Growing pains: time to reassess models of growth?

A central aim of the Centre for Geometric Biology is understanding how and why organisms grow. In a recent opinion piece, published in the journal Trends in Ecology & Evolution, Dustin Marshall and Craig White suggest that it might be time to take another look at the ways we currently understand and model growth.

In the past, growth has mainly been considered in two different ways. Mechanistic models of growth emphasise identifying the physiological processes driving growth. This group of models includes the von Bertalanffy Growth Function, which is perhaps the best-known growth model. It estimates the rate of increase in mass (growth) as the difference between anabolism (energy-consuming processes) and catabolism (energy-producing processes). Other models of this type include the Ontogenetic Growth Model and the Dynamic Energy Budget model.

In contrast, phenomenological models of growth are based on life-history theory and work from the assumption that organisms evolve to maximise their fitness. Theories and models under this framework revolve around the trade-offs between maximising reproduction against the risk of mortality.

The von Bertalanffy Growth Function and more recent mechanistic models do an excellent job of describing how the growth of most organisms slows as they approach their final size. Models such as these assume that growth slows or stops because the organism cannot acquire, distribute or use resources faster than it has to expend them on self-maintenance.

There is a problem however. Mechanistic models do not adequately consider reproduction — an energetically expensive undertaking. Most mechanistic models make the simple but crucial assumption that reproduction is proportional to body size, and that allocation to reproduction begins at birth and remains a constant fraction of total body size throughout an individual’s life. While this assumption seems unrealistic, it is essential for these models to describe growth well.

Phenomenological models tend to have different dynamics for juvenile and mature phases; after maturity, increasing allocation of resources to reproduction reduces growth. But again, most of these models assume that reproductive output is directly proportional to body size.

We now know that, for marine fish at least, reproductive output is disproportionally higher in bigger females. Dustin and Craig suspect that this pattern is the rule for most taxa but that it has been overlooked (see Figure 1).  If this does occur more generally, what does it mean for our understanding of growth?

Figure 1. Data from a range of taxa (A = marine invertebrates and B = other taxa) showing the disproportionate increase in reproductive output at larger sizes. The dotted line in each case shows what a 1:1 (or isometric) scaling of reproduction with growth would look like.

Dustin and Craig argue that many of the mechanistic models of growth are trying to explain dynamics that are driven by increasing allocation to reproduction but they do not allow for it.  Instead, these models assume that resource supply decreases as individuals get bigger so that if the allocation to reproduction is allowed to increase then organisms will shrink once they start to reproduce.

A common feature of both theoretical approaches is that they assume that the relative amount of energy available for total production decreases with size.  If we instead assume that resource acquisition and usage both change in the same proportions in relation to size, and combine those parameters with the disproportionate increase in reproductive output (hyperallometry), then we can predict growth trajectories remarkably well (see Figure 2).

Dustin and Craig posit that the growth dynamics that biologists have long sought to understand emerge simply from hyperallometric scaling of reproduction.

Figure 2. Dustin and Craig propose a simple model that allows energy intake and expenditure to scale consistently with size, but reproductive output to increase disproportionately with size. To illustrate that this “hyperallometric reproduction” model describes growth patterns on par with well-known mechanistic models, Dustin and Craig have used data for cod Gadus morhua and have fitted model outputs for the van Bertanalaffy Growth Function (orange), the Ontogentic Growth Model (blue) and the hyperallometric reproduction model shown in red.