Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats

Chiun-Chuan Chen; Yin-Liang Huang; Li-Chang Hung; Chang-Hong Wu

doi:10.3934/cpaa.2020001

Article Contents

2020, Volume 19, Issue 1: 1-18. Doi: 10.3934/cpaa.2020001

This issue Previous Article Next Article Entire subsolutions of Monge-Ampère type equations

Semi-exact solutions and pulsating fronts for Lotka-Volterra systems of two competing species in spatially periodic habitats

1.
Department of Mathematics, National Taiwan University, National Center for Theoretical Sciences, Taipei, Taiwan
2.
Department of Applied Mathematics, National University of Tainan, Tainan, Taiwan
3.
Department of Mathematics, National Taiwan University, Taipei, Taiwan
4.
Department of Applied Mathematics, National Chiao Tung University, Taiwan

^* Corresponding author
^* Corresponding author

Received: November 30, 2017

Revised: February 28, 2019

Published: July 2019

The research of C.-C. Chen is partly supported by the grant 102-2115-M-002-011-MY3 of Ministry of Science and Technology, Taiwan. The research of L.-C. Hung is partly supported by the grant 104EFA0101550 of Ministry of Science and Technology, Taiwan. The research of C.-H. Wu is partly supported by the grant MOST 105-2628-M-024-001-MY2 and 107-2636-M-024-001 of Ministry of Science and Technology, Taiwan and National Center for Theoretical Science (NCTS)

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Abstract

We are concerned with the coexistence states of the diffusive Lotka-Volterra system of two competing species when the growth rates of the two species depend periodically on the spacial variable. For the one-dimensional problem, we employ the generalized Jacobi elliptic function method to find semi-exact solutions under certain conditions on the parameters. In addition, we use the sine function to construct a pair of upper and lower solutions and obtain a solution of the above-mentioned system. Next, we provide a sufficient condition for the existence of pulsating fronts connecting two semi-trivial states by applying the abstract theory regarding monotone semiflows. Some numerical simulations are also included.

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Mathematics Subject Classification: Primary: 35K57; Secondary: 35C07, 83C15.

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Figure 1. The profile of $ \phi(x) $

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Figure 2. The profiles of $ u $ (red), $ v $ (green), $ m_1 $ (blue) and $ m_2 $ (cyan)

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Figure 3. The profiles of $ u $ (red), $ v $ (green), $ m_1 $ (blue) and $ m_2 $ (cyan)

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Figure 4. $ \frac{c_{22}}{c_{12}} = \frac{1}{3} $ (red), $ \frac{m_2(x)}{m_1(x)} $ (green) and $ \frac{c_{21}}{c_{11}} = 2 $ (blue), where $ \frac{m_2(x)}{m_1(x)} = \frac{18 \left(20+3 \sqrt{3}\right) \phi (x)+8 \left(26+9\sqrt{17}\right)}{27 \left(24+\sqrt{3}+2 \sqrt{51}\right) \phi(x)+36 \left(18+\sqrt{17}\right)} $ and $ \phi = \phi(x) $ is the solution of (17) with $ \phi'(0) = \frac{2\sqrt{2}}{9} $

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Figure 5. The profile of the long time behavior of the solution with initial data in (42)

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Figure 6. The profile of the long time behavior of the solution with initial data in (43)

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Figure 7. The spreading of $ u $ occurs

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Figure 8. The profile of the long time behavior of Example 2 shows $ u $ invades $ v $ eventually

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Figure 9. The profile of the long time behavior of Example 3 with $ h = 0.01 $

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