Itô versus Stratonovich calculus in random population growth

Paper that resolves the controversy on whether to use Itô or Stratonovich calculus on stochastic differential equation models applied to poulation growth in random environments

Population growth in random environments: which stochastic calculus?

Refereed scientific paper on stochastic differential equation models of population growth in random environments with resolution of the controversy on the use of Itô or Stratonovich calculus (extension to density-dependent noise intensities). The paper is in press in the Bulletin of ISI containing the Proceedings of the 56th Session of the ISI (2007). An electronic version is available

Weak Allee effects population growth models in a random environment

Based on a deterministic model of population growth with weak Allee e ffects, we propose a general stochastic model that incorporates environmental random fluctuations in the growth process. We study the model properties, existence and uniqueness of solution, the stationary behavior and mean and variance of the time to extinction of the population. We then consider as an example the particular case of a stochastic model with Allee e ffects based on the classic logistic model.Fundação para a Ciência e a Tecnologia e CIMA, Projeto UID/MAT/04674/201

Equações diferenciais estocásticas e aplicações biológicas

Invited dissemination paper on biological applications of stochastic differential equations

Biomatemática

Dissemination paper on Biomathematics, particularly population growth and fishing models, in the proceedings of a Summer School

A SDE growth model: Nonparametric Estimation of the Drift and the Diffusion Coefficients

We study a stochastic differential equation (SDE) growth model to describe individual growth in random environments. In particular, in this work, we discuss the estimation of the drift and the diffusion coefficients using non-parametric methods. We illustrate the methodology by using bovine growth data. Considering the diffusion process X_{t}, describing the weight of an animal at age t, characterized by the stochastic differential equation dX(t)=a(X(t))dt+b(X(t))dW(t), with W(t) being the Wiener process, we estimate the infinitesimal coefficients a(x) and b(x) nonparametrically. Our goal was to analyse which of the parametric models (with specific functional forms for a(x) and b(x)) previously used by us to describe the evolution of bovine weight were good choices and also to see whether some alternative specific parametrized functional forms of a(x) and b(x) might be suggested for further parametric analysis of this data

Modelling Individual Growth in Random Environments

We have considered, as general models for the evolution of animal size in a random environment, stochastic differential equations of the form dY(t)=b( A-Y(t))dt+\sigma dW(t), where Y(t)=g(X(t)), X(t) is the size of an animal at time t, g is a strictly increasing function, A=g(a) where a is the asymptotic size, b>0 is a rate of approach to A, s measures the effect of random environmental fluctuations on growth, and W(t) is the Wiener process. The transient and stationary behaviours of this stochastic differential equation model are well-known. We have considered the stochastic Bertalanffy-Richards model (g(x)=x^c with c>0) and the stochastic Gompertz model (g(x)=ln x). We have studied the problems of parameter estimation for one path and also considered the extension of the estimation methods to the case of several paths, assumed to be independent. We used numerical techniques to obtain the parameters estimates through maximum likelihood methods as well as bootstrap methods. The data used for illustration is the weight of "mertolengo" cattle of the "rosilho" strand

Modelling individual animal growth in random environments

We have considered, as general models for the evolution of animal size in a random environment, stochastic differential equations of the form dY(t)=b( A-Y(t))dt+sdW(t), where Y(t)=g(X(t)), X(t) is the size of an animal at time t, g is a strictly increasing function, A=g(a) where a is the asymptotic size, s measures the effect of random environmental fluctuations on growth, and W(t) is the Wiener process. We have considered the stochastic Bertalanffy-Richards model (g(x)=x^c with c>0) and the stochastic Gompertz model (g(x)=ln x). We have studied the problems of parameter estimation for one path and also considered the extension to several paths. We also used bootstrap methods. Results and methods are illustrated using bovine growth data

Modelos de Crescimento de Bovinos Mertolengos em Ambiente Aleatório

Apresentamos modelos de crescimento individual em ambiente aleatório para descrever a evolução do peso de bovinos mertolengos da estirpe rosilho. Tendo como objectivo obter modelos que incluam o efeito das variações aleatórias do ambiente na evolução do peso, recorremos a equações diferenciais estocásticas. Os modelos usados para o crescimento individual de animais em termos do tamanho X(t) no instante t têm geralmente a forma dY(t)/dt = b(A-Y(t)), onde se fez a mudança de variável Y(t)=g(X(t)) com g estritamente crescente. Aqui A=g(a), onde a representa o tamanho assintótico do animal, e b é o coeficiente de crescimento que regula a velocidade de aproximação a A. No caso de haver flutuações aleatórias do ambiente, considerámos o modelo dY(t) = b(A-Y(t))dt + dW(t), onde mede a intensidade das flutuações e W(t) é um processo de Wiener padrão. Aplicámos o modelo e estudámos os problemas de estimação e de previsão para uma trajectória (um animal). Foi também estudada a extensão a várias trajectórias (vários animais) .Considerámos o caso do modelo de Bertalanffy-Richards (g(x)=xc com c>0) e do modelo de Gompertz (g(x)=ln x). Foram também utilizados métodos bootstrap para estudar o problema de estimação

Modelos Multifásicos de Crescimento de Animais em Ambiente Aleatório

Em trabalhos anteriores, estudámos um modelo geral de crescimento individual de animais em ambiente aleatório da forma dY(t) = b(g(a)-Y(t))dt+sdW(t), com Y(t)=g(X(t)), sendo g estritamente crescente (g(x)=x^c e g(x)=ln x são casos particulares típicos), X(t) o peso do animal no instante t, W(t) um processo de Wiener padrão, b o coeficiente de crescimento e a o peso assintótico. Este modelo, que usa uma equação diferencial estocástica, já não sofre do problema dos modelos clássicos de regressão em que um atraso de crescimento num determinado momento não se repercute nos pesos futuros. Aplicámos este modelo monofásico (uma única forma funcional descreve a dinâmica média para toda a curva de crescimento) e estudámos os problemas de estimação. Na literatura têm sido propostos modelos determinísticos multifásicos para o crescimento de bovinos, já que a curva de crescimento observada sugere a existência de pelo menos duas fases. Aqui estudamos a generalização do modelo estocástico referido ao caso multifásico, em que admitimos que o coeficiente de crescimento b tem valores diferentes para diferentes fases da vida do animal. Por simplicidade, consideramos duas fases com coeficientes de crescimento b1 e b2. Aplicamos a dados de bovinos