## Roffs Model

The Roff model is derived from a simple bioenergetics relationship. First, the total body mass W is divided into somatic mass M and gonads G, that is, W = M + G. Next, the available energy E (in mass equivalents) is assumed divided between the somatic mass and the gonads:

Here the time step t is the duration of one reproductive cycle, commonly 1 year for temperate species. By assuming W a: L3 and support from empirical data showing that length growth for immature fish is linear with an annual length increment of h0 per time, length can be modeled for immature fish as

TO C ffi Figure 2 Observed length at age (•) of American plaice averaged over immature and mature individuals, and predicted growth curves according to the Roff model. The fitted curves all have growth rate (h in equations 6 and 7) of 2.28 cm yr-1, and the GSI (R in eqn ) is 0.103 for the solid line fitting the observed pattern best, and 0.08 and 0.12 for the lower and upper dashed lines, respectively. The observed mean length at age t is calculated as: 2=11 paLat, where pa is the proportion maturing at age a (maturation is occurring over age classes 11-20) and La is the length at age t of fish that mature at age a. Adapted from Roff DA (1983) An allocation model of growth and reproduction in fish. Canadian Journal of Fisheries and Aquatic Sciences 40: 1395-1404, figure 3.

Here R is the gonado-somatic index (GSI), equal to gonad mass divided by the somatic mass. Therefore, there is a

Figure 2 Observed length at age (•) of American plaice averaged over immature and mature individuals, and predicted growth curves according to the Roff model. The fitted curves all have growth rate (h in equations 6 and 7) of 2.28 cm yr-1, and the GSI (R in eqn ) is 0.103 for the solid line fitting the observed pattern best, and 0.08 and 0.12 for the lower and upper dashed lines, respectively. The observed mean length at age t is calculated as: 2=11 paLat, where pa is the proportion maturing at age a (maturation is occurring over age classes 11-20) and La is the length at age t of fish that mature at age a. Adapted from Roff DA (1983) An allocation model of growth and reproduction in fish. Canadian Journal of Fisheries and Aquatic Sciences 40: 1395-1404, figure 3.

Although GSI could change with age or length, a constant GSI is often assumed in order to parametrize the growth model. Roff found relatively good agreement between predicted and actual size at age when assuming a constant GSI for American plaice Hippoglossoides platessoides (Figure 2).

The Roff model is based on more physiologically sound mechanisms than the von Bertalanffy model: namely, that after maturation, available energy should be allocated to reproduction as well as to growth. The parameters of the Roff model (i.e., growth rate, GSI, age at maturation) are also easier to measure and are easier to give a biological interpretation than the von Bertalanffy parameters. Related to this last point is that the GSI, growth rate, and age at maturation are fitness determining, genetically controlled life-history traits which makes the Roff model a good candidate also for addressing questions in life-history evolution.

### Critique of the Roff model

Although the Roff growth model is based on more physiologically relevant mechanisms than the von Bertalanffy growth model, it has several simplifications that under some circumstances could be violated. First, Roff assumes that immature growth is linear rather than having this feature emerge from underlying mechanisms. Second, although it is not necessary to always do so, the GSI is commonly assumed constant throughout life; this assumption allows a smooth growth curve to be drawn and facilitates parametrization. Third, the simple allo-metric relationships that are assumed (e.g., Wa: L3) might not always hold. 