## Stuck on this tricky question

Hi all,

Was wondering if anyone knew how to deal with these questions? I think I might know (a) (because if N=K then N over K will always equal 1, then 1-1=0, and r0X0 would make the population growth rate 0- I think that might be correct?), but on (b) and (e) I'm completely stumped.

Suppose that a country’s rate of population growth, 𝑟, decreases as its population, 𝑁, increases according to the formula: 𝑟 = 𝑟0 (1− 𝑁/𝐾 ), where 𝑟0 and 𝐾 are constants that have yet to be determined.

(a) The constant 𝐾 is the population’s carrying capacity. By considering what happens when the country’s population is equal to the carrying capacity, i.e. when 𝑁 = 𝐾, explain why we can consider 𝐾 a steady state level for the population.

(b) Suppose that when the population is half the carrying capacity, i.e. 𝑁 = 0.5𝐾 , the rate of population growth is 1/4. Determine the value of 𝑟0 .

(e) Rewrite the relationship in equation (*) in the form 𝑟 = 𝑎𝑁 + 𝑏, identifying what a and b are.