A mathematical analysis of marine size spectra

Aquatic ecosystems are observed to follow regular patterns in abundance. The frequency distribution of all individuals across the spectrum of body mass, irrespective of their taxonomic identity (known as a ’size spectrum’), follows a power law and this has mathematically been explained by the processes of growth and mortality primarily driven by predation. In this theory of the size spectrum, predation is driven by body size: as organisms grow bigger the size of their prey also increases. This process is thought to be particularly important for marine organisms such as fish, where individual body size is an important determinant for what they eat because they are mostly limited by the size of their mouths.

Models need to capture the behaviour of real systems if reliable predictions are to emerge from them. Here, new equations for size-based predation are derived from a
stochastic process, allowing variability in organism growth. The new equations are postulated to capture real feeding behaviour better than classical models often used to simulate size spectra. Marine systems are often perturbed by seasonal processes, environmental factors and exploitation. I show how models with diffusive growth stabilise the observed power-law steady state in marine systems, and stability is explicitly linked to parameters involved in feeding.

Seasonal plankton blooms are introduced into the model, along with time-dependent reproduction, both of which are widely observed in aquatic systems. The population dynamics, along with growth and survival rates during blooms are investigated, and preliminary results are reflected in empirical data. The match/mismatch hypothesis is tested, with theoretical findings in agreement with observed seasonal trends. Adding factors such as these will make the behaviour of size-based models more indicative of real ecosystems, and thus well-informed management decisions about exploitation can be made.