American Institute of Mathematical Sciences

Advanced
July  2012, 17(5): 1407-1425. doi: 10.3934/dcdsb.2012.17.1407

Time-averaging and permanence in nonautonomous competitive systems of PDEs via Vance-Coddington estimates

Joanna Balbus 1, and  Janusz Mierczyński 1,

1. 

Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław, Poland, Poland

Received  February 2011 Revised  January 2012 Published  March 2012

One of the mathematically challenging problems in the population dynamics is finding conditions under which all of the populations coexist. A mathematical formulation of this notion is the concept of permanence, sometimes called also uniform persistence. In this article we give conditions for permanence in nonautonomous competitive Kolmogorov systems of reaction-diffusion equations. Those conditions are in a form of inequalities involving time-averages of intrinsic growth rates, as well as interaction coefficients, migration rates and principal eigenvalues. The proofs use estimates due to R. R. Vance and E. A. Coddington. Connections with invasibility via the principal spectrum theory are also investigated.
Keywords: Reaction-diffusion system of Kolmogorov type, time-averaging, principal spectrum., invasibility, permanence, competitive.
Mathematics Subject Classification: Primary: 35K57, 35Q92, 92D25; Secondary: 34C9.
Citation: Joanna Balbus, Janusz Mierczyński. Time-averaging and permanence in nonautonomous competitive systems of PDEs via Vance-Coddington estimates. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1407-1425. doi: 10.3934/dcdsb.2012.17.1407
References:
[1]

S. Ahmad and A. C. Lazer, Necessary and sufficient average growth in a Lotka-Volterra system,, Nonlinear Anal., 34 (1998), 191.  doi: 10.1016/S0362-546X(97)00602-0.  Google Scholar

[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, 40 (2000), 37.  doi: 10.1016/S0362-546X(00)85003-8.  Google Scholar

[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).   Google Scholar

[4]

H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13.   Google Scholar

[5]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,'', Wiley Ser. Math. Comput. Biol., (2003).   Google Scholar

[6]

A. Friedman, "Partial Differential Equations of Parabolic Type,'', Prentice-Hall, (1964).   Google Scholar

[7]

B. M. Garay and J. Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations,, SIAM J. Math. Anal., 34 (2003), 1007.  doi: 10.1137/S0036141001392815.  Google Scholar

[8]

K. Gopalsamy, Persistence in periodic and almost periodic Lotka-Volterra systems,, J. Math. Biol., 21 (1984), 145.  doi: 10.1007/BF00277666.  Google Scholar

[9]

K. Gopalsamy, Global asymptotic stability in a periodic Lotka-Volterra system,, J. Austral. Math. Soc. Ser. B, 27 (1985), 66.  doi: 10.1017/S0334270000004768.  Google Scholar

[10]

K. Gopalsamy, Global asymptotic stability in an almost-periodic Lotka-Volterra system,, J. Austral. Math. Soc. Ser. B, 27 (1986), 346.  doi: 10.1017/S0334270000004975.  Google Scholar

[11]

T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations,, J. Math. Biol., 24 (1986), 327.  doi: 10.1007/BF00275641.  Google Scholar

[12]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'', Lecture Notes in Math., 840 (1981).   Google Scholar

[13]

J. Hofbauer and S. J. Schreiber, Robust permanence for interacting structured populations,, J. Differential Equations, 248 (2010), 1955.  doi: 10.1016/j.jde.2009.11.010.  Google Scholar

[14]

V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients,, J. Differential Equations, 211 (2005), 135.  doi: 10.1016/j.jde.2004.06.003.  Google Scholar

[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.  doi: 10.1090/S0002-9939-00-05808-1.  Google Scholar

[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 (2008).   Google Scholar

[17]

J. Mierczyński and W. Shen, Spectral theory for forward nonautonomous parabolic equations and applications,, in, (2008), 24.   Google Scholar

[18]

J. Mierczyński and W. Shen, Persistence in forward nonautonomous competitive systems of parabolic equations,, J. Dynam. Differential Equations, 23 (2011), 551.  doi: 10.1007/s10884-010-9181-2.  Google Scholar

[19]

J. Mierczyński, W. Shen and X. Zhao, Uniform persistence for nonautonomous and random parabolic Kolmogorov systems,, J. Differential Equations, 204 (2004), 471.  doi: 10.1016/j.jde.2004.02.014.  Google Scholar

[20]

J. Pętela (J. Balbus), Average conditions for Kolmogorov systems,, Appl. Math. Comput., 215 (2009), 481.  doi: 10.1016/j.amc.2009.05.031.  Google Scholar

[21]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'', Prentice-Hall, (1967).   Google Scholar

[22]

R. Redheffer, Nonautonomous Lotka-Volterra systems. I,, J. Differential Equations, 127 (1996), 519.  doi: 10.1006/jdeq.1996.0081.  Google Scholar

[23]

R. Redheffer, Nonautonomous Lotka-Volterra systems. II,, J. Differential Equations, 132 (1996), 1.  doi: 10.1006/jdeq.1996.0168.  Google Scholar

[24]

R. Redheffer, Generalized monotonicity, integral conditions and partial survival,, J. Math. Biol., 40 (2000), 295.  doi: 10.1007/s002850050182.  Google Scholar

[25]

R. Redheffer, Mean values and the nonautonomous May-Leonard equations,, Nonlinear Anal. Real World Appl., 4 (2003), 301.  doi: 10.1016/S1468-1218(02)00021-4.  Google Scholar

[26]

R. Redheffer and R. R. Vance, An equivalence theorem concerning population growth in a variable environment,, Int. J. Math. Math. Sci., 2003 (): 2747.   Google Scholar

[27]

S. J. Schreiber, Criteria for $C^r$ robust permanence,, J. Differential Equations, 162 (2000), 400.  doi: 10.1006/jdeq.1999.3719.  Google Scholar

[28]

H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows,, Proc. Amer. Math. Soc., 127 (1999), 2395.  doi: 10.1090/S0002-9939-99-05034-0.  Google Scholar

[29]

H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology,, Math. Biosci., 166 (2000), 173.  doi: 10.1016/S0025-5564(00)00018-3.  Google Scholar

[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.  doi: 10.1016/0022-247X(91)90220-T.  Google Scholar

[31]

R. R. Vance and E. A. Coddington, A nonautonomous model of population growth,, J. Math. Biol., 27 (1989), 491.  doi: 10.1007/BF00288430.  Google Scholar

[32]

F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems,, Results Math., 21 (1992), 224.   Google Scholar

[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.  doi: 10.1016/S1468-1218(03)00038-5.  Google Scholar

[34]

X. Zhao, The qualitative analysis of $n$-species Lotka-Volterra periodic competition systems,, Math. Comput. Modelling, 15 (1991), 3.  doi: 10.1016/0895-7177(91)90100-L.  Google Scholar

[35]

X. Zhao, Uniform persistence in processes with application to nonautonomous competitive models,, J. Math. Anal. Appl., 258 (2001), 87.  doi: 10.1006/jmaa.2000.7361.  Google Scholar

[36]

X. Zhao, "Dynamical Systems in Population Biology,'', CMS Books Math./Ouvrages Math. SMC, 16 (2003).   Google Scholar

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.  doi: 10.1016/S0362-546X(97)00602-0.  Google Scholar

[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, 40 (2000), 37.  doi: 10.1016/S0362-546X(00)85003-8.  Google Scholar

[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).   Google Scholar

[4]

H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13.   Google Scholar

[5]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations,'', Wiley Ser. Math. Comput. Biol., (2003).   Google Scholar

[6]

A. Friedman, "Partial Differential Equations of Parabolic Type,'', Prentice-Hall, (1964).   Google Scholar

[7]

B. M. Garay and J. Hofbauer, Robust permanence for ecological differential equations, minimax, and discretizations,, SIAM J. Math. Anal., 34 (2003), 1007.  doi: 10.1137/S0036141001392815.  Google Scholar

[8]

K. Gopalsamy, Persistence in periodic and almost periodic Lotka-Volterra systems,, J. Math. Biol., 21 (1984), 145.  doi: 10.1007/BF00277666.  Google Scholar

[9]

K. Gopalsamy, Global asymptotic stability in a periodic Lotka-Volterra system,, J. Austral. Math. Soc. Ser. B, 27 (1985), 66.  doi: 10.1017/S0334270000004768.  Google Scholar

[10]

K. Gopalsamy, Global asymptotic stability in an almost-periodic Lotka-Volterra system,, J. Austral. Math. Soc. Ser. B, 27 (1986), 346.  doi: 10.1017/S0334270000004975.  Google Scholar

[11]

T. G. Hallam and Z. Ma, Persistence in population models with demographic fluctuations,, J. Math. Biol., 24 (1986), 327.  doi: 10.1007/BF00275641.  Google Scholar

[12]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,'', Lecture Notes in Math., 840 (1981).   Google Scholar

[13]

J. Hofbauer and S. J. Schreiber, Robust permanence for interacting structured populations,, J. Differential Equations, 248 (2010), 1955.  doi: 10.1016/j.jde.2009.11.010.  Google Scholar

[14]

V. Hutson, Y. Lou and K. Mischaikow, Convergence in competition models with small diffusion coefficients,, J. Differential Equations, 211 (2005), 135.  doi: 10.1016/j.jde.2004.06.003.  Google Scholar

[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.  doi: 10.1090/S0002-9939-00-05808-1.  Google Scholar

[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 (2008).   Google Scholar

[17]

J. Mierczyński and W. Shen, Spectral theory for forward nonautonomous parabolic equations and applications,, in, (2008), 24.   Google Scholar

[18]

J. Mierczyński and W. Shen, Persistence in forward nonautonomous competitive systems of parabolic equations,, J. Dynam. Differential Equations, 23 (2011), 551.  doi: 10.1007/s10884-010-9181-2.  Google Scholar

[19]

J. Mierczyński, W. Shen and X. Zhao, Uniform persistence for nonautonomous and random parabolic Kolmogorov systems,, J. Differential Equations, 204 (2004), 471.  doi: 10.1016/j.jde.2004.02.014.  Google Scholar

[20]

J. Pętela (J. Balbus), Average conditions for Kolmogorov systems,, Appl. Math. Comput., 215 (2009), 481.  doi: 10.1016/j.amc.2009.05.031.  Google Scholar

[21]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'', Prentice-Hall, (1967).   Google Scholar

[22]

R. Redheffer, Nonautonomous Lotka-Volterra systems. I,, J. Differential Equations, 127 (1996), 519.  doi: 10.1006/jdeq.1996.0081.  Google Scholar

[23]

R. Redheffer, Nonautonomous Lotka-Volterra systems. II,, J. Differential Equations, 132 (1996), 1.  doi: 10.1006/jdeq.1996.0168.  Google Scholar

[24]

R. Redheffer, Generalized monotonicity, integral conditions and partial survival,, J. Math. Biol., 40 (2000), 295.  doi: 10.1007/s002850050182.  Google Scholar

[25]

R. Redheffer, Mean values and the nonautonomous May-Leonard equations,, Nonlinear Anal. Real World Appl., 4 (2003), 301.  doi: 10.1016/S1468-1218(02)00021-4.  Google Scholar

[26]

R. Redheffer and R. R. Vance, An equivalence theorem concerning population growth in a variable environment,, Int. J. Math. Math. Sci., 2003 (): 2747.   Google Scholar

[27]

S. J. Schreiber, Criteria for $C^r$ robust permanence,, J. Differential Equations, 162 (2000), 400.  doi: 10.1006/jdeq.1999.3719.  Google Scholar

[28]

H. R. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows,, Proc. Amer. Math. Soc., 127 (1999), 2395.  doi: 10.1090/S0002-9939-99-05034-0.  Google Scholar

[29]

H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology,, Math. Biosci., 166 (2000), 173.  doi: 10.1016/S0025-5564(00)00018-3.  Google Scholar

[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.  doi: 10.1016/0022-247X(91)90220-T.  Google Scholar

[31]

R. R. Vance and E. A. Coddington, A nonautonomous model of population growth,, J. Math. Biol., 27 (1989), 491.  doi: 10.1007/BF00288430.  Google Scholar

[32]

F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems,, Results Math., 21 (1992), 224.   Google Scholar

[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.  doi: 10.1016/S1468-1218(03)00038-5.  Google Scholar

[34]

X. Zhao, The qualitative analysis of $n$-species Lotka-Volterra periodic competition systems,, Math. Comput. Modelling, 15 (1991), 3.  doi: 10.1016/0895-7177(91)90100-L.  Google Scholar

[35]

X. Zhao, Uniform persistence in processes with application to nonautonomous competitive models,, J. Math. Anal. Appl., 258 (2001), 87.  doi: 10.1006/jmaa.2000.7361.  Google Scholar

[36]

X. Zhao, "Dynamical Systems in Population Biology,'', CMS Books Math./Ouvrages Math. SMC, 16 (2003).   Google Scholar

[1]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493
[2]

Takashi Kajiwara. The sub-supersolution method for the FitzHugh-Nagumo type reaction-diffusion system with heterogeneity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2441-2465. doi: 10.3934/dcds.2018101
[3]

Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106
[4]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631
[5]

Qi An, Weihua Jiang. Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 487-510. doi: 10.3934/dcdsb.2018183
[6]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245
[7]

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 & Continuous Dynamical Systems - B, 2014, 19 (10) : 3267-3281. doi: 10.3934/dcdsb.2014.19.3267
[8]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[9]

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
[10]

Wei Feng, Xin Lu. Global periodicity in a class of reaction-diffusion systems with time delays. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 69-78. doi: 10.3934/dcdsb.2003.3.69
[11]

Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169
[12]

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
[13]

W. E. Fitzgibbon, M. Langlais, J.J. Morgan. A reaction-diffusion system modeling direct and indirect transmission of diseases. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 893-910. doi: 10.3934/dcdsb.2004.4.893
[14]

José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299
[15]

Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039
[16]

Sebastian Aniţa, Vincenzo Capasso. Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1673-1684. doi: 10.3934/dcdsb.2012.17.1673
[17]

Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817
[18]

José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85
[19]

Qiang Liu, Zhichang Guo, Chunpeng Wang. Renormalized solutions to a reaction-diffusion system applied to image denoising. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1839-1858. doi: 10.3934/dcdsb.2016025
[20]

Bo Li, Xiaoyan Zhang. Steady states of a Sel'kov-Schnakenberg reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1009-1023. doi: 10.3934/dcdss.2017053

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences