Chaotic dynamics are no more exceptions nor nuisance phenomena in real
ecological systems. Recent computational studies on complex systems
have challenged the conventional view of static equilibrium, and tried
to replace it with that in which more dynamic behaviors are the rules
in real ecological systems. Several useful concepts, such as edge of
chaos (Langton 1991) and homeochaos (Kaneko and Ikegami 1992), also
encourage the idea that ecological systems are wildly seeking their
tentative destinies along their evolutionary scenario.
Most theoretical studies on the topic discussed above assume
that the units of a complex system are chaotic. They often use
well-defined chaos generators, such as discrete logistic equations and
etc. Coupling multiple such generators emerge high-dimensional chaotic
behaviors, keeping chaotic characteristic of each unit, but taming the
overall fluctuation of the system. Doebeli (1995) proposes that
selections against higher moments than mean (variance, skewness,
kurtosis, and etc.) favors such high-dimensional chaotic behaviors. In
real ecological systems, however, it is not easy to verify that the
unit of a complex system is a chaos generator. The unit described with
an appropriate equation often behaves far from chaotic within
reasonable parameter region.
The gap is bridged by studies on gene-for-gene systems. The
gene-for-gene systems are well know ecological phenomena since the
middle of the 20th century. Thompson and Burdon (1992) reported that
population dynamic of a gene-for-gene system is highly chaotic when
the involving genes are a few. On the contrary, the fluctuation is
tamed down when the number of genes and number of patches (units)
increase. Doebeli (1997) propose a discrete prey-predator system in
which both preys and predators are avoiding each other in
one-dimensional parameter space that defines the population dynamics
of Nicholson-Bailey model. He showed that inherit unstable dynamics
of the discrete prey-predator systems were tamed down when the number
of genes of the parameters increased. Toquenaga et al. (1997) also
observed similar phenomena in artificial life systems where individual
fitness was determined by a bit-matching rule between lower and higher
trophic creatures.
The bit-matching rule is one of the simplest mathematical
expressions of gene-for-gene interactions. Gene-for-gene systems, or
systems governed by a bit-matching rule are good starts for studying
evolution of chaotic dynamics in real ecological systems. In this
study I show the scenario of taming chaotic behaviors in gene-for-gene
systems described by a bit-matching rule. I further show the
relationship between the system dynamics and the selection against
higher moments in the course of evolution.