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

Chang-Yuan Cheng; Shyan-Shiou Chen; Xingfu Zou; Chang-Yuan Cheng; Shyan-Shiou Chen; Xingfu Zou

doi:10.3934/mbe.2019251

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 4976-4998. doi: 10.3934/mbe.2019251

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

1.
Department of Applied Mathematics, National Pingtung University, Pingtung, ROC 90003, Taiwan
2.
Department of Mathematics, National Taiwan Normal University, Taipei, ROC11677, Taiwan
3.
Department of Applied Mathematics, University of Western Ontario, London, ON, N6A 5B7, Canada

Received: 21 October 2018 Accepted: 13 May 2019 Published: 31 May 2019

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.
- age structure,
- patch,
- density-dependent dispersal,
- delay differential equation,
- uniform persistence,
- global convergence
Citation: Chang-Yuan Cheng, Shyan-Shiou Chen, Xingfu Zou. On an age structured population model with density-dependent dispersals between two patches[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 4976-4998. doi: 10.3934/mbe.2019251

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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.

References

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