Question about population growth and limits.

Question

Suppose a pair of rabbits always has 4 offsprings. Suppose further that each mating season (the time it takes to produce offspring) is half the length of the previous mating season. Consider a den of 100 rabbits where the first mating season is one year. Let x be time in years and f(x) the population of the den, ignoring other environmental factors, determine the limit as x->2.

Answer 1

assuming the offspring are evenly divided half male, half female,no rabbits die, and older rabbits remain as fertile as young ones

then

x=0 f(x)= 100

x=1 yr, f(1) = 100+4(50) = 300

x=2 yrs, f(2) = 300 + 4(150) = 900

x=3 yrs, f(3)= 900 + 4(450) = 2700

x=4 yrs, f(4) = 2700 + 4(1350) = 8100

as x approaches 2, f(x) approaches 900

f(x) = 100(3^x)

that would be true, except each mating season is half the previous season,

then

x=0, f(0) = 100

x=1/2 f(1/2) = 300

x= 1 f(1) = 900

x= 3/2 f(3/2) =2700

x= 2 f(2) = 8100

as x approaches 2,f(x) approaches 8100

f(x) = 100(3^x)^2 = 100(3^2x)