Let w be the number of worms (in millions) and r the number of robins (in thousa

Question

Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island. Suppose w and r satisfy the following differential equations:

dw/dt = w - wr
dr/dt = -r + wr

At t = 0 there are 2.6 million worms and 1 thousand robins.
(a) Use the differential equations to calculate the derivatives: dw/dt and dr/dt at t = 0 .

Enter the exact answers.

dw/dt = _________

dr/dt = _________

(b) Use the initial values and your answer to part (a) to estimate the number of robins and worms at t =0.1

Round your answers to two decimal places.
Number of worms = _______ million

Number of robins = _______ thousand

(c) Using the method of part (a) and (b), estimate the number of robins and worms at t = 0.2 and 0.3 .

Round your answers to one decimal place.

At t =0.2 :

Number of worms = ________ million

Number of robins = ________ thousand


At t = 0.3 :

Number of worms = _________ million

Number of robins = _________ thousand

Explanation / Answer

a) Let w be the number of worms (in millions) and r the number of robins (in thousands) living on an island

dw/dt=w-wr, dr/dt=wr-r

dw/w(1-r)=dt, or both sides int.

log(w)(1-r)-1=t+c, at t=0

c=log(w)(1-r)-1,    w=2.6*106, r=103

dr/(w-1)r=dt, both sides int.

log(r)(w-1)-1=t+c1, at t=o

c1=log(r)(w-1)-1

b)t=0.1

no.s of worm, log(w)(1-r)-1=t+c

c=log(w)(1-r)-1-0.1

no.s of robins,log(r)(w-1)-1=t+c1

c1=log(r)(w-1)-1-0.1

c)again put up the values t=0.2, t=0.3

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