12. The number of new individuals in a population of animals is sometimes approximated by calculating the expected number of survivors of the new born animals from the next genera- tion. The idea is that the number of new animals born follows some probability distribution, and the number of animals that survive depend on how many are present.¹ One example of a survivor function is the following: S₁(k) = b.ck, where b and c are real numbers with 0
12. The number of new individuals in a population of animals is sometimes approximated by calculating the expected number of survivors of the new born animals from the next genera- tion. The idea is that the number of new animals born follows some probability distribution, and the number of animals that survive depend on how many are present.¹ One example of a survivor function is the following: S₁(k) = b.ck, where b and c are real numbers with 0
A First Course in Probability (10th Edition)
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![12. The number of new individuals in a population of animals is sometimes approximated
calculating the expected number of survivors of the new born animals from the next genera-
tion. The idea is that the number of new animals born follows some probability distribution,
and the number of animals that survive depend on how many are present.¹ One example of
a survivor function is the following:
S₁(k) = b.ck,
where b and c are real numbers with 0 <c< 1 based on the idea that more new born.
individuals increases competition and reduces the success rate to survive. Another example
of a survivor function is
S₂(k)
=
k
w
0≤k≤ 2,
otherwise.
The idea is that once the number of newborns are above a certain level there are not enough
resources for any to survive.
(a)
Determine E[S₁] if the number of newborns follows a Poisson distribution with
mean A. (This is the derivation of the Ricker model which is used to model parasitic
species.)
(b)
Determine E[S2] if the number of newborns follows a Negative Binomial distri-
bution with probability of success is p and number of trials before stopping is r. (You
may have to look up the distribution function (PMF - probability mass function) on
Wikipedia.) (This is similar to the method used to derive the Hassel model which use
used to model species who directly compete with each other.)
¹Brännström Å and Sumpter D.J.T 2005 The role of competition and clustering in population dynamicsProc.
R. Soc. B.2722065-2072 http://doi.org/10.1098/rspb.2005.3185](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf3928cb-73bf-4270-84ca-805354b4b59c%2Fec094bf9-eb8f-4b9c-a6af-280272c2a16f%2F6yvj6bb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:12. The number of new individuals in a population of animals is sometimes approximated
calculating the expected number of survivors of the new born animals from the next genera-
tion. The idea is that the number of new animals born follows some probability distribution,
and the number of animals that survive depend on how many are present.¹ One example of
a survivor function is the following:
S₁(k) = b.ck,
where b and c are real numbers with 0 <c< 1 based on the idea that more new born.
individuals increases competition and reduces the success rate to survive. Another example
of a survivor function is
S₂(k)
=
k
w
0≤k≤ 2,
otherwise.
The idea is that once the number of newborns are above a certain level there are not enough
resources for any to survive.
(a)
Determine E[S₁] if the number of newborns follows a Poisson distribution with
mean A. (This is the derivation of the Ricker model which is used to model parasitic
species.)
(b)
Determine E[S2] if the number of newborns follows a Negative Binomial distri-
bution with probability of success is p and number of trials before stopping is r. (You
may have to look up the distribution function (PMF - probability mass function) on
Wikipedia.) (This is similar to the method used to derive the Hassel model which use
used to model species who directly compete with each other.)
¹Brännström Å and Sumpter D.J.T 2005 The role of competition and clustering in population dynamicsProc.
R. Soc. B.2722065-2072 http://doi.org/10.1098/rspb.2005.3185
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