plz help me solve this question by using mathematica At day 0, there are n0 flie
ID: 3565655 • Letter: P
Question
plz help me solve this question by using mathematica
At day 0, there are n0 flies in an otherwise unpopulated forest. Each fly produces r offspring per day, but the environment only has enough food for k flies. Their population n[t] on day t is governed by the logistic growth equation:
dn/dt=r n(1-n/k)
r is called the growth rate, and k is called the carrying capacity.
(a) Find the symbolic solution of this ordinary differential equation, using the initial condition n = n0 at t = 0.
(b) If k = 100, n0 = 1 and r = 0.1, how long is it until there are 50 flies?
Explanation / Answer
a)
dn/dt=r*n(1-n/k)
=>dn/[n*(k-n)]=dt.r/k
=>(1/k)*[(dn/n)-(dn/(k-n))]=dt. r/k {by resolving 1/[n*(k-n)] into sum of its partial fractions}
(dn/n)-(dn/(k-n))= r.dt
integrating both sides,
ln(n)-ln(k-n)=rt+c {c is some constant}
ln(n/(k-n))=rt+c
n/(k-n)=ert+c
n=k.ert+c/(1-ert+c)
given n=n0 at t=0
=>ec= n0/(k-n0)
n=k.n0.ert/(k-n0(1+ert))
b)If k = 100, n0 = 1 and r = 0.1, how long is it until there are 50 flies?
n/(k-n)=ert+c
=>n/(k-n)=n0.ert/(k-n0)
50/(100-50)=1.e0.1*t/99
e0.1*t=99
0.1*t=ln99
t=45.95 days
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