# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020056

## Monotone and nonmonotone clines with partial panmixia across a geographical barrier

 1 Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China 2 Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China

* Corresponding author: sull@sustech.edu.cn

Received  July 2019 Revised  August 2019 Published  October 2019

The number of clines (i.e., nonconstant equilibria) maintained by viability selection, migration, and partial global panmixia in a step-environment with a geographical barrier is investigated. Our results extend the results of T. Nagylaki (2016, Clines with partial panmixia across a geographical barrier, Theor. Popul. Biol. 109) from the no dominance case to arbitrary dominance and to various other selection functions. Unexpectedly, besides the usual monotone clines, we discover nonmonotone clines with both equal and unequal limits at $\pm\infty$.

Citation: Yantao Wang, Linlin Su. Monotone and nonmonotone clines with partial panmixia across a geographical barrier. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020056
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A sketch of the situation in Remark 4(ⅱ); where (A) shows that the straight line $\beta(p-\bar p)$ is tangent to the graph of $f(p)$ at $p_1$, and (B) shows the corresponding phase portraits of (15)
Same as Fig. 4 except that the straight line $\beta(p-\bar p)$ is tangent to the graph of $f(p)$ at $p_2$
Phase portraits of (15) in Lemma 3.10 with $I<II$, $I = II$, and $I>II$, respectively; where $I$ and $II$ stand for the areas enclosed by the graphs of $L(p)$ and $f(p)$ as shown in Fig. 6
Graph of $f(p)$ and the straight line $\beta(p-\bar{p})$ if $m = 3$
Superimposition of the phase portrait $C^-$ of (18) and its image under $\Phi_{\theta_{-}, \theta_{+ }}$ with the phase portraits of (15)
Nonmonotone cline with $f(u) = 2u^2(1-u)$
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