Asymmetric diffusion in a two-patch mutualism system characterizing exchange of resource for resource

Figure 1.  Phase-plane diagram of system (5). Stable and unstable equilibria are identified by solid and open circles, respectively. Vector fields are shown by green arrows. Isoclines of species $ u,v_1 $ are represented by red and blue lines, respectively. Fix $ r = r_1 = c = 1 $. (a) Let $ a_{12} = 2.5, a_{21} = 1.3 $. Equilibrium $ E^+(2.7,2.5) $ is globally asymptotically stable. (b) Let $ a_{12} = 4.5, a_{21} = 0.9 $. There are two positive equilibria $ E^-(1.25,0.14) $ and $ E^+(2.79,1.5) $. (c) Let $ a_{12} = 3.5, a_{21} = 0.9 $. The equilibria $ E^- $ and $ E^+ $ coincide and form a saddle-node point $ E^\pm(1.6,0.45) $. In the cases of (b-c), the separatrices (the black line) of $ E^- $ subdivide the first quadrant into two regions: one is the basin of attraction of $ E_1 $ while the other is that of $ E^+ $. (d) Let $ a_{12} = 2.5, a_{21} = 0.5 $. All positive solutions converge to equilibrium $ E_1(1,0) $

Figure 4.  Comparison of $ T_1(s, 0) $ and $ T_2(s, D) $ when there is diffusion $ D $ as shown in Theorem 4.1(ⅰ). The solid blue line represents $ T_2(s, D) $, while the dash-dot red line represents $ T_1(s, 0) $. Let $ r = 0.2, c = 0.05, a_{12} = 0.1, b = b_1 = 1, s = 1 $, $ a_{21} = 0.5, r_1 = 0.8, r_2 = 0.1 $. (a) When $ s = 1 $, we have $ T_2>T_1 $ for $ D>0 $. (b) When $ s = 10 $, we have $ T_2>T_1 $ as $ 0<D< 0.0135 $ while $ T_2<T_1 $ as $ D> 0.0135 $

Figure 5.  Comparison of $ T_1(s, 0) $ and $ T_2(s,100) $ when there is asymmetry $ s $ in large diffusion as shown in Theorem 4.2(ⅰ). The solid blue line represents $ T_2(s,100) $, while the dash-dot red line represents $ T_1(s, 0) $. Let $ r = 0.2, c = 0.05, a_{12} = 0.1, b = b_1 = 1, D = 100 $, $ a_{21} = 0.5, r_1 = 0.8, r_2 = 0.1 $. Then we have $ T_1(s, 0) = 1.9233 $ and $ T_2(s,100)>T_1(s, 0) $ as $ 0.1935< s< 9.3201 $. Numerical computation shows that the function $ T_2 = T_2(s,100) $ is convex upward with $ T_2(s,100) = 0 $ as $ s\ge 12.1537 $

Figure 2.  Comparison of $ T_1(s, D) $ and $ T_1(s, 0), T_2(s, D) $ and $ T_2(s, 0) $ when there is a small diffusion rate $ D $, as shown in Theorem 4.1(ⅰ). The solid red and blue lines represent $ T_1(s, D) $ and $ T_2(s, D) $, while the dash-dot red and blue lines represent $ T_1(s, 0) $ and $ T_2(s, 0) $, respectively. Let $ r = 0.2, c = 0.05, a_{12} = 0.1, b = b_1 = 1, D = 0.1, s = 1 $, $ a_{21} = 0.5, r_1 = 0.8, r_2 = 0.1 $. Then we have $ T_1(1, 0.1)<T_1(1,0) $ but $ T_2(1,0.1)>T_2(1,0) $

Figure 3.  Comparison of $ T_1(s, D) $ and $ T_1(s, 0), T_2(s, D) $ and $ T_2(s, 0) $ when there is a large diffusion rate $ D $, as shown in Theorem 4.2(ⅰ). The solid red and blue lines represent $ T_1(s, D) $ and $ T_2(s, D) $, while the dash-dot red and blue lines represent $ T_1(s, 0) $ and $ T_2(s, 0) $, respectively. Let $ r = 0.2, c = 0.05, a_{12} = 0.1, b = b_1 = 1, D = 100, s = 0.1 $, $ a_{21} = 0.5, r_1 = 0.8, r_2 = 0.1 $. Then we have $ T_1(0.1,100)<T_1(0.1,0) $ but $ T_2(0.1,100)>T_2(0.1,0) $

Figure 6.  The surface of $ T_2 = T_2(s,D) $ when both of $ s $ and $ D $ varies. Let $ r = 0.2, c = 0.05, a_{12} = 0.1, b = 4, b_1 = 1 $, $ a_{21} = 0.5, r_1 = 0.8, r_2 = 0.1, 0<s<6, 0<D<6 $. Numerical computation shows that for fixed $ s $, the surface decreases monotonically, which is consistent with Fig. 4. For fixed $ D $, the surface is convex upward, which is consistent with Fig. 5