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 and 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), 269-286. doi: doi:10.1006/jmaa.2000.7381.

[2]

N. F. Britton, Spatial structures and periodic traveling waves in an integrodifferential reaction-diffusion population-model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: doi:10.1137/0150099.

[3]

J. Cui, L. Chen and W. Wang, The effect of dispersal on population growth with stage-structure, Comput. Math. Appl., 39 (2000), 91-102. doi: doi:10.1016/S0898-1221(99)00316-8.

[4]

S. A. Gourley, Instability in a predator-prey system with delay and spatial averaging, IMA J. Appl. Math., 56 (1996), 121-132. doi: doi:10.1093/imamat/56.2.121.

[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), 299-310. doi: doi:10.1093/imamat/51.3.299.

[6]

S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996), 297-333.

[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), 806-822. doi: doi:10.1137/S003614100139991.

[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), 391-412. doi: doi:10.1016/S0096-3003(02)00494-0.

[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), 49-78. doi: doi:10.1007/s002850100109.

[10]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin, New York, (1993).

[11]

M. W. Hirsch, The dynamical systems approach to differential equations, Bull. American Math. Soc., 11 (1984), 1-64. doi: doi:10.1090/S0273-0979-1984-15236-4.

[12]

Z. Lin, Time delayed parabolic system in a two-species competitive model with stage structure, J. Math. Anal. Appl., 315 (2006), 202-215. doi: doi:10.1016/j.jmaa.2005.06.012.

[13]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779. doi: doi:10.1006/jmaa.1996.0111.

[14]

C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal. TMA, 48 (2002), 349-362. doi: doi:10.1016/S0362-546X(00)00189-9.

[15]

C. V. Pao, Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays, J. Math. Anal. Appl., 281 (2003), 186-204.

[16]

C. V. Pao, Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays, Nonlinear Anal. RWA, 5 (2004), 91-104. doi: doi:10.1016/S1468-1218(03)00018-X.

[17]

M. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Physica D, 196 (2004), 172-192. doi: doi:10.1016/j.physd.2004.05.007.

[18]

W. Wang and L. Chen, A predator-prey system with stage-structure for predator, Comput. Math. Appl., 33 (1997), 83-91. doi: doi:10.1016/S0898-1221(97)00056-4.

[19]

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

[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), 1444-1460. doi: doi:10.1016/j.nonrwa.2007.03.015.

[21]

Y. Yamada, On a certain class of semilinear Volterra diffusion equations, J. Math. Anal. Appl., 88 (1982), 433-451. doi: doi:10.1016/0022-247X(82)90205-0.

[22]

Y. Yamada, Asymptotic stability for some systems of semilinear Volterra diffusion equations, J. Differential Equations, 52 (1984), 295-326. doi: doi:10.1016/0022-0396(84)90165-7.

[23]

X. Zhang, L. Chen and A.U. Neumann, The stage-structured predator-prey model and optimal havesting policy, Math. Biosci., 168 (2000), 201-210. doi: doi:10.1016/S0025-5564(00)00033-X.

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), 269-286. doi: doi:10.1006/jmaa.2000.7381.

[2]

N. F. Britton, Spatial structures and periodic traveling waves in an integrodifferential reaction-diffusion population-model, SIAM J. Appl. Math., 50 (1990), 1663-1688. doi: doi:10.1137/0150099.

[3]

J. Cui, L. Chen and W. Wang, The effect of dispersal on population growth with stage-structure, Comput. Math. Appl., 39 (2000), 91-102. doi: doi:10.1016/S0898-1221(99)00316-8.

[4]

S. A. Gourley, Instability in a predator-prey system with delay and spatial averaging, IMA J. Appl. Math., 56 (1996), 121-132. doi: doi:10.1093/imamat/56.2.121.

[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), 299-310. doi: doi:10.1093/imamat/51.3.299.

[6]

S. A. Gourley and N. F. Britton, A predator-prey reaction-diffusion system with nonlocal effects, J. Math. Biol., 34 (1996), 297-333.

[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), 806-822. doi: doi:10.1137/S003614100139991.

[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), 391-412. doi: doi:10.1016/S0096-3003(02)00494-0.

[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), 49-78. doi: doi:10.1007/s002850100109.

[10]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, Vol. 840, Springer-Verlag, Berlin, New York, (1993).

[11]

M. W. Hirsch, The dynamical systems approach to differential equations, Bull. American Math. Soc., 11 (1984), 1-64. doi: doi:10.1090/S0273-0979-1984-15236-4.

[12]

Z. Lin, Time delayed parabolic system in a two-species competitive model with stage structure, J. Math. Anal. Appl., 315 (2006), 202-215. doi: doi:10.1016/j.jmaa.2005.06.012.

[13]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779. doi: doi:10.1006/jmaa.1996.0111.

[14]

C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal. TMA, 48 (2002), 349-362. doi: doi:10.1016/S0362-546X(00)00189-9.

[15]

C. V. Pao, Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays, J. Math. Anal. Appl., 281 (2003), 186-204.

[16]

C. V. Pao, Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays, Nonlinear Anal. RWA, 5 (2004), 91-104. doi: doi:10.1016/S1468-1218(03)00018-X.

[17]

M. Wang, Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Physica D, 196 (2004), 172-192. doi: doi:10.1016/j.physd.2004.05.007.

[18]

W. Wang and L. Chen, A predator-prey system with stage-structure for predator, Comput. Math. Appl., 33 (1997), 83-91. doi: doi:10.1016/S0898-1221(97)00056-4.

[19]

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

[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), 1444-1460. doi: doi:10.1016/j.nonrwa.2007.03.015.

[21]

Y. Yamada, On a certain class of semilinear Volterra diffusion equations, J. Math. Anal. Appl., 88 (1982), 433-451. doi: doi:10.1016/0022-247X(82)90205-0.

[22]

Y. Yamada, Asymptotic stability for some systems of semilinear Volterra diffusion equations, J. Differential Equations, 52 (1984), 295-326. doi: doi:10.1016/0022-0396(84)90165-7.

[23]

X. Zhang, L. Chen and A.U. Neumann, The stage-structured predator-prey model and optimal havesting policy, Math. Biosci., 168 (2000), 201-210. doi: doi:10.1016/S0025-5564(00)00033-X.

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