## Abstract

Natural rivers connect to each other to form river networks. The geometric structure of a river network can significantly influence spatial dynamics of populations in the system. We consider a process-oriented model to describe population dynamics in river networks of trees, establish the fundamental theories of the corresponding parabolic problems and elliptic problems, derive the persistence threshold by using the principal eigenvalue of the corresponding eigenvalue problem, and define the net reproductive rate to describe population persistence or extinction. By virtue of numerical simulations, we investigate the effects of hydraulic, physical, and biological factors, especially the structure of the river network, on population persistence.

## Keywords

River network Population persistence Eigenvalue problems Net reproductive rate## Mathematics Subject Classification

35K57 47A75 92D25## Notes

### Acknowledgements

Y.J. gratefully acknowledges NSF Grant DMS-1411703. R.P. is supported by the NSF of China (Nos. 11671175, 11571200), the Priority Academic Program Development of Jiangsu Higher Education Institutions, Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (No. PPZY2015A013), and Qing Lan Project of Jiangsu Province. J.S. is partially supported by NSF grant DMS-1715651. The authors are also very grateful to the anonymous referee for insightful comments that helped improve the manuscript.

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