Question 1 A single breeding pair of rabbits is introduced t

Question 1 A single breeding pair of rabbits is introduced to Australia in 1900. Assume (for simplicity) that each pair can produce 4 offspring each year (2 male, 2 female). Also assume, that all births happen in the last hour of the last day each year and the parents die immediately after offspring are born.

a) How many rabbits are there after 10 years?

b) How many rabbits are there after 25 years?

If they have 20 offspring each year,

c) How many rabbits are there after 10 years?

d) How many rabbits are there after 25 years?

Question 2 Now, assume that rabbits reach sexual maturity instantly after being born and all can reproduce continuously (generations overlap). If each pair has 4 offspring and both parents die by the end of the year:

a)How many rabbits are there after 10 years?

b)How many rabbits are there after 25 years?

Question 3 Now, assume that the carrying capacity of the environment is K=1000. There are 6 offspring, both parents die as before. What is the rabbit population when the growth rate of the rabbits each year is the largest? Use discrete logistic to answer:

a) How many rabbits are there after 2 years?

b) How many rabbits are there after 3 years?

Solution

Answer

1)

a. Since, the population is increasiing in geometric progression and the intial number of rabbits were two, then after 10 years the total number of rabbits will be 2¹⁰ = 1024 rabbits after 10 years

b. After 25 years the total number of rabbits will be 2²⁵ = 33554432 rabbits.

c. If there are 20 offsprings each year, then the number of rabbits after one generation or year will be 20 (10 pairs).

Number of offsprings after one year is 10 pairs

Number of offpsrings after 2 years is 10² = 100

Number of offsprings after 3 eyars will be 10³ = 1000

the number of offsprings after 10 yeras will be 10 ¹⁰ = 10000000000

d. Nummber of offsprings after 25 years will be 10 ²⁵

2. It is given that there are 4 births every year in the population, before the death of the parents at the end of the year the population will be 6.

First calculating birth rate:

Birth rate = 4/6= 0.667

Now calculating death rate:

Birth rate = 2/6= 0.333

Now calculating r,

r = birth rate - death rate

r = 0.667- 0.333

r = 0.334

t= 10

Now putting all values in the equation of Nt you can easily calculate the number of rabbits in 10 years as well as 25 years.

For 10 years Nt can be calculated as:

Nt = No e ^rt

Nt = 6 e ^{(0.334) (10)}

For 25 years Nt can be calculated as:

Nt = No e ^rt

Nt = 6 e ^{(0.334) (25)}