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doi: 10.3934/dcdsb.2020069

Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition

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

* Corresponding author: Weihua Jiang

Received  July 2019 Revised  October 2019 Published  January 2020

Fund Project: The authors are supported by the National Natural Science Foundation of China (No.11871176).

A diffusive predator-prey model with nonlocal prey competition and the homogeneous Neumann boundary conditions is considered, to explore the effects of nonlocal reaction term. Firstly, conditions of the occurrence of Hopf, Turing, Turing-Turing and double zero bifurcations, are established. Then, several concise formulas of computing normal form at a double zero singularity for partial functional differential equations, are provided. Next, via analyzing normal form derived by utilizing these formulas, we find that diffusive predator-prey system admits interesting spatiotemporal dynamics near the double zero singularity, like tristable phenomenon that a stable spatially inhomogeneous periodic solution with the shape of $ \cos\omega_0 t\cos\frac{x}{l}- $like which is unstable in model without nonlocal competition and also greatly different from these with the shape of $ \cos\omega_0 t+\cos\frac{x}{l}- $like resulting from Turing-Hopf bifurcation, coexists with a pair of spatially inhomogeneous steady states with the shape of $ \cos\frac{x}{l}- $like. At last, numerical simulations are shown to support theory analysis. These investigations indicate that nonlocal reaction term could stabilize spatially inhomogeneous periodic solutions with the shape of $ \cos\omega_0 t\cos\frac{kx}{l}- $like for reaction-diffusion systems subject to the homogeneous Neumann boundary conditions.

Citation: Xun Cao, Weihua Jiang. Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020069
References:
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W. Jiang, Q. An and J. Shi, Formulation of the normal form of Turing-Hopf bifurcation in partial functional differential equations, J. Differential Equations. doi: 10.1016/j.jde.2019.11.039.  Google Scholar

[24]

W. JiangH. Wang and X. Cao, Turing instability and Turing-Hopf bifurcation in diffusive Schnakenberg systems with gene expression time delay, J. Dynam. Differential Equations, 31 (2019), 2223-2247.  doi: 10.1007/s10884-018-9702-y.  Google Scholar

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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.  Google Scholar

[26]

Y. LamontagneC. Coutu and C. Rousseau, Bifurcation analysis of a predator-prey system with generalised Holling type Ⅲ functional response, J. Dynam. Differential Equations, 20 (2008), 535-571.  doi: 10.1007/s10884-008-9102-9.  Google Scholar

[27]

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[29]

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

[30]

Z. Ma and W. Li, Bifurcation analysis on a diffusive Holling-Tanner predator-prey model, Appl. Math. Model., 37 (2013), 4371-4384.  doi: 10.1016/j.apm.2012.09.036.  Google Scholar

[31]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[32]

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.  Google Scholar

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F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 47–100. doi: 10.1007/BF02684366.  Google Scholar

[34]

J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867.  doi: 10.2307/1936296.  Google Scholar

[35]

J. Wang, Spatiotemporal patterns of a homogeneous diffusive predator-prey system with Holling type Ⅲ functional response, J. Dynam. Differential Equations, 29 (2017), 1383-1409.  doi: 10.1007/s10884-016-9517-7.  Google Scholar

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J. WangJ. LiangY. Liu and J. Wang, Zero singularities of codimension two in a delayed predator-prey diffusion system, Neurocomputing, 227 (2017), 10-17.  doi: 10.1016/j.neucom.2016.07.060.  Google Scholar

[37]

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.  Google Scholar

[38]

X. Xu and J. Wei, Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 765-783.  doi: 10.3934/dcdsb.2018042.  Google Scholar

[39]

Y. Yamada, On logistic diffusion equations with nonlocal interaction terms, Nonlinear Anal., 118 (2015), 51-62.  doi: 10.1016/j.na.2015.01.016.  Google Scholar

[40]

R. Yang and Y. Song, Spatial resonance and Turing-Hopf bifurcations in the Gierer-Meinhardt model, Nonlinear Anal. Real World Appl., 31 (2016), 356-387.  doi: 10.1016/j.nonrwa.2016.02.006.  Google Scholar

[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.  Google Scholar

[42]

J. ZhangW. Li and X. 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.  Google Scholar

[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 (2002), 636-682.  doi: 10.1137/S0036139901397285.  Google Scholar

show all references

References:
[1]

C. O. AlvesM. DelgadoM. A. S. Souto and A. Suarez, Existence of positive solution of a nonlocal logistic population model, Z. Angew. Math. Phys., 66 (2015), 943-953.  doi: 10.1007/s00033-014-0458-x.  Google Scholar

[2]

Q. An and W. Jiang, Turing-Hopf bifurcation and spatio-temporal patterns of a ratio-dependent Holling-Tanner model with diffusion, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 22pp. doi: 10.1142/S0218127418501080.  Google Scholar

[3]

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.  Google Scholar

[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.  Google Scholar

[5]

R. I. Bogdanov, Versal deformation of a singular point of a vector field on the plane in the case of zero eigenvalues, Funkcional Anal. i Prilžoen., 9 (1975), 144–145. doi: 10.1007/BF01075453.  Google Scholar

[6]

N. F. Britton, Spatial structures and periodic traveling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.  Google Scholar

[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.  Google Scholar

[8]

S. ChenY. Lou and J. Wei, Hopf bifurcation in a delayed reaction-diffusion-advection population model, J. Differential Equations, 264 (2018), 5333-5359.  doi: 10.1016/j.jde.2018.01.008.  Google Scholar

[9]

S. Chen, J. Wei and K. Yang, Spatial nonhomogeneous periodic solutions induced by nonlocal prey competition in a diffusive predator-prey model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 19pp. doi: 10.1142/S0218127419500433.  Google Scholar

[10]

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.  Google Scholar

[11]

B. S. Choudhury and B. Nasipuri, Self-organized spatial patterns due to diffusion in a Holling-Tanner predator-prey model, Comput. Appl. Math., 34 (2015), 177-195.  doi: 10.1007/s40314-013-0111-x.  Google Scholar

[12]

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

[13]

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.  Google Scholar

[14]

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

[15]

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.  Google Scholar

[16]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[17]

S. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Anal. Real World Appl., 42 (2018), 448-477.  doi: 10.1016/j.nonrwa.2018.01.011.  Google Scholar

[18]

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.  Google Scholar

[19]

F. Hamel and L. Ryzhik, On the nonlocal Fisher-KPP equation: Steady states, spreading speed and global bounds, Nonlinearity, 27 (2014), 2735-2753.  doi: 10.1088/0951-7715/27/11/2735.  Google Scholar

[20]

B. HanZ. Wang and Z. Feng, Traveling waves for the nonlocal diffusive single species model with Allee effect, J. Math. Anal. Appl., 443 (2016), 243-264.  doi: 10.1016/j.jmaa.2016.05.031.  Google Scholar

[21]

Z. HuP. BiW. Ma and S. Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 93-112.  doi: 10.3934/dcdsb.2011.15.93.  Google Scholar

[22]

J. HuangS. Ruan and J. Song, Bifurcations in a predator-prey system of Leslie type with generalized Holling type Ⅲ functional response, J. Differential Equations, 257 (2014), 1721-1752.  doi: 10.1016/j.jde.2014.04.024.  Google Scholar

[23]

W. Jiang, Q. An and J. Shi, Formulation of the normal form of Turing-Hopf bifurcation in partial functional differential equations, J. Differential Equations. doi: 10.1016/j.jde.2019.11.039.  Google Scholar

[24]

W. JiangH. Wang and X. Cao, Turing instability and Turing-Hopf bifurcation in diffusive Schnakenberg systems with gene expression time delay, J. Dynam. Differential Equations, 31 (2019), 2223-2247.  doi: 10.1007/s10884-018-9702-y.  Google Scholar

[25]

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.  Google Scholar

[26]

Y. LamontagneC. Coutu and C. Rousseau, Bifurcation analysis of a predator-prey system with generalised Holling type Ⅲ functional response, J. Dynam. Differential Equations, 20 (2008), 535-571.  doi: 10.1007/s10884-008-9102-9.  Google Scholar

[27]

C. LiJ. Li and Z. Ma, Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1107-1116.  doi: 10.3934/dcdsb.2015.20.1107.  Google Scholar

[28]

X. LiW. 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.  Google Scholar

[29]

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

[30]

Z. Ma and W. Li, Bifurcation analysis on a diffusive Holling-Tanner predator-prey model, Appl. Math. Model., 37 (2013), 4371-4384.  doi: 10.1016/j.apm.2012.09.036.  Google Scholar

[31]

P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[32]

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.  Google Scholar

[33]

F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 47–100. doi: 10.1007/BF02684366.  Google Scholar

[34]

J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 56 (1975), 855-867.  doi: 10.2307/1936296.  Google Scholar

[35]

J. Wang, Spatiotemporal patterns of a homogeneous diffusive predator-prey system with Holling type Ⅲ functional response, J. Dynam. Differential Equations, 29 (2017), 1383-1409.  doi: 10.1007/s10884-016-9517-7.  Google Scholar

[36]

J. WangJ. LiangY. Liu and J. Wang, Zero singularities of codimension two in a delayed predator-prey diffusion system, Neurocomputing, 227 (2017), 10-17.  doi: 10.1016/j.neucom.2016.07.060.  Google Scholar

[37]

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.  Google Scholar

[38]

X. Xu and J. Wei, Turing-Hopf bifurcation of a class of modified Leslie-Gower model with diffusion, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 765-783.  doi: 10.3934/dcdsb.2018042.  Google Scholar

[39]

Y. Yamada, On logistic diffusion equations with nonlocal interaction terms, Nonlinear Anal., 118 (2015), 51-62.  doi: 10.1016/j.na.2015.01.016.  Google Scholar

[40]

R. Yang and Y. Song, Spatial resonance and Turing-Hopf bifurcations in the Gierer-Meinhardt model, Nonlinear Anal. Real World Appl., 31 (2016), 356-387.  doi: 10.1016/j.nonrwa.2016.02.006.  Google Scholar

[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.  Google Scholar

[42]

J. ZhangW. Li and X. 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.  Google Scholar

[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 (2002), 636-682.  doi: 10.1137/S0036139901397285.  Google Scholar

Figure 4.  For $ \left(d_1,c\right) = (0.91475,0.018418)\in \mathcal{D}_5 $, a large spatially inhomogeneous periodic solution with the shape of $ \cos\omega_0 t\cos\frac{x}{l}- $like and a pair of spatially inhomogeneous steady states with the shape of $ \cos\frac{x}{l}- $like, are stable
Figure 1.  Hopf bifurcation curve, Turing bifurcation curves and bifurcation set (a), and local bifurcation set in $ d_1\text{-}c $ plane and phase portraits (b)
Figure 2.  For $ \left(d_1,c\right) = (1.17475,0.017226)\in \mathcal{D}_1 $, the coexistence $ E_* $ is stable. And, the initial values are $ u(0,x) = 0.61803-0.1\cos \frac{x}{4}, v(0,x) = 0.61803-0.1\cos \frac{x}{4} $
Figure 3.  For $ \left(d_1,c\right) = (0.97475,0.017776)\in \mathcal{D}_3 $, a large spatially inhomogeneous periodic solution with the shape of $ \cos\omega_0 t\cos\frac{x}{l}- $like, is stable, where $ \frac{2\pi}{\omega_0} $ is the temporal period. The initial values are $ u(0,x) = 0.61803-0.1\cos \frac{x}{4}, v(0,x) = 0.61803-0.1\cos \frac{x}{4} $
Figure 5.  For $ \left(d_1,c\right) = (1.17475,0.013226)\in \mathcal{D}_6 $, a pair of spatially inhomogeneous steady states with the shape of $ \cos\frac{x}{l}- $like, are stable
Figure 6.  Numerical simulations show the differences between $ \cos\omega_0 t\cos \frac{x}{l} $ and $ \cos\omega_0 t+\cos \frac{x}{l} $, where $ \omega_0 = 0.1\pi,l = 4 $
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