Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions

Abstract

<p style='text-indent:20px;'>We consider the ecological and evolutionary dynamics of a reaction-diffusion-advection model for populations residing in a one-dimensional advective homogeneous environment, with emphasis on the effects of boundary conditions and domain size. We assume that there is a net loss of individuals at the downstream end with rate <inline-formula><tex-math id="M1">\begin{document}$ b \geq 0 $\end{document}</tex-math></inline-formula>, while the no-flux condition is imposed on the upstream end. For the single species model, it is shown that the critical patch size is a decreasing function of the dispersal rate when <inline-formula><tex-math id="M2">\begin{document}$ b \leq 3/2 $\end{document}</tex-math></inline-formula>; whereas it first decreases and then increases when <inline-formula><tex-math id="M3">\begin{document}$ b &gt;3/2 $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>For the two-species competition model, we show that the infinite dispersal rate is evolutionarily stable for <inline-formula><tex-math id="M4">\begin{document}$ b &lt; 3/2 $\end{document}</tex-math></inline-formula> and, when dispersal rates of both species are large, the population with larger dispersal rate always displaces the population with the smaller rate. For certain specific population loss rate <inline-formula><tex-math id="M5">\begin{document}$ b&lt;3/2 $\end{document}</tex-math></inline-formula>, it is also shown that there can be up to three evolutionarily stable strategies. For <inline-formula><tex-math id="M6">\begin{document}$ b&gt;3/2 $\end{document}</tex-math></inline-formula>, it is proved that the infinite random dispersal rate is not evolutionarily stable, and that, for some specific <inline-formula><tex-math id="M7">\begin{document}$ b&gt;3/2 $\end{document}</tex-math></inline-formula>, a finite dispersal rate is evolutionarily stable. Furthermore, for the intermediate domain size, this dispersal rate is optimal in the sense that the species adopting this rate is able to displace its competitor with a similar but different rate. Finally, nine qualitatively different pairwise invasibility plots are obtained by varying the parameter <inline-formula><tex-math id="M8">\begin{document}$ b $\end{document}</tex-math></inline-formula> and the domain size.