December  2014, 19(10): 3267-3281. doi: 10.3934/dcdsb.2014.19.3267

Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream

1. 

Department of Mathematics, University of Louisville, Louisville, KY 40292, United States, United States

2. 

Department of Biology, The University of Maryland, College Park, MD 20742

Received  July 2013 Revised  February 2014 Published  October 2014

We propose a reaction-advection-diffusion model to study competition between two species in a stream. We divide each species into two compartments, individuals inhabiting the benthos and individuals drifting in the stream. We assume that the growth of and competitive interactions between the populations take place on the benthos and that dispersal occurs in the stream. Our system consists of two linear reaction-advection-diffusion equations and two ordinary differential equations. Here, we provide a thorough study for the corresponding single species model, which has been previously proposed. We next give formulas for the rightward spreading and leftward spreading speed for the model. We show that rightward spreading speed can be characterized as is the slowest speed of a class of traveling wave speeds. We provide sharp conditions for the spreading speeds to be positive. For the two species competition model, we investigate how a species spreads into its competitor's environment. Formulas for the spreading speeds are provided under linear determinacy conditions. We demonstrate that under certain conditions, the invading species can spread upstream. Lastly, we study the existence of traveling wave solutions for the two species competition model.
Citation: Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267
References:
[1]

B. R. Anholt, Density dependence resolves the stream drift paradox,, Ecology, 76 (1995), 2235. doi: 10.2307/1941697. Google Scholar

[2]

S. Flöder and C. Kilroy, Didymosphenia geminata (Protista, Bacillariophyceae) invasion, resistance of native periphyton communities, and implications for dispersal and management,, Biodiversity and Conservation, 18 (2009), 3809. Google Scholar

[3]

A. E. Hershey, J. Pastor, B. J. Peterson and G. W. Kling, Stable isotopes resolve the drift paradox for Baetis mayflies in an arctic river,, Ecology, 74 (1993), 2315. doi: 10.2307/1939584. Google Scholar

[4]

S. Humphries and G. D. Ruxton, Is there really a drift paradox?,, J. Anim. Ecol., 71 (2002), 151. doi: 10.1046/j.0021-8790.2001.00579.x. Google Scholar

[5]

A. C. Krist and C. C. Charles, The invasive New Zealand mudsnail, Potamopyrgus antipodarum, is an effective grazer of algae and altered the assemblage of diatoms more than native grazers,, Hydrobiologia, 694 (2012), 143. doi: 10.1007/s10750-012-1138-5. Google Scholar

[6]

B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Mathematical biosciences, 196 (2005), 82. doi: 10.1016/j.mbs.2005.03.008. Google Scholar

[7]

B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions,, J. Math. Biol., 58 (2009), 323. doi: 10.1007/s00285-008-0175-1. Google Scholar

[8]

B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion systems,, Journal of Differential Equations, 252 (2012), 4842. doi: 10.1016/j.jde.2012.01.018. Google Scholar

[9]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications,, Comm. Pure Appl. Math., 60 (2007), 1. doi: 10.1002/cpa.20154. Google Scholar

[10]

R. Lui, Biological growth and spread modeled by systems of recursions,, I Mathematical theory. Math. Biosci., 93 (1989), 269. doi: 10.1016/0025-5564(89)90026-6. Google Scholar

[11]

F. Lutscher, E. McCauley and M. A. Lewis, Spatial patterns and coexistence mechanisms in systems with unidirectional flow,, Theoretical Population Biology, 71 (2007), 267. doi: 10.1016/j.tpb.2006.11.006. Google Scholar

[12]

F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations,, SIAM Appl. Math., 65 (2005), 1305. doi: 10.1137/S0036139904440400. Google Scholar

[13]

K. Müller, Investigations on the organic drift in North Swedish streams,, Report of the Institute of Freshwater Research, 34 (1954), 133. Google Scholar

[14]

K. Müller, The colonization cycle of freshwater insects,, Oecologia, 53 (1982), 202. Google Scholar

[15]

A. Okubo, P. K. Maini, M. H. Williamson and J. D. Murray, The spatial spread of the grey squirrel in Britain,, Proceedings of the Royal Society of London Series B, 238 (1989), 113. doi: 10.1098/rspb.1989.0070. Google Scholar

[16]

E. Pachepsky, F. Lutscher, R. M. Nisbet and M. A. Lewis, Persistence, spread and the drift paradox,, Theoretical Population Biology, 67 (2005), 61. doi: 10.1016/j.tpb.2004.09.001. Google Scholar

[17]

D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries,, Ecology, 82 (2001), 1219. doi: 10.2307/2679984. Google Scholar

[18]

O. Vasilyeva and F. Lutscher, How flow speed alters competitive outcome in advective environments,, Bull. Math. Biol., 74 (2012), 2935. doi: 10.1007/s11538-012-9792-3. Google Scholar

[19]

O. Vasilyeva and F. Lutscher, Competition of three species in an advective environment,, Nonlinear Anal. Real World Appl., 13 (2012), 1730. doi: 10.1016/j.nonrwa.2011.12.004. Google Scholar

[20]

C. Wang, A Stage-Structured Delayed Reaction-Diffusion Model for Competition Between Two Species,, Ph.D Thesis. University of Louisville, (2013). Google Scholar

[21]

Q. Wang and X. -Q. Zhao, Spreading speed and traveling waves for the diffusive logistic equation with a sedentary compartment,, Dyn. Cont. Discrete Impulsive Syst. (Ser. A), 13 (2006), 231. Google Scholar

[22]

R. F. Waters, The drift of stream insects,, Annu. Rev. Entomol., 17 (1972), 253. doi: 10.1146/annurev.en.17.010172.001345. Google Scholar

[23]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183. doi: 10.1007/s002850200145. Google Scholar

show all references

References:
[1]

B. R. Anholt, Density dependence resolves the stream drift paradox,, Ecology, 76 (1995), 2235. doi: 10.2307/1941697. Google Scholar

[2]

S. Flöder and C. Kilroy, Didymosphenia geminata (Protista, Bacillariophyceae) invasion, resistance of native periphyton communities, and implications for dispersal and management,, Biodiversity and Conservation, 18 (2009), 3809. Google Scholar

[3]

A. E. Hershey, J. Pastor, B. J. Peterson and G. W. Kling, Stable isotopes resolve the drift paradox for Baetis mayflies in an arctic river,, Ecology, 74 (1993), 2315. doi: 10.2307/1939584. Google Scholar

[4]

S. Humphries and G. D. Ruxton, Is there really a drift paradox?,, J. Anim. Ecol., 71 (2002), 151. doi: 10.1046/j.0021-8790.2001.00579.x. Google Scholar

[5]

A. C. Krist and C. C. Charles, The invasive New Zealand mudsnail, Potamopyrgus antipodarum, is an effective grazer of algae and altered the assemblage of diatoms more than native grazers,, Hydrobiologia, 694 (2012), 143. doi: 10.1007/s10750-012-1138-5. Google Scholar

[6]

B. Li, H. F. Weinberger and M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems,, Mathematical biosciences, 196 (2005), 82. doi: 10.1016/j.mbs.2005.03.008. Google Scholar

[7]

B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions,, J. Math. Biol., 58 (2009), 323. doi: 10.1007/s00285-008-0175-1. Google Scholar

[8]

B. Li, Traveling wave solutions in partially degenerate cooperative reaction-diffusion systems,, Journal of Differential Equations, 252 (2012), 4842. doi: 10.1016/j.jde.2012.01.018. Google Scholar

[9]

X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications,, Comm. Pure Appl. Math., 60 (2007), 1. doi: 10.1002/cpa.20154. Google Scholar

[10]

R. Lui, Biological growth and spread modeled by systems of recursions,, I Mathematical theory. Math. Biosci., 93 (1989), 269. doi: 10.1016/0025-5564(89)90026-6. Google Scholar

[11]

F. Lutscher, E. McCauley and M. A. Lewis, Spatial patterns and coexistence mechanisms in systems with unidirectional flow,, Theoretical Population Biology, 71 (2007), 267. doi: 10.1016/j.tpb.2006.11.006. Google Scholar

[12]

F. Lutscher, E. Pachepsky and M. A. Lewis, The effect of dispersal patterns on stream populations,, SIAM Appl. Math., 65 (2005), 1305. doi: 10.1137/S0036139904440400. Google Scholar

[13]

K. Müller, Investigations on the organic drift in North Swedish streams,, Report of the Institute of Freshwater Research, 34 (1954), 133. Google Scholar

[14]

K. Müller, The colonization cycle of freshwater insects,, Oecologia, 53 (1982), 202. Google Scholar

[15]

A. Okubo, P. K. Maini, M. H. Williamson and J. D. Murray, The spatial spread of the grey squirrel in Britain,, Proceedings of the Royal Society of London Series B, 238 (1989), 113. doi: 10.1098/rspb.1989.0070. Google Scholar

[16]

E. Pachepsky, F. Lutscher, R. M. Nisbet and M. A. Lewis, Persistence, spread and the drift paradox,, Theoretical Population Biology, 67 (2005), 61. doi: 10.1016/j.tpb.2004.09.001. Google Scholar

[17]

D. C. Speirs and W. S. C. Gurney, Population persistence in rivers and estuaries,, Ecology, 82 (2001), 1219. doi: 10.2307/2679984. Google Scholar

[18]

O. Vasilyeva and F. Lutscher, How flow speed alters competitive outcome in advective environments,, Bull. Math. Biol., 74 (2012), 2935. doi: 10.1007/s11538-012-9792-3. Google Scholar

[19]

O. Vasilyeva and F. Lutscher, Competition of three species in an advective environment,, Nonlinear Anal. Real World Appl., 13 (2012), 1730. doi: 10.1016/j.nonrwa.2011.12.004. Google Scholar

[20]

C. Wang, A Stage-Structured Delayed Reaction-Diffusion Model for Competition Between Two Species,, Ph.D Thesis. University of Louisville, (2013). Google Scholar

[21]

Q. Wang and X. -Q. Zhao, Spreading speed and traveling waves for the diffusive logistic equation with a sedentary compartment,, Dyn. Cont. Discrete Impulsive Syst. (Ser. A), 13 (2006), 231. Google Scholar

[22]

R. F. Waters, The drift of stream insects,, Annu. Rev. Entomol., 17 (1972), 253. doi: 10.1146/annurev.en.17.010172.001345. Google Scholar

[23]

H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models,, J. Math. Biol., 45 (2002), 183. doi: 10.1007/s002850200145. Google Scholar

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