Page 169 Figure 3. Space-time diagram for the coupled map lattice on a hypercubic lattice with k = 5 (i.e., N = 25). For local dynamics f(x) = 1 1.52x2 is adopted, while the coupling strength is set at 0.3. On the corresponding pixel at a given time and element, a bar with a length proportional to (xn(i, j) 0.1) is painted if xn(i, j) > .1. Every fourth time step is plotted from 10,000 to 12,000. Elements are aligned according to its binary representation, that is, 0 = 00000, 1 = 00001,2 = 00010, ..., 63 = 11111. At Stage A, elements split into two clusters 0***** and I****, while they split into *0*** and * 1 *** at Stage B, **0** and ** 1 ** at Stage C, 0***** and 1 **** at Stage D, ****0 and **** I at Stage E, and again into 0***** and I **** at Stage F.

In an ecological system, many species are under strong nonlinear interaction and keep some kind of stability with diversity. This is not easily sustained. We also have to mention that static equilibria with many species are usually unstable, as studied by May in a random network model. Thus, it is interesting to search for a dynamical mechanism to allow for the diversity in a system with interacting population dynamics. Ikegami and the author [11, 17] have studied a population dynamics model with interaction among species, mutation, and mutation of mutation rates. In particular, a model with interaction among hosts and parasites has been studied. Each species is coded by a bit sequence as in section 4, whose fitness has a rugged or flat (neutral) landscape. The interaction between a host and a parasite is assumed to depend on the Hamming distance between their bit sequences.

When the interaction between hosts and parasites is weak, the mutation rates of species decrease with evolution. The dynamics of the whole species is reduced to a direct product of isolated sets of host-parasite population dynamics. When the interaction is strong, on the other hand, mutation rates are sustained at a high level, where many species form a network of population dynamics. This network consists of species connected by single point mutations. Many species are percolated in the gene space. Note that this network is dynamically sustained. The population of each species oscillates chaotically in time. The oscillation is high-dimensional chaos with small

Page 170 positive Lyapunov exponents. ("High-dimensional" here means that the number of positive exponents is large.) If the mutation rate were zero, the dynamics of each species would be essentially disconnected. Then some host-parasite pairs would show strong chaos, while others would show periodic or fixed point dynamics. By sustaining a high mutation rate, chaotic instability is shared by almost all species, leading to weak high-dimensional chaos. By the term weak, we mean that the maximum Lyapunov exponent is close to zero and that the amplitude of oscillation of each species is small. Our system has a tendency to evolve toward such weak, high-dimensional chaos. Here we propose a conjecture that diversity in an evolutionary system with interaction of many replicating units maintains its dynamical stability by forming a weak high-dimensional chaotic state, rather than in a fixed point or in strong chaos. We have coined the term homeochaos for this homeodynamic state.