According to the topic, the number of animals in 0~5 years old, 6~ 10 years old and1~15 years old is as follows:

x 1(0)= 1000，x2(0)= 1000，x3(0)= 1000

Taking five years as an age group, the number of animals in three age groups at a certain moment can be used as a vector.

X=[ X 1 X2 X3 ]T

Said. Take five years as a time period, remember

X(k)=[ x 1(k) x2(k) x3(k) ]T

It is the distribution vector of the number of animals in the k-th time period. When k=0, 1, 2, 3, X(k) represents the distribution vectors of animal numbers at present, five years later, ten years later and fifteen years later respectively. According to the reproductive ability of the animals in the second and third age groups, in the k period, the animals in the second age group have an average of 4 offspring, while the animals in the third age group have an average of 4 offspring.

x 1(k+ 1)= 4x 2(k)+3x 3(k)

Similarly, according to the survival rate of the first and second age groups, the equation can be obtained.

x2(k+ 1)=0.5x 1(k)

x3(k+ 1)=0.25 x2(k)

Establish the following mathematical model

( 1)

Also written in matrix form.

(k=0， 1，2，3 ) (2)

From this, the recurrence relation of vectors X(k) and X(k+ 1) is obtained.

X(k+ 1) =L X (k)

Wherein the matrix

L=

X(k+ 1) =L k+ 1 X (0) can be obtained from the above formula.

We can fill in the data results in the following table according to the mathematical model calculation:

k

(now)

K= 1 (five years later)

K=2 (ten years later)

K=3 (fifteen years later)

K=4 (twenty years later)

X 1

1000

7000

2750

14375

8 125

X2

1000

500

3500

1375

7 187.5

X3

1000

250

125

875

343.8

From the data changes in the table (if there are no other reasons), it can be estimated that the total number of animals on the farm will gradually increase.

After 15 years, there will be 14375,1375,875 animals.