Pairs of coupled, first-order difference equations (equivalent to
a single second-order equation) have been investigated in
several contexts ^{4,
44 -
46},
particularly in the study of temperate
zone arthropod prey-predator systems
^{2 -
4,
23,
47}. In these
two-dimensional systems, the complications in the dynamical
behaviour are further compounded by such facts as: (1) even
for analytical functions, there can be truly chaotic behaviour
(as for equations (14) and (15)), corresponding to so-called
"strange attractors"; and (2) two or more different stable states
(for example, a stable point and a stable cycle of period 3) can
occur together for the same parameter values
^{4}. In addition, the
manifestation of these phenomena usually requires less severe
nonlinearities (less steeply humped *F(X)*) than for the
one-dimensional case.

Similar systems of first-order ordinary differential equations,
or two coupled first-order differential equations, have much
simpler dynamical behaviour, made up of stable and unstable
points and limit cycles
^{48}. This is
basically because in continuous
two-dimensional systems the inside and outside of closed curves
can be distinguished; dynamic trajectories cannot cross each
other. The situation becomes qualitatively more complicated
and in many ways analogous to first-order difference equations
when one moves to systems of three or more coupled, first-order
ordinary differential equations (that is, three-dimensional
systems of ordinary differential equations). Scanlon (personal
communication) has argued that chaotic behaviour and
"strange attractors", that is solutions which are neither points
nor periodic orbits
^{48}, are
typical of such systems. Some well
studied examples arise in models for reaction-diffusion systems
in chemistry and biology
^{49}, and in the
models of Lorenz
^{15}
(three dimensions) and Ruelle and Takens
^{40} (four
dimensions)
referred to above. The analysis of these systems is, by virtue
of their higher dimensionality, much less transparent than for
equation (1).

An explicit and rather surprising example of a system which
has recently been studied from this viewpoint is the ordinary
differential equations used in ecology to describe competing
species. For one or two species these systems are very tame:
dynamic trajectories will converge on some stable equilibrium
point (which may represent coexistence, or one or both species
becoming extinct). As Smale
^{50} has
recently shown, however,
for 3 or more species these general equations can, in a certain
reasonable and well-defined sense, be compatible with any
dynamical behaviour. Smale's
^{50} discussion
is generic and
abstract: a specific study of the very peculiar dynamics which
can be exhibited by the familiar Lotka-Volterra equations
once there are 3 competitors is given by May and Leonard
^{51}.