# American Institute of Mathematical Sciences

April  2021, 26(4): 1991-2010. doi: 10.3934/dcdsb.2020362

## 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  April 2021 Early access  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.

Erratum: The name of the second author has been corrected from Xiang-Qiang Zhao to Xiao-Qiang Zhao. We apologize for any inconvenience this may cause.

Citation: Yu Jin, Xiao-Qiang Zhao. The spatial dynamics of a Zebra mussel model in river environments. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 1991-2010. 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. [2] E. Brian Davies, Linear Operators and their Spectra,, Cambridge University Press, 2007.  doi: 10.1017/CBO9780511618864. [3] H. Caswell, Matrix Population Models, Sinauer Associates Inc, 2nd edition, 2000. [4] K. Deimling, Nonlinear Functional Analysis, , Springer-Verlag, Berlin, Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7. [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. [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. [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. [8] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Search Notes in Mathematics Series, Vol.247, Longman Scientific Technical, Harlow, UK, 1991. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [26] X.-Q. Zhao, Dynamical Systems in Population Biology, , second edition, Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.

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##### 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. [2] E. Brian Davies, Linear Operators and their Spectra,, Cambridge University Press, 2007.  doi: 10.1017/CBO9780511618864. [3] H. Caswell, Matrix Population Models, Sinauer Associates Inc, 2nd edition, 2000. [4] K. Deimling, Nonlinear Functional Analysis, , Springer-Verlag, Berlin, Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7. [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. [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. [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. [8] P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Search Notes in Mathematics Series, Vol.247, Longman Scientific Technical, Harlow, UK, 1991. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [26] X.-Q. Zhao, Dynamical Systems in Population Biology, , second edition, Springer, New York, 2017. doi: 10.1007/978-3-319-56433-3.
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