# Proportionate growth rate & equilibrium size

A population of beavers was introduced into a reserve on 1 January in a particular year, and the size of the population was estimated on the same date in each subsequent year. The size of the initial population was 100, and it had grown to approximately 180 after one year. After one further year, the size of the population was approximately 288. Assume that the behavior of this population satisfies the logistic model.

(a)

Show that the annual proportionate growth rate for the population of size 100 was approximately 0.8, and that the annual proportionate growth rate for the population of size 180 was 0.6.

(b)

Find the corresponding exact values of the annual proportionate growth rate for low population levels r, and the equilibrium population size E.

Answer to (a)

Year # 1 = 100 (P0)

Year # 2 = 180 (P1)

Year # 3 = 288 (P2)

Pn = (1 - r)nP0 (n=0,1,2,...)

P1 = (1 + r)1100 P2 = (1 + r)nP1

180 = (1 - r)1100 288 = (1 + r)n 180

1 + r = 180/100 1 + r = 288/180

r = 180/100 - 1 r = 288/180 - 1

r = 0.8 r = 0.6

Answer to (b)

Probably need use:

Pn+1 - Pn = rPn ( 1 - Pn/E )

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#### Solution Preview

a.

Initial population, Po = 100.

Let us assume population growth rate, r1 = 80% == 0.80

Hence, population after one year,

P1 = Po + r1*Po

= Po*(1+r1)

= 100*(1+0.8)

= 100*1.8 = 180 == given number --Proved

For next year,

Initial population, Po = 180

Let us ...

#### Solution Summary

We solve a problem related to population growth rate different two different years, followed by estimation of equilibrium population size & proportionate growth rate.