Speed sign of traveling waves to a competitive Lotka-Volterra recursive system with bistable nonlinearity

Yahui Wang; Guo Lin; Chunhua Ou

doi:10.3934/cpaa.2022144

Article Contents

2022, Volume 21, Issue 12: 4287-4310. Doi: 10.3934/cpaa.2022144

This issue Previous Article Existence of solutions for the Finsler-Liouville type equation in two dimensions Next Article Renormalized oscillation theory for regular linear non-Hamiltonian systems

Speed sign of traveling waves to a competitive Lotka-Volterra recursive system with bistable nonlinearity

1.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
2.
Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada

^* Corresponding author
^* Corresponding author

Received: April 2022

Revised: August 2022

Early access: September 2022

Published: December 2022

The first author is supported by the China Scholarship Council (202006180063). Research of the second author was supported by NSF of China (No. 11971213). Research of the third author was supported the NSERC discovery grant of Canada (grant No. RGPIN2016-04709)

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Abstract

Traveling wave propagation is a significant phenomenon observed in population biology. Due to the occurrence of nonlocal effect in recursive systems, a deep understanding of the wavefront in the propagation direction (speed sign) is challenging. In this paper we study the sign of wave speed for bistable traveling waves to a two-species competitive system that biologically models the dynamics of two species in competition for a common resource. By a proper choice of the kernel functions, we transfer our model into a coupled localized differential system and shed a new light on how to determine the wave speed sign. Sufficient conditions with symmetry are obtained on the propagation directions of the wavefronts. This symmetry is further verified in the final analysis and numerical simulations are provided to illustrate our theoretical results.

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Mathematics Subject Classification: Primary: 39A70, 34K16; Secondary: 92D25.

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Figure 1. The evolution of the solution $ u_n(x) $ and $ v_n(x) $ $ (n = 0,50,100,150) $ with $ \alpha_1 = \frac{8}{3} $, $ \alpha_2 = \frac{3}{2} $, $ \rho_1 = \frac{1}{9} $, $ \rho_2 = \frac{6}{5} $, $ r_1 = 2 $ and $ r_2 = 3 $ in (3.1)

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Figure 2. The evolution of the solution $ u_n(x) $ and $ v_n(x) $ $ (n = 0,50,100,150) $ with $ \alpha_1 = \frac{3}{2} $, $ \alpha_2 = \frac{8}{3} $, $ \rho_1 = \frac{6}{5} $, $ \rho_2 = \frac{1}{9} $, $ r_1 = 3 $ and $ r_2 = 2 $ in (3.1)

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Figure 3. The evolution of the solution $ u_n(x) $ and $ v_n(x) $ $ (n = 0,50,100,150) $ with $ \alpha_1 = 2.1 $, $ \alpha_2 = 1.1 $, $ \rho_1 = \rho_2 = 0.6 $ and $ r_1 = r_2 = 3 $ in (3.1)

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Figure 4. The evolution of the solution $ u_n(x) $ and $ v_n(x) $ $ (n = 0,50,100,150) $ with $ \alpha_1 = 1.1 $, $ \alpha_2 = 2.1 $, $ \rho_1 = \rho_2 = 0.6 $ and $ r_1 = r_2 = 3 $ in (3.1)

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