| Citation: |
Qiannan Song, Ruizhi Yang, Chunrui Zhang, Lei Wang. BIFURCATION ANALYSIS OF A DIFFUSIVE PREDATOR-PREY MODEL WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 920-936. doi: 10.11948/20200119
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BIFURCATION ANALYSIS OF A DIFFUSIVE PREDATOR-PREY MODEL WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE
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Abstract
In this paper, we consider a diffusive predator-prey model with Beddington-DeAngelis functional response. The Turing instability and Hopf bifurcation of the coexisting equilibrium are investigated. We also use bifurcation parameters $ m, {d_2} $ to study the Turing-Hopf bifurcation. In addition, we compute the normal form for the Turing-Hopf bifurcation. On the basis of the corresponding normal form, there exists complex spatiotemporal dynamics near Turing-Hopf bifurcation point. Finally, Some numerical simulations are given to illustrate our theoretical results.
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Keywords:
- Predator-prey /
- Turing instability /
- Hopf bifurcation /
- Turing-Hopf bifurcation
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References
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Qiannan Song, Ruizhi Yang, Chunrui Zhang, Lei Wang. BIFURCATION ANALYSIS OF A DIFFUSIVE PREDATOR-PREY MODEL WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE[J]. Journal of Applied Analysis & Computation, 2021, 11(2): 920-936. doi: 10.11948/20200119
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Figure 1. Left: the bifurcation diagram. Right: Stable region for
$ (u_*, v_*) $ , Turing-Hopf bifurcation point$ (m^*, d_2^{k_*}) $ in$ m-d_2 $ plane. -
Figure 2. For
$ \left( {{\mu _1}, {\mu _2}} \right) = \left( {0.1, 1.0} \right) \in {D_1} $ , the positive equilibrium$ (u_*, v_*) $ of the system (2.1) is asymptotically stable. -
Figure 3. For
$ \left( {{\mu _1}, {\mu _2}} \right) = \left( {1.0, 0.1} \right) \in {D_2} $ , there is a pair of stable spatially inhomogeneous steady states. -
Figure 4. For
$ \left( {{\mu _1}, {\mu _2}} \right) = \left( {1.0, -0.1} \right) \in {D_3} $ , the system (2.1) has a pair of stable spatially inhomogeneous steady states and an unstable spatially homogeneous periodic solution. -
Figure 5. When
$ \left( {{\mu _1}, {\mu _2}} \right) = \left( {1.0, -0.2} \right) \in {D_4} $ , there exists a stable spatially homogeneous periodic solution and a pair of stable spatially inhomogeneous steady states, as well as a pair of unstable spatially inhomogeneous periodic solutions. -
Figure 6. For
$ \left( {{\mu _1}, {\mu _2}} \right) = \left( {1.0, -2.0} \right)\in {D_5} $ , the system (2.1) has a stable spatially homogeneous periodic solution and a pair of unstable spatially inhomogeneous steady states. The initial values are$ {u\left( {x, 0} \right) = 2.0 + 0.2\cos\left( {4x} \right), v\left( {x, 0} \right) = 2.0 - 0.2\cos\left( {4x} \right)} $ . -
Figure 7. When
$ \left( {{\mu _1}, {\mu _2}} \right) = \left( {-1.0, -0.5} \right) \in {D_6} $ , there is a stable spatially homogeneous periodic solution. The initial values are$ {u\left( {x, 0} \right) = 2.0 + 0.01, v\left( {x, 0} \right) = 2.0 - 0.01} $ .