2 Discrete Population Models
t.
It
is usually given in percentages. Population dynamics depends on how this
per capita growth rate
at
time
t
depends on the actual size of the population.
The simplest assumption is that this rate is constant. If this constant is nega-
tive then this means that there are fewer in each successive generation. If this
negative rate is constant, the obvious consequence is
a
population that dies out
rapidly. If this constant is positive then the equation that governs the dynamics
[Nt+i
-
Nt)
/Nt = r,
where the constant
r > 0
is now the per capita growth rate of the population.
This equation can be written in the form
Nt+i
=
{I + r) Nt
.
(1.1.1)
If we express the number at time
t
+ 2 by the number at time
t
+ 1, and then the
number
at
time
t +
3 by the number
at
time
t
+ 2 and so on, then the number
of the generation
at
time
t
+
n
will be
Nt+n
=
(1
+ rf
Nt
.
As
r >
0, this clearly means that the numbers go to infinity as time increases in-
definitely. If the per capita growth rate is, for example, 2%, then the hundredth
generation numbers
1.02^'^'^
=
7.24 times as much as the original one. In Nature
such exponential growth cannot go on indefinitely because some limiting factor
of the environment, lack of food, oxygen, space etc. or simply the adverse effects
of overcrowding, slows down growth sooner or later. We arrive at
a
more realis-
tic model if we assume that the per capita growth rate is
a
decreasing function
of the abundance of the population, which equals zero when the size of the pop-
ulation reaches the maximum that can be maintained by the environment. The
simplest way to do this is to set the per capita growth rate as
a
linear function
of the quantity with negative slope. In
a
graph of this function, the point where
this line intersects the horizontal axis of the quantity is the maximum amount
the environment can maintain. This value is called the carrying capacity and is
denoted by K> 0. Accordingly, Eq. (1.1.1) is modified to
[Nt+i-Nt)/Nt = r{l-Nt/K) or
Nt+i = Ntil
+
r-rNt/K] .
(1.1.2)
Here
r > 0 is
called the intrinsic growth rate of the population.
It
prevails if
Nt is small; then the per capita growth rate is approximately equal to
r.
If we
look
at
Eq. (1.1.2) we see that in case
Nt
is less than the carrying capacity
K
then Nt+i will be larger than Nt, while if Nt is larger than
K
then Nt+i will be
smaller than Nt- If Nt is equal to
K
then Nt+i will be the same. The variation
of the size of the population according to Eq. (1.1.2) is called logistic dynamics.
Besides
N
= 0 (when there is no population present),
N = K
is its equilibrium
point. For certain values of the intrinsic growth rate this point is stable in the
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