| Citation: |
Yuxin Ma, Ruizhi Yang. BIFURCATION ANALYSIS IN A MODIFIED LESLIE-GOWER WITH NONLOCAL COMPETITION AND BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2152-2184. doi: 10.11948/20240415
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BIFURCATION ANALYSIS IN A MODIFIED LESLIE-GOWER WITH NONLOCAL COMPETITION AND BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE
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Abstract
In this paper, a diffusive predator-prey system with nonlocal competition and Beddington-DeAngelis functional response is considered. After analyzing the influence of the selected parameters on the existence, multiplicity and stability of the nonhomogeneous steady-state solution, it is obtained that there is an unstable positive nonconstant steady-state in the neighborhood of the positive constant steady-state. Compared with the system without nonlocal competition, the system with nonlocal competition can generate Hopf-Hopf bifurcation under certain conditions. Through the qualitative analysis, the normal form at the Hopf-Hopf bifurcation singularity is calculated to analyze the different dynamic properties exhibited by the system in different parameter regions. In order to illustrate the feasibility of the obtained results and the dependence of the dynamic behavior on the nonlocal competition, numerical simulations are carried out. Through the numerical simulations, it is further shown that under certain conditions, the nonlocal competition will lead to the generation of stable spatially inhomogeneous periodic solutions and stable spatially inhomogeneous quasi-periodic solutions.
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References
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Yuxin Ma, Ruizhi Yang. BIFURCATION ANALYSIS IN A MODIFIED LESLIE-GOWER WITH NONLOCAL COMPETITION AND BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE[J]. Journal of Applied Analysis & Computation, 2025, 15(4): 2152-2184. doi: 10.11948/20240415
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Figure 1.
Hopf bifurcation curves of models with and without nonlocal competition.
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Figure 2.
Left: the bifurcation regions of the system (5.1) near
$ ({d_2}^*, c_0 ) $ $ (d_2, c) $ $ O_1 $ $ O_7 $ $ \mathcal{R} $ -
Figure 3.
Numerical simulations of (1.4) for parameters
$ (d_2,c)=(0.7707693, $ $ 0.00267237)\in {O_5} $ $ u(x,0)=u^*+0.0045000cos(\frac{3}{5}x) $ $ v(x,0)=v^*+0.0005000cos(\frac{3}{5}x) $ $ O_5 $ -
Figure 4.
Numerical simulations of the system without nonlocal competition for parameters
$ (d_2,c)=(0.7707693,0.00267237)\in {O_5} $ $ u(x,0)=u^*+0.0045000cos(\frac{3}{5}x) $ $ v(x,0)=v^*+0.0005000cos(\frac{3}{5}x) $ $ (u,x,t) $ $ (v,x,t) $ $ O_5 $ -
Figure 5.
Numerical simulations of (1.4) for parameters
$ (d_2,c)=(0.5807693, $ $ 0.1857237) \in {O_6} $ $ u(x,0)=u^*+0.0004000cos(\frac{2}{5}x) $ $ v(x,0)=v^*+0.0002000cos(\frac{2}{5}x) $ $ O_6 $ -
Figure 6.
Numerical simulations of (1.4) for parameters
$ (d_2,c)=(0.5450693, $ $ 0.1862237)\in {O_7} $ $ u(x,0)=u^*+0.0004500cos(\frac{1}{5}x) $ $ v(x,0)=v^*+0.0005000cos(\frac{1}{5}x) $ $ (u,x,t) $ $ (v,x,t) $ $ O_7 $ -
Figure 7.
Numerical simulations of the system without nonlocal competition for parameters in
$ O_6 $ $ O_7 $ $ O_6 $ $ O_7 $