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sufficiently close [ 6 ]. However, if the prey is cryptic such that predators can pass
reasonably close to it without detecting it (e.g. a juvenile gazelle lying motionless
in long grass) then the Broom and Ruxton [ 2 ] model is better. This will especially
apply to cases where predation attempts (or at least proximate predators) are less
frequent, so that avoiding predation makes up a small part of an animal's time or
energy budget or if the prey animal can quickly return to its previous feeding be-
haviour as soon as the predator has passed (e.g. many grazers). It may be that more
sophisticated models that take into account different costs to the prey are needed.
We then looked at a new model similar to that developed in [ 2 ], inspired by ex-
periments carried out in [ 8 ]. In the new model, the predator remains still and has to
decide how long to search for prey before giving up; the prey has to decide whether
to run, and if so when. We found some circumstances, where a particular area was
just too difficult to search, where the predator should give up immediately (and of
course the prey does not run). As the ease of search in a particular area improves,
there comes a critical point when it is worthwhile for the predator to search, but
where the prey should still always stay in cover. As searching becomes easier the
length of time the predator should stay increases. This continues until a point where
it becomes optimal for the prey to play a mixed strategy, sometimes staying in cover,
but sometimes running immediately. As searching becomes easier still, the proba-
bility that the prey runs immediately should increase, and the search time of the
predator starts to decrease again. This is because, given that prey often run immedi-
ately, conditional on not having found prey or seen them run, the probability of there
actually being no prey to find increases with ease of search through the increased
chance of early running.
Why did we not get the results of [ 8 ], that the prey sometimes wait before run-
ning? The most obvious reason is that we focused on finding a different kind of
solution, which we considered to be the natural solution to the problem that we con-
sidered. But why did they not find such solutions, but instead have prey that ran
after some time, rather than immediately or not at all? One possible explanation is
that the lost foraging cost of [ 14 ]or[ 3 ] is sufficiently important, and that the lizards
cannot afford to wait it out.
Interestingly, the “predators” of [ 8 ] effectively carried on searching forever, so
did not behave like strategic predators. This provides another possible explanation.
In our model, if prey knew there were a small fraction
of predators that waited
forever, then if a predator stays beyond time t , they would know that one of the new
predators was present, thus fleeing is then optimal.
Alternatively, what if prey can observe which area is being searched? Thus for
the prey the search rate may be
ε
ν L < ν H , depending on whether the predator
is searching in its vicinity. There would be transitions between these two rates, as
the predator's search continued. If these transition rates were r HL and r LH from high
risk to low risk and low risk to high risk respectively, then we can show that the
mean search rate
ν H or
ν
is found from
r HL
r HL +
r LH
r HL +
ν =
r LH ν L +
r LH ν H .
(15.31)
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