On an age structured population model with density-dependent dispersals between two patches

Motivated by an age-structured population model over two patches that assumes constant dispersal rates, we derive a modified model that allows density-dependent dispersal, which contains both nonlinear dispersal terms and delayed non-local birth terms resulted from the mobility of the immature individuals between the patches. A biologically meaningful assumption that the dispersal rate during the immature period depends only on the mature population enables us investigate the model theoretically. Well-posedness is confirmed, criteria for existence of a positive equilibrium are obtained, threshold for extinction/persistence is established. Also addressed are a positive invariant set and global convergence of solutions under certain conditions. Although the levels of the density-dependent dispersals play no role in determining extinction/persistence, our numerical results show that they can affect, when the population is persistent, the long term dynamics including the temporal-spatial patterns and the final population sizes.