 UBC Calculus Online Course Notes 
On this page we will investigate one example which illustrates the importance
of the inverse trigonometric functions. This example is motivated by a problem
in biology, and originates from a paper written by
Larry Dill,
a biologist at Simon Fraser University in Burnaby, BC.
The zebra danio, a tasty little morsel for hungry predators
The Zebra danio is a small tropical fish, which has many predators (larger fish) eager to have it for dinner. Surviving through the day means being able to sense danger quickly enough to escape from a hungry pair of jaws. However, the danio cannot spend all its time escaping. It too, must find food, mates, and carry on activities that sustain it. Thus, a finely tuned mechanism which allows it to react to danger but avoid overreacting is definitely an advantage.
In this diagram, we show the part of the world that is within the visual field of the Zebra danio. (We have shown only what one eye is seeing, and we will assume that the predator is approaching from this direction.) The eye of the danio is located at the sharp corner of the triangle and is said to subtend a visual angle of . In this case, the object seen is a predator, and its apparent size, as viewed from the vantage point of the prey is . The distance between the prey and the predator (which forms the horizontal leg) is labeled . The quantities in this diagram are linked through trigonometric functions.  The visual field of the danio.

The top half of the triangle, forms a
Pythagorean triangle, so
or
In the last step, we have simply used the inverse tan function to express
the same relationship, and then just multiplied both sides by 2.
Linking the visual angle to the escape response
What sort of visual input should the danio respond to, if it is to
be efficient at avoiding the predator? In principle, we would like to consider
a response that has the following features
The escape response is triggered when the predator approaches so quickly, that the rate of change of the visual angle is larger than some critical value.
The hypothesis proposed above, restated in terms of symbols used in our
diagram is that
The escape response is triggered when
where in the last step we have applied the chain rule.
Now suppose that the predator is approaching the prey at speed
,
decreasing the distance to its prey, so that
Using the computed expression for and setting this equal to we find that
In this equation, the quantities which are characteristic of the predator and its size and swimming speed are assumed constant, and so is , the prey's visual sensitivity to a "looming" threat. We are to solve for the distance at which this critical value of is reached. We will call that distance the reaction distance, . Taking reciprocals and multiplying both sides by , we find that
We find that for a predator of size moving at speed towards the prey, the escape response will be triggered at the reaction distance
This result has interesting implications which we hope you will explore below. In particular, the reaction distance depends on various constants associated with the problem. We ask you to look more closely at this dependence and to see what it implies about success and failure of this type of response based on the rapidity of change of the visual angle. Is the response always successful ? How can it be made more sensitive? How might a predator try to outwit this kind of response.
Lawrence M Dill (1974) The escape response of the Zebra Danio (Brachydanio rerio). I. The stimulus for escape. Animal Behaviour 22, 711722.
Home page of Lawrence M Dill
Information about Behavioural Ecology at Simon Fraser University