Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion

Figure 1.  Stability region and bifurcation diagram for system (5) at the unique positive equilibrium $E^*$ in the parameter plane, where $f(u) = \frac{Au^2}{B+u^2}$, $A = 1, B = 0.0025, l = 1$. (a):$d_1 = 0.05, d_2 = 0.33$. (b):$d_1 = 0.05, d_2 = 0.28.$

Figure 2.  Bifurcation diagrams and dynamical classification near the Turing-Hopf point $P^*$

Figure 3.  When $(\mu_1, \mu_2) = (-0.01, 0.02)$ lies in region ①, the positive constant equilibrium $E^*(0.1296, 0.0167)$ is asymptotically stable. The initial value is $u(x, 0) = 0.1296+0.005\cos x, v(x, 0) = 0.0167+0.01\cos x$

Figure 4.  When $(\mu_1, \mu_2) = (0.022, 0.014)$ lies in region ②, the positive constant equilibrium $E^*(0.1296, 0.0208)$ is unstable and there are two stable spatially inhomogeneous steady states like $\cos x$. (a) and (b) The initial value is $u(x, 0) = 0.1296-0.02\cos x, v(x, 0) = 0.0208+0.01\cos x$; (c) and (d) the initial value is $u(x, 0) = 0.1296+0.02\cos x, v(x, 0) = 0.0208-0.01\cos x$

Figure 5.  When $(\mu_1, \mu_2) = (0.02, 0.01)$ lies in region ③, the positive constant equilibrium $E^*(0.1296, 0.0206)$ is unstable and there is a heteroclinic orbit connecting the unstable spatially homogeneous periodic solution to stable spatially inhomogeneous steady state. The initial value is $u(x, 0) = 0.1576-0.002\cos x, v(x, 0) = 0.0234$. (a) and (b) are transient behaviours for $u$ and $v$, respectively; (c) and (d) are middle-term behaviours for $u$ and $v$, respectively; (e) and (f) are long-term behaviours for $u$ and $v$, respectively

Figure 6.  When $(\mu_1, \mu_2) = (0.4, 0.12)$ lies in region ④, the positive constant equilibrium $E^*(0.1296, 0.0698)$ is unstable and there are stable spatially inhomogeneous periodic solution. The initial value is $u(x, 0) = 0.1306-0.001\cos x, v(x, 0) = 0.0691+0.001\cos x$. (a) and (b) are transient behaviours for $u$ and $v$, respectively; (c) and (d) are long-term behaviours for $u$ and $v$, respectively

Figure 7.  When $(\mu_1, \mu_2) = (-0.01, -0.015)$ lies in region ⑤, the positive constant equilibrium $E^*(0.1296, 0.0167)$ is unstable and there are heteroclinic solution connecting the unstable spatially inhomogeneous steady state to stable spatially homogeneous periodic solution. The initial value is $u(x, 0) = 0.1526-0.065\cos x, v(x, 0) = 0.0189-0.0015\cos x$. (a) and (b) are transient behaviours for $u$ and $v$, respectively; (c) and (d) are long-term behaviours for $u$ and $v$, respectively

Figure 8.  When $(\mu_1, \mu_2) = (-0.02, -0.022)$ lies in region ⑥, the positive constant equilibrium $E^*(0.1296, 0.0154)$ is unstable and there is a stable spatially homogeneous periodic solution. The initial value is $u(x, 0) = 0.1296, v(x, 0) = 0.0154-0.001\cos x$