Transitions between metastable long-run consumption behaviors in a stochastic peer-driven consumer network

Figure 1.  Bird's eye view of the parameter plane $ D $, where remaining parameters have been fixed at $ (p_x, p_y) = \left(\frac{1}{4}, 1\right), \ $$ (b_1, b_2) = (10,20), \ \alpha_1 = 0.0002, \ \alpha_2 = 0.00052$

Figure 2.  Bifurcation diagram for $ D^N $ with $ (p_x, p_y) = \left(\frac{1}{4}, 1\right), \ (b_1, b_2) = (10,20), \ \alpha_1 = 0.0002, \ \alpha_2 = 0.00052 $. $ NS $ indicates the Neimark-Sacker bifurcation curve related to the fixed point. $ SN_3 $ curve gives the loci at which a saddle 3-cycle is born together with the attracting 3-cycle ($ C_3 $) via a saddle-node bifurcation. $ NS_3 $ designates the Neimark-Sacker bifurcation curve of the 3-cycle. The horizontal line through $ D_{21} = 0.0075 $ indicates the interval of parameter values for which our study of transitions between coexisting attractors focuses on. The $ NS $ and $ NS_3 $ curves are crossed twice at $ \star $ (red star) and $ \star $ (green star). Also the saddle node bifurcation curve $ SN_3 $ is intersected twice. The intersection points are indicated by $ \bullet $ (blue circles). Related details are revealed in Figure 3

Figure 3.  For $ D_{21} = 0.0075 $, we give bifurcation diagrams for $ 0 \leq D_{12} \leq 0.00245 $ linked to the horizontal black line in Figure 2(a) and an enlargement (b) focussing on the interval $ 0.00145 \leq D_{12} \leq 0.001975 $ over which two attractors coexists

Figure 4.  Bifurcation diagram for the case of additive noise with $ \varepsilon = 0.1 $ ($ D_{21} = 0.0075 $). If the initial value $ (x_{1,0},x_{2,0}) $ lies on the deterministic blue (red) attractor, then elements of the trajectory are colored light blue (red)

Figure 5.  Bifurcation diagram for the case of parametric noise with $ \varepsilon = 0.1 $ ($ D_{21} = 0.0075 $). If $ (x_{1,0},x_{2,0}) $ lies on the deterministic blue (red) attractor, then elements of the trajectory are colored light blue (red)

Figure 6.  Confidence sets for fixed point $ E $ ($ \bullet $) and 3-cycle $ C_3 $ ($ \bullet $) at $ D_{12} = 0.00195 $, $ D_{21} = 0.0075 $ with trajectories superimposed ($ \varepsilon = 0.1 $ (white), $ \varepsilon = 0.05 $ (grey))

Figure 7.  The top panel shows the graph of the sensitivity function for $ \Gamma $, i.e. a plot of the maximum eigenvalue ($ \lambda $) of the sensitivity matrix at a point on $ \Gamma $ versus the angle $ \phi $ identifying the point on the attractor. The subfigures on the bottom give the confidence sets $ \mathcal{C}(\Gamma, \varepsilon = 0.1) $ at $ D_{12} = 0.00157 $ for additive (a) and parametric noise (b)

Figure 8.  The figure shows the attractor $ \Gamma_3 $ at $ D_{12} = 0.0017 $ (a), the sensitivity functions for $ \Gamma_3 $ (b) as well as the related confidence sets $ \mathcal{C}(\Gamma_3, \varepsilon = 0.1) $ for additive (c) and parametric noise (d)

Figure 9.  1D bifurcation diagrams ((a),(b)) and critical intensities for coexisting attractors (c) with additive (solid lines) and parametric (dashed lines) noise for $ D_{12} \in D^{ms} $

Figure 10.  For $ (D_{12}, D_{21}) = (0.001706,0.0075) $ we show the state space representation of the coexisting attractors $ \Gamma_3 $ (dark red curves) and $ \Gamma $ (blue curve) together with their immediate basins $ \mathcal{B}(\Gamma_3) $ (light red) and $ \mathcal{B}(\Gamma) $ (light blue). The confidence sets are superimposed ($ \varepsilon \in \{ 0.1, 0.2, 0.3\} $). Periodic points (red triangles) of the 3-saddle cycle are exhibited together with its stable (black lines) and unstable (red lines) manifolds. In addition, the unstable fixed point $ E $ (blue circle) and the unstable 3-cylce (periodic point given by red circles) are given

Figure 11.  ( $ \Gamma $, $ \Gamma_3 $) at $ (D_{12}, D_{21}) = (0.001706,0.0075) $ with sample trajectory (single simulation run with $ \varepsilon = 0.2 $) superimposed