Biology Reference
In-Depth Information
several mathematical advantages. First, it decreases the number of model
parameters, as the carrying capacity parameter K in Eq. (1-25) is now
scaled to 1. Second, the results obtained for Eq. (1-26) will be independent
of the units of measurement.
An interesting aspect of Eq. (1-26) is its sensitivity to the initial value x 0
and different values of a. Figure 1-19 illustrates several different
scenarios. The ability of these models to generate oscillating trajectories
is particularly interesting.
The mathematics required for understanding and classifying all the
trajectories of Eq. (1-26) are not trivial. This may seem surprising,
given that our continuous logistic model was simple to understand. It
was not until the latter half of the twentieth century that mathematicians
began to discover the peculiarities of discrete models, such as the
Verhulst model. Decades of interdisciplinary work involving
mathematicians, ecologists, biologists, physicists, and computer
scientists were necessary before some satisfactory answers were found,
and the theory is still far from complete. Equation (1-26) is one of the
seeds from which the mathematical theory of chaos grew. We refer the
reader to Gleick (1987) for the fascinating history behind ''discovering
chaos'' and to Hirsch et al. (2003) for an introduction to the mathematical
theory.
To get a heuristic impression of why oscillations occur for the discrete
models [Eqs. (1-25) and (1-26)], notice that the net per capita growth
rate r
¼
ð
Þ
in Eq. (1-25) uses current population size to predict
growth during the next generation. As changes in population size
occur only at designated, equally spaced time intervals, there is a lag
that may cause overshooting or undershooting, similar to the inertial
effect in physical systems. Mathematically, the following argument
provides quantitative insight. When ax n >
r
P n
1, Eq. (1-26) implies that
j
; that is, the distance between the current level
of the population x n and the maximum level is smaller than the
distance between the current level x n and the level x n þ 1 of the next
generation (see Figure 1-20). Thus, in this case, x n and x n þ 1 will always
be on opposite sides of the maximum level 1, causing oscillatory
behavior.
x n þ 1
x n j > j
1
x n j
Table 1-5 presents several values from Eq. (1-26) of x n , for n
100
with x 0 ¼
0.8, and for different values of a to demonstrate the
dependence of the long-term behavior of the process upon the value
of a. For a
1.5, the system oscillates above and below 1 before settling
into the equilibrium state of 1. For a
¼
2.10, the system oscillates between
two values. As a increases further, the system will oscillate among four
values. This is an example of period doubling.Ifa were to continue
increasing, the system would be driven to chaos.
¼
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