given dp/dt = rp(1-P/k) In this model r = net growth rate (birth rate - death ra

ID: 2982246 • Letter: G

Question

given dp/dt = rp(1-P/k)

In this model

r = net growth rate (birth rate - death rate)

K= the maximum sustainable population

Find an explicit solution to the differential equation using seperation of variables.

Suppose P is a typical solution between the tow critical points (P(0)=K/2 and P(0)= 2K)

Calculate analytically the limits at infinity: limP(t) as t goes to infinity and limP(t) as t goes to -infinity

find the points of inflection for P

For which values of P is the population growth rate increasing? Decreasing?

(note: The population growth rate is dp/dt, not P)

A) dp/dt = rp(1-p/k)

or dp/dt = rp(k-p)/k

or [1/p(p-k)]*dp = r/k*dt

or (1/k)*ln[(p-k)/p] = r/k*t + c

or ln[1- k/p] = r*t + c

or [1- k/p] = c1*e^(rt)

B) for p(0) = k/2 c1 = -1 [1- k/p] = -e^(rt)

for p(0) =2k c1 = 1/2 [1- k/p] = 0.5e^(rt)

C) limP(t) as t goes to infinity = 0

limP(t) as t goes to -infinity = k

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