Persistence and positive steady states of a two-stage structured population model with mixed dispersals

We study a two-stage structured population model for which the juveniles diffuse purely by random walk while the adults exhibit long range dispersal. Questions on the persistence or extinction of the species are examined. It is shown that the population eventually dies out if the principal spectrum point ${\lambda }_p$ of the linearized system at the trivial solution is nonpositive. However, the species persists if ${\lambda }_p>0$. Moreover, at least one positive steady state exists when ${\lambda }_p>0$. The uniqueness and global stability of the positive steady-state solution is obtained under some special cases. We also establish a sup/inf characterization of ${\lambda }_p$.