Monotone and nonmonotone clines with partial panmixia across a geographical barrier

Yantao Wang; Linlin Su

doi:10.3934/dcds.2020056

Article Contents

2020, Volume 40, Issue 6: 4019-4037. Doi: 10.3934/dcds.2020056 open access

This issue Previous Article On space-time periodic solutions of the one-dimensional heat equation Next Article Refined regularity and stabilization properties in a degenerate haptotaxis system

Monotone and nonmonotone clines with partial panmixia across a geographical barrier

Yantao Wang^1,2, and
Linlin Su^2, ,

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
^* Corresponding author: sull@sustech.edu.cn

Received: July 2019

Revised: August 2019

Published: March 2020

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Abstract

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 $.

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Mathematics Subject Classification: 35K57, 92D10, 35A01, 35A02, 35A24.

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Figure 1.

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Figure 2.

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Figure 3.

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Figure 4. 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)

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Figure 5. Same as Fig. 4 except that the straight line $ \beta(p-\bar p) $ is tangent to the graph of $ f(p) $ at $ p_2 $

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Figure 7. 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

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Figure 6. Graph of $ f(p) $ and the straight line $ \beta(p-\bar{p}) $ if $ m = 3 $

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Figure 8. Superimposition of the phase portrait $ C^- $ of (18) and its image under $ \Phi_{\theta_{-}, \theta_{+ }} $ with the phase portraits of (15)

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Figure 9. Nonmonotone cline with $ f(u) = 2u^2(1-u) $

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