How Does the Logistic Equation Model the Growth of the Pacific Halibut Biomass?

[FritoTaco]

Homework Statement

The Pacific halibut fishery has been modeled by the differential equation.

\displaystyle\dfrac{dy}{dt}=ky\left(1-\dfrac{y}{M} \right)

where y(t) is the biomass (the total mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be M = 7\times 10^7 kg, and k=0.78 per year.

(a) If y(0)= 2\times 10^7 kg, find the biomass a year later. (Round your answer to two decimal places.)
(b) How long will it take for the biomass to reach 4\times 10^7 kg? (Round your answer to two decimal places.)

Homework Equations

\displaystyle P= \dfrac{K}{1+Ce^{-kt}}

The Attempt at a Solution

K = carrying capacity \implies 7\times 10^7 kg
k = 0.78 per year

At time 0, biomass is 2\times 10^7 kg \impliesy(0)= 2\times 10^7 kg

C = the difference between the carrying capacity and the initial capacity subtracted by 1.

C= \dfrac{7\times 10^7}{2 \times 10^7}-1=\dfrac{5}{2}
P=\dfrac{7 \times 10^7}{1+\dfrac{5}{2}e^{-0.78\cdot1}}= 70000001.15 kg

I'm trying to solve for the biomass (P) after 1 year. This answer doesn't seem correct. Am I using the wrong number for variable t? Or I'm not solving for P right away?

 

[mfb]

\dfrac{7 \times 10^7}{1+\dfrac{5}{2}e^{-0.78\cdot1}}= 70000001.15 kg

The two sides here are not equal. The denominator is larger than 1, the fraction cannot be larger than the numerator.

 

[Ray Vickson]

Homework Statement

The Pacific halibut fishery has been modeled by the differential equation.

\displaystyle\dfrac{dy}{dt}=ky\left(1-\dfrac{y}{M} \right)

where y(t) is the biomass (the total mass of the members of the population) in kilograms at time t (measured in years), the carrying capacity is estimated to be M = 7\times 10^7 kg, and k=0.78 per year.

(a) If y(0)= 2\times 10^7 kg, find the biomass a year later. (Round your answer to two decimal places.)
(b) How long will it take for the biomass to reach 4\times 10^7 kg? (Round your answer to two decimal places.)

Homework Equations

\displaystyle P= \dfrac{K}{1+Ce^{-kt}}

P=\dfrac{7 \times 10^7}{1+\dfrac{5}{2}e^{-0.78\cdot1}}= 70000001.15 kg

I'm trying to solve for the biomass (P) after 1 year. This answer doesn't seem correct. Am I using the wrong number for variable t? Or I'm not solving for P right away?

Is $P$ the same as $y$? I assume so.

Your final equation is incorrect; it should be
$$ \frac{7 \times 10^7}{1 +\displaystyle \frac{5}{2} e^{-0.78 \times 1}} = 4 \times 10^7$$

 

[FritoTaco]

Thanks, guys, I got the right answer now. Major parentheses mistake for the calculator.

I can solve b, but for a.) I got 3.26\times10^7

 

[Ray Vickson]

Thanks, guys, I got the right answer now. Major parentheses mistake for the calculator.

I can solve b, but for a.) I got 3.26\times10^7

Why the "but"? Your answer for (a) is correct.

 

[FritoTaco]

Sorry for my bad English. For (a) I got 3.26 \times 10^7