Abstract : (Joint work with Gonçalo Jacinto and Patricia A. Filipe.) The effect of random fluctuations of internal and external environmental conditions on the growth dynamics of individual animals is not captured by the regression model typical approach. We use stochastic differential equation (SDE) versions of a general class of models that includes the classical growth curves as particular cases. Namely, we use models of the form $d Y_t=\beta\left(\alpha-Y_t\right) d t+\sigma d W_t$, with $X_t$ being the animal size at age $t$ and $Y_t=h\left(X_t\right)$ being the transformed size by a $C^1$ monotonous function $h$ specific of the appropriate underlying growth curve model. $\alpha$ is the average transfomed maturity size of the animal, $\beta>0$ is the rate of approach to it and $\sigma>0$ measures the intensity of the effect on the growth rate of $Y_t$ of environmental fluctuations. These models can be applied to the growth of wildlife animals and also to plant growth, particularly tree growth, but, due to data availability (data furnished by the Associação dos Produtores de Bovinos Mertolengos - ACBM) and economica interest, we have applied them to cattle growth.
We briefly mention the extensive work of this team on parameter simulation methods based on data from several animals, including alternatives to maximum likelihood to correct biases and improve confidence intervals when, as usually happens, there is shortage of data for animals at older ages. We also mention mixed SDE models, in which model parameters may vary randomly from animal to animal (due, for instance, to their different genetical values and other individual characteristics), including a new approximate parameter estimation method. The dependence on genetic values opens the possibility of evolutionary studies on the parameters.
In our application to mertolengo cattle growth, the issue of profit optimization in cattle raising is very important. For that, we have obtained expressions for the expected value and the standard deviation of the profit on raising an animal as a function of the selling age for quite complex and market realistic raising cost structures and selling prices. These results were used to determine the selling age that maximizes the expected profit. A user friendly and flexible computer app for the use of farmers was developed by Ruralbit based on our results.

Keywords : Stochastic differential equations; cattle growth; profit optimization; sensitivity analysis; maximum likelihood estimators