# American Institute of Mathematical Sciences

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

## 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

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.
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. 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. 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. doi: 10.1017/S0956792504005716. [4] N. F. Britton, Aggregation and competitive exclusion principle,, J. Theoret. Biol., 136 (1989), 57. 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. 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. 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. 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. 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. 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. 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., 459 (2003), 1563. 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. 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. 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. 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. [16] W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited,, Nature, 287 (1980), 17. doi: 10.1038/287017a0. [17] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surveys and Monographs, (1988). [18] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (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. 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. 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. 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. doi: 10.1090/S0002-9947-1990-0967316-X. [23] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations,, Springer, (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, (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. 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. 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. 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. 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. 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. 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. doi: 10.1017/S0308210509000262. [32] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Applied Math. Sci., (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. 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. [35] R. Xu, A reaction-diffusion predator-prey model with stage structure and nonlocal delay,, Appl. Math. Comput., 175 (2006), 984. 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. 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. 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. 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. [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. 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. doi: 10.1016/j.jde.2008.03.007. [42] X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (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. [44] X.-Q. Zhao, Spatial dynamics of some evolution system in biology,, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, (2009), 332. doi: 10.1142/9789812834744_0015.

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##### 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. 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. 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. doi: 10.1017/S0956792504005716. [4] N. F. Britton, Aggregation and competitive exclusion principle,, J. Theoret. Biol., 136 (1989), 57. 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. 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. 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. 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. 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. 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. 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., 459 (2003), 1563. 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. 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. 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. 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. [16] W. Gurney, S. Blythe and R. Nisbet, Nicholson's blowflies revisited,, Nature, 287 (1980), 17. doi: 10.1038/287017a0. [17] J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Math. Surveys and Monographs, (1988). [18] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (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. 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. 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. 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. doi: 10.1090/S0002-9947-1990-0967316-X. [23] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations,, Springer, (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, (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. 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. 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. 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. 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. 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. 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. doi: 10.1017/S0308210509000262. [32] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Applied Math. Sci., (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. 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. [35] R. Xu, A reaction-diffusion predator-prey model with stage structure and nonlocal delay,, Appl. Math. Comput., 175 (2006), 984. 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. 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. 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. 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. [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. 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. doi: 10.1016/j.jde.2008.03.007. [42] X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (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. [44] X.-Q. Zhao, Spatial dynamics of some evolution system in biology,, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, (2009), 332. doi: 10.1142/9789812834744_0015.
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