• Previous Article
    Maximal regularity for time-stepping schemes arising from convolution quadrature of non-local in time equations
  • DCDS Home
  • This Issue
  • Next Article
    Global solutions of semilinear parabolic equations with drift term on Riemannian manifolds
August  2022, 42(8): 3747-3785. doi: 10.3934/dcds.2022031

Bogdanov-Takens bifurcation with $ Z_2 $ symmetry and spatiotemporal dynamics in diffusive Rosenzweig-MacArthur model involving nonlocal prey competition

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Weihua Jiang

Received  July 2021 Revised  February 2022 Published  August 2022 Early access  March 2022

Fund Project: The authors are supported by the National Natural Science Foundation of China (No.11871176) and the Fundamental Research Funds for the Central Universities (FRFCU5630103121)

A diffusive Rosenzweig-MacArthur model involving nonlocal prey competition is studied. Via considering joint effects of prey's carrying capacity and predator's diffusion rate, the first Turing (Hopf) bifurcation curve is precisely described, which can help to determine the parameter region where coexistence equilibrium is stable. Particularly, coexistence equilibrium can lose its stability through not only codimension one Turing (Hopf) bifurcation, but also codimension two Bogdanov-Takens, Turing-Hopf and Hopf-Hopf bifurcations, even codimension three Bogdanov-Takens-Hopf bifurcation, etc., thus the concept of Turing (Hopf) instability is extended to high codimension bifurcation instability, such as Bogdanov-Takens instability. To meticulously describe spatiotemporal patterns resulting from $ Z_2 $ symmetric Bogdanov-Takens bifurcation, the corresponding third-order normal form for partial functional differential equations (PFDEs) involving nonlocal interactions is derived, which is expressed concisely by original PFDEs' parameters, making it convenient to analyze effects of original parameters on dynamics and also to calculate normal form on computer. With the aid of these formulas, complex spatiotemporal patterns are theoretically predicted and numerically shown, including tri-stable nonuniform patterns with the shape of $ \cos \omega t\cos \frac{x}{l}- $like or $ \cos \frac{x}{l}- $like, which reflects the effects of nonlocal interactions, such as stabilizing spatiotemporal nonuniform patterns.

Citation: Xun Cao, Xianyong Chen, Weihua Jiang. Bogdanov-Takens bifurcation with $ Z_2 $ symmetry and spatiotemporal dynamics in diffusive Rosenzweig-MacArthur model involving nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3747-3785. doi: 10.3934/dcds.2022031
References:
[1]

Q. An and W. Jiang, Turing-Hopf bifurcation and spatio-temporal patterns of a ratio-dependent Holling-Tanner model with diffusion, Int. J. Bifurcation Chaos, 28 (2018), 1850108, 22 pp. doi: 10.1142/S0218127418501080.

[2]

M. BaurmannT. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229.  doi: 10.1016/j.jtbi.2006.09.036.

[3]

H. BerestyckiG. NadinB. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.  doi: 10.1088/0951-7715/22/12/002.

[4]

J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313-346.  doi: 10.1088/0951-7715/17/1/018.

[5]

N. F. Britton, Aggregation and the competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66.  doi: 10.1016/S0022-5193(89)80189-4.

[6]

J. Cao, P. Wang, R. Yuan and Y. Mei, Bogdanov-Takens bifurcation of a class of delayed reaction-diffusion system, Int. J. Bifurcation Chaos, 25 (2015), 1550082, 11 pp. doi: 10.1142/S0218127415500820.

[7]

X. Cao and W. Jiang, Turing-Hopf bifurcation and spatiotemporal patterns in a diffusive predator-prey system with Crowley-Martin functional response, Nonlinear Anal. Real World Appl., 43 (2018), 428-450.  doi: 10.1016/j.nonrwa.2018.03.010.

[8]

X. Cao and W. Jiang, Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 3461-3489.  doi: 10.3934/dcdsb.2020069.

[9]

J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences, 35 Springer-Verlag, New York-Berlin, 1981.

[10]

S. Chen and J. Shi, Stability and hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 253 (2012), 3440-3470.  doi: 10.1016/j.jde.2012.08.031.

[11]

S. Chen and J. Yu, Stability and bifurcation on predator-prey systems with nonlocal prey competition, Discrete Contin. Dyn. Syst., 38 (2018), 43-62.  doi: 10.3934/dcds.2018002.

[12]

M. Chirilus-BrucknerP. Van HeijsterH. Ikeda and J. D. M. Rademacher, Unfolding symmetric Bogdanov-Takens bifurcations for front dynamics in a reaction-diffusion system, J. Nonlinear Sci., 29 (2019), 2911-2953.  doi: 10.1007/s00332-019-09563-2.

[13]

G. FanS. A. CampbellG. S. K. Wolkowicz and H. Zhu, The bifurcation study of 1:2 resonance in a delayed system of two coupled neurons, J. Dynam. Differential Equations, 25 (2013), 193-216.  doi: 10.1007/s10884-012-9279-9.

[14]

J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.  doi: 10.1088/0951-7715/24/11/002.

[15]

T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7.

[16]

T. Faria, Normal forms for semilinear functional differential equations in Banach spaces and applications. II, Discrete Contin. Dyn. Syst., 7 (2001), 155-176.  doi: 10.3934/dcds.2001.7.155.

[17]

T. Faria and L. T. Magalhaes, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations, 122 (1995), 201-224.  doi: 10.1006/jdeq.1995.1145.

[18]

H. FujiiM. Mimura and Y. Nishiura, A picture of the global bifurcation diagram in ecological interacting and diffusing systems, Phys. D, 5 (1982), 1-42.  doi: 10.1016/0167-2789(82)90048-3.

[19]

J. Furter and M. Grinfeld, Local vs non-local interactions in population-dynamics, J. Math. Biol., 27 (1989), 65-80.  doi: 10.1007/BF00276081.

[20]

M. Golubitsky and I. Stewart, The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Progress in Mathematics, 200. Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8167-8.

[21]

S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996), 297-333.  doi: 10.1007/BF00160498.

[22]

S. A. GourleyM. A. J. Chaplain and F. A. Davidson, Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation, Dyn. Syst., 16 (2001), 173-192.  doi: 10.1080/14689360116914.

[23]

S. A. Gourley and J. W.-H. So, Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, J. Math. Biol., 44 (2002), 49-78.  doi: 10.1007/s002850100109.

[24]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1990.

[25]

S. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differential Equations, 259 (2015), 1409-1448.  doi: 10.1016/j.jde.2015.03.006.

[26]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[27]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41. Cambridge University Press, Cambridge-New York, 1981.

[28]

W. JiangQ. An and J. Shi, Formulation of the normal form of Turing-Hopf bifurcation in partial functional differential equations, J. Differential Equations, 268 (2020), 6067-6102.  doi: 10.1016/j.jde.2019.11.039.

[29]

W. Jiang and Y. Yuan, Bogdanov-Takens singularity in Van der Pol's oscillator with delayed feedback, Phys. D, 227 (2007), 149-161.  doi: 10.1016/j.physd.2007.01.003.

[30]

Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993.

[31]

Y. Liu, D. Duan and B. Niu, Spatiotemporal dynamics in a diffusive predator prey model with group defense and nonlocal competition, Appl. Math. Lett., 103 (2020), 106175, 6 pp. doi: 10.1016/j.aml.2019.106175.

[32]

Z. Liu, P. Magal and D. Xiao, Bogdanov-Takens bifurcation in a predator-prey model, Z. Angew. Math. Phys., 67 (2016), Art. 137, 29 pp. doi: 10.1007/s00033-016-0724-1.

[33]

S. M. Merchant and W. Nagata, Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition, Theor. Popul. Biol., 80 (2011), 289-297.  doi: 10.1016/j.tpb.2011.10.001.

[34]

W. NiJ. Shi and M. Wang, Global stability and pattern formation in a nonlocal diffusive Lotka-Volterra competition model, J. Differential Equations, 264 (2018), 6891-6932.  doi: 10.1016/j.jde.2018.02.002.

[35]

F. Paquin-Lefebvre, B. Xu, K. L. Dipietro, A. E. Lindsay and A. Jilkine, Pattern formation in a coupled membrane-bulk reaction-diffusion model for intracellular polarization and oscillations, J. Theoret. Biol., 497 (2020), 110242, 23 pp. doi: 10.1016/j.jtbi.2020.110242.

[36]

J. W.-H. SoJ. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I. Travelling wavefronts on unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.

[37]

Y. Su and X. Zou, Transient oscillatory patterns in the diffusive non-local blowfly equation with delay under the zero-flux boundary condition, Nonlinearity, 27 (2014), 87-104.  doi: 10.1088/0951-7715/27/1/87.

[38]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, 119. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[39]

S. Wu and Y. Song, Stability and spatiotemporal dynamics in a diffusive predator-prey model with nonlocal prey competition, Nonlinear Anal. Real World Appl., 48 (2019), 12-39.  doi: 10.1016/j.nonrwa.2019.01.004.

[40]

D. Xiao and S. Ruan, Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response, J. Differential Equations, 176 (2001), 494-510.  doi: 10.1006/jdeq.2000.3982.

[41]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.

[42]

J.-F. ZhangW.-T. Li and X.-P. Yan, Multiple bifurcations in a delayed predator-prey diffusion system with a functional response, Nonlinear Anal. Real World Appl., 11 (2010), 2708-2725.  doi: 10.1016/j.nonrwa.2009.09.019.

[43]

H. ZhuS. A. Campbell and G. S. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63 (2003), 636-682.  doi: 10.1137/S0036139901397285.

show all references

References:
[1]

Q. An and W. Jiang, Turing-Hopf bifurcation and spatio-temporal patterns of a ratio-dependent Holling-Tanner model with diffusion, Int. J. Bifurcation Chaos, 28 (2018), 1850108, 22 pp. doi: 10.1142/S0218127418501080.

[2]

M. BaurmannT. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations, J. Theoret. Biol., 245 (2007), 220-229.  doi: 10.1016/j.jtbi.2006.09.036.

[3]

H. BerestyckiG. NadinB. Perthame and L. Ryzhik, The non-local Fisher-KPP equation: Travelling waves and steady states, Nonlinearity, 22 (2009), 2813-2844.  doi: 10.1088/0951-7715/22/12/002.

[4]

J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313-346.  doi: 10.1088/0951-7715/17/1/018.

[5]

N. F. Britton, Aggregation and the competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66.  doi: 10.1016/S0022-5193(89)80189-4.

[6]

J. Cao, P. Wang, R. Yuan and Y. Mei, Bogdanov-Takens bifurcation of a class of delayed reaction-diffusion system, Int. J. Bifurcation Chaos, 25 (2015), 1550082, 11 pp. doi: 10.1142/S0218127415500820.

[7]

X. Cao and W. Jiang, Turing-Hopf bifurcation and spatiotemporal patterns in a diffusive predator-prey system with Crowley-Martin functional response, Nonlinear Anal. Real World Appl., 43 (2018), 428-450.  doi: 10.1016/j.nonrwa.2018.03.010.

[8]

X. Cao and W. Jiang, Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 3461-3489.  doi: 10.3934/dcdsb.2020069.

[9]

J. Carr, Applications of Centre Manifold Theory, Applied Mathematical Sciences, 35 Springer-Verlag, New York-Berlin, 1981.

[10]

S. Chen and J. Shi, Stability and hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 253 (2012), 3440-3470.  doi: 10.1016/j.jde.2012.08.031.

[11]

S. Chen and J. Yu, Stability and bifurcation on predator-prey systems with nonlocal prey competition, Discrete Contin. Dyn. Syst., 38 (2018), 43-62.  doi: 10.3934/dcds.2018002.

[12]

M. Chirilus-BrucknerP. Van HeijsterH. Ikeda and J. D. M. Rademacher, Unfolding symmetric Bogdanov-Takens bifurcations for front dynamics in a reaction-diffusion system, J. Nonlinear Sci., 29 (2019), 2911-2953.  doi: 10.1007/s00332-019-09563-2.

[13]

G. FanS. A. CampbellG. S. K. Wolkowicz and H. Zhu, The bifurcation study of 1:2 resonance in a delayed system of two coupled neurons, J. Dynam. Differential Equations, 25 (2013), 193-216.  doi: 10.1007/s10884-012-9279-9.

[14]

J. Fang and X.-Q. Zhao, Monotone wavefronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.  doi: 10.1088/0951-7715/24/11/002.

[15]

T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7.

[16]

T. Faria, Normal forms for semilinear functional differential equations in Banach spaces and applications. II, Discrete Contin. Dyn. Syst., 7 (2001), 155-176.  doi: 10.3934/dcds.2001.7.155.

[17]

T. Faria and L. T. Magalhaes, Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity, J. Differential Equations, 122 (1995), 201-224.  doi: 10.1006/jdeq.1995.1145.

[18]

H. FujiiM. Mimura and Y. Nishiura, A picture of the global bifurcation diagram in ecological interacting and diffusing systems, Phys. D, 5 (1982), 1-42.  doi: 10.1016/0167-2789(82)90048-3.

[19]

J. Furter and M. Grinfeld, Local vs non-local interactions in population-dynamics, J. Math. Biol., 27 (1989), 65-80.  doi: 10.1007/BF00276081.

[20]

M. Golubitsky and I. Stewart, The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Progress in Mathematics, 200. Birkhäuser Verlag, Basel, 2002. doi: 10.1007/978-3-0348-8167-8.

[21]

S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996), 297-333.  doi: 10.1007/BF00160498.

[22]

S. A. GourleyM. A. J. Chaplain and F. A. Davidson, Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation, Dyn. Syst., 16 (2001), 173-192.  doi: 10.1080/14689360116914.

[23]

S. A. Gourley and J. W.-H. So, Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, J. Math. Biol., 44 (2002), 49-78.  doi: 10.1007/s002850100109.

[24]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1990.

[25]

S. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differential Equations, 259 (2015), 1409-1448.  doi: 10.1016/j.jde.2015.03.006.

[26]

J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[27]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41. Cambridge University Press, Cambridge-New York, 1981.

[28]

W. JiangQ. An and J. Shi, Formulation of the normal form of Turing-Hopf bifurcation in partial functional differential equations, J. Differential Equations, 268 (2020), 6067-6102.  doi: 10.1016/j.jde.2019.11.039.

[29]

W. Jiang and Y. Yuan, Bogdanov-Takens singularity in Van der Pol's oscillator with delayed feedback, Phys. D, 227 (2007), 149-161.  doi: 10.1016/j.physd.2007.01.003.

[30]

Y. Kuang, Delay Differential Equations: With Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993.

[31]

Y. Liu, D. Duan and B. Niu, Spatiotemporal dynamics in a diffusive predator prey model with group defense and nonlocal competition, Appl. Math. Lett., 103 (2020), 106175, 6 pp. doi: 10.1016/j.aml.2019.106175.

[32]

Z. Liu, P. Magal and D. Xiao, Bogdanov-Takens bifurcation in a predator-prey model, Z. Angew. Math. Phys., 67 (2016), Art. 137, 29 pp. doi: 10.1007/s00033-016-0724-1.

[33]

S. M. Merchant and W. Nagata, Instabilities and spatiotemporal patterns behind predator invasions with nonlocal prey competition, Theor. Popul. Biol., 80 (2011), 289-297.  doi: 10.1016/j.tpb.2011.10.001.

[34]

W. NiJ. Shi and M. Wang, Global stability and pattern formation in a nonlocal diffusive Lotka-Volterra competition model, J. Differential Equations, 264 (2018), 6891-6932.  doi: 10.1016/j.jde.2018.02.002.

[35]

F. Paquin-Lefebvre, B. Xu, K. L. Dipietro, A. E. Lindsay and A. Jilkine, Pattern formation in a coupled membrane-bulk reaction-diffusion model for intracellular polarization and oscillations, J. Theoret. Biol., 497 (2020), 110242, 23 pp. doi: 10.1016/j.jtbi.2020.110242.

[36]

J. W.-H. SoJ. Wu and X. Zou, A reaction-diffusion model for a single species with age structure. I. Travelling wavefronts on unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.

[37]

Y. Su and X. Zou, Transient oscillatory patterns in the diffusive non-local blowfly equation with delay under the zero-flux boundary condition, Nonlinearity, 27 (2014), 87-104.  doi: 10.1088/0951-7715/27/1/87.

[38]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, 119. Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[39]

S. Wu and Y. Song, Stability and spatiotemporal dynamics in a diffusive predator-prey model with nonlocal prey competition, Nonlinear Anal. Real World Appl., 48 (2019), 12-39.  doi: 10.1016/j.nonrwa.2019.01.004.

[40]

D. Xiao and S. Ruan, Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response, J. Differential Equations, 176 (2001), 494-510.  doi: 10.1006/jdeq.2000.3982.

[41]

F. YiJ. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024.

[42]

J.-F. ZhangW.-T. Li and X.-P. Yan, Multiple bifurcations in a delayed predator-prey diffusion system with a functional response, Nonlinear Anal. Real World Appl., 11 (2010), 2708-2725.  doi: 10.1016/j.nonrwa.2009.09.019.

[43]

H. ZhuS. A. Campbell and G. S. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63 (2003), 636-682.  doi: 10.1137/S0036139901397285.

Figure 1.  Turing and Hopf bifurcation curves, their locations and intersections in the first quadrant of $ K \text{-}d_2 $ plane, and the parameter region described by Theorem 2.13
Figure 2.  The zooming of Figure 1(a), 1(b), 1(c) and 1(d) respectively, focus on showing the first bifurcation curve and the parameter region where $ \left(\zeta, v_{\zeta}\right) $ is stable
Figure 3.  Bifurcation set in the first quadrant of $ K\text{-}d_2 $ plane for $ d_1 = 0.125, d = 0.6, \gamma = 0.72, l = 1.5 $
Figure 4.  Local bifurcation set near $ \left(K_1^{BT}, d_1^{BT}\right) $ and the corresponding phase portraits
Figure 5.  For $ \left(K, d_2\right) = (6.5591, 0.36298)\in \mathcal{D}_1 $, the coexistence equilibrium $ \left(\zeta, v_{\zeta}\right) $ is asymptotically stable. And, the initial values are $ u(0, x) = 5.0-0.1\cos \frac{2x}{3}, v(0, x) = 2.1688-0.1\cos \frac{2x}{3} $
Figure 6.  For $ \left(K, d_2\right) = \left(6.759, 0.33673\right)\in \mathcal{D}_3 $, a large-amplitude spatially inhomogeneous periodic solution with the shape of $ \cos\omega_0 t\cos\frac{x}{l}- $like, is stable. And the initial values are $ u(0, x) = 5.0-0.1\cos \frac{2x}{3}, v(0, x) = 2.1688-0.1\cos \frac{2x}{3} $
Figure 7.  For $ \left(K, d_2\right) = \left(7.259, 0.34983\right)\in \mathcal{D}_5 $, a large-amplitude spatially inhomogeneous periodic solution with the shape of $ \cos\omega_0 t\cos\frac{x}{l}- $like and a pair of nonconstant steady states with the shape of $ \cos\frac{x}{l}- $like, coexist stably
Figure 8.  For $ \left(K, d_2\right) = (7.2591, 0.37603)\in \mathcal{D}_6 $, the pair of nonconstant steady states with the shape of $ \cos\frac{x}{l}- $like, are stable
[1]

Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3253-3269. doi: 10.3934/dcdss.2020130

[2]

Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041

[3]

Jinfeng Wang, Hongxia Fan. Dynamics in a Rosenzweig-Macarthur predator-prey system with quiescence. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 909-918. doi: 10.3934/dcdsb.2016.21.909

[4]

Xiaoqing Lin, Yancong Xu, Daozhou Gao, Guihong Fan. Bifurcation and overexploitation in Rosenzweig-MacArthur model. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022094

[5]

Wei Feng, Nicole Rocco, Michael Freeze, Xin Lu. Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain. Discrete and Continuous Dynamical Systems - S, 2014, 7 (6) : 1215-1230. doi: 10.3934/dcdss.2014.7.1215

[6]

Xun Cao, Weihua Jiang. Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3461-3489. doi: 10.3934/dcdsb.2020069

[7]

Min Lu, Chuang Xiang, Jicai Huang. Bogdanov-Takens bifurcation in a SIRS epidemic model with a generalized nonmonotone incidence rate. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3125-3138. doi: 10.3934/dcdss.2020115

[8]

Hebai Chen, Xingwu Chen, Jianhua Xie. Global phase portrait of a degenerate Bogdanov-Takens system with symmetry. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1273-1293. doi: 10.3934/dcdsb.2017062

[9]

Hebai Chen, Xingwu Chen. Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ). Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4141-4170. doi: 10.3934/dcdsb.2018130

[10]

Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035

[11]

Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807

[12]

Wonlyul Ko, Inkyung Ahn. Pattern formation of a diffusive eco-epidemiological model with predator-prey interaction. Communications on Pure and Applied Analysis, 2018, 17 (2) : 375-389. doi: 10.3934/cpaa.2018021

[13]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002

[14]

Dingyong Bai, Jianshe Yu, Yun Kang. Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 2949-2973. doi: 10.3934/dcdss.2020132

[15]

Xiaoling Li, Guangping Hu, Zhaosheng Feng, Dongliang Li. A periodic and diffusive predator-prey model with disease in the prey. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021

[16]

Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure and Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026

[17]

Liang Zhang, Zhi-Cheng Wang. Spatial dynamics of a diffusive predator-prey model with stage structure. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1831-1853. doi: 10.3934/dcdsb.2015.20.1831

[18]

Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979

[19]

Yu-Xia Hao, Wan-Tong Li, Fei-Ying Yang. Traveling waves in a nonlocal dispersal predator-prey model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3113-3139. doi: 10.3934/dcdss.2020340

[20]

Hongyong Zhao, Daiyong Wu. Point to point traveling wave and periodic traveling wave induced by Hopf bifurcation for a diffusive predator-prey system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3271-3284. doi: 10.3934/dcdss.2020129

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (204)
  • HTML views (222)
  • Cited by (0)

Other articles
by authors

[Back to Top]