A census of quail in Conecuh National Forest is taken in the fall and it is foun

Question

A census of quail in Conecuh National Forest is taken in the fall and it is found that there are ao adult birds and jo juvenile birds. Studies have shown that for each adult one can expect to find one year later 0.2 adults (due to survival) and 1.6 juveniles (due to reproduction), and for each juvenile one can expect to find one year later 0.4 adults (due to maturation and survival) and 1.4 juveniles (due to reproduction) (a) Write a system of equations that expresses the next year's adult a and juvenile ji populations in terms of the current year's numbers, ao and jo- (b) If the next year's populations are to be 380 adults and 1600 juveniles, what must the current populations be?

Explanation / Answer

(a) Adults in next year = Adults from adults + Adults from juveniles = 0.2 * a0+ 0.4 * j0 = a1 --------- equation (1)

Juveniles in next year = Juveniles from adults + Juveniles from juveniles = 1.6* a0+ 1.4 * j0 = j1 -------- equation(2)

(b) According to question , a1 = 380 and j1 = 1600 , we have to find a0 and j0 .

equation (1) becomes

0.2 a0 + 0.4 j0 = 380

=> 0.2 a0 = (380 - 0.4 j0)

=> a0 = (380 - 0.4 j0)/0.2

=> a0 = 1900 - 2 j0 ------- equation (3)

Put j1 = 1600 in (2) , (2) becomes

1.6* a0+ 1.4 * j0 = 1600 --------- equation (4)

Put a0 from (3) in (4) , we get

1.6 * (1900 - 2 j0)+1.4 j0 = 1600

=> 1.6* 1900 - 1.6*2 j0 + 1.4 j0 = 1600

=> 3040 - 3.2 j0 + 1.4 j0 = 1600

=> - 1.8 j0 = 1600 - 3040

=> j0 = (-1440) / (-1.8)

=> j0 = 800

Put j0 = 800 in (3) we get

a0 = 1900 - 2 j0 = 1900 - 2 * 800 = 1900 - 1600 = 300 => a0 = 300

The current population is total number of adults + juveniles = 800 + 300 = 1100 Answer

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