Applications of Search in Biology: Some Open Problems - Search Theory: A Game Theoretic Perspective - page 286

tions listed here, this last is likely to require the smallest modifications to existing
theory, but has the potential to provide the most revealing new insights.
18.3 Some Open Problems
These problems benefit from being both practically relevant and mathematically
tractable. The author, with co-workers, has made attempts to solve each of them but
there is wide scope for fresh thinking and new approaches.
Problem 1: How Fast Should a Fish Swim? (Complications 1, 2, 3, and 5)
Fish are important; we eat them, and they play crucial roles in wider ecosystem
function. In a sustainable fishery one would hope that each fish removed from the
adult population (the “stock”) is replaced by a juvenile entering that population
(a “recruit”). Unfortunately, the stock-recruitment relationship is notoriously unpre-
dictable [ 5 , 19 ]. One of the main reasons for this is the peculiar reproductive strategy
used by many pelagic species, where each female will produce millions of small and
seemingly useless eggs over her lifetime. Only a tiny proportion of these survive the
egg and the larval stages to reach adulthood and contribute to future generations.
Most die of starvation or are devoured by anything with a larger mouth - including
their own brothers and sisters.
So, how fast should a fish larva swim? The problem is ostensibly a simple bal-
ance between energy costs and foraging benefits. For example, swimming twice as
fast might double the predator-prey encounter rate, but may also incur a four-fold
increase in the cost of swimming (assuming Stokes drag, appropriate for small for-
agers in water). Simple optimisation would reveal a quadratic fitness landscape with
a fixed optimal swimming speed.
Reality, however, is more interesting. For a small animal in a large ocean, prey
are not homogeneously distributed and turbulence cannot be ignored. Effectively,
local random stirring brings the predator into contact with prey regardless of active
swimming - perhaps the best strategy is to sit and wait?
The temporal mean-field problem was tackled by Pitchford et al. [ 22 , 23 ]. They
use a simple multi-scale approach using (appropriately enough) Poisson processes:
the forager, the patches of prey, and the prey individuals are regarded as independent
spheres, with their relative speeds and encounter rates governed by physiology and
physics. A predator finds, forages within, and leaves a patch at Poisson rates which
combine swimming speeds with turbulent speeds at the appropriate length scales
(visual range for forager-prey encounters, patch size for forager-patch encounters).
Pitchford et al. derive analytical expressions for the optimal swimming speeds, and
extend the study to ask when a predator should alter its behaviour according to
whether it thinks it is in a patch.