Similar to the California \"Gold Rush, \" the \"Pearl Rush\" of the mid 1850\'s

Question

Similar to the California "Gold Rush, " the "Pearl Rush" of the mid 1850's has threatened the population of Freshwater Mussels. The population percent P(t) is declining at a rate proportional to the difference between the population percent and 3. The population percent was at most 100 right before the pearl rush. Today only 20 percent of the population remains. Set up a differential equation for the rate of change of the population percent. Find the particular solution and use it to determine the population percent 50 years from now.

Explanation / Answer

given p(t) is the population

let us consider the population is p

now given the rate of change of poplulation is given by dp/dt which is proportional to decrease in popluation given by (p-3)

now dp/dt p(p-3)

=> dp/dt = kp(p-3) since k is the proportional constant

now dp/p(p-3) =k dt

dp(1/p(p-3) = k.dt

by partial fractions A/p +B/p-3 = 1/p(p-3) => A+B = 0 , A= -1/3 and B = 1/3

integrating both sides

-1/3p+1/3(p-3) .dp = k.dt

-ln(3p)+ln(3p-9) = k.dt

=> ln(3p-9/3p) = kt+C

=> 1-9/3p = Ce^kt

given p =100 at t=0

1-1/300 = C.e^0

=> C = 291/300

now given present p =20 and => t =1

=>1-9/60 = 291/300.e^(k)

=> k = ln(255/291) = -0.13

1-9/3p = 291/300e^(-0.13t)

now population after 50 yrs from now is given by

1-9/3p= 291/300e^(-0.13*50)

=> 1 - 9/3p = 291/300(1.5*10^(-3))

=> 1-9/3p = 1.45*10^(-3)

=> 3p = 9/0.9985

=>p = 3%

population after 50years is 3%

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