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Transversality for time-periodic competitive-cooperative tridiagonal systems
Spatial dynamics of a diffusive predator-prey model with stage structure
1. | School of Mathematics and Statistics, Lanzhou University , and Key Laboratory of Applied Mathematics and Complex Systems of Gansu province, Lanzhou, Gansu 730000, China |
2. | School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000 |
References:
[1] |
W. G. Aiello and H. I. Freedman, A time-delay model of single species growth with stage structure, Math. Biosci., 101 (1990), 139-153.
doi: 10.1016/0025-5564(90)90019-U. |
[2] |
J. Al-Omari and S. A. Gourley, Monotone travelling fronts in an age-structured reaction-diffusion model of a single species, J. Math. Biol., 45 (2002), 294-312.
doi: 10.1007/s002850200159. |
[3] |
J. Al-Omari and S. A. Gourley, A nonlocal reaction-diffusion model for a single species with stage structure and distributed maturation delay, European J. Appl. Math., 16 (2005), 37-51.
doi: 10.1017/S0956792504005716. |
[4] |
N. F. Britton, Aggregation and competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66.
doi: 10.1016/S0022-5193(89)80189-4. |
[5] |
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. |
[6] |
L. Berezansky, E. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Model., 34 (2010), 1405-1417.
doi: 10.1016/j.apm.2009.08.027. |
[7] |
D. Duehring and W. Huang, Periodic traveling waves for diffusion equations with time delayed and non-local responding reaction, J. Dynam. Differential Equations, 19 (2007), 457-477.
doi: 10.1007/s10884-006-9048-8. |
[8] |
T. Faria, W. Huang and J. Wu, Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229-261.
doi: 10.1098/rspa.2005.1554. |
[9] |
T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376.
doi: 10.1016/j.jde.2006.05.006. |
[10] |
S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with non-local effects, J. Math. Biol., 34 (1996), 297-333.
doi: 10.1007/BF00160498. |
[11] |
S. A. Gourley and Y. Kuang, Wavefronts and global stability is a time-delayed population model with stage structure, R. Soc. Lond. Proc., Ser. A: Math. phys. Eng. Sci., 459 (2003), 1563-1579.
doi: 10.1098/rspa.2002.1094. |
[12] |
S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.
doi: 10.1137/S003614100139991. |
[13] |
S. A. Gourley and J. H.-W. So, Extinction and wavefront propagation in a reaction-diffusion model of a structured population with distributed maturation delay, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 527-548.
doi: 10.1017/S0308210500002523. |
[14] |
S. A. Gourley, J. H.-W. 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. |
[15] |
S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, Fields Institute Communications, 48 (2006), 137-200. |
[16] |
W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.
doi: 10.1038/287017a0. |
[17] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, 25, Amer. Math. Soc., Providence, RI, 1988. |
[18] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[19] |
K. Hong and P. Weng, Stability and traveling waves of a stage-structured predator-prey model with Holling type-II functional response and harvesting, Nonlinear Anal. Real World Appl., 14 (2013), 83-103.
doi: 10.1016/j.nonrwa.2012.05.004. |
[20] |
W. Huang, Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response, J. Differential Equations, 244 (2008), 1230-1254.
doi: 10.1016/j.jde.2007.10.001. |
[21] |
Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.
doi: 10.1007/s00285-010-0346-8. |
[22] |
R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.1090/S0002-9947-1990-0967316-X. |
[23] |
J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, New York, 1986.
doi: 10.1007/978-3-662-13159-6. |
[24] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative systems, Math. Surveys and Monographs, 41, Amer. Math. Soc., Providence, RI, 1995. |
[25] |
J. W. -H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure I. Traveling wavefronts on unbounded domains, Proc. R. Soc. Lond., 475 (2001), 1841-1853.
doi: 10.1098/rspa.2001.0789. |
[26] |
J. W.-H. So and Y. Yang, Dirichlet problem for the diffusive Nicholson's blowflies equation, J. Differential Equations, 150 (1998), 317-348.
doi: 10.1006/jdeq.1998.3489. |
[27] |
H. R. Thieme, Convergence results and Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
doi: 10.1007/BF00173267. |
[28] |
H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.
doi: 10.1137/0524026. |
[29] |
H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.
doi: 10.1016/S0362-546X(00)00112-7. |
[30] |
H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.
doi: 10.1016/S0022-0396(03)00175-X. |
[31] |
Z.-C. Wang and W.-T. Li, Dynamics of a non-local delayed reaction-diffusion equation without quasi-monotonicity, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1081-1109.
doi: 10.1017/S0308210509000262. |
[32] |
J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Math. Sci., 119, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[33] |
J. Wu and X.-Q. Zhao, Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations, J. Differential Equations, 186 (2002), 470-484.
doi: 10.1016/S0022-0396(02)00012-8. |
[34] |
D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q., 11 (2003), 303-319. |
[35] |
R. Xu, A reaction-diffusion predator-prey model with stage structure and nonlocal delay, Appl. Math. Comput., 175 (2006), 984-1006.
doi: 10.1016/j.amc.2005.08.014. |
[36] |
R. Xu, Global convergence of a predator-prey model with stage structure and spatio-temporal delay, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 273-291.
doi: 10.3934/dcdsb.2011.15.273. |
[37] |
R. Xu, M. A. J. Chaplain and F. A. Davidson, Travelling wave and convergence in stage-structured reaction-diffusion competitive models with nonlocal delays, Chaos Solitons Fractals, 30 (2006), 974-992.
doi: 10.1016/j.chaos.2005.09.022. |
[38] |
R. Xu, M. A. J. Chaplain and F. A. Davidson, Global convergence of a reaction-diffusion predator-prey model with stage structure and nonlocal delays, Comput. Math. Appl., 53 (2007), 770-788.
doi: 10.1016/j.camwa.2007.02.002. |
[39] |
Y. Yang and J. W.-H. So, Dynamics for the diffusive Nicholson's blowflies equation. Dynamical systems and differential equations, Vol. II (Springfield, MO, 1996), Discrete Contin. Dynam. Systems, 2 (1998), 333-352. |
[40] |
T. Yi. Y. Chen and J. Wu, Threshold dynamics of a delayed reaction diffusion equation subject to the Dirichlet condition, J. Biol. Dyn., 3 (2009), 331-341.
doi: 10.1080/17513750802425656. |
[41] |
T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case, J. Differential Equations, 245 (2008), 3376-3388.
doi: 10.1016/j.jde.2008.03.007. |
[42] |
X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
[43] |
X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equation with time delay, Canad. Appl. Math. Quart., 17 (2009), 271-281. |
[44] |
X.-Q. Zhao, Spatial dynamics of some evolution system in biology, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific Publishing Co. Pvt. Ltd., Singapore, 2009, 332-363.
doi: 10.1142/9789812834744_0015. |
show all references
References:
[1] |
W. G. Aiello and H. I. Freedman, A time-delay model of single species growth with stage structure, Math. Biosci., 101 (1990), 139-153.
doi: 10.1016/0025-5564(90)90019-U. |
[2] |
J. Al-Omari and S. A. Gourley, Monotone travelling fronts in an age-structured reaction-diffusion model of a single species, J. Math. Biol., 45 (2002), 294-312.
doi: 10.1007/s002850200159. |
[3] |
J. Al-Omari and S. A. Gourley, A nonlocal reaction-diffusion model for a single species with stage structure and distributed maturation delay, European J. Appl. Math., 16 (2005), 37-51.
doi: 10.1017/S0956792504005716. |
[4] |
N. F. Britton, Aggregation and competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66.
doi: 10.1016/S0022-5193(89)80189-4. |
[5] |
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. |
[6] |
L. Berezansky, E. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Model., 34 (2010), 1405-1417.
doi: 10.1016/j.apm.2009.08.027. |
[7] |
D. Duehring and W. Huang, Periodic traveling waves for diffusion equations with time delayed and non-local responding reaction, J. Dynam. Differential Equations, 19 (2007), 457-477.
doi: 10.1007/s10884-006-9048-8. |
[8] |
T. Faria, W. Huang and J. Wu, Travelling waves for delayed reaction-diffusion equations with global response, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 229-261.
doi: 10.1098/rspa.2005.1554. |
[9] |
T. Faria and S. Trofimchuk, Nonmonotone travelling waves in a single species reaction-diffusion equation with delay, J. Differential Equations, 228 (2006), 357-376.
doi: 10.1016/j.jde.2006.05.006. |
[10] |
S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with non-local effects, J. Math. Biol., 34 (1996), 297-333.
doi: 10.1007/BF00160498. |
[11] |
S. A. Gourley and Y. Kuang, Wavefronts and global stability is a time-delayed population model with stage structure, R. Soc. Lond. Proc., Ser. A: Math. phys. Eng. Sci., 459 (2003), 1563-1579.
doi: 10.1098/rspa.2002.1094. |
[12] |
S. A. Gourley and S. Ruan, Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. Anal., 35 (2003), 806-822.
doi: 10.1137/S003614100139991. |
[13] |
S. A. Gourley and J. H.-W. So, Extinction and wavefront propagation in a reaction-diffusion model of a structured population with distributed maturation delay, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 527-548.
doi: 10.1017/S0308210500002523. |
[14] |
S. A. Gourley, J. H.-W. 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. |
[15] |
S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, Fields Institute Communications, 48 (2006), 137-200. |
[16] |
W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.
doi: 10.1038/287017a0. |
[17] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, 25, Amer. Math. Soc., Providence, RI, 1988. |
[18] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[19] |
K. Hong and P. Weng, Stability and traveling waves of a stage-structured predator-prey model with Holling type-II functional response and harvesting, Nonlinear Anal. Real World Appl., 14 (2013), 83-103.
doi: 10.1016/j.nonrwa.2012.05.004. |
[20] |
W. Huang, Traveling waves connecting equilibrium and periodic orbit for reaction-diffusion equations with time delay and nonlocal response, J. Differential Equations, 244 (2008), 1230-1254.
doi: 10.1016/j.jde.2007.10.001. |
[21] |
Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. Math. Biol., 62 (2011), 543-568.
doi: 10.1007/s00285-010-0346-8. |
[22] |
R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.1090/S0002-9947-1990-0967316-X. |
[23] |
J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, New York, 1986.
doi: 10.1007/978-3-662-13159-6. |
[24] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative systems, Math. Surveys and Monographs, 41, Amer. Math. Soc., Providence, RI, 1995. |
[25] |
J. W. -H. So, J. Wu and X. Zou, A reaction-diffusion model for a single species with age structure I. Traveling wavefronts on unbounded domains, Proc. R. Soc. Lond., 475 (2001), 1841-1853.
doi: 10.1098/rspa.2001.0789. |
[26] |
J. W.-H. So and Y. Yang, Dirichlet problem for the diffusive Nicholson's blowflies equation, J. Differential Equations, 150 (1998), 317-348.
doi: 10.1006/jdeq.1998.3489. |
[27] |
H. R. Thieme, Convergence results and Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.
doi: 10.1007/BF00173267. |
[28] |
H. R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24 (1993), 407-435.
doi: 10.1137/0524026. |
[29] |
H. R. Thieme and X.-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.
doi: 10.1016/S0362-546X(00)00112-7. |
[30] |
H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.
doi: 10.1016/S0022-0396(03)00175-X. |
[31] |
Z.-C. Wang and W.-T. Li, Dynamics of a non-local delayed reaction-diffusion equation without quasi-monotonicity, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 1081-1109.
doi: 10.1017/S0308210509000262. |
[32] |
J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Math. Sci., 119, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[33] |
J. Wu and X.-Q. Zhao, Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations, J. Differential Equations, 186 (2002), 470-484.
doi: 10.1016/S0022-0396(02)00012-8. |
[34] |
D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q., 11 (2003), 303-319. |
[35] |
R. Xu, A reaction-diffusion predator-prey model with stage structure and nonlocal delay, Appl. Math. Comput., 175 (2006), 984-1006.
doi: 10.1016/j.amc.2005.08.014. |
[36] |
R. Xu, Global convergence of a predator-prey model with stage structure and spatio-temporal delay, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 273-291.
doi: 10.3934/dcdsb.2011.15.273. |
[37] |
R. Xu, M. A. J. Chaplain and F. A. Davidson, Travelling wave and convergence in stage-structured reaction-diffusion competitive models with nonlocal delays, Chaos Solitons Fractals, 30 (2006), 974-992.
doi: 10.1016/j.chaos.2005.09.022. |
[38] |
R. Xu, M. A. J. Chaplain and F. A. Davidson, Global convergence of a reaction-diffusion predator-prey model with stage structure and nonlocal delays, Comput. Math. Appl., 53 (2007), 770-788.
doi: 10.1016/j.camwa.2007.02.002. |
[39] |
Y. Yang and J. W.-H. So, Dynamics for the diffusive Nicholson's blowflies equation. Dynamical systems and differential equations, Vol. II (Springfield, MO, 1996), Discrete Contin. Dynam. Systems, 2 (1998), 333-352. |
[40] |
T. Yi. Y. Chen and J. Wu, Threshold dynamics of a delayed reaction diffusion equation subject to the Dirichlet condition, J. Biol. Dyn., 3 (2009), 331-341.
doi: 10.1080/17513750802425656. |
[41] |
T. Yi and X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case, J. Differential Equations, 245 (2008), 3376-3388.
doi: 10.1016/j.jde.2008.03.007. |
[42] |
X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003.
doi: 10.1007/978-0-387-21761-1. |
[43] |
X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equation with time delay, Canad. Appl. Math. Quart., 17 (2009), 271-281. |
[44] |
X.-Q. Zhao, Spatial dynamics of some evolution system in biology, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific Publishing Co. Pvt. Ltd., Singapore, 2009, 332-363.
doi: 10.1142/9789812834744_0015. |
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