# American Institute of Mathematical Sciences

March  2018, 23(2): 765-783. doi: 10.3934/dcdsb.2018042

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

 1 Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China 2 Department of Mathematics, Northeast Forestry University, Harbin, Heilongjiang 150040, China

* Corresponding author: Junjie Wei

Received  November 2016 Revised  October 2017 Published  December 2017

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

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.

Citation: Xiaofeng Xu, Junjie Wei. Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 765-783. doi: 10.3934/dcdsb.2018042
##### References:

show all references

##### References:
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):$d_1 = 0.05, d_2 = 0.28.$
Bifurcation diagrams and dynamical classification near the Turing-Hopf point $P^*$
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 $u(x, 0) = 0.1296+0.005\cos x, v(x, 0) = 0.0167+0.01\cos x$
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 $u(x, 0) = 0.1296+0.02\cos x, v(x, 0) = 0.0208-0.01\cos x$
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
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
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
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 $u(x, 0) = 0.1296, v(x, 0) = 0.0154-0.001\cos x$
 [1] Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045 [2] Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183 [3] Hongmei Cheng, Rong Yuan. Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5433-5454. doi: 10.3934/dcds.2017236 [4] Hongwei Yin, Xiaoyong Xiao, Xiaoqing Wen. Analysis of a Lévy-diffusion Leslie-Gower predator-prey model with nonmonotonic functional response. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2121-2151. doi: 10.3934/dcdsb.2018228 [5] Mingxin Wang, Qianying Zhang. Dynamics for the diffusive Leslie-Gower model with double free boundaries. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2591-2607. doi: 10.3934/dcds.2018109 [6] Andrei Korobeinikov, William T. Lee. Global asymptotic properties for a Leslie-Gower food chain model. Mathematical Biosciences & Engineering, 2009, 6 (3) : 585-590. doi: 10.3934/mbe.2009.6.585 [7] Yunfeng Liu, Zhiming Guo, Mohammad El Smaily, Lin Wang. A Leslie-Gower predator-prey model with a free boundary. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2063-2084. doi: 10.3934/dcdss.2019133 [8] Yunshyong Chow, Kenneth Palmer. On a discrete three-dimensional Leslie-Gower competition model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4367-4377. doi: 10.3934/dcdsb.2019123 [9] Jun Zhou, Chan-Gyun Kim, Junping Shi. Positive steady state solutions of a diffusive Leslie-Gower predator-prey model with Holling type II functional response and cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3875-3899. doi: 10.3934/dcds.2014.34.3875 [10] Walid Abid, Radouane Yafia, M.A. Aziz-Alaoui, Habib Bouhafa, Azgal Abichou. Global dynamics on a circular domain of a diffusion predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional type. Evolution Equations & Control Theory, 2015, 4 (2) : 115-129. doi: 10.3934/eect.2015.4.115 [11] C. R. Zhu, K. Q. Lan. Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 289-306. doi: 10.3934/dcdsb.2010.14.289 [12] Wen-Bin Yang, Yan-Ling Li, Jianhua Wu, Hai-Xia Li. Dynamics of a food chain model with ratio-dependent and modified Leslie-Gower functional responses. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2269-2290. doi: 10.3934/dcdsb.2015.20.2269 [13] Wenjie Ni, Mingxin Wang. Dynamical properties of a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3409-3420. doi: 10.3934/dcdsb.2017172 [14] Jun Zhou. Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1127-1145. doi: 10.3934/cpaa.2015.14.1127 [15] Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203 [16] Changrong Zhu, Lei Kong. Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1187-1206. doi: 10.3934/dcdss.2017065 [17] Safia Slimani, Paul Raynaud de Fitte, Islam Boussaada. Dynamics of a prey-predator system with modified Leslie-Gower and Holling type Ⅱ schemes incorporating a prey refuge. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5003-5039. doi: 10.3934/dcdsb.2019042 [18] Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325 [19] Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 [20] Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031

2018 Impact Factor: 1.008