This series of papers is concerned with principal Lyapunov exponents and principal Floquet subspaces of positive random dynamical systems in ordered Banach spaces. The current part of the series focuses on the development of general theory. First, the notions of generalized principal Floquet subspaces, generalized principal Lyapunov exponents, and generalized exponential separations for general positive random dynamical systems in ordered Banach spaces are introduced, which extend the classical notions of principal Floquet subspaces, principal Lyapunov exponents, and exponential separations for strongly positive deterministic systems in strongly ordered Banach to general positive random dynamical systems in ordered Banach spaces. Under some quite general assumptions, it is then shown that a positive random dynamical system in an ordered Banach space admits a family of generalized principal Floquet subspaces, a generalized principal Lyapunov exponent, and a generalized exponential separation. We will consider in the forthcoming part(s) the applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations, including random Leslie matrix models, random cooperative systems of ordinary differential equations, and random parabolic equations.

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The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C 1 submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the C r (r 1) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C 1 but not neatly embedded, the other in which S is not of class C 1 .