# American Institute of Mathematical Sciences

January  2018, 38(1): 43-62. doi: 10.3934/dcds.2018002

## Stability and bifurcation on predator-prey systems with nonlocal prey competition

 1 School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong, 510006, China 2 Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong, 264209, China

* Corresponding author

Received  November 2016 Revised  July 2017 Published  September 2017

Fund Project: The authors are supported by the National Natural Science Foundation of China (Nos. 11471085,11771109).

In this paper, we investigate diffusive predator-prey systems with nonlocal intraspecific competition of prey for resources. We prove the existence and uniqueness of positive steady states when the conversion rate is large. To show the existence of complex spatiotemporal patterns, we consider the Hopf bifurcation for a spatially homogeneous kernel function, by using the conversion rate as the bifurcation parameter. Our results suggest that Hopf bifurcation is more likely to occur with nonlocal competition of prey. Moreover, we find that the steady state can lose the stability when conversion rate passes through some Hopf bifurcation value, and the bifurcating periodic solutions near such bifurcation value can be spatially nonhomogeneous. This phenomenon is different from that for the model without nonlocal competition of prey, where the bifurcating periodic solutions are spatially homogeneous near such bifurcation value.

Citation: 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
##### References:
 [1] C.O. Alves, M. Delgado, M.A.S. Souto and A. Suárez, 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] H. Berestycki, G. Nadin, B. 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. [3] J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313-346.  doi: 10.1088/0951-7715/17/1/018. [4] N.F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099. [5] C.-C. Chen and L.-C. Hung, Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species, Commun. Pure Appl. Anal., 15 (2016), 1451-1469.  doi: 10.3934/cpaa.2016.15.1451. [6] S. Chen and J. Yu, Dynamics of a diffusive predator-prey system with a nonlinear growth rate for the predator, J. Differential Equations, 260 (2016), 7923-7939.  doi: 10.1016/j.jde.2016.02.007. [7] S. Chen and J. Yu, Stability analysis of a reaction-diffusion equation with spatiotemporal delay and Dirichlet boundary condition, J. Dyn. Diff. Equat., 28 (2016), 857-866.  doi: 10.1007/s10884-014-9384-z. [8] S. Chen and J. Yu, Stability and bifurcations in a nonlocal delayed reaction-diffusion population model, J. Differential Equations, 260 (2016), 218-240.  doi: 10.1016/j.jde.2015.08.038. [9] F.J. S.A. Corrêa, M. Delgado and A. Suárez, Some nonlinear heterogeneous problems with nonlocal reaction term, Advances in Differential Equations, 16 (2011), 623-641. [10] Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010. [11] Y. Du and S.-B. Hsu, On a nonlocal reaction-diffusion problem arising from the modeling of phytoplankton growth, SIAM J. Math. Anal., 42 (2010), 1305-1333.  doi: 10.1137/090775105. [12] Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475.  doi: 10.1090/S0002-9947-97-01842-4. [13] Y. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144 (1998), 390-440.  doi: 10.1006/jdeq.1997.3394. [14] Y. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349.  doi: 10.1017/S0308210500000895. [15] 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. [16] G. Faye and M. Holzer, Modulated traveling fronts for a nonlocal Fisher-KPP equation: A dynamical systems approach, J. Differential Equations, 258 (2015), 2257-2289.  doi: 10.1016/j.jde.2014.12.006. [17] J. Furter and M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80.  doi: 10.1007/BF00276081. [18] S.A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047. [19] 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. [20] 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. [21] S.A. Gourley, J.W.-H. So and J. Wu, Nonlocality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics, J. Math. Sci., 124 (2004), 5119-5153.  doi: 10.1023/B:JOTH.0000047249.39572.6d. [22] 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. [23] B.-S. Han, Z.-C. 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. [24] J. Jin, J. Shi, J. Wei and F. Yi, Bifurcations of patterned solutions in diffusive Lengyel-Epstein system of CIMA chemical reaction, Rocky Moun. J. Math., 43 (2013), 1637-1674.  doi: 10.1216/RMJ-2013-43-5-1637. [25] W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge, J. Differential Equations, 231 (2006), 534-550.  doi: 10.1016/j.jde.2006.08.001. [26] A. Leung, Limiting behaviour for a prey-predator model with diffusion and crowding effects, J. Math. Biol., 6 (1978), 87-93.  doi: 10.1007/BF02478520. [27] G.M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary $p$ in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406.  doi: 10.1137/S003614100343651X. [28] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7. [29] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157. [30] Y. Lou, W.-M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete Cont. Dyn. Syst., 35 (2015), 1589-1607.  doi: 10.3934/dcds.2015.35.1589. [31] A. Madzvamuse, H.S. Ndakwo and R. Barreira, Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion, Discrete Cont. Dyn. Syst., 36 (2016), 2133-2170.  doi: 10.3934/dcds.2016.36.2133. [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] R. Peng and J. Shi, Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differential Equations, 247 (2009), 866-886.  doi: 10.1016/j.jde.2009.03.008. [34] R. Peng, J. Shi and M. Wang, Stationary pattern of a ratio-dependent food chain model with diffusion, SIAM J. Appl. Math., 67 (2007), 1479-1503.  doi: 10.1137/05064624X. [35] R. Peng, J. Shi and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488.  doi: 10.1088/0951-7715/21/7/006. [36] R. Peng, F.-Q. Yi and X.-Q. Zhao, Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differential Equations, 254 (2013), 2465-2498.  doi: 10.1016/j.jde.2012.12.009. [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] L. Sun, J. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267-1278.  doi: 10.1007/s00033-012-0286-9. [39] C. Wang, R. Liu, J. Shi and C.M. del Rio, Traveling waves of a mutualistic model of mistletoes and birds, Discrete Cont. Dyn. Syst., 35 (2015), 1743-1765.  doi: 10.3934/dcds.2015.35.1743. [40] Y. Yamada, On logistic diffusion equations with nonlocal interaction terms, Nonlinear Anal., 118 (2015), 51-62.  doi: 10.1016/j.na.2015.01.016. [41] W.-b. Yang, J.-H. Wu and H. Nie, Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate, Commun. Pure Appl. Anal., 14 (2015), 1183-1204.  doi: 10.3934/cpaa.2015.14.1183. [42] F. Yi, J. 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. [43] J. Zhou, Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses, Commun. Pure Appl. Anal., 14 (2015), 1127-1145.  doi: 10.3934/cpaa.2015.14.1127. [44] J. Zhou and C. Mu, Coexistence states of a Holling type-II predator-prey system, J. Math. Anal. Appl., 369 (2010), 555-563.  doi: 10.1016/j.jmaa.2010.04.001.

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##### References:
 [1] C.O. Alves, M. Delgado, M.A.S. Souto and A. Suárez, 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] H. Berestycki, G. Nadin, B. 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. [3] J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313-346.  doi: 10.1088/0951-7715/17/1/018. [4] N.F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099. [5] C.-C. Chen and L.-C. Hung, Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive Lotka-Volterra systems of three competing species, Commun. Pure Appl. Anal., 15 (2016), 1451-1469.  doi: 10.3934/cpaa.2016.15.1451. [6] S. Chen and J. Yu, Dynamics of a diffusive predator-prey system with a nonlinear growth rate for the predator, J. Differential Equations, 260 (2016), 7923-7939.  doi: 10.1016/j.jde.2016.02.007. [7] S. Chen and J. Yu, Stability analysis of a reaction-diffusion equation with spatiotemporal delay and Dirichlet boundary condition, J. Dyn. Diff. Equat., 28 (2016), 857-866.  doi: 10.1007/s10884-014-9384-z. [8] S. Chen and J. Yu, Stability and bifurcations in a nonlocal delayed reaction-diffusion population model, J. Differential Equations, 260 (2016), 218-240.  doi: 10.1016/j.jde.2015.08.038. [9] F.J. S.A. Corrêa, M. Delgado and A. Suárez, Some nonlinear heterogeneous problems with nonlocal reaction term, Advances in Differential Equations, 16 (2011), 623-641. [10] Y. Du and S.-B. Hsu, A diffusive predator-prey model in heterogeneous environment, J. Differential Equations, 203 (2004), 331-364.  doi: 10.1016/j.jde.2004.05.010. [11] Y. Du and S.-B. Hsu, On a nonlocal reaction-diffusion problem arising from the modeling of phytoplankton growth, SIAM J. Math. Anal., 42 (2010), 1305-1333.  doi: 10.1137/090775105. [12] Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475.  doi: 10.1090/S0002-9947-97-01842-4. [13] Y. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations, 144 (1998), 390-440.  doi: 10.1006/jdeq.1997.3394. [14] Y. Du and Y. Lou, Qualitative behaviour of positive solutions of a predator-prey model: Effects of saturation, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 321-349.  doi: 10.1017/S0308210500000895. [15] 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. [16] G. Faye and M. Holzer, Modulated traveling fronts for a nonlocal Fisher-KPP equation: A dynamical systems approach, J. Differential Equations, 258 (2015), 2257-2289.  doi: 10.1016/j.jde.2014.12.006. [17] J. Furter and M. Grinfeld, Local vs. non-local interactions in population dynamics, J. Math. Biol., 27 (1989), 65-80.  doi: 10.1007/BF00276081. [18] S.A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047. [19] 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. [20] 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. [21] S.A. Gourley, J.W.-H. So and J. Wu, Nonlocality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics, J. Math. Sci., 124 (2004), 5119-5153.  doi: 10.1023/B:JOTH.0000047249.39572.6d. [22] 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. [23] B.-S. Han, Z.-C. 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. [24] J. Jin, J. Shi, J. Wei and F. Yi, Bifurcations of patterned solutions in diffusive Lengyel-Epstein system of CIMA chemical reaction, Rocky Moun. J. Math., 43 (2013), 1637-1674.  doi: 10.1216/RMJ-2013-43-5-1637. [25] W. Ko and K. Ryu, Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge, J. Differential Equations, 231 (2006), 534-550.  doi: 10.1016/j.jde.2006.08.001. [26] A. Leung, Limiting behaviour for a prey-predator model with diffusion and crowding effects, J. Math. Biol., 6 (1978), 87-93.  doi: 10.1007/BF02478520. [27] G.M. Lieberman, Bounds for the steady-state Sel'kov model for arbitrary $p$ in any number of dimensions, SIAM J. Math. Anal., 36 (2005), 1400-1406.  doi: 10.1137/S003614100343651X. [28] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7. [29] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157. [30] Y. Lou, W.-M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete Cont. Dyn. Syst., 35 (2015), 1589-1607.  doi: 10.3934/dcds.2015.35.1589. [31] A. Madzvamuse, H.S. Ndakwo and R. Barreira, Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion, Discrete Cont. Dyn. Syst., 36 (2016), 2133-2170.  doi: 10.3934/dcds.2016.36.2133. [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] R. Peng and J. Shi, Non-existence of non-constant positive steady states of two Holling type-II predator-prey systems: Strong interaction case, J. Differential Equations, 247 (2009), 866-886.  doi: 10.1016/j.jde.2009.03.008. [34] R. Peng, J. Shi and M. Wang, Stationary pattern of a ratio-dependent food chain model with diffusion, SIAM J. Appl. Math., 67 (2007), 1479-1503.  doi: 10.1137/05064624X. [35] R. Peng, J. Shi and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity, 21 (2008), 1471-1488.  doi: 10.1088/0951-7715/21/7/006. [36] R. Peng, F.-Q. Yi and X.-Q. Zhao, Spatiotemporal patterns in a reaction-diffusion model with the Degn-Harrison reaction scheme, J. Differential Equations, 254 (2013), 2465-2498.  doi: 10.1016/j.jde.2012.12.009. [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] L. Sun, J. Shi and Y. Wang, Existence and uniqueness of steady state solutions of a nonlocal diffusive logistic equation, Z. Angew. Math. Phys., 64 (2013), 1267-1278.  doi: 10.1007/s00033-012-0286-9. [39] C. Wang, R. Liu, J. Shi and C.M. del Rio, Traveling waves of a mutualistic model of mistletoes and birds, Discrete Cont. Dyn. Syst., 35 (2015), 1743-1765.  doi: 10.3934/dcds.2015.35.1743. [40] Y. Yamada, On logistic diffusion equations with nonlocal interaction terms, Nonlinear Anal., 118 (2015), 51-62.  doi: 10.1016/j.na.2015.01.016. [41] W.-b. Yang, J.-H. Wu and H. Nie, Some uniqueness and multiplicity results for a predator-prey dynamics with a nonlinear growth rate, Commun. Pure Appl. Anal., 14 (2015), 1183-1204.  doi: 10.3934/cpaa.2015.14.1183. [42] F. Yi, J. 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. [43] J. Zhou, Qualitative analysis of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses, Commun. Pure Appl. Anal., 14 (2015), 1127-1145.  doi: 10.3934/cpaa.2015.14.1127. [44] J. Zhou and C. Mu, Coexistence states of a Holling type-II predator-prey system, J. Math. Anal. Appl., 369 (2010), 555-563.  doi: 10.1016/j.jmaa.2010.04.001.
The constant steady state loses its stability through Hopf bifurcation, and the solution converges to the bifurcated spatially nonhomogeneous periodic solution. Here initial values: $u(x,0)=0.3+0.1\cos^2\frac{x}{4},v(x,0)=0.2+0.1\cos^2\frac{x}{2},x\in[0,2\pi]$. (Upper): $\gamma=4$; (Lower): $\gamma=9$.
The constant steady state loses its stability through Hopf bifurcation. (Upper): $\gamma=2.7$, and the solution converges to the bifurcated spatially nonhomogeneous periodic solution. Here initial values: $u(x,0)=0.3+0.1\cos^2\frac{x}{3},v(x,0)=0.2+0.1\cos^2\frac{x}{3},x\in[0,1.5\pi]$. (Lower): $\gamma=6$, and the solution converges to the bifurcated spatially homogeneous periodic solution. Here initial values: $u(x,0)=0.7+0.5\cos^2\frac{x}{3},v(x,0)=0.7+0.5\cos^2\frac{x}{3},x\in[0,1.5\pi]$.
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