Geoscience Reference
In-Depth Information
represents natural factors such as the rate at which births and deaths occur, avail-
ability of food, or threat of predation. This equation is used iteratively or, in other
words, the answer from one time step (
t
) becomes the input for the next step (
t
+
1). This population model is simple because it uses very straightforward mathemat-
ics, but it can capture complex population dynamics that are highly sensitive to the
value of
representing the rate of growth. Figure 5.1 illustrates these complex
dynamics by showing a single end-point or 'attractor' for thousands of different
iterations. The value of each system end point, given by the y-axis, varies widely
with changes in
α
along the
x
-axis.
Systems often have feedback. In the population model, the use of iteration creates
feedback between the present population
X
t
and future population
X
t+1
. When
α
is
between one and three, for example, negative feedback causes the population to settle
over time to a single value of 1
α
. This value is an example of a stable attractor,
or the value within a mathematical system towards which a variable inevitably tends
or reaches. When
−
1/
α
α
=
3, for example, the population value settles down to become
1
ranges between zero and one.
In this case, the population dies out over time due to negative feedback. The popula-
tion can also expand endlessly via positive feedback when
−
3
or
X
=
3
. Another stable attractor occurs when
α
is greater than four.
Deterministic systems can be both sensitive, in that changes in their overall
behaviour may occur as a result of small changes or perturbations in one of their
α
Figure 5.1
Bifurcations in the system attractor as a function of alpha in the popula-
tion growth model.
