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Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting
Traveling wave solutions in a nonlocal reaction-diffusion population model
1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China, China |
References:
[1] |
S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133.
doi: 10.1016/j.jde.2006.08.015. |
[2] |
M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099.
doi: 10.1016/j.aml.2012.05.006. |
[3] |
M. Alfaro, J. Coville and G. Raoul, Traveling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait, Comm. Partial Differential Equations, 38 (2013), 2126-2154.
doi: 10.1080/03605302.2013.828069. |
[4] |
M. Alfaro, J. Coville and G. Raoul, Bistable traveling waves for nonlocal reaction diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 1775-1791.
doi: 10.3934/dcds.2014.34.1775. |
[5] |
N. Apreutesei, A. Ducrot and V. Volpert, Traveling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 541-561.
doi: 10.3934/dcdsb.2009.11.541. |
[6] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[7] |
P. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Traveling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122.
doi: 10.1007/s00033-002-8145-8. |
[8] |
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. |
[9] |
J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313-346.
doi: 10.1088/0951-7715/17/1/018. |
[10] |
N. F. Britton, Aggregation and the competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66.
doi: 10.1016/S0022-5193(89)80189-4. |
[11] |
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. |
[12] |
I. Demin and V. Volpert, Existence of waves for a nonlocal reaction-diffusion equation, Math. Model. Nat. Phenom., 5 (2010), 80-101.
doi: 10.1051/mmnp/20105506. |
[13] |
K. Deng, On a nonlocal reaction-diffusion population model, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 65-73.
doi: 10.3934/dcdsb.2008.9.65. |
[14] |
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. |
[15] |
J. Fang and X.-Q. Zhao, Monotone wave fronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.
doi: 10.1088/0951-7715/24/11/002. |
[16] |
S. Genieys and B. Perthame, Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit, Math. Model. Nat. Phenom., 2 (2007), 135-151.
doi: 10.1051/mmnp:2008029. |
[17] |
S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82.
doi: 10.1051/mmnp:2006004. |
[18] |
A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations, 250 (2011), 1767-1787.
doi: 10.1016/j.jde.2010.11.011. |
[19] |
G. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. |
[20] |
S. A. Gourley, Traveling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.
doi: 10.1007/s002850000047. |
[21] |
S. A. Gourley and N. F. Britton, Instability of traveling wave solutions of a population model with nonlocal effects, IMA J. Appl. Math., 51 (1993), 299-310.
doi: 10.1093/imamat/51.3.299. |
[22] |
S. A. Gourley and N. F. Britton, On a modified Volterra population equation with diffusion, Nonlinear Anal., 21 (1993), 389-395.
doi: 10.1016/0362-546X(93)90082-4. |
[23] |
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. |
[24] |
S. A. Gourley, M. 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. |
[25] |
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. |
[26] |
G. Nadin, B. Perthame, L. Rossi and L. Ryzhik, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model, Math. Model. Nat. Phenom., 8 (2013), 33-41.
doi: 10.1051/mmnp/20138304. |
[27] |
G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Math. Acad. Sci. Paris, 349 (2011), 553-557.
doi: 10.1016/j.crma.2011.03.008. |
[28] |
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[29] |
C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125.
doi: 10.1137/050638011. |
[30] |
A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, Translated from the Russian manuscript by James F. Heyda. Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994. |
[31] |
Z.-C. Wang and W.-T. Li, Monotone travelling fronts of a food-limited population model with nonlocal delay, Nonlinear Anal. Real World Appl., 8 (2007), 699-712.
doi: 10.1016/j.nonrwa.2006.03.001. |
[32] |
Z.-C. Wang and W.-T. Li, Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays, Z. Angew. Math. Phys., 58 (2007), 571-591.
doi: 10.1007/s00033-006-5125-4. |
[33] |
Z.-C. Wang, W.-T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.
doi: 10.1016/j.jde.2005.08.010. |
[34] |
Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200.
doi: 10.1016/j.jde.2007.03.025. |
[35] |
Z.-C. Wang and J. Wu, Traveling waves in a bio-reactor model with stage-structure, J. Math. Anal. Appl., 385 (2012), 683-692.
doi: 10.1016/j.jmaa.2011.06.084. |
[36] |
Z.-C. Wang, J. Wu and R. Liu, Traveling waves of the spread of avian influenza, Proc. Amer. Math. Soc., 140 (2012), 3931-3946.
doi: 10.1090/S0002-9939-2012-11246-8. |
[37] |
G. X. Yang and J. Xu, Analysis of spatiotemporal patterns in a single species reaction-diffusion model with spatiotemporal delay, Nonlinear Anal. Real World Appl., 22 (2015), 54-65.
doi: 10.1016/j.nonrwa.2014.07.013. |
[38] |
Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Diffusion Equations, Science Publish, Beijing, 2011. |
show all references
References:
[1] |
S. Ai, Traveling wave fronts for generalized Fisher equations with spatio-temporal delays, J. Differential Equations, 232 (2007), 104-133.
doi: 10.1016/j.jde.2006.08.015. |
[2] |
M. Alfaro and J. Coville, Rapid traveling waves in the nonlocal Fisher equation connect two unstable states, Appl. Math. Lett., 25 (2012), 2095-2099.
doi: 10.1016/j.aml.2012.05.006. |
[3] |
M. Alfaro, J. Coville and G. Raoul, Traveling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait, Comm. Partial Differential Equations, 38 (2013), 2126-2154.
doi: 10.1080/03605302.2013.828069. |
[4] |
M. Alfaro, J. Coville and G. Raoul, Bistable traveling waves for nonlocal reaction diffusion equations, Discrete Contin. Dyn. Syst., 34 (2014), 1775-1791.
doi: 10.3934/dcds.2014.34.1775. |
[5] |
N. Apreutesei, A. Ducrot and V. Volpert, Traveling waves for integro-differential equations in population dynamics, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 541-561.
doi: 10.3934/dcdsb.2009.11.541. |
[6] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[7] |
P. Ashwin, M. V. Bartuccelli, T. J. Bridges and S. A. Gourley, Traveling fronts for the KPP equation with spatio-temporal delay, Z. Angew. Math. Phys., 53 (2002), 103-122.
doi: 10.1007/s00033-002-8145-8. |
[8] |
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. |
[9] |
J. Billingham, Dynamics of a strongly nonlocal reaction-diffusion population model, Nonlinearity, 17 (2004), 313-346.
doi: 10.1088/0951-7715/17/1/018. |
[10] |
N. F. Britton, Aggregation and the competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66.
doi: 10.1016/S0022-5193(89)80189-4. |
[11] |
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. |
[12] |
I. Demin and V. Volpert, Existence of waves for a nonlocal reaction-diffusion equation, Math. Model. Nat. Phenom., 5 (2010), 80-101.
doi: 10.1051/mmnp/20105506. |
[13] |
K. Deng, On a nonlocal reaction-diffusion population model, Discrete Contin. Dyn. Syst. Ser. B, 9 (2008), 65-73.
doi: 10.3934/dcdsb.2008.9.65. |
[14] |
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. |
[15] |
J. Fang and X.-Q. Zhao, Monotone wave fronts of the nonlocal Fisher-KPP equation, Nonlinearity, 24 (2011), 3043-3054.
doi: 10.1088/0951-7715/24/11/002. |
[16] |
S. Genieys and B. Perthame, Concentration in the nonlocal Fisher equation: the Hamilton-Jacobi limit, Math. Model. Nat. Phenom., 2 (2007), 135-151.
doi: 10.1051/mmnp:2008029. |
[17] |
S. Genieys, V. Volpert and P. Auger, Pattern and waves for a model in population dynamics with nonlocal consumption of resources, Math. Model. Nat. Phenom., 1 (2006), 65-82.
doi: 10.1051/mmnp:2006004. |
[18] |
A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations, 250 (2011), 1767-1787.
doi: 10.1016/j.jde.2010.11.011. |
[19] |
G. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. |
[20] |
S. A. Gourley, Traveling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.
doi: 10.1007/s002850000047. |
[21] |
S. A. Gourley and N. F. Britton, Instability of traveling wave solutions of a population model with nonlocal effects, IMA J. Appl. Math., 51 (1993), 299-310.
doi: 10.1093/imamat/51.3.299. |
[22] |
S. A. Gourley and N. F. Britton, On a modified Volterra population equation with diffusion, Nonlinear Anal., 21 (1993), 389-395.
doi: 10.1016/0362-546X(93)90082-4. |
[23] |
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. |
[24] |
S. A. Gourley, M. 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. |
[25] |
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. |
[26] |
G. Nadin, B. Perthame, L. Rossi and L. Ryzhik, Wave-like solutions for nonlocal reaction-diffusion equations: A toy model, Math. Model. Nat. Phenom., 8 (2013), 33-41.
doi: 10.1051/mmnp/20138304. |
[27] |
G. Nadin, B. Perthame and M. Tang, Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation, C. R. Math. Acad. Sci. Paris, 349 (2011), 553-557.
doi: 10.1016/j.crma.2011.03.008. |
[28] |
D. H. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), 312-355.
doi: 10.1016/0001-8708(76)90098-0. |
[29] |
C. Ou and J. Wu, Traveling wavefronts in a delayed food-limited population model, SIAM J. Math. Anal., 39 (2007), 103-125.
doi: 10.1137/050638011. |
[30] |
A. Volpert, V. Volpert and V. Volpert, Traveling Wave Solutions of Parabolic Systems, Translated from the Russian manuscript by James F. Heyda. Translations of Mathematical Monographs, 140. American Mathematical Society, Providence, RI, 1994. |
[31] |
Z.-C. Wang and W.-T. Li, Monotone travelling fronts of a food-limited population model with nonlocal delay, Nonlinear Anal. Real World Appl., 8 (2007), 699-712.
doi: 10.1016/j.nonrwa.2006.03.001. |
[32] |
Z.-C. Wang and W.-T. Li, Traveling fronts in diffusive and cooperative Lotka-Volterra system with nonlocal delays, Z. Angew. Math. Phys., 58 (2007), 571-591.
doi: 10.1007/s00033-006-5125-4. |
[33] |
Z.-C. Wang, W.-T. Li and S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.
doi: 10.1016/j.jde.2005.08.010. |
[34] |
Z.-C. Wang, W.-T. Li and S. Ruan, Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay, J. Differential Equations, 238 (2007), 153-200.
doi: 10.1016/j.jde.2007.03.025. |
[35] |
Z.-C. Wang and J. Wu, Traveling waves in a bio-reactor model with stage-structure, J. Math. Anal. Appl., 385 (2012), 683-692.
doi: 10.1016/j.jmaa.2011.06.084. |
[36] |
Z.-C. Wang, J. Wu and R. Liu, Traveling waves of the spread of avian influenza, Proc. Amer. Math. Soc., 140 (2012), 3931-3946.
doi: 10.1090/S0002-9939-2012-11246-8. |
[37] |
G. X. Yang and J. Xu, Analysis of spatiotemporal patterns in a single species reaction-diffusion model with spatiotemporal delay, Nonlinear Anal. Real World Appl., 22 (2015), 54-65.
doi: 10.1016/j.nonrwa.2014.07.013. |
[38] |
Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction of Reaction-Diffusion Equations, Science Publish, Beijing, 2011. |
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