Let \(A\) be an infinitesimal generator of a strongly continuous semigroup of positive linear operators in a Banach lattice \(X\) and \(F\) be a nonlinear operator in \(X_+\) satisfying \(F(cx)= cF(x)\), \(x\in X_+\), \(c\geq 0\). The author considers the abstract differential equation \[ Z_x' (t)= AZ_x (t)+ F(Z_x (t)), \qquad t\geq 0, \quad Z_x (0)= x, \] and gives some sufficient conditions under which solutions of this equation are of asynchronous exponential growth on \(U\subset X_+\), that is that there is a real constant \(\lambda\) such that \(Q_x:= \lim_{t\to \infty} e^{-\lambda t} Z_x (t)\) exists, \(Q_x\) is nonzero for all \(x\in U\), and \(R(Q_x)\) lies in a one-dimensional subspace of \(X\). Such type of behavior occurs in models of population growth. The underlying ideas of the proved results are illustrated by some examples.
For the entire collection see [Zbl 0785.00028].