American Institute of Mathematical Sciences

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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]

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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

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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

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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

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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

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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

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

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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

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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

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