Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis

Figure 1.  The stable region/unstable region of equilibrium $ (u^*, v^*) $ of system (1) in $ \xi-\eta $ parameter plane. Here $ f, g $ and other parameters are taken from (25) in Section 4. (Left): $ d = 0.06 $, $ (0, 0)\in S $; (Right): $ d = 0.01 $, $ (0, 0)\not\in S $. The thick solid curve is $ f_u+dg_v+\eta v^*f_v-\xi u^*g_u-2\sqrt{d+\xi\eta u^*v^*}\sqrt{Det J} = 0 $, the thin solid curve is $ d+\xi\eta u^*v^* = 0 $ in both panels

Figure 2.  (Left): $ D(\eta, \xi, p) = 0 $ in $ \xi-p $ plane for fixed $ \eta $; (right): there are two bifurcation value $ \xi_S^2 $(corresponding to $ \lambda_2 = 4 $) and $ \xi_S^3 $(corresponding to $ \lambda_3 = 9 $), here $ f, g $ and other parameters are taken from (25) in Section 4 with $ \eta = 0.02 $, $ d = 0.01 $, $ \Omega = (0, \pi) $

Figure 3.  Turing instability for (25) with $ \beta = 0.2 $, $ m = 2 $, $ s = 0.5 $, $ \Omega = (0, 10\pi) $, $ d = 0.01 $, $ \xi = \eta = 0 $ and initial value $ (0.7+0.1\sin(2x), 0.7+0.2\sin(3x)) $

Figure 4.  Stable constant equilibrium for (25) with $ \beta = 0.2 $, $ m = 2 $, $ s = 0.5 $, $ \Omega = (0, 10\pi) $, $ d = 0.06 $, $ \xi = \eta = 0 $ and initial value $ (0.7+0.1\sin(2x), 0.7+0.2\sin(3x)) $

Figure 5.  Turing pattern in (25) is stabilized when $ \xi = 0.9 $ and $ \eta = 0.4 $, and the same initial value $ (0.7+0.1\sin(2x), 0.7+0.2\sin(3x)) $. Other parameters are also same as in Figure 3

Figure 6.  Instability induced by taxis in (25) when $ \xi = -0.5 $ and $ \eta = -0.4 $ and initial condition $ (0.7+0.1\sin(2x), 0.7+0.2\sin(3x)) $. Other parameters are same as in Figure 4

Figure 7.  The amplitude change of non-constant equilibrium solutions of (25) for different $ (\xi, \eta) $. Here the value is the difference of maximum and minimum values of the non-constant equilibrium solution; the initial value is $ (0.7+0.1\sin(2x), 0.7+0.2\sin(3x)) $ and other parameters are the same as Figure 3