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

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

Equações diferenciais estocásticas e aplicações biológicas

Invited dissemination paper on biological applications of stochastic differential equations

Crecimiento y extinción en ambientes que varían aleatoriamente: modelamiento y optimización mediante ecuaciones diferenciales estocásticas

This paper presents a brief overview of a range of applications of stochastic differential equations (SDE) in describing the growth of wildlife populations living in randomly varying environments and the associated risks of extinction, including profit optimization issues in the particular case of fish or other populations subjected to harvesting. The same basic ideas apply also to the growth of individual animals and how to optimize the profit of the farmers that raise such animals

Biomatemática

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

Growth and extinction of populations in randomly varying environments

Review paper on stochastic differential equation models for the growth and extinction of populations in a randomly varying environment

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

Modelos de evolução do peso de animais em ambiente aleatório

Após uma breve revisão dos métodos usualmente utilizados para modelar o crescimento de animais, propõe-se como modelos descritivos gerais para a evolução do peso de animais em ambiente aleatório equações diferenciais estocásticas da forma: dX(t)=f(X(t))dt+sdW(t) (1) onde X(t) representa o peso (ou uma potência do peso) do animal na idade t, s mede a intensidade dos efeitos das perturbações aleatórias do ambiente sobre o crescimento, W(t) é o processo de Wiener e x(0) é o peso à nascença (que supomos conhecido). Partindo do modelo de Bertalanffy-Richards, foi considerado f(X(t))=b(A-X(t)), onde os parâmetros A e b representam, respectivamente, o peso assintótico (ou peso na maturidade) e a velocidade com que o animal dele se aproxima. Deste modo, (1) apresenta a forma do conhecido modelo de Vasicek utilizado na modelação da dinâmica das taxas de juro. A partir da solução de (1), é apresentada uma expressão explícita para a função de máxima verosimilhança. O modelo foi aplicado a dados de crescimento de bovinos mertolengos da estirpe rosilho. São apresentadas as estimativas dos parâmetros e intervalos de confiança assintóticos

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