Bird-Brained
A mother bird has 2kg of mass available to produce her eggs. The chance of an egg surviving to hatch (which we will call p) is greater for heavier eggs.
This chance is defined by the following function:
p = 0 if individual egg mass (which we will call s) $\le{150}$g
p = $\frac{s-150}{k+s-150}$ if s > 150g, where k is a constant
Sketch a graph of this function. Can you describe how survival chance, p, is affected by egg mass, s? What assumptions have been made in applying this model?
How does the shape of the graph change as k varies? (Try thinking about what will happen if k is very big or very small)
if k = 0.4, what is the optimal size for each egg to be? Hence, how many eggs should the bird lay to produce the greatest expected number of chicks? What can you say about this number? If k was bigger, would the optimal number of eggs increase or decrease?
Given that a bird cannot actively choose how many eggs to produce and what mass each should be, do you think that birds in nature produce the optimal number of eggs? Why do you think this?
When working out what optimal egg size is, try defining the function that you are attempting to maximise - the payout. In this case this is the expected number of chicks that will hatch, which is equal to the number of eggs multiplied by the probability that each egg has of survival. What is the total number of eggs equal to? How might you go about working out what the maximum of a function is? Once you've defined your function it might help to sketch a graph of it.