## Definition

Thanks to the preceding conceptual dissection of population dynamical processes, it is now possible to arrive at a general fitness definition by falling back on a mathematical result that is easy to state but very difficult to prove. Luckily, a suite of mathematicians have done the latter already and the result is known as the multiplicative ergodic theorem. This theorem deals with general ergodic sequences of matrices. The matrices in [1] are restricted to having nonnegative components, and a similar result applies to the solutions of [2]. From this combination the following fact emerges: under some technical conditions, in ergodic environments, lim 1ln(n(t)) = p [3]

t!1 t with p a unique real number, which mathematicians call the dominant Lyapunov exponent and which here will be called invasion fitness (on the presupposition that E is generated as the output of some community dynamics).

The technical conditions are somewhat involved, but are generally fulfilled in biological applications. For example, it suffices that all the components of the matrices A are smaller than some uniform upper bound, and that for any initial condition after sufficient time all components of N are bound to be positive.

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