These days, we know the value of , but we can still use Buffons needles for other

These days, we know the value of π, but we can still use Buffon’s needles for other applications. The kiwi, which is New Zealand’s national bird, is highly threatened by introduced predators such as stoats and ferrets. An adult kiwi can fight off a By the way, the kiwi is not a small brown furry fruit. It’s a small brown furry bird. The small brown furry fruit is called a kiwiFRUIT. You must never forget the "fruit" bit on the end.

Don’t be that person. Sorry. Just had to get that off my chest. I feel better now.

stoat, but a chick has no chance. The probability of a kiwi chick surviving to adulthood when stoats are present is approximately zero. For this reason, conservationists create safe zones where stoats are kept out as far as possible. Kiwi chicks can survive inside the safe zone, but not outside.

Unfortunately, the kiwi chicks don’t know which locations are safe and which aren’t. A chick that grows up in a safe zone will disperse at random to make its adult territory, which will then be fixed for its adult life. This territory might or might not be inside the safe zone.

Fig. 36.6 shows a model for this situation. The safe zone is a square of side L km. After a kiwi chick survives to adulthood inside the safe zone, it disperses a distance of d km in a random direction. To keep things simple, we’ll fix d at d = 5 km. If the kiwi chick disperses outside the safe zone, it will survive and breed, but its own chicks will never survive.

The question we need to answer is how large should the safe zone be to ensure the population survives? This means that we need enough kiwi chicks surviving and staying inside the safe zone to replace any adults within the safe zone that die each year. We’ll work through the calculations to find out how large the safe zone needs to be.

a. Suppose that within the safe zone there’s an average of 4 kiwi nests per km2

, corresponding to an average of 8 adult kiwi per km2

. In terms of L, what’s the average number of adult kiwi inside the safe zone?

b. Each adult kiwi dies with probability 0.12 per year. How many adult kiwi are lost from the safe zone each year, on average?

c. How many kiwi chicks do we require to survive and stay inside the safe zone each year, on average, to replenish the adult kiwi that are lost?

d. Each kiwi nest within the safe zone results in one chick that survives to adulthood with probability 0.45. What is the average number of chicks born in the safe zone that survive to adulthood?

e. We now need to consider the dispersal phase. Using Buffon’s needle model, we can show that if each surviving chick disperses distance

d, then the probability it remains inside the safe zone is P(chick remains inside the square)
= 1 πL 2 
d 2 − 4dL + πL 2 
.
Write down an expression for this probability when the dispersal distance is d = 5 km.

f. Overall, in terms of L, what is the average number of chicks per year that survive and stay inside the safe zone when d = 5 km?
g. Bringing together your answers above, show that we need L to satisfy 0.96L 2 = 1.8 π

25 − 20L + πL 2 
.
Solve this equation to find the size of safe zone that will enable the kiwi population to remain self-sustaining.