| Citation: |
Hongyu Chen, Chunrui Zhang. BIFURCATIONS AND HYDRA EFFECTS IN A REACTION-DIFFUSION PREDATOR-PREY MODEL WITH HOLLING Ⅱ FUNCTIONAL RESPONSE[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 424-444. doi: 10.11948/20220221
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BIFURCATIONS AND HYDRA EFFECTS IN A REACTION-DIFFUSION PREDATOR-PREY MODEL WITH HOLLING Ⅱ FUNCTIONAL RESPONSE
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Abstract
In this paper, through bifurcation analysis and numerical simulations, we consider a reaction-diffusion predator-prey model with Holling Ⅱ functional response to analyze the existence of hydra effect and the relationship between mortality independent of predator density and different steady-state solutions of the system. The hydra effect, which is a paradoxical result in both theoretical and applied ecology, refers to the phenomenon in which an increase in population mortality enhances its own population size. We investigate the existence of the hydra effect when the positive equilibrium point is locally asymptotically stable and Turing unstable. Meanwhile, numerical simulations verify the existence of the hydra effect when the one-dimensional reaction-diffusion system has a spatially inhomogeneous steady-state solution. In addition, we introduce the existence of the Turing bifurcation, the Hopf bifurcation, and the Turing-Hopf bifurcation with the parameters $ d_{2} $ and $ m_{C} $, respectively, as well as the normal form for the Turing-Hopf bifurcation. Based on the obtained normal form, we analyze the complex spatio-temporal dynamics near the Turing-Hopf bifurcation point. Finally, the numerical simulations are carried out to corroborate the obtained theoretical results.
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Keywords:
- Predator-prey system /
- hydra effect /
- Turing instability /
- Turing-Hopf bifurcation /
- normal form
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References
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Hongyu Chen, Chunrui Zhang. BIFURCATIONS AND HYDRA EFFECTS IN A REACTION-DIFFUSION PREDATOR-PREY MODEL WITH HOLLING Ⅱ FUNCTIONAL RESPONSE[J]. Journal of Applied Analysis & Computation, 2023, 13(1): 424-444. doi: 10.11948/20220221
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Figure 1.
The existence of the hydra effect is depicted when the system is in a stable state. The parameters value are
$ r=5.01, K=6.01, a_{CR}=1.01, Th_{CR}=1, ef_{RC}=1, q_{C}=0.01 $ -
Figure 2.
Spatial distribution of predator population size when the other parameters are fixed at values:
$ d_{1}=0.01, d_{2}=2.51, r=0.25, K=14.95, a_{CR}=1.01, Th_{CR}=1, \alpha=2.61, \beta=0.61, l=1, x\in(0,2\pi) $ $ m_{C} $ -
Figure 3.
Turing-Hopf bifurcation point
$ (m_{C}^{*},d_{2}^{n_{*}}) $ $ m_{C}-d_{2} $ -
Figure 4.
The bifurcation set
$ (a) $ $ (b) $ -
Figure 5.
When
$ (\sigma_{1},\sigma_{2})=(1,0) $ $ D_{1} $ $ E_{*}(u_{*},v_{*})=(0.9483,0.4539) $ $ u(x,0)=0.94, v(x,0)=0.45 $ -
Figure 6.
When
$ (\sigma_{1},\sigma_{2})=(1,1) $ $ D_{2} $ $ E_{*}(u_{*},v_{*})=(0.9483,0.4539) $ $ cos(2x) $ -
Figure 7.
When
$ (\sigma_{1},\sigma_{2})=(-0.1,0) $ $ D_{3} $ $ E_{*}(u_{*},v_{*})=(0.9483,0.4539) $ $ cos(2x) $ -
Figure 8.
When
$ (\sigma_{1},\sigma_{2})=(-0.1,-0.04) $ $ D_{4} $ $ E_{*}(u_{*},v_{*})=(0.9483,0.4539) $ $ u(x,0)=0.948292462+sin(2x), v(x,0)=0.453859373 $ $ u $ $ v $ $ u $ $ v $ -
Figure 9.
$ (a) $ $ (b) $ $ x-t $ -
Figure 10.
When
$ (\sigma_{1},\sigma_{2})=(-0.1,-0.1) $ $ D_{5} $ $ E_{*}(u_{*},v_{*})=(0.9483,0.4539) $ $ u(x,0)=0.948292462+sin(3x), v(x,0)=0.453859373 $ $ u $ $ v $ $ u $ $ v $ -
Figure 11.
$ (a) $ $ (b) $ $ x-t $ -
Figure 12.
When
$ (\sigma_{1},\sigma_{2})=(-0.1,-0.2) $ $ D_{6} $ $ E_{*}(u_{*},v_{*})=(0.9483,0.4539) $ $ u(x,0)=0.948292462+0.1, v(x,0)=0.453859373-0.1 $