# American Institute of Mathematical Sciences

January  2021, 26(1): 367-400. doi: 10.3934/dcdsb.2020283

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

 1 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA 2 Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

* Corresponding author: lam.184@math.ohio-state.edu

Received  March 2020 Revised  August 2020 Published  September 2020

Fund Project: The first author is supported by NSF grant DMS-1818769. The second and third authors are supported by NSF grant DMS-1853561

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 $b \geq 0$, 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 $b \leq 3/2$; whereas it first decreases and then increases when $b >3/2$.

For the two-species competition model, we show that the infinite dispersal rate is evolutionarily stable for $b < 3/2$ 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 $b<3/2$, it is also shown that there can be up to three evolutionarily stable strategies. For $b>3/2$, it is proved that the infinite random dispersal rate is not evolutionarily stable, and that, for some specific $b>3/2$, 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 $b$ and the domain size.

Citation: Wenrui Hao, King-Yeung Lam, Yuan Lou. Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 367-400. doi: 10.3934/dcdsb.2020283
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##### References:
Normal form diagrams of $\ell^*$ against $\mu$ for different cases of $b$, as the illustrations of Proposition 1.3. The value of $\mu_{\min}$ is given in (1.4)
The above normal form diagrams summarize the analytical results from Theorems 1.7, 1.9 and 1.11. They illustrate the transition of 9 qualitatively different pairwise invasibility plots, i.e. nullclines of $\Lambda(\xi,\tau)$, as parameters $b$ and $\ell$ vary. For a pair of strategies $(\xi,\tau)$, if it lies on a region marked with a plus (resp. minus) sign, then it indicates that the species with strategy $\tau$ can (resp. cannot) invade the species with strategy $\xi$ when rare. A red circle stands for an ESS and CvSS; a red square stands for an ESS and non-CvSS; a green square stands for a non-ESS and non-CvSS
Numerical simulation of the pairwise invasibility plots (i.e. the nullclines of $\Lambda(\xi,\tau)$) for parameter values $b = 1.49, 1.5, 1.51$ and $\ell = 10, 20, 50$. The horizontal axis is $\xi$ and the vertical axis is $\tau$
Signs of the second derivatives of $\Lambda$ at $(\xi, \tau) = (0,0)$ when $b = \frac{3}{2}$
 $\Lambda_{\tau\tau}(0,0)$ $(\Lambda_{\tau\tau} + \Lambda_{\tau\xi})(0,0)$ $\Lambda_{\xi\xi}(0,0)$ $\ell> {51}/{2}$ $<0$ (ESS) $>0$ (not CvSS) $<0$ ${27}/{2}<\ell< {51}/{2}$ $<0$ (ESS) $<0$ (CvSS) $<0$ ${3}/{2}<\ell< {27}/{2}$ $<0$ (ESS) $<0$ (CvSS) $>0$
 $\Lambda_{\tau\tau}(0,0)$ $(\Lambda_{\tau\tau} + \Lambda_{\tau\xi})(0,0)$ $\Lambda_{\xi\xi}(0,0)$ $\ell> {51}/{2}$ $<0$ (ESS) $>0$ (not CvSS) $<0$ ${27}/{2}<\ell< {51}/{2}$ $<0$ (ESS) $<0$ (CvSS) $<0$ ${3}/{2}<\ell< {27}/{2}$ $<0$ (ESS) $<0$ (CvSS) $>0$
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