# American Institute of Mathematical Sciences

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January  2019, 18(1): 33-50. doi: 10.3934/cpaa.2019003

## 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

Received  March 2017 Revised  January 2018 Published  August 2018

Fund Project: 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 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.

Citation: Chiun-Chuan Chen, Li-Chang Hung, Chen-Chih Lai. An N-barrier maximum principle for autonomous systems of $n$ species and its application to problems arising from population dynamics. Communications on Pure & Applied Analysis, 2019, 18 (1) : 33-50. doi: 10.3934/cpaa.2019003
##### References:

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##### References:
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
N-barrier for cases (ⅰ), (ⅱ) and (ⅲ) (from the left to the right)
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