A nonlinear ergodic theorem for discrete systems
Consider the discrete-time evolution model of a structured population \(x(t+1)= P(t)x(t)\), where \(x\in {\mathcal R}^ m\) and \(P(t)= \sum_{i=0}^ r a_ i(t){\mathcal L}^ i\) with \({\mathcal L}\) a nonnegative \(m\times m\) matrix and scalar functions \(a_ i(t)\), such that \(0\leq a_ 0(t) \leq a\) and \(0<b\leq a_ i(t)\), \(i=1,\dots,r\), for some uniform \(a\) and \(b\). If \(0\leq w\in {\mathcal R}^ m\) is a weight vector then \(x(t)/ w^ T x(t)\to v^ +\), where \(v^ +\) is a positive eigenvector associated to a strictly positive, dominant, simple eigenvalue of \({\mathcal L}\), normalized by \(w^ T v^ +=1\). This gives a knowledge of the limit distribution of the structure, while the magnitude could be estimated by a one- dimensional model. An application is given to a class of size-structured (intraspecific) competition models with resources in limited supply (zoo plankton, mollusks or anemones).
strictly positive dominant simple eigenvalue
scientific article; zbMATH DE number 554361