13a. A country has a carrying capacity of 1,000,000 people. It is said that the

Question

13a. A country has a carrying capacity of 1,000,000 people. It is said that the doubling time for the population is 91.65 yr and the growth rate has been constant at 0.016 yr-1.The year is 2000, what is the initial population for this country?

13b. To increase stability, the country decided to adopt a closed border policy for 50 years, but at the end of this 50 year period the government will allow immigration at a rate of 5/100 people and an emigration rate 10x lower than the immigration rate. What will the population of the country be at the end of the 50 year period if the population growth rate and the carrying capacity remain constant?

13c. Describe in words how dP/dt changes from when P is very small compared to K and when P is similar to K.

13d. If P = K, then what is dP/dt?

Explanation / Answer

(a)

The population after t years will be double now we require the initial population

P(t)=2P0

As per the logistic growth method

P(91.65)=2P0

K/(1+Ae-rt)=2P0

A=K-P0/P0

P0K/(P0+(K-P0)e-91.65x0.016)=2P0

K=1000000

P0(1000000)/(P0+(1000000-P0)0.230754)=2P0

500000=P0+(1000000-P0)x0.230754

P0=3,50,012   (There might be some variation due to decimals)

(b)

The rate of increase of population=0.016

The emigration rate and immigration rate will not be considered because it is clearly specified that the emigration and immigration are after the 50 years, so they will not be counted in the 50 year period

P0=350012

P(50)=K/(1+Ae-rt)

A=K-P0/P0

A=(1,000000-350012)/350012

A=1.857

P(50)=K/(1+1.857e-0.016x50)

P(50)=100000/(1.83)

P(50)=546449

(c)

When P(t) is very small, then P(t)/K is close to 0, so the entire factor [1-P(t)/K] is close to 1 and the equation itself is then close to P'(t) = r P(t), we then expect that the population grows approximately at an exponential rate when the population is small. On the other hands, if P(t) gets to be near K, then P(t)/K will be approximately 1, so [1-P(t)/K] will be approximately 0, and the logistic differential equation will then say approximately P'(t) =r P(t) 0 = 0. The growth rate will be essentially 0, so the population will not grow significantly more.

(d) dP/(P[1-P/K])=dt is the differential equation

dP/dt=P(1-P/K)

IF P=K

dP/dt=K(1-K/K)

dP/dt=0 when P=K

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