Biphasic growth in fish I: theoretical foundations.

We develop the theory of biphasic somatic growth in fish using models based on the distinction between pre- and post-maturation growth and an explicit description of energy allocation within a growing season. We define a 'generic biphasic' (GB) model that assumes post-maturation growth has a von Bertalanffy (vB) form. For this model we derive an explicit expression for the gonad weight/somatic weight ratio (g) which may either remain fixed or vary with size. Optimal biphasic models are then developed with reproductive strategies that maximise lifetime reproductive output. We consider two optimal growth models. In the first (fixed g optimal), gonad weight is constrained to be proportional to somatic weight. In the second (variable g optimal) model, allocation to reproduction is unconstrained and g increases with size. For the first of these two models, adult growth in a scaled measure of length has the exact vB form. When there are no constraints on allocation, growth is vB to a very good approximation. In both models, pre-maturation growth is linear. In a companion paper we use growth data from lake trout (Salvelinus namaycush) to test the bioenergetics assumptions used to develop these models, and demonstrate that they have advantages over the vB model, both in quality of fit, and in the information contained in the fitted parameters.