Extending the work of Pitchford et al. (MEPS 2003) and Preston et al. (Interface 2010), we are investigating the ideas surrounding larval fish foraging and stock recruitment as probabilistic models. The trade-offs between starvation, recruitment and predation are modelled as competing deterministic and stochastic energy sources and expenditures. We use Markov chains to predict growth patterns through a discrete energy level model. Working with colleagues at the University of York.

We utilise as much empirical research and data as possible to construct a realistic model of how a larval fish grows in the ocean, from ~5mm to ~30mm. The aim is to find not only a relationship for "optimal" swimming speed and larval size but to quantify how each external factor influences the probabilities of three possible outcomes for the larval fish: starvation, recruitment into the next life history sub-stage and being eaten by a predator. The effects that we account for include turbulence on encounter rates, turbulence on probabilities of success in encounters, predation ecounters and energy expenditure due to swimming.
Our novel approach to this problem is to only model the energy fluctuations (and not how long it takes) a larval fish to gain the requisite amount of energy or die trying. To do this we utilse a discrete Markov chain with each state representing an energy level for the larva and solve to determine the probabilities of reaching each terminal (in two cases literally!) states.

The graph above shows how the probabilities of starvation, survival and predation can change with increasing swimming speed. The red indicates predation, blue starvation and green survival, with the black line showing the optimal speed (maximising survival probability).

Pitchford et al.: Optimal foraging in patchy turbulent environments, MEPS Vol. 256, 2003
Preston et al.: Evolutionary optimality in stochastic search problems, J Royal Society: Interface, Vol. 7 No. 50, 2010