Essay

Applied Math essays

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Course
Applied Math 1201B (APPLMATH1201B)

Institution
University Of Western Ontario (UWO )

This document has both essays that are to be written in Applied math 1201B and provides a good model for them as they both received perfect grades. The professors are very vague when they assign this task and so I think it would be very beneficial to see examples of the essays.

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Uploaded on May 13, 2021
Number of pages 2
Written in 2020/2021
Type Essay
Professor(s) Unknown
Grade A+

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The task presented asks for the creation of a model representing the relationship between
the fraction of islands occupied by the species and time, while also considering that this species
will occupy a large and fixed number of islands. To solve this, Levin’s model for metapopulation
ecology that describes changes over time in the regional abundance of a species by the
colonization of lesser populations and extinction must be used.

To create an accurate model, certain assumptions must be made, and variables must be
identified. The variable p represents the fraction of islands that are already occupied and 1-p
represents available islands, both of which affect the rate of colonization. The first assumption is
that the rate of colonization and extinction are constant and cannot be equal, as the equilibrium
point is zero (p1). The extinction threshold, p2 (1-e/c), is the second equilibrium point for
metapopulation size or the extinction threshold. The first point is stable and the second is not.
Second, all the islands are homogenous and equal in size. Finally, there are many individuals in
the species which move between identical islands at random.

Using the law of conservation and Levin’s model we can relate the two and create

dN
=cN ( 1−N )−e N , or fraction of rate of change of occupied islands = colonization rate -
dt
extinction rate. From Levin’s equation, the following autonomous ordinary differential equation
can be integrated to solve the problem (please note that some variables were changed):

dp dp 1 A B
=cp ( 1− p )−mp ∫ =∫ dt  = +
dt cp ( 1−p )−mp p [c ( 1− p )−m] p [c ( 1− p )−m]

1 c
If p=0, then A= and if p=1, then B=
c−m m−c

A B B
∫ ( + )dp=¿ Aln| p|− ln |c ( 1− p ) −m|¿
p [ c (1− p )−m ] c

1 1 ¿ =(t+ C)¿
ln| p|+ ln |c ( 1− p )−m|=t+ C  ln∨ p(c (1−p)−m)∨
c−m c−m ( c−m)

By looking at this final equation, the function of the fraction of occupied islands and
time, it is understood that population will either reach near 0 or 1. If the rate of extinction is zero
the population will reach near 1 and approach complete colonization. If extinction rate is equal to
colonization rate, the population will remain unchanged. If the number of species that is being

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