Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Applications of the multi-sigmoidal deterministic and stochastic logistic models for plant dynamics

Antonio Di Crescenzo, Paola Paraggio, Patricia Román-Román, Francisco Torres-Ruiz

TL;DR

A generalization of the classical logistic growth model introducing more than one inflection point, called multi-sigmoidal, is considered, with the aim of obtaining a more manageable stochastic description of the growth.

Abstract

We consider a generalization of the classical logistic growth model introducing more than one inflection point. The growth, called multi-sigmoidal, is firstly analyzed from a deterministic point of view in order to obtain the main properties of the curve, such as the limit behavior, the inflection points and the threshold-crossing-time through a fixed boundary. We also present an application in population dynamics of plants based on real data. Then, we define two different birth-death processes, one with linear birth and death rates and the other with quadratic rates, and we analyze their main features. The conditions under which the processes have a mean of multi-sigmoidal logistic type and the first-passage-time problem are also discussed. Finally, with the aim of obtaining a more manageable stochastic description of the growth, we perform a scaling procedure leading to a lognormal diffusion process with mean of multi-sigmoidal logistic type. We finally conduct a detailed probabilistic analysis of this process

Applications of the multi-sigmoidal deterministic and stochastic logistic models for plant dynamics

TL;DR

A generalization of the classical logistic growth model introducing more than one inflection point, called multi-sigmoidal, is considered, with the aim of obtaining a more manageable stochastic description of the growth.

Abstract

We consider a generalization of the classical logistic growth model introducing more than one inflection point. The growth, called multi-sigmoidal, is firstly analyzed from a deterministic point of view in order to obtain the main properties of the curve, such as the limit behavior, the inflection points and the threshold-crossing-time through a fixed boundary. We also present an application in population dynamics of plants based on real data. Then, we define two different birth-death processes, one with linear birth and death rates and the other with quadratic rates, and we analyze their main features. The conditions under which the processes have a mean of multi-sigmoidal logistic type and the first-passage-time problem are also discussed. Finally, with the aim of obtaining a more manageable stochastic description of the growth, we perform a scaling procedure leading to a lognormal diffusion process with mean of multi-sigmoidal logistic type. We finally conduct a detailed probabilistic analysis of this process
Paper Structure (13 sections, 1 theorem, 94 equations, 14 figures, 3 tables)

This paper contains 13 sections, 1 theorem, 94 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The multi-sigmoidal logistic function
  3. Inflection points
  4. Threshold crossing problem
  5. An application
  6. Analysis of a linear birth-death process
  7. First-passage-time problem
  8. A non linear birth-death process
  9. Asymptotic behavior
  10. A diffusive approximation
  11. A diffusion process with multi-sigmoidal logistic mean
  12. First-passage-time problem
  13. Conclusions

Key Result

Proposition 4.1

The linear birth-death process $N(t)$ with rates specified by 3.1 has conditional mean of multi-sigmoidal logistic type if, and only if, the net growth rate A is given by where $h_\theta(t)$ is defined in htheta.

Figures (14)

  • Figure 1: The multi-sigmoidal logistic function for (a) $p=2$ and (b) $p=3$. The values of the parameters are $\eta=e^{-0.5}, e^{-1}, e^{-2}$ (from bottom to top), $l_0=(\eta+1)^{-1}$ and (a) $Q_\beta(t)=-0.1t+0.09t^2$, (b) $Q_\beta(t)=0.1t-0.009t^2+0.0002t^3$.
  • Figure 2: The derivative of the multi-sigmoidal logistic function for (a) $p=2$ and (b) $p=3$. The values of the parameters are $\eta=e^{-0.5}$, $e^{-1}$, $e^{-2}$ (from bottom to top), $l_0=(\eta+1)^{-1}$ and (a) $Q_\beta(t)=-0.1t+0.09t^2$, (b) $Q_\beta(t)=0.1t-0.009t^2+0.0002t^3$.
  • Figure 3: The function $h_\theta$ for (a) $p=2$ and (b) $p=3$. The values of the parameters are $\eta=e^{-0.5}$, $e^{-1}$, $e^{-2}$ (from bottom to top for large $t$) and (a) $Q_\beta(t)=-0.1t+0.09t^2$, (b) $Q_\beta(t)=0.1t-0.009t^2+0.0002t^3$.
  • Figure 4: The function $l"_m$ for (a) $p=2$ and (b) $p=3$. The values of the parameters are $\eta=e^{-0.5}$, $e^{-1}$, $e^{-2}$ (from bottom to top near the origin), $l_0=(\eta+1)^{-1}$ and (a) $Q_\beta(t)=-0.1t+0.09t^2$, (b) $Q_\beta(t)=0.1t-0.009t^2+0.0002t^3$.
  • Figure 5: The threshold crossing times for $\eta=0.01$ and (a) $\theta_U$, for $Q_\beta(t)=-t+0.01t^2$, $Q_\beta(t)=-2t+0.02t^2$, $Q_\beta(t)=-3t+0.03t^2$ (for top to bottom), and (b) $\theta_L$ (for the same choices of $Q_\beta(t)$, in reversed order).
  • ...and 9 more figures

Theorems & Definitions (6)

  • Remark 2.1
  • Remark 2.2
  • Proposition 4.1
  • Example 4.1
  • Remark 5.1
  • Example 5.1