Applications of Search in Biology: Some Open Problems - Search Theory: A Game Theoretic Perspective - page 288

This is because, although the environment contains no information per se, the fact
that a patch has been acquired provides implicit information that a competitor has
not already visited this location. In a patchy world the evolutionary pressure for
directed proliferation becomes stronger.
The results in Croft et al. are mainly simulation-driven, and therefore rather nar-
row in scope and limited by the available computational power. Initial analytical
understanding of the foraging problems faced by roots may be possible by analogy
with the fish foraging results above. Perhaps more exciting is the opportunity to ex-
pand the remit to encompass more realism: The soil environment is complex. Mod-
els based on three spatial dimensions allow competitors to overlap. Alternatively,
network-based models of the soil environment may better capture the fractal-like
structure in which roots grow. Roots can also form symbioses with soil fungi (my-
corrhizas), allowing the plant to forage cheaply across a larger area at the cost of
providing carbon to the fungal partner. Finally, while agriculture may rely on mono-
cultures, the plants themselves have typically evolved within complex competitive
communities. How can knowledge of the strategies evolved in the wild be exploited
so as to more efficiently exploit managed crop systems? These issues are described
more thoroughly in [ 10 ], but a firm mathematical grasp remains elusive and the
practical problems are unsolved.
Problem 3: Safety in Numbers, or Presenting
a Bigger Target? (Complications 1, 2, 3, 4)
The notion of “safety in numbers” imagines princesses gathering in groups of size
n so that, even if the monster finds the group, each individual only has a probability
1/n of meeting a grisly demise. This verbal reasoning is used to explain many nat-
ural instances of prey aggregation, from minnows to gazelles. There are additional
factors such as the increased vigilance afforded by many sets of eyes, and complica-
tions such as individuals only visiting the periphery of the group when physiology
dictates [ 3 ].
These problems have received attention from ecologists and strong theory ex-
ists, but perhaps in directions somewhat orthogonal to the mathematics presented
in this volume. The simple question of when it pays for two princesses to collab-
orate in hiding is naturally game-theoretic, and will generalise naturally to larger
groups, complex environments, or group-induced changes in speed of movement or
detectability.
For small wet princesses, the turbulent encounter theories of Problem 1 allow
quantification of the processes involved. A group of fish larvae or zooplankton is
larger than an individual and therefore subject to faster random turbulent advection;
encounter rates with predators therefore increase. When grouping confers another
advantage, for example navigational precision via a “many wrongs” principle [ 7 ],
then further trade-offs emerge; depending on the levels of turbulence a group of lar-
vae reach safety of a coral reef more quickly, but present a larger target whilst doing