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

  • * Corresponding author: Junjie Wei

    * Corresponding author: Junjie Wei

The corresponding author is supported by National Natural Science Foundation of China (Nos.11371111 and 11771109)

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  • In this paper, the dynamics of a class of modified Leslie-Gower model with diffusion is considered. The stability of positive equilibrium and the existence of Turing-Hopf bifurcation are shown by analyzing the distribution of eigenvalues. The normal form on the centre manifold near the Turing-Hopf singularity is derived by using the method of Song et al. Finally, some numerical simulations are carried out to illustrate the analytical results. For spruce budworm model, the dynamics in the neighbourhood of the bifurcation point can be divided into six categories, each of which is exactly demonstrated by the numerical simulations. Then according to this dynamical classification, a stable spatially inhomogeneous periodic solution has been found, which can be used to explain the phenomenon of periodic outbreaks of spruce budworm.

    Mathematics Subject Classification: Primary: 35B32, 37G15; Secondary: 92D25.


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  • 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 $u(x, 0) = 0.1296, v(x, 0) = 0.0154-0.001\cos x$

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