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September  2020, 25(9): 3461-3489. 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  September 2020 Early access  April 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 and Continuous Dynamical Systems - B, 2020, 25 (9) : 3461-3489. doi: 10.3934/dcdsb.2020069
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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.

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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.

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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 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.

[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.

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.

[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.

[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.

[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]

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.

[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.

[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]

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[33]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 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.

[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.

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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