Abstract
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.
Highlights
Among the various features for population dynamics of a single species are the heterogeneity of different habitats for the species and the age structure of species
Combining the two aspects in [2] and [3] and assuming constant dispersal rates between patches, So et al [4] derived a system of delayed differential equations (DDEs) to describe the adult populations of a single species living on two patches; and by analyzing the derived model, they were able to illustrate some effect of the immature death rate on global dynamics
The improved model turns out to be a system of delay differential equation with spatial non-local birth terms resulted from the dispersals of the immature individual
Summary
Among the various features for population dynamics of a single species are the heterogeneity of different habitats for the species and the age structure of species. Combining the two aspects in [2] and [3] and assuming constant dispersal rates between patches, So et al [4] derived a system of delayed differential equations (DDEs) to describe the adult populations of a single species living on two patches; and by analyzing the derived model, they were able to illustrate some effect of the immature death rate on global dynamics. Xu [6] investigated a population model in two identical patches and attacked this problem by seeking an attractor for the system and applying the monotone dynamical system theory; and for a species living in a general patchy environment with heterogeneity between two patches, Terry [15] examined the population dynamics of the model with impulsive culling of the adults by the comparison with certain linear DDE system.
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