Using the supplied matrix and information, determine if the blue whale populatio
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Question
Using the supplied matrix and information, determine if the blue whale population is becoming extinct in this model. if it is not becoming extinct, determine the percentage of each class in the stable population.
to soh 31 I Computer Project 21) ro 883-dt-content rid-15169446 1/courses/2017-FALL-TRICH MATH-220-8532 LEC/Computer Project 2%281% 1. In the 1930's a researcher studied the blue whale population (dat a from references 2, 4, an 5, slightly modified, are used here). Due to the long gestation period, mating habits, and migration of the blue whale, a female can produce a calf only once in a two-year period. Thus t years, 2 or 3 years, 4 or 5 he age classes for the whale were assumed to be: less than 2 6 or 7 years, 8 or 9 years, 10 or 11 years, and 12 or more years. The matrix for the model is years 0 0 19 44 .50 .50 45 87 0 0 0 0 0 0 0 87 0 0 0 0 0 0 0.87 0 0 00 00 0 87 0 0 0 0 0 0 0 87 0 0 0 0 0 0 0 87 8S Det ermine whether the blue whale population is becoming extinet in this model. If the population is not becoming extinct, deternine the percentage of each class in the stable populationExplanation / Answer
At any given time t, the size of a population (Zt) is the number of individuals in the branching process. We set Z0 1 unless otherwise specified. The probability of extinction of a branching process is q limt P(Zt= 0|Z0 = 1). If the starting size of the population is 2 greater than one, then the overall probability of extinction can be defined as
lim t P(Zt = 0|Z0 = N) = q N So we can solve for extinction in the case of Z0 = 1 and extend the results to larger starting populations if necessary.
The recursive formula for finding q can be found through a first step analysis (Kimmel and Axelrod, 2002). The probability that the lineage of a single individual eventually goes extinct is the probability that it dies without offspring (p0) plus the probability that it produces a single offspring whose lineage dies out (p1q) plus the probability that it produces two offspring whose joint lineages die out (p2q2 ), and so on. This leads to the formal definition of the probability generating function: f(q) = E[qX] = p0 + p1.q + p2q2 + p3q3 . . . pnqn ----------------------------(1)
The probability of extinction of a branching process starting with a single individual is the smallest root of the equation f(q) = q for q [0, 1]. The solution q = 1 is always a root of (1) and is not necessarily the smallest positive root. In some cases, the probability of extinction is trivially obvious. For instance, if p0 = 0 individuals always produces at least one offspring, therefore q = 0. Furthermore, cases where E[X] 1 always yield q = 1 (Kimmel and Axelrod, 2002).
coming to our problem i have attached a file.
Probability generating function for first and last rows are
f(q) = p0+p1q1+p1q2+p3q3+p4q4+p5q5+p6q6+p7q7
(1)age classes for the whale is to be extinct for less than 2 years is
=(0.19)(0.81)+(0.44)(0.56)+(0.50)(0.50)+(0.50)(0.50)+(0.45)(0.55)
=1.1478
it is more than one.
it is not extinct.
we need to calculate average
Average = 0.19 + 0.44 + 0.5 + 0.5 + 0.45 = 2.08 = 2.08/7 = 0.297143 = 30%
from the above matrix, 2nd row to 6th row we found that there is only element ,
whale population becoming extinct when = 0
= 0
with help of excel sheet, n=67 iterations, the value is 8.87E-05(approximately 0,took 5 digits accuracy)
(7) age classes for the whale is to be extinct for 12 or more years is
= (0.87)(0.13) + (0.88)(0.12) = 0.2187
whale population becoming extinct when = 0
= 0
n = 7 iterations , the value is 2.39E-05(approximately 0, took 5 digits accuracy)
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