Partial survival and extinction of species in discrete nonautonomous Lotka-Volterra systems. (English) Zbl 1081.39014

Muroya, Yoshiaki

Tokyo J. Math. 28, No. 1, 189-200 (2005).

Consider the following discrete model of a nonautonomous Lotka-Volterra system, \[ N_i(p+1)=N_i(p)\exp \biggl\{c_i(p)-\sum_{j=1}^n \sum_{l=0}^m a_{ij}^l (p)N_j(p-k_l)\biggr\}, \quad p\geq 0, \;1\leq i\leq n, \] with \[ N_i(p)=N_{ip}\geq 0, \;p\leq 0, \quad \text{and}\quad N_{i0}>0, \;1\leq i\leq n, \] which is the discrete version of the system considered by Y. Muroya [Dyn. Syst. Appl. 12, No. 3–4, 295–306 (2003; Zbl 1044.92049)]. The author studies the partial survial and extinction of species. These results extend those of S. Ahmad [Proc. Am. Math. Soc. 127, No. 10, 2905–2910 (1999; Zbl 0924.34040)] to discrete nonautonomous Lotka-Volterra systems.

Reviewer: Yuming Chen (Waterloo)

MSC:

39A12	Discrete version of topics in analysis
92D25	Population dynamics (general)

Keywords:

discrete model; nonautonomous Lotka-Volterra system; partial survial; extinction; delay

Citations:

Zbl 1044.92049; Zbl 0924.34040

References:

[1]	S. Ahmad, On the nonautonomous Volterra-Lotka competition equations , Proc. Amer. Math. Soc. 117 (1993), 199-204. · Zbl 0848.34033
[2]	S. Ahmad, Extinction of species in nonautonomous Lotka-Volterra systems , Proc. Amer. Math. Soc. 127 (1999), 2905-2910. · Zbl 0924.34040
[3]	S. Ahmad and A. C. Lazer, On the nonautonomous N-competing species problem , Applicable Anal. 57 (1995), 209-323. · Zbl 0859.34033
[4]	S. Ahmad and A. C. Lazer, Necessary and sufficient average growth in a Lotka-Volterra system , Nonlinear Analysis TMA 34 (1998), 191-228. · Zbl 0934.34037
[5]	S. Ahmad and A. C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka-Volterra system , Nonlinear Analysis TMA 40 (2000), 37-49. · Zbl 0955.34041
[6]	S. Ahmad and F. Montes de Oca, Extinction in nonautonomous T-periodic competitive Lotka-Volterra systems , Appl. Math. Comput. 90 (1998), 155-166. · Zbl 0906.92024
[7]	C. Alvarez and A. C. Lazer, An application of topological degree to the periodic competing species problem , J. Austral. Math. Soc. Ser. B 28 (1986), 202-219. · Zbl 0625.92018
[8]	A. Battauz and F. Zanolin, Coexistence states for periodic competitive Kolmogorov systems , J. Math. Anol. Appl. 219 (1998), 179-199. · Zbl 0911.34037
[9]	K. Gopalsamy, Exchange of equilibria in two species Lotka-Volterra competition models , J. Austral. Math. Soc. Ser B 24 (1982), 160-170. · Zbl 0498.92016
[10]	K. Gopalsamy, Global asymptotic stability in a periodic Lotka-Volterra system , J. Austral. Math. Soc. Ser B 27 (1985), 66-72. · Zbl 0588.92019
[11]	Y. Muroya, Persistence and global stability for discrete models of nonautonomous Lotka-Volterra type , J. Math. Anal. Appl. 273 (2002), 492-511. · Zbl 1033.39013
[12]	Y. Muroya, Averaged growth and global stability in nonautonomous Lotka-Volterra system with delays , Communications on Applied Nonlinear Analysis 10 (2003), 35-54. · Zbl 1058.34102
[13]	Y. Muroya, Partial survival and extinction of species in nonautonomous Lotka-Volterra systems with delays , Dynamic Systems and Applications 12 (2003), 295-306. · Zbl 1044.92049
[14]	F. Montes de Oca and M. L. Zeeman, Extinction in nonautonomous competitive Lotka-Volterra systems , Proc. Amer. Math. Soc. 124 (1996), 3677-3687. · Zbl 0866.34029
[15]	R. Ortega and A. Tineo, An exclusion principle for periodic competitive Lotka-Volterra systems in three dimensions , Nonlinear Anal. TMA 31 (1998), 883-893. · Zbl 0901.34049
[16]	R. Redheffer, Nonautonomous Lotka-Volterra system I , J. Differential Equations 127 (1996), 519-540. · Zbl 0856.34056
[17]	R. Redheffer, Nonautonomous Lotka-Volterra system II , J. Differential Equations 127 (1996), 1-20. · Zbl 0864.34043
[18]	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), 44-60. · Zbl 0729.92025