We study the asymptotic behavior of effective degrees of freedom (i.e., the number of species or the diversity) of large-dimensional replicator equations (RE) with quenched antisymmetric random interspecies interactions. It is found that the diversity is maintained at the order of initial diversity N if the random interactions are antisymmetric, while it is known that the diversity rapidly decreases to order 1 via ''extinction'' in the RE with symmetric or asymmetric random quenched interactions. In the antisymmetric case, the global stability of a unique attracting set is contrasted to the symmetric and asymmetric cases which have many attractors in general. The present result suggests a new mechanism of maintaining biodiversity which resolves the ''paradox of ecology''. Large-scale nonlinear systems with complex interactions have a complex landscape of phase space as spin glasses, 1) of which the complex dynamics has recently become a subject of interest in nonlinear and nonequilibrium statistical physics. In particular, evolutionary systems with random interactions have attracted the attention of many theorists 2-4) but analysis of their global behavior still presents many open problems because the dynamics often exhibits complex behavior.In tropical rain forests and in coral reefs, a number of biological species coexist and maintain complex interspecies interactions. Ecologists in the 1960s believed that the stability of such ecosystems is related to the diversity of the species and the complexity of their interactions. 5,6) At the same time, opposite results 4,7) were obtained using mathematical models, which is the ''paradox of ecology (diversity)''. 8) Since the model has a fully random asymmetric interaction matrix where individual matrix elements have no correlation with each other, it has been believed that some ''unknown correlation'' must be embedded in the interactions of real ecosystems. 9,10) Many researchers in the field and theorists have offered solutions for the paradox in specific cases, for example, many species can coexist in a spatially heterogeneous habitat. [11][12][13] No explicit solution, however, has been presented yet in the case of the sympatric coexistence of one hundred species, in the sense of the original question posed by May. 4) In this letter, we report a conclusive condition for interactions, which yields and stabilizes a large complex ecosystem. The condition is random antisymmetry which models the situation in which many species interact with each other in prey-predator or parasitic relationships. This is a simple and biologically relevant hypothesis on interactions because prey-predator relationships are the most common in nature and are accepted to be the most important by ecologists. The system with the random antisymmetric interactions maintains large diversity (number of species) at the same order of initial diversity N, while only order 1 of species survive under the random symmetric or asymmetric interactions. The diversity is typically maintained by chaotic dynami...

We consider the effect of microscopic external noise on the collective motion of a globally coupled map in fully desynchronized states. Without the external noise a macroscopic variable shows high-dimensional chaos distinguishable from random motion. With the increase of external noise intensity, the collective motion is successively simplified. The number of effective degrees of freedom in the collective motion is found to decrease as $-\log{\sigma^2}$ with the external noise variance $\sigma^2$. It is shown how the microscopic noise can suppress the number of degrees of freedom at a macroscopic level.Comment: 9 pages RevTex file and 4 postscript figure

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A new type of asymptotic behavior in a game dynamics system is discovered. The system exhibits behavior which combines chaotic motion and attraction to heteroclinic cycles; the trajectory visits several unstable stationary states repeatedly with an irregular order, and the typical length of the stay near the steady states grows exponentially with the number of visits. The dynamics underlying this irregular motion is analyzed by introducing a dynamically rescaled time variable, and its relation to the low-dimensional chaotic dynamics is thus uncovered. The relation of this irregular motion with a strange type of instability of heteroclinic cycles is also examined.Comment: 7 pages (Revtex) + 4 figures (postscript