Advertisement

Journal of Nonlinear Science

, Volume 29, Issue 6, pp 2501–2545 | Cite as

Population Dynamics in River Networks

  • Yu JinEmail author
  • Rui Peng
  • Junping Shi
Research

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.

References

  1. Anderson, K.E., Paul, A.J., Mccauley, E., Jackson, L.J., Post, J.R., Nisbet, R.M.: Instream flow needs in streams and rivers: the importance of understanding ecological dynamics. Front. Ecol. Environ. 4, 309–318 (2006)Google Scholar
  2. Arendt, W., Dier, D., Fijavz̆, M.K.: Diffusion in networks with time-dependent transmission conditions. Appl. Math. Optim. 69, 315–336 (2014)MathSciNetzbMATHGoogle Scholar
  3. Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs. Mathematical Surveys and Monographs, vol. 186. American Mathematical Society, Providence (2013)zbMATHGoogle Scholar
  4. Bertuzzo, E., Casagrandi, R., Gatto, M., Rodriguez-Iturbe, I., Rinaldo, A.: On spatially explicit models of cholera epidemics. J. R. Soc. Interface 7, 321–333 (2010)Google Scholar
  5. Chaudhry, M.H.: Open-Channel Flow. Prentice-Hall, Englewood Cliffs (1993)zbMATHGoogle Scholar
  6. Cuddington, K., Yodzis, P.: Predator-prey dynamics and movement in fractal environments. Am. Nat. 160, 119–134 (2002)Google Scholar
  7. Du, Y.: Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Maximum Principles and Applications, vol. 1. World Scientific Publishing Co. Pte. Ltd., Singapore (2006)zbMATHGoogle Scholar
  8. Du, Y., Lou, B., Peng, R., Zhou, M.: The fisher-KPP equation over simple graphs: Varied persistence states in river networks (2018). arXiv:1809.06961
  9. Du, K., Peng, R., Sun, N.: The role of protection zone on species spreading governed by a reaction–diffusion model with strong Allee effect. J. Differ. Equ. 266, 7327–7356 (2019)MathSciNetzbMATHGoogle Scholar
  10. Fagan, W.F.: Connectivity, fragmentation, and extinction risk in dendritic metapopulations. Ecology 83(12), 3243–3249 (2002)Google Scholar
  11. Fijavz̆, M.K., Mugnolo, D., Sikolya, E.: Variational and semigroup methods for waves and diffusion in networks. Appl. Math. Optim. 55, 219–240 (2007)MathSciNetzbMATHGoogle Scholar
  12. Goldberg, E.E., Lynch, H.J., Neubert, M.G., Fagan, W.F.: Effects of branching spatial structure and life history on the asymptotic growth rate of a population. Theor. Ecol. 3, 137–152 (2010)Google Scholar
  13. Grant, E.H.C., Lowe, W.H., Fagan, W.F.: Living in the branches: population dynamics and ecological processes in dendritic networks. Ecol. Lett. 10, 165–175 (2007)Google Scholar
  14. Grant, E.H.C., Nichols, J.D., Lowe, W.H., Fagan, W.F.: Use of multiple dispersal pathways facilitates amphibian persistence in stream networks. Proc. Natl. Acad. Sci. USA 107, 6936–6940 (2010)Google Scholar
  15. Huang, Q., Jin, Y., Lewis, M.A.: \(R_0\) analysis of a Benthic-drift model for a stream population. SIAM J. Appl. Dyn. Syst. 15(1), 287–321 (2016)MathSciNetzbMATHGoogle Scholar
  16. Jin, Y., Lewis, M.A.: Seasonal influences on population spread and persistence in streams: critical domain size. SIAM J. Appl. Math. 71(4), 1241–1262 (2011)MathSciNetzbMATHGoogle Scholar
  17. Jin, Y., Lewis, M.A.: Seasonal invasion dynamics in a spatially heterogeneous river with fluctuating flows. Bull. Math. Biol. 76(7), 1522–1565 (2014)MathSciNetzbMATHGoogle Scholar
  18. Jin, Y., Lutscher, F., Pei, Y.: Meandering rivers: how important is lateral variability for species persistence? Bull. Math. Biol. 79(12), 2954–2985 (2017)MathSciNetzbMATHGoogle Scholar
  19. Ladyzenskaja, O.A., Solonnikov, V.A., Uralceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type. AMS, Providence (1968)Google Scholar
  20. Lam, K.Y., Lou, Y., Lutscher, F.: The emergence of range limits in advective environments. SIAM J. Appl. Math. 76(2), 641–662 (2016)MathSciNetzbMATHGoogle Scholar
  21. Lumer, G.: Connecting of local operators and evolution equations on networks. In: Proceedings of the Colloquium on Convexity, Copenhagen, 1979. Potential theory. Copenhagen 1979, volume 787 of Lecture Notes in Mathematics, pp. 219–234. Springer, Berlin (1980)Google Scholar
  22. Lutscher, F., Pachepsky, E., Lewis, M.A.: The effect of dispersal patterns on stream populations. SIAM Rev. 47(4), 749–772 (2005)MathSciNetzbMATHGoogle Scholar
  23. Lutscher, F., Lewis, M.A., McCauley, E.: Effects of heterogeneity on spread and persistence in rivers. Bull. Math. Biol. 68, 2129–2160 (2006)MathSciNetzbMATHGoogle Scholar
  24. Mari, L., Casagrandi, R., Bertuzzo, E., Rinaldo, A., Gatto, M.: Metapopulation persistence and species spread in river networks. Ecol. Lett. 17, 426–434 (2014)Google Scholar
  25. Mckenzie, H.W., Jin, Y., Jacobsen, J., Lewis, M.A.: \(R_0\) analysis of a spatiotemporal model for a stream population. SIAM J. Appl. Dyn. Syst. 11(2), 567–596 (2012)MathSciNetzbMATHGoogle Scholar
  26. Mugnolo, D.: Gaussian estimates for a heat equation on a network. Netw. Heterog. Media 2, 55–79 (2012)MathSciNetzbMATHGoogle Scholar
  27. Müller, K.: Investigations on the organic drift in North Swedish streams. Rep. Inst. Freshw. Res. Drottningholm 35, 133–148 (1954)Google Scholar
  28. Müller, K.: The colonization cycle of freshwater insects. Oecologia 52, 202–207 (1982)Google Scholar
  29. Pachepsky, E., Lutscher, F., Nisbet, R.M., Lewis, M.A.: Persistence, spread and the drift paradox. Theor. Popul. Biol. 67(1), 61–73 (2005)zbMATHGoogle Scholar
  30. Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)zbMATHGoogle Scholar
  31. Peterson, E.E., Ver Hoef, J.M., Isaak, D.J., Falke, J.A., Fortin, M.J., Jordan, C.E., McNyset, K., Monestiez, P., Ruesch, A.S., Sengupta, A., Som, N., Steel, E.A., Theobald, D.M., Torgersen, C.E., Wenger, S.J.: Modelling dendritic ecological networks in space: an integrated network perspective. Ecol. Lett. 16, 707–719 (2013)Google Scholar
  32. Protter, M.H., Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice Hall, Englewood Cliffs (1967)zbMATHGoogle Scholar
  33. Ramirez, J.M.: Population persistence under advection-diffusion in river networks. J. Math. Biol. 65(5), 919–942 (2012)MathSciNetzbMATHGoogle Scholar
  34. Sarhad, J.J., Anderson, K.E.: Modeling population persistence in continuous aquatic networks using metric graphs. Fundam. Appl. Limnol. 186, 135–152 (2015)Google Scholar
  35. Sarhad, J.J., Carlson, R., Anderson, K.E.: Population persistence in river networks. J. Math. Biol. 69(2), 401–448 (2014)MathSciNetzbMATHGoogle Scholar
  36. Solonnikov, V.A.: On boundary value problems for linear parabolic systems of differential equations of general form. In: Boundary Value Problems of Mathematical Physics, Volume 83 of Trudy Mathematicheskogo instituta im. V. A. Steklova RAN, pp. 3–163. Proceedings of the Steklov Institute of Mathematics, vol. 83, pp. 1–184 (1965)Google Scholar
  37. Speirs, D.C., Gurney, W.S.C.: Population persistence in rivers and estuaries. Ecology 82(5), 1219–1237 (2001)Google Scholar
  38. Thieme, H.: Spectral bound and reproductive number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math. 70, 188–211 (2009)MathSciNetzbMATHGoogle Scholar
  39. Vasilyeva, O.: Population dynamics in river networks: analysis of steady states. J. Math. Biol. (2019).  https://doi.org/10.1007/s00285-019-01350-7 MathSciNetCrossRefzbMATHGoogle Scholar
  40. von Below, J.: Classical solvability of linear parabolic equations on networks. J. Differ. Equ. 72(2), 316–337 (1988a)MathSciNetzbMATHGoogle Scholar
  41. von Below, J.: Sturm-Liouville eigenvalue problems on networks. Math. Methods Appl. Sci. 10(4), 383–395 (1988b)MathSciNetzbMATHGoogle Scholar
  42. von Below, J.: A maximum principle for semilinear parabolic network equations. In: Differential Equations with Applications in Biology, Physics, and Engineering (Leibnitz, 1989), volume 133 of Lecture Notes in Pure and Applied Mathematics, pp. 37–45. Dekker, New York (1991)Google Scholar
  43. von Below, J.: Nonlinear and dynamical node transition in network diffusion problems. In: Evolution equations, control theory, and biomathematics (Han sur Lesse, 1991), Volume 155 of Lecture Notes in Pure and Applied Mathematics, pp. 1–10. Dekker, New York (1994)Google Scholar
  44. von Below, J., Lubary, J.A.: Instability of stationary solutions of reaction–diffusion–equations on graphs. Results Math. 68(1–2), 171–201 (2015)MathSciNetzbMATHGoogle Scholar
  45. Wang, W., Zhao, X.-Q.: Basic reproduction numbers for reaction-diffusion epidemic models. SIAM J. Appl. Dyn. Syst. 11(4), 1652–1673 (2012)MathSciNetzbMATHGoogle Scholar
  46. Yanagida, E.: Stability of nonconstant steady states in reaction–diffusion systems on graphs. Jpn. J. Ind. Appl. Math. 18, 25–42 (2001)MathSciNetzbMATHGoogle Scholar
  47. Ye, Q., Li, Z., Wang, M., Wu, Y.: An Introduction to Reaction–Diffusion Equations, 2nd edn. Science Press, Beijing (2011)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Nebraska-LincolnLincolnUSA
  2. 2.Department of MathematicsJiangsu Normal UniversityXuzhouChina
  3. 3.Department of MathematicsCollege of William and MaryWilliamsburgUSA

Personalised recommendations