André Dumas

A review of mathematical functions for the analysis of growth in poultry

Poultry industries face various decisions in the production cycle that affect the profitability of an operation. Predictions of growth when the birds are ready for sale are important factors that contribute to the economy of poultry operations. Mathematical functions called ‘growth functions’ have been used to relate body weight (W) to age or cumulative feed intake. These can also be used as response functions to predict daily energy and protein dietary requirements for maintenance and growth (France et al., 1989). When describing growth versus age in poultry, a fixed point of inflexion can be a limitation with equations such as the Gompertz and logistic. Inflexion points vary depending on age, sex, breed and type of animal, so equations such as the Richards and López are generally recommended. For describing retention rate against daily intake, which generally does not exhibit an inflexion point, the monomolecular would appear the function of choice.

Modelling the ontogeny of ectotherms exhibiting indeterminate growth.

Numerous growth functions exist to describe the ontogeny of animals. Such functions (e.g., von Bertalanffys equation, thermal-unit growth coefficient) are currently applied to ectotherms even though they fail to provide analytical expressions that adapt to a wide range of fluctuating temperatures. The underlying mechanisms responsible for the ontogeny of ectotherms exhibiting indeterminate growth have not yet been summarised in terms of a simple but meaningful mathematical equation. Here, a growth function is developed, with parameters having physical or biological interpretation that accommodates indeterminate growth under fluctuating temperatures assuming the latter vary seasonally. The equation is derived as a special case of von Bertalanffys equation providing realistic growth trajectories throughout the ontogeny of several groups of ectotherms (R(2)>0.90). The results suggest that the effect of temperature on growth trajectory supersedes that of reproduction in an environment with fluctuating temperature. Furthermore, values of the allometric weight exponent (0<b<0.75) indicate that the rules of body surface and body weight do not apply under certain circumstances. Finally, the growth function circumvents problems associated with models based on thermodynamic and chemical kinetic principles (e.g., inability to predict growth of organisms in which ontogeny exceeds 3 months) and on rule of thermal summation (e.g., reliable only in a certain range of temperature). The growth function can handle a wide range of temperature fluctuations, encompass life stages and apply to key organisms in ecology, fisheries and agriculture.

Mathematical descriptions of indeterminate growth

Two models were derived in an effort to better describe the indeterminate nature of growth exhibited by ectotherms. The models are characterized by their non-sigmoidal shape and are based on three assumptions: quantity of growth machinery works at a rate dependent on feed intake; the relationship between growth rate and intake level follows the law of diminishing returns; and growth is irreversible. The Michaelis-Menten and Mitscherlich equations are used in their formulation. To investigate their potential, the models were fitted to six datasets, representing repeated measures of live body weights of two species: rainbow trout (Oncorhynchus mykiss) and Nile tilapia (Oreochromis niloticus). The models were evaluated on the basis of fitting behaviour, examination of residuals, along with measures of goodness-of-fit. Agreement between predicted and observed body weights, and flexibility to mimic growth patterns given varying species and culture conditions, affirm the ability of both models to describe indeterminate growth in fish.