-
Previous Article
A mathematical and numerical analysis of the Maxwell-Stefan diffusion equations
- DCDS-B Home
- This Issue
-
Next Article
Convergence results for the vector penalty-projection and two-step artificial compressibility methods
Time-averaging and permanence in nonautonomous competitive systems of PDEs via Vance-Coddington estimates
1. | Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław, Poland, Poland |
References:
[1] |
S. Ahmad and A. C. Lazer, Necessary and sufficient average growth in a Lotka-Volterra system, Nonlinear Anal., 34 (1998), 191-228.
doi: 10.1016/S0362-546X(97)00602-0. |
[2] |
S. Ahmad and A. C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Lakshmikantham's Legacy: A Tribute on his 75th Birthday, Nonlinear Anal. Ser A: Theory Methods, 40 (2000), 37-49.
doi: 10.1016/S0362-546X(00)85003-8. |
[3] |
S. Ahmad and A. C. Lazer, Average growth and total permanence in a competitive Lotka-Volterra system, Ann. Mat. Pura Appl. (4), 185 (2006), suppl., S47-S67. |
[4] |
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. |
[5] |
R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,'' Wiley Ser. Math. Comput. Biol., John Wiley & Sons, Ltd., Chichester, 2003. |
[6] |
A. Friedman, "Partial Differential Equations of Parabolic Type,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[7] |
B. M. Garay and J. Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations, SIAM J. Math. Anal., 34 (2003), 1007-1039.
doi: 10.1137/S0036141001392815. |
[8] |
K. Gopalsamy, Persistence in periodic and almost periodic Lotka-Volterra systems, J. Math. Biol., 21 (1984), 145-148.
doi: 10.1007/BF00277666. |
[9] |
K. Gopalsamy, Global asymptotic stability in a periodic Lotka-Volterra system, J. Austral. Math. Soc. Ser. B, 27 (1985), 66-72.
doi: 10.1017/S0334270000004768. |
[10] |
K. Gopalsamy, Global asymptotic stability in an almost-periodic Lotka-Volterra system, J. Austral. Math. Soc. Ser. B, 27 (1986), 346-360.
doi: 10.1017/S0334270000004975. |
[11] |
T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339.
doi: 10.1007/BF00275641. |
[12] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'' Lecture Notes in Math., 840, Springer, Berlin-New York, 1981. |
[13] |
J. Hofbauer and S. J. Schreiber, Robust permanence for interacting structured populations, J. Differential Equations, 248 (2010), 1955-1971.
doi: 10.1016/j.jde.2009.11.010. |
[14] |
V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients, J. Differential Equations, 211 (2005), 135-161.
doi: 10.1016/j.jde.2004.06.003. |
[15] |
V. Hutson, W. Shen and G. T. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc., 129 (2001), 1669-1679.
doi: 10.1090/S0002-9939-00-05808-1. |
[16] |
J. Mierczyński and W. Shen, "Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications,'' Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., 139, CRC Press, Boca Raton, FL, 2008. |
[17] |
J. Mierczyński and W. Shen, Spectral theory for forward nonautonomous parabolic equations and applications, in "International Conference on Infinite Dimensional Dynamical Systems," York University, Toronto, September 24-28, 2008, dedicated to Professor George Sell on the occasion of his 70th birthday, Fields Inst. Commun., in press. |
[18] |
J. Mierczyński and W. Shen, Persistence in forward nonautonomous competitive systems of parabolic equations, J. Dynam. Differential Equations, 23 (2011), 551-571.
doi: 10.1007/s10884-010-9181-2. |
[19] |
J. Mierczyński, W. Shen and X. Zhao, Uniform persistence for nonautonomous and random parabolic Kolmogorov systems, J. Differential Equations, 204 (2004), 471-510.
doi: 10.1016/j.jde.2004.02.014. |
[20] |
J. Pętela (J. Balbus), Average conditions for Kolmogorov systems, Appl. Math. Comput., 215 (2009), 481-494.
doi: 10.1016/j.amc.2009.05.031. |
[21] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. |
[22] |
R. Redheffer, Nonautonomous Lotka-Volterra systems. I, J. Differential Equations, 127 (1996), 519-541.
doi: 10.1006/jdeq.1996.0081. |
[23] |
R. Redheffer, Nonautonomous Lotka-Volterra systems. II, J. Differential Equations, 132 (1996), 1-20.
doi: 10.1006/jdeq.1996.0168. |
[24] |
R. Redheffer, Generalized monotonicity, integral conditions and partial survival, J. Math. Biol., 40 (2000), 295-320.
doi: 10.1007/s002850050182. |
[25] |
R. Redheffer, Mean values and the nonautonomous May-Leonard equations, Nonlinear Anal. Real World Appl., 4 (2003), 301-306.
doi: 10.1016/S1468-1218(02)00021-4. |
[26] |
R. Redheffer and R. R. Vance, An equivalence theorem concerning population growth in a variable environment, Int. J. Math. Math. Sci., 2003, 2747-2758. |
[27] |
S. J. Schreiber, Criteria for $C^r$ robust permanence, J. Differential Equations, 162 (2000), 400-426.
doi: 10.1006/jdeq.1999.3719. |
[28] |
H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows, Proc. Amer. Math. Soc., 127 (1999), 2395-2403.
doi: 10.1090/S0002-9939-99-05034-0. |
[29] |
H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000), 173-201.
doi: 10.1016/S0025-5564(00)00018-3. |
[30] |
A. Tineo and C. Alvarez, A different consideration about the globally asymptotically stable solution of the periodic $n$-competing species problem, J. Math. Anal. Appl., 159 (1991), 44-50.
doi: 10.1016/0022-247X(91)90220-T. |
[31] |
R. R. Vance and E. A. Coddington, A nonautonomous model of population growth, J. Math. Biol., 27 (1989), 491-506.
doi: 10.1007/BF00288430. |
[32] |
F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems, Results Math., 21 (1992), 224-250. |
[33] |
J. Zhao, J. Jiang and A. C. Lazer, The permanence and global attractivity in a nonautonomous Lotka-Volterra system, Nonlinear Anal. Real World Appl., 5 (2004), 265-276.
doi: 10.1016/S1468-1218(03)00038-5. |
[34] |
X. Zhao, The qualitative analysis of $n$-species Lotka-Volterra periodic competition systems, Math. Comput. Modelling, 15 (1991), 3-8.
doi: 10.1016/0895-7177(91)90100-L. |
[35] |
X. Zhao, Uniform persistence in processes with application to nonautonomous competitive models, J. Math. Anal. Appl., 258 (2001), 87-101.
doi: 10.1006/jmaa.2000.7361. |
[36] |
X. Zhao, "Dynamical Systems in Population Biology,'' CMS Books Math./Ouvrages Math. SMC, 16, Springer-Verlag, New York, 2003. |
show all references
References:
[1] |
S. Ahmad and A. C. Lazer, Necessary and sufficient average growth in a Lotka-Volterra system, Nonlinear Anal., 34 (1998), 191-228.
doi: 10.1016/S0362-546X(97)00602-0. |
[2] |
S. Ahmad and A. C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system, Lakshmikantham's Legacy: A Tribute on his 75th Birthday, Nonlinear Anal. Ser A: Theory Methods, 40 (2000), 37-49.
doi: 10.1016/S0362-546X(00)85003-8. |
[3] |
S. Ahmad and A. C. Lazer, Average growth and total permanence in a competitive Lotka-Volterra system, Ann. Mat. Pura Appl. (4), 185 (2006), suppl., S47-S67. |
[4] |
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. |
[5] |
R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,'' Wiley Ser. Math. Comput. Biol., John Wiley & Sons, Ltd., Chichester, 2003. |
[6] |
A. Friedman, "Partial Differential Equations of Parabolic Type,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. |
[7] |
B. M. Garay and J. Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations, SIAM J. Math. Anal., 34 (2003), 1007-1039.
doi: 10.1137/S0036141001392815. |
[8] |
K. Gopalsamy, Persistence in periodic and almost periodic Lotka-Volterra systems, J. Math. Biol., 21 (1984), 145-148.
doi: 10.1007/BF00277666. |
[9] |
K. Gopalsamy, Global asymptotic stability in a periodic Lotka-Volterra system, J. Austral. Math. Soc. Ser. B, 27 (1985), 66-72.
doi: 10.1017/S0334270000004768. |
[10] |
K. Gopalsamy, Global asymptotic stability in an almost-periodic Lotka-Volterra system, J. Austral. Math. Soc. Ser. B, 27 (1986), 346-360.
doi: 10.1017/S0334270000004975. |
[11] |
T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations, J. Math. Biol., 24 (1986), 327-339.
doi: 10.1007/BF00275641. |
[12] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'' Lecture Notes in Math., 840, Springer, Berlin-New York, 1981. |
[13] |
J. Hofbauer and S. J. Schreiber, Robust permanence for interacting structured populations, J. Differential Equations, 248 (2010), 1955-1971.
doi: 10.1016/j.jde.2009.11.010. |
[14] |
V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients, J. Differential Equations, 211 (2005), 135-161.
doi: 10.1016/j.jde.2004.06.003. |
[15] |
V. Hutson, W. Shen and G. T. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators, Proc. Amer. Math. Soc., 129 (2001), 1669-1679.
doi: 10.1090/S0002-9939-00-05808-1. |
[16] |
J. Mierczyński and W. Shen, "Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications,'' Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., 139, CRC Press, Boca Raton, FL, 2008. |
[17] |
J. Mierczyński and W. Shen, Spectral theory for forward nonautonomous parabolic equations and applications, in "International Conference on Infinite Dimensional Dynamical Systems," York University, Toronto, September 24-28, 2008, dedicated to Professor George Sell on the occasion of his 70th birthday, Fields Inst. Commun., in press. |
[18] |
J. Mierczyński and W. Shen, Persistence in forward nonautonomous competitive systems of parabolic equations, J. Dynam. Differential Equations, 23 (2011), 551-571.
doi: 10.1007/s10884-010-9181-2. |
[19] |
J. Mierczyński, W. Shen and X. Zhao, Uniform persistence for nonautonomous and random parabolic Kolmogorov systems, J. Differential Equations, 204 (2004), 471-510.
doi: 10.1016/j.jde.2004.02.014. |
[20] |
J. Pętela (J. Balbus), Average conditions for Kolmogorov systems, Appl. Math. Comput., 215 (2009), 481-494.
doi: 10.1016/j.amc.2009.05.031. |
[21] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. |
[22] |
R. Redheffer, Nonautonomous Lotka-Volterra systems. I, J. Differential Equations, 127 (1996), 519-541.
doi: 10.1006/jdeq.1996.0081. |
[23] |
R. Redheffer, Nonautonomous Lotka-Volterra systems. II, J. Differential Equations, 132 (1996), 1-20.
doi: 10.1006/jdeq.1996.0168. |
[24] |
R. Redheffer, Generalized monotonicity, integral conditions and partial survival, J. Math. Biol., 40 (2000), 295-320.
doi: 10.1007/s002850050182. |
[25] |
R. Redheffer, Mean values and the nonautonomous May-Leonard equations, Nonlinear Anal. Real World Appl., 4 (2003), 301-306.
doi: 10.1016/S1468-1218(02)00021-4. |
[26] |
R. Redheffer and R. R. Vance, An equivalence theorem concerning population growth in a variable environment, Int. J. Math. Math. Sci., 2003, 2747-2758. |
[27] |
S. J. Schreiber, Criteria for $C^r$ robust permanence, J. Differential Equations, 162 (2000), 400-426.
doi: 10.1006/jdeq.1999.3719. |
[28] |
H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows, Proc. Amer. Math. Soc., 127 (1999), 2395-2403.
doi: 10.1090/S0002-9939-99-05034-0. |
[29] |
H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci., 166 (2000), 173-201.
doi: 10.1016/S0025-5564(00)00018-3. |
[30] |
A. Tineo and C. Alvarez, A different consideration about the globally asymptotically stable solution of the periodic $n$-competing species problem, J. Math. Anal. Appl., 159 (1991), 44-50.
doi: 10.1016/0022-247X(91)90220-T. |
[31] |
R. R. Vance and E. A. Coddington, A nonautonomous model of population growth, J. Math. Biol., 27 (1989), 491-506.
doi: 10.1007/BF00288430. |
[32] |
F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems, Results Math., 21 (1992), 224-250. |
[33] |
J. Zhao, J. Jiang and A. C. Lazer, The permanence and global attractivity in a nonautonomous Lotka-Volterra system, Nonlinear Anal. Real World Appl., 5 (2004), 265-276.
doi: 10.1016/S1468-1218(03)00038-5. |
[34] |
X. Zhao, The qualitative analysis of $n$-species Lotka-Volterra periodic competition systems, Math. Comput. Modelling, 15 (1991), 3-8.
doi: 10.1016/0895-7177(91)90100-L. |
[35] |
X. Zhao, Uniform persistence in processes with application to nonautonomous competitive models, J. Math. Anal. Appl., 258 (2001), 87-101.
doi: 10.1006/jmaa.2000.7361. |
[36] |
X. Zhao, "Dynamical Systems in Population Biology,'' CMS Books Math./Ouvrages Math. SMC, 16, Springer-Verlag, New York, 2003. |
[1] |
Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete and Continuous Dynamical Systems, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493 |
[2] |
Isabel Coelho, Carlota Rebelo, Elisa Sovrano. Extinction or coexistence in periodic Kolmogorov systems of competitive type. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5743-5764. doi: 10.3934/dcds.2021094 |
[3] |
Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106 |
[4] |
Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101 |
[5] |
Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631 |
[6] |
Yansu Ji, Jianwei Shen, Xiaochen Mao. Pattern formation of Brusselator in the reaction-diffusion system. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022103 |
[7] |
Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183 |
[8] |
Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245 |
[9] |
Bingtuan Li, William F. Fagan, Garrett Otto, Chunwei Wang. Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267 |
[10] |
Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189 |
[11] |
Ming Mei. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Conference Publications, 2009, 2009 (Special) : 526-535. doi: 10.3934/proc.2009.2009.526 |
[12] |
Wei Feng, Xin Lu. Global periodicity in a class of reaction-diffusion systems with time delays. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 69-78. doi: 10.3934/dcdsb.2003.3.69 |
[13] |
Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169 |
[14] |
Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032 |
[15] |
Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227 |
[16] |
José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299 |
[17] |
W. E. Fitzgibbon, M. Langlais, J.J. Morgan. A reaction-diffusion system modeling direct and indirect transmission of diseases. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 893-910. doi: 10.3934/dcdsb.2004.4.893 |
[18] |
Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039 |
[19] |
Sebastian Aniţa, Vincenzo Capasso. Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 1673-1684. doi: 10.3934/dcdsb.2012.17.1673 |
[20] |
Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]