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Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition
Global convergence of a predator-prey model with stage structure and spatio-temporal delay
1. | Institute of Applied Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China |
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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