Suppose an animal lives three years. The first year it is immature and does not

Question

Suppose an animal lives three years. The first year it is immature and does not reproduce. The second year it is an adolescent and reproduces at a rate of .8 female offspring per female individual. The last year it is an adult and produces 3.5 female offspring per female individual. Further suppose that 80% of the first-year females survive to become second-year females, and 90% of second-year females survive to become third-year females. All third-year females die. We are interested in modeling only the female potion of this population.

a) Draw a state diagram for this scenario.

b) construct the leslie matrix.

c) compute the eigenvalues for this matrix. From these determine if the population will eventually grow or decline. What is the rate of this growth(or decline)?

d) Suppose that a population of 100 first-year females are released into a study area along with a sufficient number of males for reproductive needs. Track the female population over 10 years.

e) Compute the eigenvector associated with the dominant eigenvalue. Normalize both this eigenvector and the population distribution after 10 years. Compare these two vectors.

Explanation / Answer

a) State Diagram If y0, y1 and y2 are the population of animal in their zero, first and second year.

Then, the population in the next year can be modelled as,

y0new= 0*y0 + 0.8*y1 + 3.5*y2

y1new = 0.8*y0

y2new = 0.9*y1

b) As modeled in part (a),

Now we can construct Lesie matrix as

egin{bmatrix} y0_{new}\ y1_{new}\ y2_{new} end{bmatrix}

= egin{bmatrix} 0 & 0.8 & 3.5 \ 0.8 & 0 & 0\ 0 & 0.9 & 0 end{bmatrix}

egin{bmatrix} y0\ y1\ y2 end{bmatrix}

Therefore, Lesie matrix is

L= egin{bmatrix} 0 & 0.8 & 3.5 \ 0.8 & 0 & 0\ 0 & 0.9 & 0 end{bmatrix}

c) We can calculate eigen value using matlab by typing following command

eig(L)

ans =

   1.5170        
-0.7585 + 1.0421i
-0.7585 - 1.0421i

From this we can see that dominant eigen value is greater than 1 and is 1.517. Therefore, population will grow at a rate of 51.7% per annum.

d) We can use following formula to calculate final population after 10 years.

Population= egin{bmatrix} 0 & 0.8 & 3.5 \ 0.8 & 0 & 0\ 0 & 0.9 & 0 end{bmatrix}^{10}

egin{bmatrix} 100\ 0\ 0 end{bmatrix}

Thus, Population= egin{bmatrix} 1571 \ 1492\ 890 end{bmatrix}

Please don't forget to rate positively if you found this response helpful. Feel free to comment on the answer if some part is not clear or you would like to be elaborated upon. Thanks and have a good day!

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