Permanence of sparse catalytic networks.

Some global dynamical properties of catalytic networks, in particular permanence, are closely related with a directed graph representing the differential equation. It can be shown that for every directed graph with a Hamiltonian circuit there is a choice of rate constants such that the system is permanent. On the other hand, one can find properties of the graphs, for example, reducibility or the presence of endpoints, that are incompatible with permanence.

[1]  Christian V. Forst,et al.  Full characterization of a strange attractor: Chaotic dynamics in low-dimensional replicator system , 1991 .

[2]  P. Stadler,et al.  The probability of permanence. , 1993, Mathematical biosciences.

[3]  K. Schmitt,et al.  Permanence and the dynamics of biological systems. , 1992, Mathematical biosciences.

[4]  V. Hutson,et al.  Persistence of species obeying difference equations , 1982 .

[5]  H. I. Freedman,et al.  Mathematical analysis of some three-species food-chain models , 1977 .

[6]  Peter Schuster,et al.  Dynamical systems under constant organiza-tion III: Cooperative and competitive behaviour of hypercy , 1979 .

[7]  P. Schuster,et al.  Permanence and Uninvadability for Deterministic Population Models , 1984 .

[8]  H. I. Freedman,et al.  Persistence definitions and their connections , 1990 .

[9]  W. Jansen,et al.  A permanence theorem for replicator and Lotka-Volterra systems , 1987 .

[10]  Josef Hofbauer,et al.  A general cooperation theorem for hypercycles , 1981 .

[11]  P. Taylor,et al.  Evolutionarily Stable Strategies and Game Dynamics , 1978 .

[12]  P. Schuster,et al.  Dynamical Systems Under Constant Organization II: Homogeneous Growth Functions of Degree $p = 2$ , 1980 .

[13]  D. Friedman EVOLUTIONARY GAMES IN ECONOMICS , 1991 .

[14]  Josef Hofbauer,et al.  Competition and cooperation in catalytic selfreplication , 1981 .

[15]  P. Schuster,et al.  Dynamics of small autocatalytic reaction networks--I. Bifurcations, permanence and exclusion. , 1990, Bulletin of mathematical biology.

[16]  Dmitriĭ Olegovich Logofet,et al.  Matrices and Graphs Stability Problems in Mathematical Ecology , 1993 .