American Institute of Mathematical Sciences

Advanced
January  2011, 15(1): 273-291. doi: 10.3934/dcdsb.2011.15.273

Global convergence of a predator-prey model with stage structure and spatio-temporal delay

Rui Xu 1,

1. 

Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China

Received  May 2009 Revised  February 2010 Published  October 2010

In this paper, a predator-prey model with stage structure for the predator and a spatio-temporal delay describing the gestation period of the predator under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states is established. Sufficient conditions are derived for the global attractiveness of the positive steady state and the global stability of the semi-trivial steady state of the proposed problem by using the method of upper-lower solutions and its associated monotone iteration scheme. Numerical simulations are carried out to illustrate the main results.
Keywords: stability., global convergence, Spatio-temporal delay, stage structure.
Mathematics Subject Classification: Primary: 35B35, 35K57; Secondary: 92D2.
Citation: Rui Xu. Global convergence of a predator-prey model with stage structure and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 273-291. doi: 10.3934/dcdsb.2011.15.273
References:
[1]

K. Boshaba and S. Ruan, Instability in diffusive ecological models with nonlocal delay effects,, J. Math. Anal. Appl., 258 (2001), 269.  doi: doi:10.1006/jmaa.2000.7381.  Google Scholar

[2]

N. F. Britton, Spatial structures and periodic traveling waves in an integrodifferential reaction-diffusion population-model,, SIAM J. Appl. Math., 50 (1990), 1663.  doi: doi:10.1137/0150099.  Google Scholar

[3]

J. Cui, L. Chen and W. Wang, The effect of dispersal on population growth with stage-structure,, Comput. Math. Appl., 39 (2000), 91.  doi: doi:10.1016/S0898-1221(99)00316-8.  Google Scholar

[4]

S. A. Gourley, Instability in a predator-prey system with delay and spatial averaging,, IMA J. Appl. Math., 56 (1996), 121.  doi: doi:10.1093/imamat/56.2.121.  Google Scholar

[5]

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.  doi: doi:10.1093/imamat/51.3.299.  Google Scholar

[6]

S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects,, J. Math. Biol., 34 (1996), 297.   Google Scholar

[7]

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: doi:10.1137/S003614100139991.  Google Scholar

[8]

S. A. Gourley and S. Ruan, Spatio-temporal delays in a nutrient-plankton model on a finite domain: Linear stability and bifurcations,, Appl. Math. Comput., 145 (2003), 391.  doi: doi:10.1016/S0096-3003(02)00494-0.  Google Scholar

[9]

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.  doi: doi:10.1007/s002850100109.  Google Scholar

[10]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1993).   Google Scholar

[11]

M. W. Hirsch, The dynamical systems approach to differential equations,, Bull. American Math. Soc., 11 (1984), 1.  doi: doi:10.1090/S0273-0979-1984-15236-4.  Google Scholar

[12]

Z. Lin, Time delayed parabolic system in a two-species competitive model with stage structure,, J. Math. Anal. Appl., 315 (2006), 202.  doi: doi:10.1016/j.jmaa.2005.06.012.  Google Scholar

[13]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays,, J. Math. Anal. Appl., 198 (1996), 751.  doi: doi:10.1006/jmaa.1996.0111.  Google Scholar

[14]

C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays,, Nonlinear Anal. TMA, 48 (2002), 349.  doi: doi:10.1016/S0362-546X(00)00189-9.  Google Scholar

[15]

C. V. Pao, Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays,, J. Math. Anal. Appl., 281 (2003), 186.   Google Scholar

[16]

C. V. Pao, Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays,, Nonlinear Anal. RWA, 5 (2004), 91.  doi: doi:10.1016/S1468-1218(03)00018-X.  Google Scholar

[17]

M. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion,, Physica D, 196 (2004), 172.  doi: doi:10.1016/j.physd.2004.05.007.  Google Scholar

[18]

W. Wang and L. Chen, A predator-prey system with stage-structure for predator,, Comput. Math. Appl., 33 (1997), 83.  doi: doi:10.1016/S0898-1221(97)00056-4.  Google Scholar

[19]

J. Wu, "Theory and Applications of Partial Functional Differential Equations,", Springer-Verlag, (1996).   Google Scholar

[20]

R. Xu and Z. Ma, Stability and Hopf bifurcation in a predator-prey model with stage structure for the predator,, Nonlinear Anal. RWA, 9 (2008), 1444.  doi: doi:10.1016/j.nonrwa.2007.03.015.  Google Scholar

[21]

Y. Yamada, On a certain class of semilinear Volterra diffusion equations,, J. Math. Anal. Appl., 88 (1982), 433.  doi: doi:10.1016/0022-247X(82)90205-0.  Google Scholar

[22]

Y. Yamada, Asymptotic stability for some systems of semilinear Volterra diffusion equations,, J. Differential Equations, 52 (1984), 295.  doi: doi:10.1016/0022-0396(84)90165-7.  Google Scholar

[23]

X. Zhang, L. Chen and A.U. Neumann, The stage-structured predator-prey model and optimal havesting policy,, Math. Biosci., 168 (2000), 201.  doi: doi:10.1016/S0025-5564(00)00033-X.  Google Scholar

show all references

References:
[1]

K. Boshaba and S. Ruan, Instability in diffusive ecological models with nonlocal delay effects,, J. Math. Anal. Appl., 258 (2001), 269.  doi: doi:10.1006/jmaa.2000.7381.  Google Scholar

[2]

N. F. Britton, Spatial structures and periodic traveling waves in an integrodifferential reaction-diffusion population-model,, SIAM J. Appl. Math., 50 (1990), 1663.  doi: doi:10.1137/0150099.  Google Scholar

[3]

J. Cui, L. Chen and W. Wang, The effect of dispersal on population growth with stage-structure,, Comput. Math. Appl., 39 (2000), 91.  doi: doi:10.1016/S0898-1221(99)00316-8.  Google Scholar

[4]

S. A. Gourley, Instability in a predator-prey system with delay and spatial averaging,, IMA J. Appl. Math., 56 (1996), 121.  doi: doi:10.1093/imamat/56.2.121.  Google Scholar

[5]

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.  doi: doi:10.1093/imamat/51.3.299.  Google Scholar

[6]

S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects,, J. Math. Biol., 34 (1996), 297.   Google Scholar

[7]

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: doi:10.1137/S003614100139991.  Google Scholar

[8]

S. A. Gourley and S. Ruan, Spatio-temporal delays in a nutrient-plankton model on a finite domain: Linear stability and bifurcations,, Appl. Math. Comput., 145 (2003), 391.  doi: doi:10.1016/S0096-3003(02)00494-0.  Google Scholar

[9]

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.  doi: doi:10.1007/s002850100109.  Google Scholar

[10]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1993).   Google Scholar

[11]

M. W. Hirsch, The dynamical systems approach to differential equations,, Bull. American Math. Soc., 11 (1984), 1.  doi: doi:10.1090/S0273-0979-1984-15236-4.  Google Scholar

[12]

Z. Lin, Time delayed parabolic system in a two-species competitive model with stage structure,, J. Math. Anal. Appl., 315 (2006), 202.  doi: doi:10.1016/j.jmaa.2005.06.012.  Google Scholar

[13]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays,, J. Math. Anal. Appl., 198 (1996), 751.  doi: doi:10.1006/jmaa.1996.0111.  Google Scholar

[14]

C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays,, Nonlinear Anal. TMA, 48 (2002), 349.  doi: doi:10.1016/S0362-546X(00)00189-9.  Google Scholar

[15]

C. V. Pao, Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays,, J. Math. Anal. Appl., 281 (2003), 186.   Google Scholar

[16]

C. V. Pao, Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays,, Nonlinear Anal. RWA, 5 (2004), 91.  doi: doi:10.1016/S1468-1218(03)00018-X.  Google Scholar

[17]

M. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion,, Physica D, 196 (2004), 172.  doi: doi:10.1016/j.physd.2004.05.007.  Google Scholar

[18]

W. Wang and L. Chen, A predator-prey system with stage-structure for predator,, Comput. Math. Appl., 33 (1997), 83.  doi: doi:10.1016/S0898-1221(97)00056-4.  Google Scholar

[19]

J. Wu, "Theory and Applications of Partial Functional Differential Equations,", Springer-Verlag, (1996).   Google Scholar

[20]

R. Xu and Z. Ma, Stability and Hopf bifurcation in a predator-prey model with stage structure for the predator,, Nonlinear Anal. RWA, 9 (2008), 1444.  doi: doi:10.1016/j.nonrwa.2007.03.015.  Google Scholar

[21]

Y. Yamada, On a certain class of semilinear Volterra diffusion equations,, J. Math. Anal. Appl., 88 (1982), 433.  doi: doi:10.1016/0022-247X(82)90205-0.  Google Scholar

[22]

Y. Yamada, Asymptotic stability for some systems of semilinear Volterra diffusion equations,, J. Differential Equations, 52 (1984), 295.  doi: doi:10.1016/0022-0396(84)90165-7.  Google Scholar

[23]

X. Zhang, L. Chen and A.U. Neumann, The stage-structured predator-prey model and optimal havesting policy,, Math. Biosci., 168 (2000), 201.  doi: doi:10.1016/S0025-5564(00)00033-X.  Google Scholar

[1]

Jingdong Wei, Jiangbo Zhou, Wenxia Chen, Zaili Zhen, Lixin Tian. Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2853-2886. doi: 10.3934/cpaa.2020125
[2]

Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082
[3]

Zhi-Xian Yu, Rong Yuan. Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 709-728. doi: 10.3934/dcdsb.2010.13.709
[4]

Cicely K. Macnamara, Mark A. J. Chaplain. Spatio-temporal models of synthetic genetic oscillators. Mathematical Biosciences & Engineering, 2017, 14 (1) : 249-262. doi: 10.3934/mbe.2017016
[5]

Francesca Sapuppo, Elena Umana, Mattia Frasca, Manuela La Rosa, David Shannahoff-Khalsa, Luigi Fortuna, Maide Bucolo. Complex spatio-temporal features in meg data. Mathematical Biosciences & Engineering, 2006, 3 (4) : 697-716. doi: 10.3934/mbe.2006.3.697
[6]

Noura Azzabou, Nikos Paragios. Spatio-temporal speckle reduction in ultrasound sequences. Inverse Problems & Imaging, 2010, 4 (2) : 211-222. doi: 10.3934/ipi.2010.4.211
[7]

Xiaoying Chen, Chong Zhang, Zonglin Shi, Weidong Xiao. Spatio-temporal keywords queries in HBase. Big Data & Information Analytics, 2016, 1 (1) : 81-91. doi: 10.3934/bdia.2016.1.81
[8]

Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017
[9]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020228
[10]

Buddhi Pantha, Judy Day, Suzanne Lenhart. Investigating the effects of intervention strategies in a spatio-temporal anthrax model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1607-1622. doi: 10.3934/dcdsb.2019242
[11]

Zhong Li, Maoan Han, Fengde Chen. Global stability of a predator-prey system with stage structure and mutual interference. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 173-187. doi: 10.3934/dcdsb.2014.19.173
[12]

Wenjia Jing, Panagiotis E. Souganidis, Hung V. Tran. Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 915-939. doi: 10.3934/dcdss.2018055
[13]

Raimund BÜrger, Gerardo Chowell, Elvis GavilÁn, Pep Mulet, Luis M. Villada. Numerical solution of a spatio-temporal gender-structured model for hantavirus infection in rodents. Mathematical Biosciences & Engineering, 2018, 15 (1) : 95-123. doi: 10.3934/mbe.2018004
[14]

Thomas Hillen, Jeffery Zielinski, Kevin J. Painter. Merging-emerging systems can describe spatio-temporal patterning in a chemotaxis model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2513-2536. doi: 10.3934/dcdsb.2013.18.2513
[15]

Lin Wang, James Watmough, Fang Yu. Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with Dirichlet boundary conditions. Mathematical Biosciences & Engineering, 2015, 12 (4) : 699-715. doi: 10.3934/mbe.2015.12.699
[16]

Zelik S.. Formally gradient reaction-diffusion systems in Rn have zero spatio-temporal topological. Conference Publications, 2003, 2003 (Special) : 960-966. doi: 10.3934/proc.2003.2003.960
[17]

Rodrigo A. Garrido, Ivan Aguirre. Emergency logistics for disaster management under spatio-temporal demand correlation: The earthquakes case. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019058
[18]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[19]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[20]

Cui-Ping Cheng, Wan-Tong Li, Zhi-Cheng Wang. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 559-575. doi: 10.3934/dcdsb.2010.13.559

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (24)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences