Asymmetric dispersal and evolutional selection in two-patch system

Yong-Jung Kim; Hyowon Seo; Changwook Yoon

doi:10.3934/dcds.2020043

Article Contents

2020, Volume 40, Issue 6: 3571-3593. Doi: 10.3934/dcds.2020043 open access

This issue Previous Article On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates Next Article Hysteresis-driven pattern formation in reaction-diffusion-ODE systems

Asymmetric dispersal and evolutional selection in two-patch system

1.
Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, Korea
2.
Department of Applied Mathematics and the Institute of Natural Sciences, Kyung Hee University, Yongin, 446-701, South Korea
3.
College of Science & Technology, Korea University, Sejong 30019, Republic of Korea

^* Corresponding author: Changwook Yoon
^* Corresponding author: Changwook Yoon

Received: February 28, 2019

Revised: April 30, 2019

Published: March 2020

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Abstract

Biological organisms leave their habitat when the environment becomes harsh. The essence of a biological dispersal is not in the rate, but in the capability to adjust to the environmental changes. In nature, conditional asymmetric dispersal strategies appear due to the spatial and temporal heterogeneity in the environment. Authors show that such a dispersal strategy is evolutionary selected in the context two-patch problem of Lotka-Volterra competition model. They conclude that, if a conditional asymmetric dispersal strategy is taken, the dispersal is not necessarily disadvantageous even for the case that there is no temporal fluctuation of environment at all.

Keywords:

Mathematics Subject Classification: Primary: 35F50, 92D25, 97M60, 74G30, 34D20; Secondary: 34D05.

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Figure 5. Asymptotic behavior of numerical computation of (31)-(32) with $ \epsilon = 0.02 $. The legends are ordered by the size of asymptotic limits

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Figure 1. A mega patch is a collection of many smaller patches. Dispersal across mega patches are counted in a two-patch system

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Figure 2. Steady state solutions of (14). In the left figure, $ (K_1,K_2) = (2,5) $ and $ \theta_i $'s are monotone. In the right one, $ (K_1,K_2) = (0.2,5) $ and $ \theta_1 $ has maximum at $ d = 0.6613 $

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Figure 3. Diagrams for $ y = -x(1-\frac{x}{K_1}) $ and $ y = x(1-\frac{x}{K_2}) $. Steady states are intersection points with $ y = R $. See (19)

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Figure 4. The graph of motility function $ \gamma(s) $ without uniqueness. We have chosen a piecewise linear motility $ \gamma $ which takes the values in (20). Since $ s_i $ and $ \tilde s_i $ are close to each other, $ \gamma $ increases steeply for $ s\in(s_i,\tilde s_i) $

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Figure 6. Asymptotic behavior ($ h = 0.1, \ell = 0.01, d = 0.02 $). The legends are ordered by the size of asymptotic limits

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