Logistic Growth Word Problem

Question

A population of 20 deer are introduced into a wildlife sanctuary. It is estimated that the sanctuary can sustain up to 200 deer. Absent constraints, the population would grow by 40% per year.

Estimate the population after one year

P1=28

Estimate the population after two years

P2=36.60032

i have the answers just not sure how they got the second answer considering the formula would make the answer 39.2 right?

Answer 1

exponential growth has geometric sequencing, meaning

growth in ratio per time period is constant.

"change in P"=g*P ,,,,,g is "grows by" constant

"new P"=P+gP=P(!+g),,,per time period specified in "grows by" ratio

since (1+g)=r where r is growth coefficient

P_new=rP per time period

It is not that way with logistical equation.

the discrete time period to time period (year to year in this problem)

logistic equation is:

"change in P"=g*P(1-P/K)

P_new=P+g*P(1-P/K),,,,g is growth rate and K carrying capacity

P_new=P(1+g)(1-P/K)

P_new=rP(1-P/K) ,,,,where r=1+g

I kind of disagree after one year population is 28,

I calculate P₁=27.2=20+20*.4(1-20/200)

for population after two years

P₂=27.2+27.2*.4(1-(27.2/200))

=36.60032