Math
Sequences & Series Solutions
[8 Points) Sequences used in Ecology
Suppose we want to describe the size of a fish population as Pn where n
is the year after some initial time. A simple model that describes the size
of the fish population with a sequence is the following,
bpn
Pn+1 =
a + Pn
Note that a, b are positive constants that should be chosen based on the
species to be studied and it's environment. In this question assume that
initially, there are fish present so that at year zero, Po > 0.
a) [2 Points) Show that if Pn is convergent, then the only possible limits
are 0 and b - a.
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b) (2 Points) Show that Pn+1 < (6/a)Pn.
c) (2 Points] Use part b) to show that if a > b, then limn00 Pn = 0. Give
a biological interpretation of what this means about the community.
d) (2 Points) Assume a < b and show that if pn < b - a, then Pn is
increasing and 0 < Pn < b - a. Show also that if pn > b - a the Pn is
decreasing and Pn > b-a. Deduce that if a <b, then limno Pn = b-a.
Note that this is referred to as a stable equilibrium as the population tends
to this for almost all initial conditions.
