| Citation: |
Xun Cao, Weihua Jiang. INTERACTIONS OF TURING AND HOPF BIFURCATIONS IN AN ADDITIONAL FOOD PROVIDED DIFFUSIVE PREDATOR-PREY MODEL[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1277-1304. doi: 10.11948/2156-907X.20180224
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INTERACTIONS OF TURING AND HOPF BIFURCATIONS IN AN ADDITIONAL FOOD PROVIDED DIFFUSIVE PREDATOR-PREY MODEL
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Abstract
Complex spatiotemporal dynamics of a diffusive predator-prey system involving additional food supply to predator and intra-specific competition among predator, are investigated. We establish critical conditions of the occurrence of Turing instability, which are necessary and sufficient. Furthermore, we also establish conditions of the occurrence of codimension-2 Turing-Hopf bifurcation and Turing-Turing bifurcation, by exploring interactions of Turing bifurcations and Hopf bifurcation. For Turing-Hopf bifurcation, by analyzing normal form truncated to order 3 which are derived by applying normal form method, it is shown that under proper conditions, diffusive predator-prey system generates interesting spatial, temporal and spatiotemporal patterns, including a pair of spatially inhomogeneous steady states, a spatially homogeneous periodic solution and a pair of spatially inhomogeneous periodic solutions. And numerical simulations are also shown to support theory analysis. Moreover, it is found that proper intra-specific competition among predator helps generate complex spatiotemporal dynamics. And, proper additional food supply to predator helps control the population fluctuations of predator and prey, while large quantity and high quality of additional food supply will lead to the extinction of prey and make predator change the source of food, which finally destroy the ecosystem. These investigations might help understand complex spatiotemporal dynamics of predator-prey system involving additional food supply to predator and intra-specific competition among predator, and help conserve species in an ecosystem via supplying suitable additional food. -
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References
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Xun Cao, Weihua Jiang. INTERACTIONS OF TURING AND HOPF BIFURCATIONS IN AN ADDITIONAL FOOD PROVIDED DIFFUSIVE PREDATOR-PREY MODEL[J]. Journal of Applied Analysis & Computation, 2019, 9(4): 1277-1304. doi: 10.11948/2156-907X.20180224
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Figure 1. Equilibrium bifurcation:
$K=50, \eta=1, \beta=0.8, \gamma=0.4, d=0.2, l=1$ and$\xi=0.1, \xi=0.7, \xi=0.9$ respectively. And,$g$ is bifurcation parameter. System (2.1) might have no positive equilibrium, one positive equilibrium, two positive equilibria or three positive equilibria, depending on different combinations of parameters$\xi, g$ . -
Figure 2. Parameter region of the stability for coexistence equilibrium, and bifurcation curves in
$d_2\text{-}r$ plane. And$HT$ stands for$\left(k_0^*, 0\right)$ -mode Turing-Hopf bifurcation point$\left(d_{k_0^*}^*, r_0\right)$ . Moreover,$TT_k$ represents$(k, k+1)$ -mode Turing-Turing bifurcation point$\left(d_{k, k+1}^*, r_{k, k+1}^*\right)$ for$k\in\mathbb{N}$ , which locates on the critical Turing bifurcation curve$\mathcal{T}$ . -
Figure 3. Parameter region of the stability for coexistence equilibrium
$E_*$ and global bifurcation set in$d_2\text{-}r$ plane. -
Figure 4. Local bifurcation set (Left) and phase portraits (Right) at Turing-Hopf point
$\left(d_{k_0^*}^*, r_0\right)$ . -
Figure 5. Global bifurcation set with local bifurcation curves
$\mathcal{T}_1, \mathcal{T}_2$ at Turing-Hopf bifurcation point$\left(d_{k_0^*}^*, r_0\right)$ . -
Figure 6. For
$(d_2, r)=(1.0290, 0.05690)\in \mathcal{D}_2$ , a spatially homogeneous periodic solution is stable. The initial values are$u(0, x)=0.4058560903+0.1, v(0, x)=1.313932244-0.2$ . -
Figure 7. For
$(d_2, r)=(1.3290, 0.0659)\in\mathcal{D}_4$ , a pair of spatially inhomogeneous periodic solutions are stable. Graphs in the middle reflect transient patterns, which indicate that the spatially homogeneous periodic solution and a pair of spatially inhomogeneous steady states are unstable. -
Figure 8. For
$(d_2, r)=(1.3290, 0.0549)\in\mathcal{D}_6$ , a pair of spatially inhomogeneous steady states are stable. -
Figure 9. For intra-specific competition parameter
$g=0.04$ , there exist multiple stable coexistence equilibria for diffusive system (2.1). -
Figure 10. For the quantity of additional food supply
$\xi=0.5$ , spatially homogeneous steady states of diffusive system (2.1) are stable. The initial values are$u(0, x)=0.4058560903+0.1, v(0, x)=1.313932244-0.2$ .