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

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) $