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

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    * Corresponding author 

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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  • 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.

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


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

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

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

    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} $

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

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

    Figure 7.  The spreading of $ u $ occurs

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

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

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