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## The spatial dynamics of a Zebra mussel model in river environments

 1 Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA 2 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C5S7, Canada

* Corresponding author: Yu Jin

(Dedicated to Professor Sze-Bi Hsu on the occasion of his retirement)

Received  August 2020 Revised  November 2020 Published  December 2020

Fund Project: X.-Q. Zhao's research is supported in part by the NSERC of Canada

Huang et al. [10] developed a hybrid continuous/discrete-time model to describe the persistence and invasion dynamics of Zebra mussels in rivers. They used a net reproductive rate $R_0$ to determine population persistence in a bounded domain and estimated spreading speeds by applying the linear determinacy conjecture and using the formula in [16]. Since the associated solution operator is non-monotonic and non-compact, it is nontrivial to rigorously establish these quantities. In this paper, we analyze the spatial dynamics of this model mathematically. We first solve the parabolic equation and rewrite the model into a fully discrete-time model. In a bounded domain, we show that the spectral radius $\hat{r}$ of the linearized operator can be used to determine population persistence and that the sign of $\hat{r}-1$ is the same as that of $R_0-1$, which confirms that $R_0$ defined in [10] can be used to determine population persistence. In an unbounded domain, we construct two monotonic operators to control the model operator from above and from below and obtain upper and lower bounds of the spreading speeds of the model.

Citation: Yu Jin, Xiang-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020362
##### References:
 [1] D. T. E. Bastviken, N. F. Caraco and J. J. Cole, Experimental measurements of zebra mussel (Dreissena polymorpha) impacts on phytoplankton community composition, Freshwater Biology, 39 (1998), 375-386.  doi: 10.1046/j.1365-2427.1998.00283.x.  Google Scholar [2] E. Brian Davies, Linear Operators and their Spectra,, Cambridge University Press, 2007.  doi: 10.1017/CBO9780511618864.  Google Scholar [3] H. Caswell, Matrix Population Models, Sinauer Associates Inc, 2nd edition, 2000. Google Scholar [4] K. Deimling, Nonlinear Functional Analysis, , Springer-Verlag, Berlin, Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [5] Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations: Maximum Principles and Applications, , Volume 1 (Partial Differential Equations and Application), World Scientific Pub Co Inc, 2006. doi: 10.1142/9789812774446.  Google Scholar [6] J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM Journal on Mathematical Analysis, 46 (2014), 3678-3704.  doi: 10.1137/140953939.  Google Scholar [7] D. W. Garton and W. R. Haag, Seasonal reproductive cycles and settlement patterns of Dreissena polymorpha in western Lake Erie, in Zebra Mussels: Biology, Impacts, and Control, T. F. Nalepa and D. W. Schloesser, eds., Lewis Publishers, Boca Raton, FL, 1993, 111-128. Google Scholar [8] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Search Notes in Mathematics Series, Vol.247, Longman Scientific Technical, Harlow, UK, 1991.  Google Scholar [9] S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for non-monotone integro-difference equations, SIAM Journal on Mathematical Analysis, 40 (2008), 776-789.  doi: 10.1137/070703016.  Google Scholar [10] Q. Huang, H. Wang and M. A. Lewis, A hybrid continudous/discrete-time model for invasion dynamics of zebra mussles in rivers, SIAM Journal on Applied Mathematics, 77 (2017), 854-880.  doi: 10.1137/16M1057826.  Google Scholar [11] X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, Journal of Differential Equations, 231 (2006), 57-77.  doi: 10.1016/j.jde.2006.04.010.  Google Scholar [12] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 61 (2008), 137-138.  doi: 10.1002/cpa.20221.  Google Scholar [13] X. Liang, L. Zhang and X.-Q. Zhao, The principal eigenvalue for degenerate periodic reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 49 (2017), 3603-3636.  doi: 10.1137/16M1108832.  Google Scholar [14] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM Journal on Mathematical Analysis, 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar [15] H. W. Mckenzie, Y. Jin, J. Jacobsen and M. A. Lewis, $R_0$ analysis of a spatiotemporal model for a stream population, SIAM Journal on Applied Dynamical Systems, 11 (2012), 567-596.  doi: 10.1137/100802189.  Google Scholar [16] M. G. Neubert and H. Caswell, Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations, Ecology, 81 (2000), 1613-1628.   Google Scholar [17] A. Ricciardi, F. G. Whoriskey and J. B. Rasmussen, Impact of the Dreissena invasion on native unionid bivalves in the upper St. Lawrence River, The Canadian Journal of Fisheries and Aquatic Sciences, 53 (1996), 1434-1444.  doi: 10.1139/f96-068.  Google Scholar [18] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Analysis, 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar [19] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM Journal on Applied Mathematics, 70 (2009), 188-211.  doi: 10.1137/080732870.  Google Scholar [20] X. Wang and X.-Q. Zhao, Target reproduction numbers for reaction-diffusion population models, Journal of Mathematical Biology, 81 (2020), 625-647.  doi: 10.1007/s00285-020-01523-9.  Google Scholar [21] H. F. Weinberger, Long-time behavior of a class of biological models, SIAM Journal on Mathematical Analysis, 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar [22] H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, Journal of Mathematical Biology, 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.  Google Scholar [23] H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds of spatially periodic integro-difference models for populations with non-monotone recruitment functions, Journal of Mathematical Biology, 57 (2008), 387-411.  doi: 10.1007/s00285-008-0168-0.  Google Scholar [24] P. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, Journal of Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar [25] R. Wu and X.-Q. Zhao, Spatial invasion of a birth pulse populatoin with nonlocal dispersal, SIAM Journal on Applied Mathematics, 79 (2019), 1075-1097.  doi: 10.1137/18M1209805.  Google Scholar [26] X.-Q. Zhao, Dynamical Systems in Population Biology, , second edition, Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

show all references

##### References:
 [1] D. T. E. Bastviken, N. F. Caraco and J. J. Cole, Experimental measurements of zebra mussel (Dreissena polymorpha) impacts on phytoplankton community composition, Freshwater Biology, 39 (1998), 375-386.  doi: 10.1046/j.1365-2427.1998.00283.x.  Google Scholar [2] E. Brian Davies, Linear Operators and their Spectra,, Cambridge University Press, 2007.  doi: 10.1017/CBO9780511618864.  Google Scholar [3] H. Caswell, Matrix Population Models, Sinauer Associates Inc, 2nd edition, 2000. Google Scholar [4] K. Deimling, Nonlinear Functional Analysis, , Springer-Verlag, Berlin, Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar [5] Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations: Maximum Principles and Applications, , Volume 1 (Partial Differential Equations and Application), World Scientific Pub Co Inc, 2006. doi: 10.1142/9789812774446.  Google Scholar [6] J. Fang and X.-Q. Zhao, Traveling waves for monotone semiflows with weak compactness, SIAM Journal on Mathematical Analysis, 46 (2014), 3678-3704.  doi: 10.1137/140953939.  Google Scholar [7] D. W. Garton and W. R. Haag, Seasonal reproductive cycles and settlement patterns of Dreissena polymorpha in western Lake Erie, in Zebra Mussels: Biology, Impacts, and Control, T. F. Nalepa and D. W. Schloesser, eds., Lewis Publishers, Boca Raton, FL, 1993, 111-128. Google Scholar [8] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Search Notes in Mathematics Series, Vol.247, Longman Scientific Technical, Harlow, UK, 1991.  Google Scholar [9] S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for non-monotone integro-difference equations, SIAM Journal on Mathematical Analysis, 40 (2008), 776-789.  doi: 10.1137/070703016.  Google Scholar [10] Q. Huang, H. Wang and M. A. Lewis, A hybrid continudous/discrete-time model for invasion dynamics of zebra mussles in rivers, SIAM Journal on Applied Mathematics, 77 (2017), 854-880.  doi: 10.1137/16M1057826.  Google Scholar [11] X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, Journal of Differential Equations, 231 (2006), 57-77.  doi: 10.1016/j.jde.2006.04.010.  Google Scholar [12] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 61 (2008), 137-138.  doi: 10.1002/cpa.20221.  Google Scholar [13] X. Liang, L. Zhang and X.-Q. Zhao, The principal eigenvalue for degenerate periodic reaction-diffusion systems, SIAM Journal on Mathematical Analysis, 49 (2017), 3603-3636.  doi: 10.1137/16M1108832.  Google Scholar [14] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM Journal on Mathematical Analysis, 37 (2005), 251-275.  doi: 10.1137/S0036141003439173.  Google Scholar [15] H. W. Mckenzie, Y. Jin, J. Jacobsen and M. A. Lewis, $R_0$ analysis of a spatiotemporal model for a stream population, SIAM Journal on Applied Dynamical Systems, 11 (2012), 567-596.  doi: 10.1137/100802189.  Google Scholar [16] M. G. Neubert and H. Caswell, Demography and dispersal: Calculation and sensitivity analysis of invasion speed for structured populations, Ecology, 81 (2000), 1613-1628.   Google Scholar [17] A. Ricciardi, F. G. Whoriskey and J. B. Rasmussen, Impact of the Dreissena invasion on native unionid bivalves in the upper St. Lawrence River, The Canadian Journal of Fisheries and Aquatic Sciences, 53 (1996), 1434-1444.  doi: 10.1139/f96-068.  Google Scholar [18] H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Analysis, 47 (2001), 6169-6179.  doi: 10.1016/S0362-546X(01)00678-2.  Google Scholar [19] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM Journal on Applied Mathematics, 70 (2009), 188-211.  doi: 10.1137/080732870.  Google Scholar [20] X. Wang and X.-Q. Zhao, Target reproduction numbers for reaction-diffusion population models, Journal of Mathematical Biology, 81 (2020), 625-647.  doi: 10.1007/s00285-020-01523-9.  Google Scholar [21] H. F. Weinberger, Long-time behavior of a class of biological models, SIAM Journal on Mathematical Analysis, 13 (1982), 353-396.  doi: 10.1137/0513028.  Google Scholar [22] H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, Journal of Mathematical Biology, 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.  Google Scholar [23] H. F. Weinberger, K. Kawasaki and N. Shigesada, Spreading speeds of spatially periodic integro-difference models for populations with non-monotone recruitment functions, Journal of Mathematical Biology, 57 (2008), 387-411.  doi: 10.1007/s00285-008-0168-0.  Google Scholar [24] P. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, Journal of Differential Equations, 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar [25] R. Wu and X.-Q. Zhao, Spatial invasion of a birth pulse populatoin with nonlocal dispersal, SIAM Journal on Applied Mathematics, 79 (2019), 1075-1097.  doi: 10.1137/18M1209805.  Google Scholar [26] X.-Q. Zhao, Dynamical Systems in Population Biology, , second edition, Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar
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