| Citation: |
Yiming Tang, Xin Wu, Rong Yuan, Fengjie Geng, Zhaohai Ma. ASYMPTOTIC BEHAVIOR OF A DELAYED NONLOCAL DISPERSAL LOTKA-VOLTERRA COMPETITIVE SYSTEM[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1453-1482. doi: 10.11948/20240262
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ASYMPTOTIC BEHAVIOR OF A DELAYED NONLOCAL DISPERSAL LOTKA-VOLTERRA COMPETITIVE SYSTEM
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Abstract
This paper investigates the asymptotic behavior of a nonlocal dispersal Lotka-Volterra competitive system with time delay across the entire $\mathbb{R}^N$. We establish $L^\infty-$decay estimates of solutions of linear systems converging to equilibria utilizing the Fourier transform method applied to the fundamental solution and the Fourier splitting technique. For the nonlinear time-delayed nonlocal dispersal Lotka-Volterra competitive system, we leverage the results from linear systems and obtain the long-time behavior of solutions of the nonlinear system manifesting as the form of time-exponential. More precisely, we further deduce $L^\infty-$decay estimates of solutions of the original nonlinear system through the properties of convolution and Hölder inequality. Additionally, numerical simulations are presented to bolster the principal theoretical results and illustrate that the time delay impedes species growth.
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Keywords:
- Asymptotic stability /
- nonlocal dispersal /
- competitive system /
- Fourier transform
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References
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Yiming Tang, Xin Wu, Rong Yuan, Fengjie Geng, Zhaohai Ma. ASYMPTOTIC BEHAVIOR OF A DELAYED NONLOCAL DISPERSAL LOTKA-VOLTERRA COMPETITIVE SYSTEM[J]. Journal of Applied Analysis & Computation, 2025, 15(3): 1453-1482. doi: 10.11948/20240262
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Figure 1.
The asymptotic behavior of the solution convergence to
$ E_1^{\ast} $ -
Figure 2.
(a)
$ L^\infty- $ $ \tau=0.5 $ $ 1.5 $ $ 2.5 $ $ 3.5 $ -
Figure 3.
The asymptotic behavior of the solution convergence to
$ E_2^{\ast} $ -
Figure 4.
(a)
$ L^\infty- $ $ \tau=1 $ $ 2 $ $ 3 $ $ 4 $ -
Figure 5.
The asymptotic behavior of the solution convergence to
$ E_3^{\ast} $ -
Figure 6.
(a)
$ L^\infty- $ $ \tau=0.75 $ $ 1.75 $ $ 2.75 $ $ 3.75 $ -
Figure 7.
The space periodic solution.
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Figure 8.
The time periodic solution.