# American Institute of Mathematical Sciences

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

 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.
Citation: Joanna Balbus, Janusz Mierczyński. Time-averaging and permanence in nonautonomous competitive systems of PDEs via Vance-Coddington estimates. Discrete and 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), 191-228. 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.

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##### 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.
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