Calculating the probability of an animal finding a resource

[killbot2000]

I'd like some help working out the probability of my model animal finding water in a day.

In my 'model' there is a square landscape 17km^2
The water pool is a circle of 500m radius (so approx 0.7854km^2)
So in a one off shot the chances of hitting the water is 0.7854/17 = 0.0462 right?

But if I suppose that the animal can search over the course of a day, how can I incorporate this into increasing its prob. of finding the water?

Assume for the sake of argument that it can travel at 1m/second and has a 500m detection range then it should 'cut out' circles the same radius as the water pool every 1000 seconds. Then I think it would cut out 86400/1000 = 86.4 in day.

If this is correct how can I convert it into a probability?

I know this is a ramble, but I've been at it all morning, so any help would be much appreciated!

 

[mfb]

So in a one off shot the chances of hitting the water is 0.7854/17 = 0.0462 right?

If the animal magically appears at a random position in the landscape, this is the probability that it will appear in the water.

But if I suppose that the animal can search over the course of a day, how can I incorporate this into increasing its prob. of finding the water?

It depends on the way it searches for water.

Neglecting effects at the edges of the landscape, your model animal can search 1000m² per second. With an ideal path, it would need roughly 17000 seconds to see the whole landscape, less than 5 hours. The probability to find the pool within a day is 1, independent of its size.

Real animals rarely walk on those ideal lines, so the probability depends on the walking algorithm.

 

[Evo]

What's the probability of an animal appearing out of thin air into an area to which it's never been?

 

[256bits]

If the animal magically appears at a random position in the landscape, this is the probability that it will appear in the water.

It depends on the way it searches for water.

Neglecting effects at the edges of the landscape, your model animal can search 1000m² per second. With an ideal path, it would need roughly 17000 seconds to see the whole landscape, less than 5 hours. The probability to find the pool within a day is 1, independent of its size.

Real animals rarely walk on those ideal lines, so the probability depends on the walking algorithm.

Pool size radius matters: ( versus a search independent of pool radius )
Even less search time for a 500 radius water pool. Search radius would be detection radius 500m plus the water pool radius giving a total 1000m search radius ( assuming pool edge detection ). A 5km by 3.5 km grid (17.5 square km ) could be searched in 2 passes. So, not exactly, but somewhere around 9000 seconds of search required.

In fact a 5 x 4 km rectangular area could be searched in less than 10000 seconds ( again assuming edge detection of the water pool ).

 

[Pythagorean]

There's a lot of association of animal search patterns with fractal properties:

The foraging ecology of spider monkeys

Foraging patterns of female reindeer

Search patterns of albatross, bumblebees, and deers

 

[mfb]

Pool size radius matters

It reduces searching time, but it does not increase the probability to find the pool.

pool edge detection

:D

If the pool is circular and fully within the search area, we can do even better. We can ignore 1km on both sides of the 5km-range, and the total travel distance is below 8.5km*.
A 3x6km-area would require at most 6km, as it can be done in a single pass.

*Edit: Below $(6.5 + \sqrt{2})km \approx 7.9km$.