Convex dominates concave: An exclusion principle in discrete-time Kolmogorov systems

Kon, Ryusuke

doi:10.1090/S0002-9939-06-08309-2

Convex dominates concave: An exclusion principle in discrete-time Kolmogorov systems
HTML articles powered by AMS MathViewer

by Ryusuke Kon PDF

Proc. Amer. Math. Soc. 134 (2006), 3025-3034 Request permission

Abstract:

We establish an exclusion principle in discrete-time Kolmogorov systems by using average Liapunov functions. The exclusion principle shows that a weakly dominant species with a convex logarithmic growth rate function eliminates species with concave logarithmic growth rate functions. A general result is applied to specific population models. This application gives an improved exclusion principle for the specific population models.

References

David M. Chan and John E. Franke, Multiple extinctions in a discrete competitive system, Nonlinear Anal. Real World Appl. 2 (2001), no. 1, 75–91. MR 1809865, DOI 10.1016/S0362-546X(99)00282-5
Oscar de Feo and Regis Ferriere, Bifurcation analysis of population invasion: on-off intermittency and basin riddling, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 10 (2000), no. 2, 443–452. MR 1754082, DOI 10.1142/S0218127400000281
Saber Elaydi and Abdul-Aziz Yakubu, Global stability of cycles: Lotka-Volterra competition model with stocking, J. Difference Equ. Appl. 8 (2002), no. 6, 537–549. MR 1903949, DOI 10.1080/10236190290027666
John E. Franke and Abdul-Aziz Yakubu, Global attractors in competitive systems, Nonlinear Anal. 16 (1991), no. 2, 111–129. MR 1090785, DOI 10.1016/0362-546X(91)90163-U
John E. Franke and Abdul-Aziz Yakubu, Mutual exclusion versus coexistence for discrete competitive systems, J. Math. Biol. 30 (1991), no. 2, 161–168. MR 1138846, DOI 10.1007/BF00160333
John E. Franke and Abdul-Aziz Yakubu, Geometry of exclusion principles in discrete systems, J. Math. Anal. Appl. 168 (1992), no. 2, 385–400. MR 1175998, DOI 10.1016/0022-247X(92)90167-C
John E. Franke and Abdul-Aziz Yakubu, Species extinction using geometry of level surfaces, Nonlinear Anal. 21 (1993), no. 5, 369–378. MR 1237128, DOI 10.1016/0362-546X(93)90080-C
S. A. H. Geritz, M. Gyllenberg, F. J. A. Jacobs, and K. Parvinen, Invasion dynamics and attractor inheritance, J. Math. Biol. 44 (2002), no. 6, 548–560. MR 1917846, DOI 10.1007/s002850100136
J. Hofbauer, V. Hutson, and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type, J. Math. Biol. 25 (1987), no. 5, 553–570. MR 915090, DOI 10.1007/BF00276199
Josef Hofbauer, A unified approach to persistence, Acta Appl. Math. 14 (1989), no. 1-2, 11–22. Evolution and control in biological systems (Laxenburg, 1987). MR 990032, DOI 10.1007/BF00046670
Hutson, V., Moran, W. and Vickers, G. T., On a criterion for survival of species in models governed by difference equations, J. Math. Biol. 18 (1983), 89-90.
V. Hutson, A theorem on average Liapunov functions, Monatsh. Math. 98 (1984), no. 4, 267–275. MR 776353, DOI 10.1007/BF01540776
Ryusuke Kon and Yasuhiro Takeuchi, Permanence of host-parasitoid systems, Proceedings of the Third World Congress of Nonlinear Analysts, Part 2 (Catania, 2000), 2001, pp. 1383–1393. MR 1970745, DOI 10.1016/S0362-546X(01)00273-5
Ryusuke Kon and Yasuhiro Takeuchi, Permanence of 2-host 1-parasitoid systems, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 10 (2003), no. 1-3, 389–402. Second International Conference on Dynamics of Continuous, Discrete and Impulsive Systems (London, ON, 2001). MR 1973439
Ryusuke Kon, Permanence of discrete-time Kolmogorov systems for two species and saturated fixed points, J. Math. Biol. 48 (2004), no. 1, 57–81. MR 2035520, DOI 10.1007/s00285-003-0224-8
Ranta, E., Kaitala, V., Alaja, S. and Tesar, D., Nonlinear dynamics and the evolution of semeloparous and iteroparous reproductive strategies, Amer. Natur. 155 (2000), 294-300.
Abdul-Aziz Yakubu, The effects of planting and harvesting on endangered species in discrete competitive systems, Math. Biosci. 126 (1995), no. 1, 1–20. MR 1317925, DOI 10.1016/0025-5564(94)00033-V