×

The existence of an equilibrium for permanent systems. (English) Zbl 0722.92012

The property of permanence in biological populations described by the ODE \(\dot x_ i=x_ i\cdot f_ i(^{\rightharpoonup})\), \(i=1,...,n\) (or by a corresponding difference equation) implies the existence of an equilibrium point in the interior of \({\mathbb{R}}^ n_+\). This known fact is proved in a simple way by application of Schauder’s fixed point theorem. Analogous results are derived for a class of differential-delay equations by means of Horn’s theorem.

MSC:

92D25 Population dynamics (general)
37C75 Stability theory for smooth dynamical systems
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
34K20 Stability theory of functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. P. Aubin and A. Cellina, Differential inclusions , Springer-Verlag, Berlin, 1984. · Zbl 0538.34007
[2] N. Bhatia and G. Szegö, Stability theory of dynamical systems , Springer-Verlag, Berlin, 1970. · Zbl 0213.10904
[3] T. Burton and V. Hutson, Repellers in systems with infinite delay , J. Math. Anal. Appl., 137 (1989), 240 -263. · Zbl 0677.92016 · doi:10.1016/0022-247X(89)90287-4
[4] G. Butler, H. Freedman and P. Waltman, Uniformly persistent systems , Proc. Amer. Math. Soc. 96 (1986), 425-430. JSTOR: · Zbl 0603.34043 · doi:10.2307/2046588
[5] H. Freedman and J. So, Persistence in discrete semi-dynamical systems , SIAM J. Math. Anal. 20 (1989), 930-938. · Zbl 0676.92011 · doi:10.1137/0520062
[6] D. Henry, Geometric theory of semilinear parabolic equations , Lecture Notes 840 , Springer-Verlag, Berlin, 1981. · Zbl 0456.35001
[7] J. Hofbauer, V. Hutson and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type , J. Math. Biol. 25 (1987), 553-570. · Zbl 0638.92019 · doi:10.1007/BF00276199
[8] J. Hofbauer and K. Sigmund, Dynamical systems and the theory of evolution , Cambridge University Press, 1988. · Zbl 0678.92010
[9] W. Horn, Some fixed point theorems for compact mappings and flows on a Banach space , Trans. Amer. Math. Soc. 149 (1970), 391-404. JSTOR: · Zbl 0201.46203 · doi:10.2307/1995402
[10] V. Hutson and W. Moran, Persistence of species obeying difference equations , Math. Biosci. 15 (1982), 203-213. · Zbl 0495.92015 · doi:10.1007/BF00275073
[11] V. Hutson, A theorem on average Liapunov functions , Monatsh. Math. 98 (1984), 267-275. · Zbl 0542.34043 · doi:10.1007/BF01540776
[12] \emdash/– and J. Pym, Repellers for generalized semidynamical systems , in Mathematics of dynamic processes (A. Kurzhanski, ed.), Lecture Notes in Economics and Mathematical Systems, Vol. 287 , Springer-Verlag, Berlin, 1987, 39-49. · Zbl 0657.34015
[13] \emdash/– and W. Moran, Repellers in reaction-diffusion systems , Rocky Mountain J. Math. 17 (1987), 301-314. · Zbl 0636.35037 · doi:10.1216/RMJ-1987-17-2-301
[14] E. Zeidler, Nonlinear functional analysis and its applications I, Springer-Verlag, New York, 1986. · Zbl 0583.47050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.