The deterministic logistic model is defined in terms of a differential equation
where the linear constant defines the growth rate, whereas the quadratic term
serves to inhibit or retard this rate. In this sense, the quadratic term is usually
smaller than the linear one. When the population is small, the quadratic term, or
inhibiting term, has little effect on the rising, so the population starts off with
almost exponential growth. However, as the population increases, the inhibiting
term eventually slows the rate of growth dramatically. Environmental fluctuations
and lack of precision measurements represent ubiquitous noises that contribute
with randomness to the logistic model (Li et al. 2011).
For processes whose mean function will be a logistic curve, it is possible to
explicitly determine the transition density function, which allows to answer
questions like the estimation of parameters through discrete sampling of paths after
finding the likelihood from the transition density functions and the initial distribution. Such improvement enables the researcher to calculate the first passage time
density, which requires an explicit form of such densities (Román-Román and
Torres Ruiz 2012).
Logistic models offer the possibility of finding explicitly transition densities.
This characteristic allows analyzing inference from the discrete sampling of trajectories. They also permit the estimation of functions, which allow predictive uses
of the model. Finally, by studying first passage times, these models help in the
determination of time variables used to locate when a preset condition is verified.
2.3.2.5 Statistical Approaches
The statistical approaching methods for modeling systems and processes consist,
mainly, in the evaluation through, and application of, statistical techniques to
datasets. Their only limitation is large number of datasets with different experimental conditions required for the application of any statistical packages. This
limitation maybe seems to be small, but it usually is a challenging task.
Some examples of statistical techniques are: Levenberge-Marquardt (LeM)
method, Nonlinear mixed effects (NLME), and first-order kinetics (Stein et al. 2007).
Since these models use a large number of datasets, it is commonly found that
the magnitude of the generated coefficients varies strongly by species.
2.3.2.6 Time-Dependent Retardation Model
Time-dependent retardation models assume that there is a continuous variation in
the studied process, which is common for every biosystem process, where time is a
common degradation variable. However, it is important to note that the performance of this kind of models is often limited by a residual outlet concentration of
certain variables in the studied system.
This kind of modeling has been considered to be one of the most efficient
methods for designing CWs, because it allows a steady-state decrease in chemical
60
M. A. Vázquez-Cruz et al.
where the linear constant defines the growth rate, whereas the quadratic term
serves to inhibit or retard this rate. In this sense, the quadratic term is usually
smaller than the linear one. When the population is small, the quadratic term, or
inhibiting term, has little effect on the rising, so the population starts off with
almost exponential growth. However, as the population increases, the inhibiting
term eventually slows the rate of growth dramatically. Environmental fluctuations
and lack of precision measurements represent ubiquitous noises that contribute
with randomness to the logistic model (Li et al. 2011).
For processes whose mean function will be a logistic curve, it is possible to
explicitly determine the transition density function, which allows to answer
questions like the estimation of parameters through discrete sampling of paths after
finding the likelihood from the transition density functions and the initial distribution. Such improvement enables the researcher to calculate the first passage time
density, which requires an explicit form of such densities (Román-Román and
Torres Ruiz 2012).
Logistic models offer the possibility of finding explicitly transition densities.
This characteristic allows analyzing inference from the discrete sampling of trajectories. They also permit the estimation of functions, which allow predictive uses
of the model. Finally, by studying first passage times, these models help in the
determination of time variables used to locate when a preset condition is verified.
2.3.2.5 Statistical Approaches
The statistical approaching methods for modeling systems and processes consist,
mainly, in the evaluation through, and application of, statistical techniques to
datasets. Their only limitation is large number of datasets with different experimental conditions required for the application of any statistical packages. This
limitation maybe seems to be small, but it usually is a challenging task.
Some examples of statistical techniques are: Levenberge-Marquardt (LeM)
method, Nonlinear mixed effects (NLME), and first-order kinetics (Stein et al. 2007).
Since these models use a large number of datasets, it is commonly found that
the magnitude of the generated coefficients varies strongly by species.
2.3.2.6 Time-Dependent Retardation Model
Time-dependent retardation models assume that there is a continuous variation in
the studied process, which is common for every biosystem process, where time is a
common degradation variable. However, it is important to note that the performance of this kind of models is often limited by a residual outlet concentration of
certain variables in the studied system.
This kind of modeling has been considered to be one of the most efficient
methods for designing CWs, because it allows a steady-state decrease in chemical
60
M. A. Vázquez-Cruz et al.