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Biodiversity and vulnerability in a 3D mutualistic system
Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients
1. | Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Calle Tarfia s/n, 41012-Seville, Spain |
2. | Departamento de Matemática y Mecánica, I.I.M.A.S - U.N.A.M., Apdo. Postal 20-726, 01000 México D. F., Mexico |
References:
[1] |
S. Ahmad, Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations,, J. Math. Anal. Appl., 127 (1987), 377.
doi: 10.1016/0022-247X(87)90116-8. |
[2] |
J. Balbus and J. Mierczyński, Time-averaging and permanence in nonautonomous competitive systems of PDE's via Vance-Coddington estiamtes,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1407.
doi: 10.3934/dcdsb.2012.17.1407. |
[3] |
T. A. Burton and V. Hutson, Permanence for nonautonomous predator-prey systems,, Differential Integral Equations, 4 (1991), 1269.
|
[4] |
T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes,, Nonlinear Analysis, 72 (2010), 1967.
doi: 10.1016/j.na.2009.09.037. |
[5] |
D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems,, World Scientific Publishing Co. Pte. Ltd, (2004).
doi: 10.1142/9789812563088. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Colloquium Publications, (2002).
|
[7] |
B. D. Coleman, Nonautonomous logistic equations as models of the adjustment of populations to environmental change,, Math. Biosci, 45 (1979), 159.
doi: 10.1016/0025-5564(79)90057-9. |
[8] |
T. G. Hallam and C. E. Clarck, Nonautonomous logistic equations as models of populations in a deteriorating environment,, J. Theoret. Biol., 93 (1981), 301.
doi: 10.1016/0022-5193(81)90106-5. |
[9] |
J. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 9 (1989), 388.
doi: 10.1137/0520025. |
[10] |
V. Hutson and K. Schmitt, Permanence in dynamical systems,, Math. Biosci., 111 (1992), 1.
doi: 10.1016/0025-5564(92)90078-B. |
[11] |
P. E. Kloeden, Pullback attractors of nonautonomous semidynamical system,, Stochastics and Dynamics, 3 (2003), 101.
doi: 10.1142/S0219493703000632. |
[12] |
J. A. Langa, J. C. Robinson, A. Rodríguez-Bernal and A. Suárez, Permanence and asymptotically stable complete trajectories for nonautonomous Lotka-Volterra models with diffusion,, SIAM J. Math. Anal., 40 (2009), 2179.
doi: 10.1137/080721790. |
[13] |
J. A. Langa, J. C. Robinson and A. Suárez, Forwards and pullback behaviour of a non-autonomous Lotka-Volterra system,, Nonlinearity, 16 (2003), 1277.
doi: 10.1088/0951-7715/16/4/305. |
[14] |
J. A. Langa, A. Rodríguez-Bernal and S. Suárez, On the long time behaviour of non-autonomous Lotka-Volterra models with diffusion via the sub-supertrajectory method,, J. Differential Equations, 249 (2010), 414.
doi: 10.1016/j.jde.2010.04.001. |
[15] |
M. N. Nkashama, Dynamics of logistic equations with non-autonomous bounded coefficients,, Electron. J. Differential Equations, 2000 (2000).
|
[16] |
J. C. Robinson, Infinite-Dimensional Dynamical System. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge Text in Applied Mathematics, (2001).
doi: 10.1007/978-94-010-0732-0. |
[17] |
G. R. Sell, Nonautonomous differential equations and dynamical systems,, Trans. Amer. Math. Soc., 127 (1967), 241.
|
[18] |
J. Sugie, Y. Saito and M. Fan, Global asymptotic stability for predator-prey systems whose prey receives time-variation of the environment,, Proc. Amer. Math. Soc., 139 (2011), 3475.
doi: 10.1090/S0002-9939-2011-11124-9. |
[19] |
R. R. Vance and E. A. Coddington, A nonautonomous model of population growth,, J. Math. Biol., 27 (1989), 491.
doi: 10.1007/BF00288430. |
show all references
References:
[1] |
S. Ahmad, Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations,, J. Math. Anal. Appl., 127 (1987), 377.
doi: 10.1016/0022-247X(87)90116-8. |
[2] |
J. Balbus and J. Mierczyński, Time-averaging and permanence in nonautonomous competitive systems of PDE's via Vance-Coddington estiamtes,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1407.
doi: 10.3934/dcdsb.2012.17.1407. |
[3] |
T. A. Burton and V. Hutson, Permanence for nonautonomous predator-prey systems,, Differential Integral Equations, 4 (1991), 1269.
|
[4] |
T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes,, Nonlinear Analysis, 72 (2010), 1967.
doi: 10.1016/j.na.2009.09.037. |
[5] |
D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems,, World Scientific Publishing Co. Pte. Ltd, (2004).
doi: 10.1142/9789812563088. |
[6] |
V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Colloquium Publications, (2002).
|
[7] |
B. D. Coleman, Nonautonomous logistic equations as models of the adjustment of populations to environmental change,, Math. Biosci, 45 (1979), 159.
doi: 10.1016/0025-5564(79)90057-9. |
[8] |
T. G. Hallam and C. E. Clarck, Nonautonomous logistic equations as models of populations in a deteriorating environment,, J. Theoret. Biol., 93 (1981), 301.
doi: 10.1016/0022-5193(81)90106-5. |
[9] |
J. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 9 (1989), 388.
doi: 10.1137/0520025. |
[10] |
V. Hutson and K. Schmitt, Permanence in dynamical systems,, Math. Biosci., 111 (1992), 1.
doi: 10.1016/0025-5564(92)90078-B. |
[11] |
P. E. Kloeden, Pullback attractors of nonautonomous semidynamical system,, Stochastics and Dynamics, 3 (2003), 101.
doi: 10.1142/S0219493703000632. |
[12] |
J. A. Langa, J. C. Robinson, A. Rodríguez-Bernal and A. Suárez, Permanence and asymptotically stable complete trajectories for nonautonomous Lotka-Volterra models with diffusion,, SIAM J. Math. Anal., 40 (2009), 2179.
doi: 10.1137/080721790. |
[13] |
J. A. Langa, J. C. Robinson and A. Suárez, Forwards and pullback behaviour of a non-autonomous Lotka-Volterra system,, Nonlinearity, 16 (2003), 1277.
doi: 10.1088/0951-7715/16/4/305. |
[14] |
J. A. Langa, A. Rodríguez-Bernal and S. Suárez, On the long time behaviour of non-autonomous Lotka-Volterra models with diffusion via the sub-supertrajectory method,, J. Differential Equations, 249 (2010), 414.
doi: 10.1016/j.jde.2010.04.001. |
[15] |
M. N. Nkashama, Dynamics of logistic equations with non-autonomous bounded coefficients,, Electron. J. Differential Equations, 2000 (2000).
|
[16] |
J. C. Robinson, Infinite-Dimensional Dynamical System. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge Text in Applied Mathematics, (2001).
doi: 10.1007/978-94-010-0732-0. |
[17] |
G. R. Sell, Nonautonomous differential equations and dynamical systems,, Trans. Amer. Math. Soc., 127 (1967), 241.
|
[18] |
J. Sugie, Y. Saito and M. Fan, Global asymptotic stability for predator-prey systems whose prey receives time-variation of the environment,, Proc. Amer. Math. Soc., 139 (2011), 3475.
doi: 10.1090/S0002-9939-2011-11124-9. |
[19] |
R. R. Vance and E. A. Coddington, A nonautonomous model of population growth,, J. Math. Biol., 27 (1989), 491.
doi: 10.1007/BF00288430. |
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