(2) How many generations of rats can be produced in a year (365 days)?
According to the problem, the offspring can be born after 3 months (about 9 1 day) of growth and development, plus 22 days of pregnancy, that is, after 9 1+22= 1 13 days. There are (365-22) ÷113 = 3.035 generations in this year. According to the meaning of the question, only three generations can be taken. These newly bred three generations of mice and their mothers participate in breeding in the same year, and the fourth generation can give birth to the fifth generation, which shows that mice can live under the same roof for five generations every year. That is, mother, son, grandson, great-grandson and great-grandson generation.
For the convenience of calculation, we call each offspring Z 1, Z2, …, each grandchild S 1, S2, …, and each great grandchild ZS 1, ZS2, …
(3). In fact, the sex ratio of each offspring of mice is different, and only female mice can give birth. It is known that the sex ratio is close to 1: 1 in most cases. For the convenience of calculation, this question also considers that their sex ratio is 1: 1, so that each female mouse can have about 6 male and female offspring. For the convenience of calculation, let's assume that n=3.25, that is, each female mouse and each fetus can produce n pairs of offspring.
(4)∵∴ There is only one female mouse, and each fetus can only give birth to N pairs of offspring; When it is the offspring's turn to give birth, the number of grandchildren born by the mother mouse will increase to n, and n×n pairs of grandchildren can be born at the same time; When it's grandchildren's turn to take part in childbirth, the number of female grandchildren who take part in childbirth at the same time increases to n×n, and N× N pairs of great-grandchildren can be born at the same time; When great-grandchildren participate in childbirth, the number of female great-grandchildren who participate in childbirth at the same birth time increases to n×n×n, and N× N× N pairs of great-grandchildren can be born at the same time. Their number can be arranged in geometric series:
N, n×n, n×n×n, N× N× N is n 1, n2, n3, n4,
(5). We take the day when the mother mouse began to conceive as the zero starting point of this problem, so as to calculate the first pregnancy date of the offspring (Z), grandchildren (S) and great-grandchildren (ZS) of that year. The dates are as follows:
0, 1× (22+9 1) = 1 13,2× (22+9 1) = = 226,3× (22+9 1) = 339.
Based on the above data, we can calculate the earliest birth date of each first-born female rat in a year, namely: 0+22 = 22,113+22 =1ZS.
Namely: 22nd, 135, 248, 36 1 day. Of course, these four dates are also the first birthdays of children, grandchildren, great-grandchildren and great-grandchildren.
(6).∵ In this year, each generation began to conceive on different days, ∴ which means that the number of days available to each generation within 365 days of this year is also different. These days should be: [365-0 (pregnancy day)]. According to the data in (5), the available childbearing days of these mothers, offspring, grandchildren and great-grandchildren should be (365-0), (365-13), (365-226) and (365-339), namely, 366. This is a decreasing sequence, the first term is 365 and the tolerance is 1 13.
According to the number of days of birth, it can be concluded that the maximum number of births per generation in that year is calculated according to a single female mouse:
(365 ÷ 22 = 16.59), (252 ÷ 22 = 1 1.45), ( 139 ÷ 22 = 6.32), (26 ÷ 22)
That is to say, the maximum number of children a mother, offspring, grandchildren and great-grandchildren can have within 365 days of that year is 16, 1, 6,1,which is also clear in the chart.
With the above six calculation results, we can calculate the number of fetuses and the logarithm of offspring that these four generations of mice can have in that year:
1. Counting the female rats first: According to (6), the female rats can give birth to 16 fetuses at most in one year.
We named the offspring rats as Z 1, Z2, Z3, Z4, Z5, Z6, Z7, Z8, Z9, Z 10, Z1,Z 12 and Z/kloc.
(2) Statistics of offspring: According to (6), there are 252 days in a year for offspring rats to give birth, and the female rats with the most firstborn Z 11can give birth.
For every twenty-two offspring in the future, new women will join the ranks of fertility. According to this method, the arithmetic decline of offspring families can be calculated, including Z 1, Z2, Z3, Z4, Z5, Z6, Z7, Z8, Z9, Z 10 and Z1.
Other offspring families, such as Z 12, Z 13, Z 14, Z 15 and Z 16, have no time to give birth this year, so this question is not counted.
The total number of births of these offspring is:11+10+9+8+7+6+5+4+3+2+1= 66.
According to (4), there are n×n= n2 females in each offspring.
The total number of offspring that can directly give birth to grandchildren in a year is n×n×66=66 n2 pairs.
③ Counting grandchildren again: According to (6), the number of days available for grandchildren to give birth in a year is 139 days, and the first-born female mouse with the most reproduction can give birth to 6 children. That is, the descendants of the Z 1 family and the grandchildren of the S 1 generation can have six children in one year.
In the future, every 22 days, a new female mouse will join the fertility ranks. According to this method, the number of great-grandchildren of offspring Z 1 family in one year is 5, 4, 3, 2, 1 respectively.
In this way, the total number of great-grandchildren born in Z 1 family is 6+5+4+3+2+ 1=2 1.
By analogy, we can calculate the other grandchildren of the offspring Z2, Z3, Z4, Z5 and Z6 families, and the total number of great-grandchildren born during the year is (5, 4, 3, 2,1) respectively; ( 4,3,2, 1); ( 3,2, 1); ( 2, 1); 1。 In this way, the great-grandson algebra generated by all the grandchildren of the families of descendants Z2, Z3, Z4, Z5 and Z6 can be calculated as follows:
(5+4+3+2+ 1= 15),(4+3+2+ 1= 10),(3+2+ 1=6),(2+ 1=3), 1 ,
Namely 15, 10, 6, 3, 1,
Grandchildren Z7, Z8, …, Z 15, and Z 16 born by other descendants were not born in this year, so they are not counted.
According to this, the total number of great-grandchildren in one year is 21+15+10+6+3+1= 56.
According to (4), there are n×n×n=n3 female rats in each fetus of grandchildren.
In ∴ years, the total number of great-grandchildren born directly by grandchildren was: n×n×n×56=56 n3.
④ Finally counting great-grandchildren: According to (6), the number of days available for great-grandson women to have children in that year was 29, and they could only have 1 child as the only great-grandson. According to (4), each great-grandson mouse has n× n× n = n4 females, so the total number of great-grandson mice born directly is:
n×n×n×n× 1=n4
According to the above calculation, it can be known that the total logarithm of annual reproduction of mice is:
[( 16n)+(66n×n)+(56n×n×n)+(n×n×n)]=[ 16n+66 N2+56n 3+n4]
= [52+697+1922+11] = 2782 pairs (only integers are taken according to the meaning of this question). That is, 2782×2=5564 offspring.
After the above calculation, we get a formula for finding the algebra of mice after one year-[(16n+66n2+56n3+n4) × 2] In fact, all generations of mice continue to reproduce, and this problem only calculates 365 days of that year.
A: After a year of breeding, a pair of Rattus norvegicus can theoretically reach 5564 offspring. This problem is calculated by taking the representative Rattus norvegicus as an example.
It should be noted that the above is only a theoretical calculation of the strong reproductive potential of mice. The strong fertility of mice makes up for its high mortality. Under the unfavorable factors such as food shortage, natural enemies, diseases, climate and human hunting, its reproductive ability is greatly reduced. After all, only a few mice can survive. The above is a theoretical data, which is actually impossible to achieve.
With the formula -[( 16n+66n2+56n3+N4) × 2] for finding the offspring of the Year of the Rat, we can easily calculate different answers in different situations:
One, if each fetus gives birth to four on average, that is, n=4÷2=2. According to the formula of "seeking the number of children and grandchildren in the Year of the Rat", children and grandchildren can reach:
( 16n+66 N2+56n 3+N4)×2 =( 16×2+66×22+56×23+24)×2 = 1520。
Second, if each fetus gives birth to six on average, that is, n=6÷2=3. According to the formula for finding the algebra after the year of the rat, the algebra after one year can be (16n+66n2+56n3+n4) × 2 = (16x3+66x32+56x33+34 )× 2 = 4470.
Three, such as seven per fetus on average, that is, n=7÷2=3.5. According to the calculation formula of algebra after the year of the rat, the algebra after one year can be: (16n+66n2+56n3+n4) × 2 = (16× 3.5+66× 3.52+56× 3.53+3.54 )× 2 = 688.
Four, such as eight per fetus on average, that is, n=8÷2=4. According to the calculation formula of the algebra after the year of the rat, the algebra after one year can be (16n+66n2+56n3+n4) × 2 = (16x4+66x42+56x43+44 )× 2 = 9920.