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.

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

Metrics

  • PDF downloads (122)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]