American Institute of Mathematical Sciences

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August  2015, 20(6): 1831-1853. doi: 10.3934/dcdsb.2015.20.1831

Spatial dynamics of a diffusive predator-prey model with stage structure

Liang Zhang 1, and  Zhi-Cheng Wang 2,

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

Received  October 2013 Revised  April 2014 Published  June 2015

In this paper, we propose a nonlocal and time-delayed reaction-diffusion predator-prey model with stage structure. It is assumed that prey individuals undergo two stages, immature and mature, and the conversion of consumed prey biomass into predator biomass has a retardation. In terms of the principal eigenvalue of nonlocal elliptic eigenvalue problems, we establish the uniform persistence and global extinction for the model. In particular, the uniform persistence implies the existence of positive steady states. Finally, we investigate a specially spatially homogeneous predator-prey system and show the complicated dynamics of the system due to the non-local delay in the prey equation.
Keywords: principal eigenvalues, fluctuation methods, persistence and extinction, Predator-prey model, non-local delays, global attractivity..
Mathematics Subject Classification: 35K57, 35B35, 35B40, 92D25, 93B6.
Citation: Liang Zhang, Zhi-Cheng Wang. Spatial dynamics of a diffusive predator-prey model with stage structure. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1831-1853. doi: 10.3934/dcdsb.2015.20.1831
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

N. F. Britton, Aggregation and competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66. doi: 10.1016/S0022-5193(89)80189-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.  Google Scholar

[17]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, 25, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q., 11 (2003), 303-319.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

N. F. Britton, Aggregation and competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66. doi: 10.1016/S0022-5193(89)80189-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.  Google Scholar

[17]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs, 25, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[18]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q., 11 (2003), 303-319.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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