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Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations

  • * Corresponding author: Ben Niu

    * Corresponding author: Ben Niu 

The authors are supported by National Natural Science Foundation of China (No.11771109), Shandong Provincial Natural Science Foundation (No.ZR2019QA020)

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  • We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.

    Mathematics Subject Classification: Primary: 35K57, 35B10; Secondary: 37G15, 58J55.

    Citation:

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  • Figure 1.  (a) The diagram of $ d_{2}(n^{2}) $ on the $ d_{2} $-$ n $ plane. In region "R", $ B_{n}+C_{n}<0 $. (b) The bifurcation curves on the $ \tau $-$ d_{2} $ plane, where "TH" is the Turing-Hopf bifurcation point

    Figure 2.  The bifurcation set on the $ (\alpha_{1}, \alpha_{2}) $ plane

    Figure 3.  The dynamical classifications in region $ \Re_{1} $-$ \Re_{6} $

    Figure 4.  Choosing $ (\alpha_{1}, \alpha_{2}) = (-0.3936, -0.0858) $ at the red dot in region $ \Re_{1} $, the positive equilibrium $ E_{\ast}(u_{\ast}, v_{\ast}) $ is asymptotically stable. The initial functions are $ u(x, 0) = 1.2+0.02\cos2x $, $ v(x, 0) = 13.8-0.02\cos2x $

    Figure 5.  Choosing $ (\alpha_{1}, \alpha_{2}) = (-0.3936, 0.0742) $ at the red dot in region $ \Re_{2} $, two nonconstant steady-states are stable. (a) The initial functions are $ u(x, 0) = 1.2+0.02\cos2x $, $ v(x, 0) = 13.8-0.02\cos2x $; (b) The initial functions are $ u(x, 0) = 1.2-0.02\cos2x $, $ v(x, 0) = 13.8+0.02\cos2x $

    Figure 6.  Choosing $ (\alpha_{1}, \alpha_{2}) = (-0.0936, 0.0742) $ at the red dot in region $ \Re_{3} $, two spatially inhomogeneous periodic solutions are stable. The initial functions of (a) and (b) are the same as those in Figure 5(a), (b), respectively

    Figure 7.  Choosing $ (\alpha_{1}, \alpha_{2}) = (0.0064, 0.0742) $ at the red dot in region $ \Re_{4} $, the spatially homogeneous periodic solution is unstable (transient state) and two spatially inhomogeneous periodic solutions are stable. The initial functions of (a) and (b) are $ u(x, 0) = 1.2+0.02\cos2x $, $ v(x, 0) = 13.8-0.02\cos2x $; The initial functions of (c) and (d) are $ u(x, 0) = 1.2-0.02\cos2x $, $ v(x, 0) = 13.8+0.02\cos2x $

    Figure 8.  The partial bifurcation set on the $ \tau $-$ r_{2} $ plane, where the colored curves stand for Hopf bifurcation curves

    Figure 9.  Complete bifurcation sets near the double Hopf point "HH" of system (2)

    Figure 10.  The dynamical classifications in region $ \textbf{D1} $-$ \textbf{D8} $

    Figure 11.  When $ (\tau, r_{2}) = (6.25, 0.23) $ (see P1 in Figure 9), the constant steady state $ (u_{\ast}, v_{\ast}) $ of system (2) is asymptotically stable with the initial functions $ u_{0}(x) = 2.18+0.02\cos2x $, $ v_{0}(x) = 9.24-0.02\cos2x $

    Figure 12.  When $ (\tau, r_{2}) = (32.43, 0.23) $ (see P2 in Figure 9), the spatially homogeneous periodic solutions of system (2) are stable with the initial functions $ u_{0}(x) = 2.18+0.02\cos2x $, $ v_{0}(x) = 9.24-0.02\cos2x $

    Figure 13.  When $ (\tau, r_{2}) = (33.58, 0.23) $ (see P3 in Figure 9), the spatially homogeneous periodic or quasi-periodic solutions of system (2) are unstable with the initial functions $ u_{0}(x) = 2.18+0.02\cos2x $, $ v_{0}(x) = 9.24-0.02\cos2x $. (a): the periodic solutions; (b): the quasi-periodic solutions

    Figure 14.  The corresponding Poincaré map on the $ u(0, t-2\tau)-v(0, t) $. (a) $ \tau = 33.58 $; (b) $ \tau = 34.2 $; (c) $ \tau = 35.41 $ (see P3, P4, P5 in Figure 9). The parameter $ r_{2} = 0.23 $ is fixed with the initial functions $ u_{0}(x) = 2.18+0.02\cos2x $, $ v_{0}(x) = 9.24-0.02\cos2x $

    Figure 15.  The local maximum map of $ u $. (a) $ (\tau, r_{2}) = (34.2, 0.23) $; (b) $ (\tau, r_{2}) = (35.41, 0.23) $

    Table 1.  The twelve unfoldings of system (20)

    $ \mathrm{Case} $ $ \mathrm{Ia} $ $ \mathrm{Ib} $ $ \mathrm{II} $ $ \mathbf{III} $ $ \mathrm{IVa} $ $ \mathrm{IVb} $ $ \mathrm{V} $ $ \mathbf{VIa} $ $ \mathrm{VIb} $ $ \mathrm{VIIa} $ $ \mathrm{VIIb} $ $ \mathrm{VIII} $
    $ d $ + + + + + + $ {-} $ $ {-} $ $ {-} $ $ {-} $ $ {-} $ $ {-} $
    $ b $ + + + $ {-} $ $ {-} $ $ {-} $ + + + $ {-} $ $ {-} $ $ {-} $
    $ c $ + + $ {-} $ + $ {-} $ $ {-} $ + $ {-} $ $ {-} $ + + $ {-} $
    $ d-bc $ + $ {-} $ + + + $ {-} $ $ {-} $ + $ {-} $ + $ {-} $ $ {-} $
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