On the Linearized Stability of Age-structured Multispecies Populations

In [13] Prüß investigated a very general nonlinear age-structured model consisting of n species. He proved the principle of linearized stability, which means that the asymptotic behavior of stationary solutions of the nonlinear system is determined by the spectrum of the linearized operator. In other words the stability is determined by the roots of a complex valued characteristic function as it is claimed in [2] for general physiologically structured population models, as well. More recently Kato [11] proved the principle of linearized stability for more general abstract nonlinear evolution equations of type (d/dt)u(t) +Au(t)= 0 where A is a quasim-accretive operator. In the present paper, we extend the approach first used in [5] then later in [6–8], to actually deduce the characteristic function and show that it can be obtained as a determinant of an n-by-nmatrix as claimed in [2]. We restrict ourselve to the frequently studied case where the vital rate functions depend on the total population quantity, but it should be clear that the method can be extended to more general cases of multispecies structured population models. In [13], Prüß discussed the stability of stationary solutions with trivial components and established stability conditions for a very special two-species system. He also derived conditions for the positivity of the governing linear semigroup and he used it to prove an instability result.