# 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:
 [1] M. Aziz, Study of a Leslie-Gower-type tritrophic population, Chaos Soliton Fract., 14 (2002), 1275-1293. doi: 10.1016/S0960-0779(02)00079-6. [2] L. Chen and F. Chen, Global stability of a Leslie-Gower predator-prey model with feedback controls, Appl. Math. Lett., 22 (2009), 1330-1334. doi: 10.1016/j.aml.2009.03.005. [3] S. Chen, J. Shi and J. Wei, Global stability and Hopf bifurcation in a delayed difusive LeslieGower predator-prey system, Int. J. Bifurcat. Chaos, 22 (2012), 1250061, 11pp-1334. [4] S. Chen, J. Shi and J. Wei, The effect of delay on a diffusive predator-prey system with Holling type-Ⅱ predator functional response, Commun. Pur. Appl. Anal., 12 (2013), 481-501. [5] J. Collings, The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol., 36 (1997), 149-168. doi: 10.1007/s002850050095. [6] T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, T. Am. Math. Soc., 352 (2000), 2217-2238. doi: 10.1090/S0002-9947-00-02280-7. [7] P. Feng and Y. Kang, Dynamics of a modified Leslie-Gower model with double Allee efects, Nonlinear Dynam., 80 (2015), 1051-1062. doi: 10.1007/s11071-015-1927-2. [8] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. [9] G. Guo, B. Li, M. Wei and J. Huang, Hopf bifurcation and steady-state bifurcation for an autocatalysis reaction-diffusion model, J. Math. Anal. Appl., 391 (2012), 265-277. doi: 10.1016/j.jmaa.2012.02.012. [10] G. Hu and W. Li, Hopf bifurcation analysis for a delayed predator-prey system with diffusion effects, Nonl. Anal. Real World Appl., 11 (2010), 819-826. doi: 10.1016/j.nonrwa.2009.01.027. [11] J. Jin, J. Shi, J. Wei and F. Yi, Bifurcations of patterned solutions in diffusive Lengyel-Epstein system of CIMA chemical reaction, Rocky Mt. J. Math., 43 (2013), 1637-1674. doi: 10.1216/RMJ-2013-43-5-1637. [12] Y. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer-Verlag, New York, 1998. [13] P. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45 (1958), 16-31. doi: 10.1093/biomet/45.1-2.16. [14] P. Leslie and J. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234. doi: 10.1093/biomet/47.3-4.219. [15] X. Li, W. Jiang and J. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA. J. Appl. Math., 78 (2013), 287-306. doi: 10.1093/imamat/hxr050. [16] P. Liu, J. Shi, W. Wang and X. Feng, Bifurcation analysis of reaction-diffusion Schnakenberg model, J. Math. Chem., 51 (2013), 2001-2019. doi: 10.1007/s10910-013-0196-x. [17] M. Liu and K. Wang, Dynamics of a Leslie-Gower Holling-type Ⅱ predator-prey system with Lévy jumps, Nonlinear Anal. Theor., 85 (2013), 204-213. doi: 10.1016/j.na.2013.02.018. [18] Y. Ma, Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays, Nonl. Anal. Real World Appl., 13 (2012), 370-375. doi: 10.1016/j.nonrwa.2011.07.045. [19] J. Murray, Mathematical Biology, 2nd edition, Springer-Verlag Berlin Heidelberg, New York, 1993. [20] Y. Song, T. Zhang and Y. Peng, Turing-Hopf bifurcation in the reaction-diffusion equations and its applications, Commun. Nonlinear Sci., 33 (2016), 229-258. doi: 10.1016/j.cnsns.2015.10.002. [21] Y. Song and X. Zhou, Bifurcation analysis of a diffusive ratio-dependent predator-prey model, Nonliner Dynam., 78 (2014), 49-70. doi: 10.1007/s11071-014-1421-2. [22] X. Tang, Y. Song and T. Zhang, Turing-Hopf bifurcation analysis of a predator-prey model with herd behavior and cross-diffusion, Nonliner Dynam., 86 (2016), 73-89. doi: 10.1007/s11071-016-2873-3. [23] J. Wollkind, J. Collings and A. Logan, Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees, B. Math. Biol., 50 (1988), 379-409. doi: 10.1007/BF02459707. [24] R. Yang and Y. Song, Spatial resonance and Turing-Hopf bifurcation in the Gierer-Meinhardt model, Nonl. Anal. Real World Appl., 31 (2016), 356-387. doi: 10.1016/j.nonrwa.2016.02.006. [25] F. Yi, J. Wei and J. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonl. Anal. Real World Appl., 9 (2008), 1038-1051. doi: 10.1016/j.nonrwa.2007.02.005. [26] F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differ. Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024. [27] J. Zhou, Positive steady state solutions of a Leslie-Gower predator-prey model with Holling type Ⅱ functional response and density-dependent difusion, Nonlinear Anal. Theor., 82 (2013), 47-65. doi: 10.1016/j.na.2012.12.014.

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##### References:
 [1] M. Aziz, Study of a Leslie-Gower-type tritrophic population, Chaos Soliton Fract., 14 (2002), 1275-1293. doi: 10.1016/S0960-0779(02)00079-6. [2] L. Chen and F. Chen, Global stability of a Leslie-Gower predator-prey model with feedback controls, Appl. Math. Lett., 22 (2009), 1330-1334. doi: 10.1016/j.aml.2009.03.005. [3] S. Chen, J. Shi and J. Wei, Global stability and Hopf bifurcation in a delayed difusive LeslieGower predator-prey system, Int. J. Bifurcat. Chaos, 22 (2012), 1250061, 11pp-1334. [4] S. Chen, J. Shi and J. Wei, The effect of delay on a diffusive predator-prey system with Holling type-Ⅱ predator functional response, Commun. Pur. Appl. Anal., 12 (2013), 481-501. [5] J. Collings, The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model, J. Math. Biol., 36 (1997), 149-168. doi: 10.1007/s002850050095. [6] T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, T. Am. Math. Soc., 352 (2000), 2217-2238. doi: 10.1090/S0002-9947-00-02280-7. [7] P. Feng and Y. Kang, Dynamics of a modified Leslie-Gower model with double Allee efects, Nonlinear Dynam., 80 (2015), 1051-1062. doi: 10.1007/s11071-015-1927-2. [8] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. [9] G. Guo, B. Li, M. Wei and J. Huang, Hopf bifurcation and steady-state bifurcation for an autocatalysis reaction-diffusion model, J. Math. Anal. Appl., 391 (2012), 265-277. doi: 10.1016/j.jmaa.2012.02.012. [10] G. Hu and W. Li, Hopf bifurcation analysis for a delayed predator-prey system with diffusion effects, Nonl. Anal. Real World Appl., 11 (2010), 819-826. doi: 10.1016/j.nonrwa.2009.01.027. [11] J. Jin, J. Shi, J. Wei and F. Yi, Bifurcations of patterned solutions in diffusive Lengyel-Epstein system of CIMA chemical reaction, Rocky Mt. J. Math., 43 (2013), 1637-1674. doi: 10.1216/RMJ-2013-43-5-1637. [12] Y. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer-Verlag, New York, 1998. [13] P. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika, 45 (1958), 16-31. doi: 10.1093/biomet/45.1-2.16. [14] P. Leslie and J. Gower, The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47 (1960), 219-234. doi: 10.1093/biomet/47.3-4.219. [15] X. Li, W. Jiang and J. Shi, Hopf bifurcation and Turing instability in the reaction-diffusion Holling-Tanner predator-prey model, IMA. J. Appl. Math., 78 (2013), 287-306. doi: 10.1093/imamat/hxr050. [16] P. Liu, J. Shi, W. Wang and X. Feng, Bifurcation analysis of reaction-diffusion Schnakenberg model, J. Math. Chem., 51 (2013), 2001-2019. doi: 10.1007/s10910-013-0196-x. [17] M. Liu and K. Wang, Dynamics of a Leslie-Gower Holling-type Ⅱ predator-prey system with Lévy jumps, Nonlinear Anal. Theor., 85 (2013), 204-213. doi: 10.1016/j.na.2013.02.018. [18] Y. Ma, Global Hopf bifurcation in the Leslie-Gower predator-prey model with two delays, Nonl. Anal. Real World Appl., 13 (2012), 370-375. doi: 10.1016/j.nonrwa.2011.07.045. [19] J. Murray, Mathematical Biology, 2nd edition, Springer-Verlag Berlin Heidelberg, New York, 1993. [20] Y. Song, T. Zhang and Y. Peng, Turing-Hopf bifurcation in the reaction-diffusion equations and its applications, Commun. Nonlinear Sci., 33 (2016), 229-258. doi: 10.1016/j.cnsns.2015.10.002. [21] Y. Song and X. Zhou, Bifurcation analysis of a diffusive ratio-dependent predator-prey model, Nonliner Dynam., 78 (2014), 49-70. doi: 10.1007/s11071-014-1421-2. [22] X. Tang, Y. Song and T. Zhang, Turing-Hopf bifurcation analysis of a predator-prey model with herd behavior and cross-diffusion, Nonliner Dynam., 86 (2016), 73-89. doi: 10.1007/s11071-016-2873-3. [23] J. Wollkind, J. Collings and A. Logan, Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees, B. Math. Biol., 50 (1988), 379-409. doi: 10.1007/BF02459707. [24] R. Yang and Y. Song, Spatial resonance and Turing-Hopf bifurcation in the Gierer-Meinhardt model, Nonl. Anal. Real World Appl., 31 (2016), 356-387. doi: 10.1016/j.nonrwa.2016.02.006. [25] F. Yi, J. Wei and J. Shi, Diffusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonl. Anal. Real World Appl., 9 (2008), 1038-1051. doi: 10.1016/j.nonrwa.2007.02.005. [26] F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differ. Equations, 246 (2009), 1944-1977. doi: 10.1016/j.jde.2008.10.024. [27] J. Zhou, Positive steady state solutions of a Leslie-Gower predator-prey model with Holling type Ⅱ functional response and density-dependent difusion, Nonlinear Anal. Theor., 82 (2013), 47-65. doi: 10.1016/j.na.2012.12.014.
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 $u(x, 0) = 0.1296, v(x, 0) = 0.0154-0.001\cos x$
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