Short note on the position of bifurcation points for the limiting system arising from the two competing species model

Yukio Kan-on

doi:10.3934/dcdsb.2023022

Article Contents

2023, Volume 28, Issue 12: 6233-6247. Doi: 10.3934/dcdsb.2023022

This issue Previous Article Spatial segregation of multiple species: A singular limit approach Next Article Effects of environmental heterogeneity on species spreading via numerical analysis of some free boundary models

Short note on the position of bifurcation points for the limiting system arising from the two competing species model

Yukio Kan-on

Department of Mathematics, Faculty of Education, Ehime University, Matsuyama, 790-8577, Japan

Dedicated to the memory of Professor Masayasu Mimura

Received: August 2022

Revised: January 2023

Early access: January 31, 2023

Published online: January 31, 2023

The author is supported by JSPS KAKENHI Grant Number JP19K03621.

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Abstract

In this paper, we treat the competition-diffusion system with nonlinear diffusion term, which was proposed by Sigesada, Kawasaki and Teramoto in 1979 to model the segregation of interacting species, and discuss the bifurcation structure of nonnegative solution for the limiting system arising from the competition-diffusion system as the interspecific competition rate tends to $ +\infty $. To do this, we employ the comparison principle and the property of the Bessel function, and study the position of bifurcation points, at which a nonconstant solution bifurcates from the constant solution, for the limiting system.

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Mathematics Subject Classification: Primary: 35B32, 35B09; Secondary: 92B05.

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Figure 1. Global bifurcation structure of solution for (6) when $ a \in (0, 1) $

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Figure 2. Local bifurcation structure of solution for (5) when $ a \in (0, 1) $ and $ \ell \ge 2 $

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Figure 3. Bifurcation points $ \varepsilon = \varepsilon_1^0 $, $ \varepsilon = \varepsilon_1^- $ and $ \varepsilon = \varepsilon_1^+ $ with respect to $ a $

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Figure 4. Global bifurcation structure of solution for (5) when $ a = 1 $ and $ k \in \mathbb{N} $

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Figure 5. Global bifurcation structure of positive solution for (5)

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Figure 6. Global bifurcation structure of solution for (2) when $ \theta > 0 $ is sufficiently large

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References

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