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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

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

Pages: 3267 - 3281, Volume 19, Issue 10, December 2014      doi:10.3934/dcdsb.2014.19.3267

 
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Bingtuan Li - Department of Mathematics, University of Louisville, Louisville, KY 40292, United States (email)
William F. Fagan - Department of Biology, The University of Maryland, College Park, MD 20742, United States (email)
Garrett Otto - Department of Mathematics, University of Louisville, Louisville, KY 40292, United States (email)
Chunwei Wang - Department of Mathematics, University of Louisville, Louisville, KY 40292, United States (email)

Abstract: 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.

Keywords:  Reaction-diffusion system, linear determinacy, spreading speed, traveling wave solution.
Mathematics Subject Classification:  Primary: 34C05, 34D20; Secondary: 92D45.

Received: July 2013;      Revised: February 2014;      Available Online: October 2014.

 References