The Logistic Model for Population GrowthDate: 03/29/98 at 02:42:58 From: Herve Duchemin Subject: High School Calculus I have a problem in my high school calculus class. It is known as the Logistic Model of Population Growth and it is: 1/P dP/dt = B - KP where B equals the birth rate, and K equals the death rate. Also, there is an initial condition that P(0) = P_0. I need to calculate P(t), which will predict the population at any time. I also need to find the limit of P(t) as t approaches infinity. Thanks for your help! Date: 03/29/98 at 07:11:08 From: Doctor Anthony Subject: Re: High School Calculus We start with the following equation: dp/dt = p(b - kp) Rearranging the equation so that p is on the left and t is on the right, we get: dp -------- = dt p(b - kp) Use partial fractions for the left hand side: 1 A B --------- = --- + ------ p(b - kp) p b - kp 1 = A(b - kp) + Bp Then: p = 0 gives 1 = Ab and so A = 1/b p = b/k gives 1 = Bb/k and so B = k/b The lefthand side can then be written: 1 k [--- + ---------] dp = dt bp b(b - kp) (1/b)[ln(p) - ln(b - kp)] = t + constant ln[p/(b - kp)] = bt + constant p/(b - kp) = Ae^(bt) where A is a constant p = (b - kp)Ae^(bt) p = Abe^(bt) - Akpe^(bt) p(1 + Ake^(bt)) = Abe^(bt) Abe^(bt) p = -------------- (Equation 1) 1 + Ake^(bt) At t=0, p = p(0), so we have: p(0)/(b - k*p(0)) = A (Equation 2) For any given problem, you can calculate A from (2) and then substitute for A in (1). Dividing (1) top and bottom by e^(bt), we get: Ab p = ------------ e^(-bt) + Ak and as t -> infinity, the term e^(-bt) -> zero, and we have: Ab b p(infinity) = ---- = --- Ak k -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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