Interactions of time delay and spatial diffusion induce the periodic oscillation of the vegetation system

Figure 1.  Time series of vegetation coverage in Gansu Province from January 2000 to December 2018

Figure 2.  Stability of equilibria of system (5) with respect to parameter $ a $ for m = 0.8, the solid line and dotted line respectively indicate that the equilibria are stable and unstable, except for pink dotted line $ a = 2m $

Figure 3.  The time series (a)-(c) and phase plane (d)-(f) diagrams of system (4) without diffusion when $ \tau>\tau_{00}^{0} $ are plotted: (a)(d) $ m = 1.4, \tau = 4.5>\tau_{00}^{0} = 4.2650; $ (b)(e) $ m = 1.6, \tau = 3.3>\tau_{00}^{0} = 2.9887; $ (c)(f) $ m = 1.8, \tau = 2.7>\tau_{00}^{0} = 2.3657. $ Red dot, blue dot and green closed loop denote the coincidence equilibrium $ E_{**} $, initial value and stable limit cycle, respectively

Figure 4.  (a) $ \tau = 0.9 <\tau_{01}^{0} $, $ E^{*} $ is asymptotically stable; (b) $ \tau = 2.8>\tau_{01}^{0} $, $ E^{*} $ becomes unstable, and Hopf bifurcation is bifurcated from $ E^{*} $. (c) intercepts a part of (b) from $ t = 3450 $ to $ t = 3500 $. Parameters are $ a = 3.61 $, $ m = 1.8 $. The initial conditions are $ n(t) = 1.2 $ and $ s(t) = 1.8 $, $ t \in[-\tau, 0] $

Figure 5.  The evolutions of vegetation biomass and the density of soil water with time and space: (a) $ \tau = 2<\tau_{2}^{0}; $ (b) $ \tau = 4.2>\tau_{2}^{0} $. Other parameters are $ a = 2.8 $, $ m = 1.4 $, $ \beta = 0.0 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $. The initial conditions are $ n(x, t) = 1.2 $ and $ s(x, t) = 1.6 $, $ (x, t)\in[0, \pi]\times[-\tau, 0] $

Figure 6.  The spatiotemporal evolutions of $ n(x, t) $ and $ w(x, t) $ with respect to $ \tau $: (a)(d) $ \tau = 2<\tau_{2}^{0}; $ (b)(c)(e)(f) $ \tau = 4.2>\tau_{2}^{0} $. Other parameters are $ a = 2.8 $, $ m = 1.4 $, $ \beta = 0.0 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $. The initial conditions are $ n(x, t) = 1.2 $ and $ s(x, t) = 1.6 $, $ (x, t)\in[0, \pi]\times[-\tau, 0] $

Figure 7.  The variation of the value of $ \tau_{c} $ for three groups $ a $ and $ m $: (a) with $ \beta $ when $ d_{1} = 0.02 $, $ d_{2} = 0.2 $; (b) with $ d_{2} $ when $ \beta = 0.0 $, $ d_{1} = 0.02 $. (c) The variation of the critical bifurcation parameter $ \tau_{c} $, the amplitude and the period of the spatially homogeneous periodic solutions with respect to $ m $. Other parameters as $ d_{1} = 0.02 $, $ d_{2} = 0.2 $, $ \beta = 0.0 $, $ a = 2m $ and $ \tau = 4.2 $

Figure 8.  Region divisions of parameter space $ a-m $: (a) for system (4) without diffusion; (b) for system (4) with parameters as $ \beta = 0.1 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $

Figure 9.  The variation of the sign of expression $ P(k) $ with $ k $: (a) under different $ a = 4.0, \ 3.8 $ and $ 3.61 $. Other parameter as $ m = 1.8 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $; (b) under different $ d_{2} = 1, \ 0.6 $ and $ 0.2 $. Other parameter as $ a = 3.61 $, $ m = 1.8 $, $ d_{1} = 0.02 $; (c) under different $ m = 1.8, \ 1.6 $ and $ 1.4 $. Other parameter as $ a = 3.61 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $

Figure 10.  The variation of the sign of expression $ Q_{2}(k) $ with $ k $: (a) under different $ \beta = 0.2, \ 0.1 $ and $ 0.0 $. Other parameter as $ a = 3.2 $, $ m = 1.6 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $; (b) under different $ d_{2} = 1, \ 0.6 $ and $ 0.2 $. Other parameter as $ \beta = 0.1 $, $ a = 3.2 $, $ m = 1.6 $, $ d_{1} = 0.02 $; (c) under different $ m = 1.8, \ 1.6 $ and $ 1.4 $. Other parameter as $ \beta = 0.1 $, $ a = 2m $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $

Figure 11.  The evolutions of vegetation biomass and the density of soil water with time and space: (a)(d) vegetation biomass, (b)(e) the density of soil water. The evolutions of vegetation biomass and the density of soil water: (c) with space at $ t = 700 $, (f) with time at $ x = 20 $. The initial conditions are $ n(x, t) = u(i, j) = 1.2+0.1\mathrm{cos}(2i) $ and $ s(x, t) = v(i, j) = 1.6-0.2\mathrm{cos}(2i) $, $ (x, t)\in[0, \pi]\times[-\tau, 0] $. Parameters: $ a = 2.8 $, $ m = 1.4 $, $ \beta = 0.0 $, $ d_{1} = 0.02 $, $ d_{2} = 0.2 $, $ \tau = 2 $

Figure 12.  (a) The influence of $ \beta $ on the amplitudes corresponding to the vegetation and soil water under different $ m $. Other parameters are $ d_{1} = 0.02 $, $ d_{2} = 0.2 $, $ \tau = 2 $; (b) The influence of $ d_{2} $ on the amplitudes corresponding to the vegetation and soil water under different $ m $. Other parameters are $ \beta = 0.0 $, $ d_{1} = 0.02 $, $ \tau = 2 $; (c) The influence of $ m $ on the amplitudes corresponding to the vegetation and soil water under different $ \beta $. Other parameters are $ d_{1} = 0.02 $, $ d_{2} = 0.2 $, $ \tau = 2 $