On a stochastic logistic population model with time-varying carrying capacity
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On a stochastic logistic population model with time-varying carrying capacity J. Calatayud1 · J.-C. Cortés1
· F. A. Dorini2 · M. Jornet1
Received: 27 May 2020 / Revised: 27 May 2020 / Accepted: 26 September 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we deal with the logistic growth model with a time-dependent carrying capacity that was proposed in the literature for the study of the total bacterial biomass during occlusion of healthy human skin. Accounting for data and model errors, randomness is incorporated into the equation by assuming that the input parameters are random variables. The uncertainty is quantified by approximations of the solution stochastic process via truncated series solution together with the random variable transformation method. Numerical examples illustrate the theoretical results. Keywords Logistic growth model · Time-dependent carrying capacity · Random parameters · Probability density function Mathematics Subject Classification 34F05 · 92D25 · 92D40
Communicated by Valeria Neves Domingos Cavalcanti. This work has been supported by the Spanish Ministerio de Economía, Industria y Competitividad (MINECO), the Agencia Estatal de Investigación (AEI), and Fondo Europeo de Desarrollo Regional (FEDER UE) Grant MTM2017-89664-P.
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J.-C. Cortés [email protected] J. Calatayud [email protected] F. A. Dorini [email protected] M. Jornet [email protected]
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Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
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Department of Mathematics, Federal University of Technology-Paraná, 80230-901 Curitiba, PR, Brazil 0123456789().: V,-vol
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1 Introduction Growth models such as the logistic equation are widely studied and applied in population and ecological modeling. Classically, the carrying capacity of the logistic equation model has been considered constant. However, some works started to consider it as a function of time, motivated by the principle that a changing environment may result in a significant change in the limiting capacity (Safuan et al. 2013). It is the case of the model proposed in Safuan et al. (2011, 2013) for the study of total bacterial biomass during occlusion of healthy human skin. The model is presented by the non-autonomous logistic equation: ⎧ N (t) ⎨ N (t) = a N (t) 1 − , t > 0, (1) K (t) ⎩ N (0) = N0 , where N0 > 0 is the initial condition and a > 0 is the growth rate parameter, driven by the time-varying capacity, K (t), that takes the form: K 0 −ct K (t) = K s 1 − 1 − e , (2) Ks where K 0 = K (0) is the initial limiting capacity, K s = limt→+∞ K (t) is the bacterial saturation (or equilibrium) level, and c > 0 is the saturation constant. It is assumed N0 < K0 < Ks . This model assumes that on the unoccluded skin, the environment is relatively constant and the density of microbes is in equilibrium with its environment (K 0 ≈ N0 ). After an occlusion is applie
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