Abstract
In the present paper forward nonautonomous competitive systems of two parabolic second order partial differential equations are studied. The concept of forward uniform persistence for such systems is introduced. Sufficient conditions, expressed in terms of the principal spectrum, are given for those systems to be forward uniformly persistent.
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Álvarez-Caudevilla P., López-Gómez J.: A dynamics of a class of cooperative systems. Discret. Contin. Dyn. Syst. 26(2), 397–415 (2010)
Cantrell R.S., Cosner C.: Should a park be an island? SIAM J. Appl. Math. 53(1), 219–252 (1993)
Cantrell R.S., Cosner C.: Spatial Ecology via Reaction–Diffusion Equations (Wiley Series in Mathematical and Computational Biology). Wiley, Chichester (2003)
Cantrell R.S., Cosner C., Hutson V.: Permanence in ecological systems with spatial heterogeneity. Proc. R. Soc. Edinb. A 123(3), 533–559 (1993)
Cantrell R.S., Cosner C., Huston V.: Ecological models, permanence and spatial heterogeneity. Rocky Mt. J. Math. 26(1), 1–35 (1996)
Hale J., Waltman P.: Persistence in infinite-dimensional systems. SIAM J. Math. Anal. 20(2), 388–395 (1989)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin, New York (1981)
Hetzer G., Shen W.: Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems. SIAM J. Math. Anal. 34(1), 204–227 (2002)
Húska J.: Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains. J. Differ. Equat. 226(2), 541–557 (2006)
Húska J., Poláčik P.: The principal Floquet bundle and exponential separation for linear parabolic equations. J. Dynam. Differ. Equat. 16(2), 347–375 (2004)
Húska J., Poláčik P., Safonov M.V.: Harnack inequality, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 24(5), 711–739 (2007)
Hutson V., Schmitt K.: Permanence and the dynamics of the biological systems. Math. Biosci. 111(1), 1–71 (1992)
Hutson V., Shen W., Vickers G.T.: Estimates for the principal spectrum point for certain time-dependent parabolic operators. Proc. Am. Math. Soc. 129(6), 1669–1679 (2000)
Johnson R.A., Palmer K.J., Sell G.R.: Ergodic properties of linear dynamical systems. SIAM J. Math. Anal. 18(1), 1–33 (1987)
Langa J.A., Robinson J.C., Rodríguez-Bernal A., Suárez A.: Permanence and asymptotically stable complete trajectories for nonautonomous Lotka-Volterra models with diffusion. SIAM J. Math. Anal. 40(6), 2179–2216 (2009)
López-Gómez J.: On the structure of the permanence region for competing species models with general diffusivities and transport effects. Discret. Contin. Dyn. Syst. 2(4), 525–542 (1996)
Mierczyński, J.: The principal spectrum for linear nonautonomous parabolic PDEs of second order: basic properties. In: Special issue in celebration of Jack K. Hale’s 70th birthday, Part 2 (Atlanta, GA/Lisbon, 1998). J. Differ. Equat. 168(2), 453–476 (2000)
Mierczyński J., Shen W.: Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations. J. Differ. Equat. 191(1), 175–205 (2003)
Mierczyński J., Shen W.: Lyapunov exponents and asymptotic dynamics in random Kolmogorov models. J. Evol. Equat. 4(3), 371–390 (2004)
Mierczyński J., Shen W.: The Faber–Krahn inequality for random/nonautonomous parabolic equations. Commun. Pure Appl. Anal. 4(1), 101–114 (2005)
Mierczyński J., Shen W.: Time averaging for nonautonomous/random parabolic equations. Discret. Contin. Dyn. Syst. Ser. B 9(3/4), 661–699 (2008)
Mierczyński, J., Shen, W.: Spectral theory for random and nonautonomous parabolic equations and applications. In: Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall/CRC, Boca Raton, FL, (2008)
Mierczyński, J., Shen, W.: 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.
Mierczyński J., Shen W., Zhao X.-Q.: Uniform persistence for nonautonomous and random parabolic Kolmogorov systems. J. Differ. Equat. 204(2), 471–510 (2004)
Poláčik P.: On uniqueness of positive entire solutions and other properties of linear parabolic equations. Discret. Contin. Dyn. Syst. 12(1), 13–26 (2005)
Thieme H.R.: Uniform weak implies uniform strong persistence for non-autonomous semiflows. Proc. Am. Math. Soc. 127(8), 2395–2403 (1999)
Thieme H.R.: Uniform persistence and permanence for non-autonomous semiflows in population biology. Math. Biosci. 166(2), 173–201 (2000)
Zhao X.-Q.: Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications Canad. Appl. Math. Quart. 3(4), 473–495 (1995)
Acknowledgment
The first-named author was supported from resources for science in years 2009–2012 as research project (grant MENII N N201 394537, Poland). The second-named author was partially supported by NSF grant DMS–0907752.
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Dedicated to Professor Russell Johnson on the occasion of his 60th birthday.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Mierczyński, J., Shen, W. Persistence in Forward Nonautonomous Competitive Systems of Parabolic Equations. J Dyn Diff Equat 23, 551–571 (2011). https://doi.org/10.1007/s10884-010-9181-2
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DOI: https://doi.org/10.1007/s10884-010-9181-2
Keywords
- Forward nonautonomous competitive system of parabolic equation
- System of parabolic equations of Kolmogorov type
- Principal spectrum
- Forward uniform persistence