Asymptotic population abundance of a two-patch system with asymmetric diffusion

Figure 1.  Phase-plane diagram of system (6). Stable equilibrium is displayed by solid circle. Vector fields are shown by gray arrows. Isoclines of nutrient $ u_1 $ and consumer $ v_1 $ are represented by red and blue lines, respectively. According to parameter values in experiments by Zhang et al. (2017), let $ N_{01} = 0.02, r = 0.1, k_1 = 0.1, \gamma = 0.01, g_1 = 0.0001 $. Then $ u_{01} = 0.2 < r^2/g_1 $. Numerical simulations show that all positive solutions of (6) converge to equilibrium $ E_1^+ $, which is consistent with Theorem 2.1(ⅱ)

Figure 2.  Numerical simulations for comparison of $ T_1 $ and $ T_0 $ when $ s $ varies, Let $ r = 0.1, u_{01} = 0.06, u_{02} = 0.0002, g_1 = 0.001, g_2 = 0.0005, D = 100 $. When $ s = 0.1 $, we obtain $ T_1 = 11.9531 >8.5095 = T_0 $ by numerical computations on (4)

Figure 3.  Numerical simulations for comparison of $ T_1 $ and $ T_2 $ when $ s $ varies, Let $ r = 0.1, u_{01} = 0.06, u_{02} = 0.0002, g_1 = 0.001, g_2 = 0.0005, D = 100 $. Then $ u_{mean} = 0.0301 $. When $ s = 0.1 $, we obtain $ T_1 = 13.4364> 13.2452 = T_2 $ by numerical computations on (4)

Figure 4.  Numerical simulations for comparison of $ T_1 $ and $ T_2 $ when $ s $ varies, Let $ r = 0.1, u_{01} = 0.06, u_{02} = 0.0002, g_1 = 0.001, g_2 = 0.0005, D = 100 $. When the initial values are $ (1.4, 1.4, 1.4, 1.4) $, $ (3.4, 3.4, 3.4, 3.4) $, $ (4, 4, 4, 4) $, $ (7, 7, 7, 7) $ and $ (8, 8, 8, 8) $, numerical computations on (4) show that all solutions converge to the same equilibrium $ (0.1549, 4.4737, 0.0002, 8.9457) $, while the component $ v_1(t) $ is displayed in this figure