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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

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