Question

Create another rabbit model to answer the question “In which generation does the rabbit population reach a level of 10,000 pairs of rabbits?”

Now go back to the original model from Part 1 and change assumption 4 (see assumptions below). Original model of Part 1 is the Fibonacci sequence:

Rabbits in Generation n+1 = Rabbits in Generation n + Offspring born in Generation n+1

Let’s assume that rabbits die after 5 generations instead of living forever. Create another rabbit model to answer the question “In which generation does the rabbit population reach a level of 10,000 pairs of rabbits?” under this modified assumption. Be sure to give an algorithmic description of this model in terms of a recurrence relation and also build your model into a spreadsheet.

Initial Assumptions (Assumption 4 was changed in this exercise):

1. The generations are distinct (making this a discrete system).

2. Each rabbit pair will be ready to reproduce at the end of the second generation (born in Generation 1, develop in Generation 2, first offspring in Generation 3).

3. Each pair of mature rabbits has one pair of offspring in each generation.

4. Rabbits will live forever. The model does not assume that there is a life expectancy for rabbits.

5. Rabbits will continue to have offspring forever. The model does not assume that rabbits will stop having offspring after a certain number of generations.

Without altering the initial assumptions, this model applies the Fibonacci sequence to rabbit breeding. Here, assumption 4 is altered so that the rabbits are dying after the 5th generation and after having had offspring. Please find the algorithm for this.

Answer 1

solution:-)

each rabbit yields two offspring in each generation and total yield is six offspring in whole life.

Therefore, each rabbit pair yields 6 offspring in 5 generations.

Growth rate, r=(N_t-N₀)/N₀

N_t is the population after t generations while N₀ is the population at the initial generation.

N_t= 8

N₀=2

Growth rate, r=(8-2)/2

Growth rate, r=3

To calculate the population of 10000 in t generations, use the following formula.

Next- generation population N^T = N_0^ert

N_t=10000

N₀= 2

10000 = 2 x e^3xt

5000 = e^3t

Taking log on both sides:

Log 5000 = log e^3t

Log 5000 = 3t log e

3.7 = 3t (0.43)

3t = 8.6

t = 2.9 years

One rabbit pair can have theoretically 12 genartations period in one year. Therefore, total generations will be as below:

Generations = 2.9 x 12

Generations = 34.8

Therefore, 35 generations are used to reach a population of 10000.

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