detailed solution pls.. The logistic model has been applied to predict the growt

Question

detailed solution pls..

The logistic model has been applied to predict the growth of the halibut population in a certain part of the Pacific Ocean. Assume that the biomass y ( t ) of the halibut population at time t (years) is measured in kilograms (kg) and that the environmental carrying capacity is 80.5 * 10^6 kg . The r parameter in the logistic equation is estimated to be r = 0.71 per year, and at time t = 0 the biomass is estimated to be 2 * 10^7 kg . (a) Determine the biomass of the halibut population two years later. (b) How long does it take for the biomass to reach 60% of the environmental carrying capacity?

Explanation / Answer

Basically the logistic function is given by

P(x)=1/(1+ex)

For population growth in a habitat the function can be transformed into

dP/dt =r P [ 1- (P/k) ]

where k-carrying capacity

          p - habitat population at a given time

          r - growth rate

a)   dP/(2-0) = 0.71*P [ 1   - P/ 80.5*106 ]

      P - 2*10 7 =2 * [0.71 * P - P2 /80.5*106]

On solving this quadratic equation

We get P as   60445545.4498

i.e. After two years population of biomass in the habitat is P = 6.04*106 kg

b) Now it is given that the amount of population is of 60% of the total initial amount

hence P=2*107 / 6 =3.33*106

Now by these terms in the above equation

(3.33 - 20)*106 = t * 0.71 * [ 1 - (3.33*106 / 80.5*106)] * 3.33 * 106

on solving for t we get t =7.346133536 years

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