Gompertz-type Diffusion Process Research Articles

Modeling Number of Trees per Hectare Dynamics for Uneven-Aged, Mixed-Species Stands Using the Copula Approach

For the monitoring and management of forest resources, the main index is the stand volume, which is determined on the basis of the tree diameter, height, and number of trees per hectare of three-dimensional distribution. The development of trees in the forest stand is dynamic and is driven by random phenomena. In this study, the tree diameter, the potentially available area, and the height are described by the mixed-effect parameters of the Gompertz-type diffusion process. A normal copula function is used to connect a three-dimensional distribution to its one-dimensional margins. The newly developed model was illustrated using empirical data from 53 permanent experimental plots (measured for seven cycles), which were characterized as follows: pine forests (Pinus sylvestris), 63.8%; spruce (Picea abies), 30.2%; silver birch (Betula pendula Roth and Betula pubescens Ehrh.), 5.8%; and others, 0.2%. An analysis of the tree diameter and height of growth, including current and mean increments and inflection points, is presented. The models for the change in the number of trees per hectare with age are presented on the basis of the probabilistic density functions of the solutions of stochastic differential equations and the copula function. The dynamics of the number of trees per hectare are visualized graphically, and the goodness of fit of the newly developed models is evaluated using standard statistical measures.

Restricted Gompertz-Type Diffusion Processes with Periodic Regulation Functions

We consider two different time-inhomogeneous diffusion processes useful to model the evolution of a population in a random environment. The first is a Gompertz-type diffusion process with time-dependent growth intensity, carrying capacity and noise intensity, whose conditional median coincides with the deterministic solution. The second is a shifted-restricted Gompertz-type diffusion process with a reflecting condition in zero state and with time-dependent regulation functions. For both processes, we analyze the transient and the asymptotic behavior of the transition probability density functions and their conditional moments. Particular attention is dedicated to the first-passage time, by deriving some closed form for its density through special boundaries. Finally, special cases of periodic regulation functions are discussed.

Some Notes about Inference for the Lognormal Diffusion Process with Exogenous Factors

Different versions of the lognormal diffusion process with exogenous factors have been used in recent years to model and study the behavior of phenomena following a given growth curve. In each case considered, the estimation of the model has been addressed, generally by maximum likelihood (ML), as has been the study of several characteristics associated with the type of curve considered. For this process, a unified version of the ML estimation problem is presented, including how to obtain estimation errors and asymptotic confidence intervals for parametric functions when no explicit expression is available for the estimators of the parameters of the model. The Gompertz-type diffusion process is used here to illustrate the application of the methodology.

Estimating a non-homogeneous Gompertz process with jumps as model of tumor dynamics

A non-homogeneous stochastic model based on a Gompertz-type diffusion process with jumps is proposed to describe the evolution of a solid tumor subject to an intermittent therapeutic program. Each therapeutic application, represented by a jump in the process, instantly reduces the tumor size to a fixed value and, simultaneously, increases the growth rate of the model to represent the toxicity of the therapy. This effect is described by introducing a time-dependent function in the drift of the process. The resulting model is a combination of several non-homogeneous diffusion processes characterized by different drifts, whose transition probability density function and main characteristics are studied. The study of the model is performed by distinguishing whether the therapeutic instances are fixed in advance or guided by a strategy based on the mean of the first-passage-time through a control threshold. Simulation studies are carried out for different choices of the parameters and time-dependent functions involved.

On the effect of a therapy able to modify both the growth rates in a Gompertz stochastic model

A Gompertz-type diffusion process characterized by the presence of exogenous factors in the drift term is considered. Such a process is able to describe the dynamics of populations in which both the intrinsic rates are modified by means of time-dependent terms. In order to quantify the effect of such terms the evaluation of the relative entropy is made. The first passage time problem through suitable boundaries is studied. Moreover, some simulation results are shown in order to capture the dependence of the involved functions on the parameters. Finally, an application to tumor growth is presented and simulation results are shown.

Ergodic method on optimal harvesting for a stochastic Gompertz-type diffusion process

In this paper, the stochastic harvesting problem is regarded as a mathematical formulation of finding the maximum sustained yield and the corresponding best sustainable harvesting strategies under uncertainty. We use a new method to solve this problem, and prove the equivalency between this method and previous methods. This paper is the first attempt to apply the ergodic theory on the optimal harvesting problem, to the best of our knowledge.

Estimating the parameters of a Gompertz-type diffusion process by means of Simulated Annealing

This paper explores the application of the Simulated Annealing algorithm for the maximum likelihood estimation of the parameters of a Gompertz-type process. Firstly, the solution space is bounded using relevant information about the process provided by the sample data. Secondly, a proposal for improvement is made, namely the application of a second cycle of the algorithm, including a refinement factor. Finally, both the specifications for the application of the algorithm and the proposed improvement are validated through their application to simulated and real data.

Inferring the effect of therapy on tumors showing stochastic Gompertzian growth

The present work deals with a Gompertz-type diffusion process, which includes in the drift term a time-dependent function C( t) representing the effect of a therapy able to modify the dynamics of the underlying process. However, in experimental studies is not immediate to deduce the functional form of C( t) from a treatment protocol. So a statistical approach is proposed in order to estimate this function when a control group and one or more treated groups are observed. In order to validate the proposed strategy a simulation study for several interesting functional forms of C( t) has been carried out. Finally, an application to infer the net effect of cisplatin and doxorubicin+cyclophosphamide in actual murine models is presented.

SOME TIME RANDOM VARIABLES RELATED TO A GOMPERTZ-TYPE DIFFUSION PROCESS

In the context of a Gompertz-type diffusion process (associated with a particular expression of the Gompertz curve for which its limit value depends on the initial value), we consider the study of two time-random variables: the inflection time of the model and the time at which it achieves a certain percentage of the total growth. Once we have shown the difficulties in the approach of the former, we deal with it as a particular case of the second. Furthermore, because of the peculiar characteristics of the considered process, we prove that these two problems can be formulated as a first-passage-time problem through a constant boundary. Finally, we conclude with an application to the growth of rabbits, in which we obtain the density functions of the inflection time and the time at which a rabbit achieves the half of its total growth.

A new Gompertz-type diffusion process with application to random growth

Stochastic models describing growth kinetics are very important for predicting many biological phenomena. In this paper, a new Gompertz-type diffusion process is introduced, by means of which bounded sigmoidal growth patterns can be modeled by time-continuous variables. The main innovation of the process is that the bound can depend on the initial value, a situation that is not provided by the models considered to date. After building the model, a comprehensive study is presented, including its main characteristics and a simulation of sample paths. With the aim of applying this model to real-life situations, and given its possibilities in forecasting via the mean function, discrete sampling based inference is developed. The likelihood equations are not directly solvable, and because of difficulties that arise with the usual numerical methods employed to solve them, an iterative procedure is proposed. The possibilities of the new process are illustrated by means of an application to real data, concretely, to growth in rabbits.