medium, Braz. J. Microbiol., 37 (2006), 90-95. [8] A. Talkington, R. Durrett, Estimating tumor growth rates in vivo, Bull. Math. Biol., 77 (2015), 1934-1954. [9] C.P. Birch, A new generalized logistic sigmoid growth equation compared with the Richards growth equation, Ann. Bot. London, 83 (1999), 713-723. [10] X. Yin, J. Goudriaan, E.A. Lantinga, J. Vos, H.J. Spiertz, A flexible sigmoid function of determinate growth, Ann. Bot. London, 91 (2003), 361-37. [11] J. Ukalska, S. Jastrzebowski, Sigmoid growth curves, a new approach to study the dynamics of the epicotyl emergence of oak, Folia For. Pol., Ser. A, 61 (2019), 30-41. [12] J.G. Garca, R. Ramrez, R. Nez, J.A. Hidalgo, Dataset on growth curves of Boer goats fitted by ten non-linear functions, Data Brief, 23 (2019), 1-10. [13] J.M. Coyne, K. Matilainen, D.P. Berry, M.L. Sevn, E.A. Mantysaari, J.Juga, et al., Estimation of genetic (co)variances of Gompertz growth function parameters in pigs, J. Anim. Breed Genet., 134 (2017), 136-143. [14] R.C. Bruce, Application of the Gompertz function in studies of growth in dusky salamanders (Plethodontidae: Desmognathus), Copeia, 104 (2016), 94-100. [15] A. Silveira, J.R. Moreno, M.J. Correia, V. Ferro, A method for the rapid evaluation of leather biodegradability during the production phase, Waste Manage., 87 (2019), 661-661. [16] H. Fujikawa, A. Kai, S. Morozumi, A new logistic model for Escherichia coli growth at constant and dynamic temperatures, Int. J. Food Microbiol., 21 (2004), 501-509. [17] P. Vadasz, A.S. Vadasz, Predictive modeling of microorganisms: LAG and LIP in monotonic growth, Int. J. Food Microbiol., 102 (2005), 257-257. [18] S. Brown, An estimate of the duration of the lag phase of the logistic growth curve, Ann. West Univ. Timisoara, 17 (2014), 25-32. [19] A. Tsoularis, J. Wallace, Analysis of logistic growth models, Math. Biosci., 179 (2002), 21-55. [20] S. Ohnishi, T. Yamakawa, T. Akamine, On the analytical solution for the Putter-Bertalanffy growth equation, J. Theor. Biol., 343 (2014), 174-177. [21] P. Vadasz, A.S. Vadasz, The neoclassical theory of population dynamics in spatially homogeneous environments. (I) Derivation of universal laws and monotonic growth, Physica A, 309 (2002), 329-359. [22] F. Poschet, K.M. Vereecken, A.H. Geeraerd, B.M. Nicolai, J.F. Van Impe, Analysis of a novel class of predictive microbial growth models and application to coculture growth, Int. J. Food Microbiol., 100 (2005), 107-124. [23] J.F. Van Impe, F. Poschet, A.H. Geeraerd, K.M. Vereecken, Towards a novel class of predictive microbial growth models, Int. J. Food Microbiol., 100 (2005), 97-105. [24] I.A. Swinnen, K. Bernaerts, E.J. Dens, A.H. Geeraerd, J.F. Van Impe, Predictive modelling of the microbial lag phase: A review, Int. J. Food Microbiol., 94 (2004), 137-159. [25] A. Di Crescenzo, S. Spina, Analysis of a growth model inspired by Gompertz and Korf laws, and an analogous birth-death process, Math. Biosci, 282 (2016), 121-134. [26] I. Mytilinaios, M. Salih, H.K. Schofield, R.J. Lambert, Growth curve prediction from optical density data, Int. J. Food Microbiol., 154 (2012), 169-176. [27] B.M. Nicolai, J.F. Van Impe, B. Verlinden, T. Martens, J. Vandewalle, J. De Baerdemaeker, Predictive modelling of surface growth of lactic acid bacteria in vacuum-packed meat, Int. J. Food Microbiol., 10 (1993), 229-238. [28] H. Fujikawa, S. Morozumi, Modeling Staphylococcus aureus growth and enterotoxin production in milk, Int. J. Food Microbiol., 23 (2006), 260-267. [29] J. Baranyi, T.A. Roberts, Mathematics of predictive food Microbiol.ogy, Int. J. Food Microbiol., 26 (1995), 199-218. [30] P.R Koya, A.T. Goshu, Generalized mathematical model for biological growths, Open J. Modell. Simul., 1 (2013), 42-53. [31] J. Baranyi, T.A. Roberts, A dynamic approach to predicting bacterial growth in food, Int. J. Food Microbiol., 23 (1994), 277-294. [32] F. Baty, M. L. Delignette-Muller, Estimating the bacterial lag time: Which model, which precision?, Int. J. Food Microbiol., 91 (2004), 261-277. [33] A.W. Mayo, M. Muraza, J. Norbert, Modelling nitrogen transformation and removal in mara river basin wetlands upstream of lake Victoria, Phys. Chem. Earth, 105 (2018), 136-146. [34] K. Dutta, V.V. Dasu, B. Mahanty, A.A. Prabhu, Substrate inhibition growth kinetics for cutinase producing pseudomonas cepacia using tomato-peel extracted cutin, Chem. Biochem. Eng. Q, 29 (2015), 437-445. [35] M. Li, Y. Li, X. Huang, G. Zhao, W. Tian, Evaluating growth models of Pseudomonas spp. in seasoned prepared chicken stored at different temperatures by the principal component analysis (PCA), Int. J. Food Microbiol., 40 (2014), 41-47.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

2.194 3.0

Article outline

Figures(11)

• On This Site