An N-barrier maximum principle for autonomous systems of $n$ species and its application to problems arising from population dynamics

Chiun-Chuan Chen; Li-Chang Hung; Chen-Chih Lai

doi:10.3934/cpaa.2019003

Article Contents

2019, Volume 18, Issue 1: 33-50. Doi: 10.3934/cpaa.2019003

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An N-barrier maximum principle for autonomous systems of $n$ species and its application to problems arising from population dynamics

1.
Department of Mathematics, National Taiwan University, National Center for Theoretical Sciences, Taipei, Taiwan
2.
College of Engineering, National Taiwan University of Science and Technology, Department of Mathematics, National Taiwan University, Taipei, Taiwan
3.
Department of Mathematics, University of British Columbia, Vancouver, Canada

* Corresponding author
* Corresponding author

Received: February 28, 2017

Revised: December 31, 2017

Published: August 2018

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.

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Abstract

The main contribution of the N-barrier maximum principle is that it provides rather generic a priori upper and lower bounds for the linear combination of the components of a vector-valued solution. We show that the N-barrier maximum principle (NBMP, C.-C. Chen and L.-C. Hung (2016)) remains true for $n$ $(n>2)$ species. In addition, a stronger lower bound in NBMP is given by employing an improved tangent line method. As an application of NBMP, we establish a nonexistence result for traveling wave solutions to the four species Lotka-Volterra system.

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Mathematics Subject Classification: Primary: 35B50; Secondary: 35C07, 35K57.

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Figure 2. u₁(x) (red); u₂(x) (green); u₃(x) (blue); u₄(x) (magenta). Figure 2(a) and Figure 2(b) show Type Ⅰ and Type Ⅱ solutions in Theorem 5.1, respectively

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Figure 1. N-barrier for cases (ⅰ), (ⅱ) and (ⅲ) (from the left to the right)

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