Concepts and problems - Nonequilibrium Ecology - page 21

1.0

1.0

1.0

x

x

t

t

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1.0

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1.0 3.9

r

Figure 1.2. Bifurcation diagram showing population size x (as the proportion of the
carrying capacity 1) plotted against reproductive rate r. Insets show change of
population size of populations with selected reproductive rates plotted against time t.
Note: the population size is at a stable equilibrium with single values of r until a
certain value of r (larger than 3) is reached. Now there are two equilibria, and at even
higher values of r there are 4, 8, and 16. At r ¼ 3.57 fluctuations in population size
become chaotic. Note: x never reaches carrying capacity.

conditions may well arise without the involvement of such disturbances.
May (e.g., 1975 ; and others), in a series of brilliant papers, have demon-
strated that chaotic fluctuations in population size simply arise as the result
of high intrinsic rates of population growth r (Figure 1.2 ) . At low rates,
populations are in equilibrium with a single value of population size,
when values exceed 3, there are at first 2, then 4, 8, and 16 values, until at
about r ¼ 3.57 fluctuations become chaotic, that is, apparently random
but in fact strictly deterministic. Also, fluctuations are extremely sensitive
to initial conditions, i.e., very small differences in initial population size
lead to very large differences in future population fluctuations, making
predictions impossible.
Considering chaos in metapopulations, Rohde and Rohde ( 2001 ) used
''fuzzy chaos'' modelling to show that the degree of chaos is reduced
when subpopulations composing a metapopulation and distinguished
by different reproductive rates are largely segregated. In such meta-
populations, the width of the chaotic band becomes much narrower, and it