Competitive exclusion in a periodic Lotka–Volterra system.(English)Zbl 1100.92070

From the paper: This paper deals with the asymptotic extinction of one species in the two-dimensional competition model: $\begin{cases} u'=u(a_1(t)-b_{11}(t)u-b_{12}(t)v),\\ v'=v(a_2(t)-b_{21}(t)u-b_{22}(t)v), \end{cases}\tag{1}$ where the coefficients $$a_i(t)$$ and $$b_{ij}(t)$$ are continuous, T-periodic, $$b_{ij}(t) > 0$$, $$i,j= 1,2$$. We also assume $m[a_i]=T^{-1}\int^T_0 a_i(s)\,ds>0,\quad i=1,2.$ In the case all coefficients are positive numbers, the inequalities $a_1>b_{12}(a_2/b_{22}),\quad a_2< b_{21}(a_2/b_{11}),$ imply that if $$u(t)$$, $$v(t)$$ is any positive solution of (1) then the species $$v(t)$$ is forced to extinction and $$u(t)$$ approaches $$a_1/b_{11}$$ as $$t\to\infty$$. This result is known as “the principle of competitive exclusion”. Introducing a suitable average conditions, we extend the known principle of competitive exclusion to the periodic case.

MSC:

 92D40 Ecology 34C60 Qualitative investigation and simulation of ordinary differential equation models
Full Text:

References:

 [1] Ahmad, S., On the nonautonomous volterra – lotka competition equations, Proc. am. math. soc., 117, 199-204, (1993) · Zbl 0848.34033 [2] Ahmad, S.; Montes de Oca, F., Extinction in non autonomous T-periodic competitive lotka – volterra system, Appl. math. comput., 90, 155-166, (1998) · Zbl 0906.92024 [3] Ahmad, S.; Montes de Oca, F., Average growth and extinction in a two dimensional lotka – volterra system, Dyn. cont. discrete impuls. syst., ser. A, math. anal., 9, 177-186, (2002) · Zbl 1081.34513 [4] de Mottoni, P.; Schiaffino, A., Competition systems with periodic coefficients: a geometric approach, J. math. anal. appl., 11, 319-335, (1981) · Zbl 0474.92015 [5] Eilbeck, J.C.; Lopez-Gomez, J., On the periodic lotka – volterra competition model, J. math. anal. appl., 210, 58-87, (1997) · Zbl 0874.34039 [6] Lisena, B., Global stability in periodic competitive system, Nonlinear anal. RWA, 5, 613-627, (2004) · Zbl 1089.34044 [7] Lopez-Gomez, J.; Ortega, R.; Tineo, A., The periodic predator-prey lotka – volterra model, Adv. differ. eq., 1, 405-423, (1996) · Zbl 0849.34026
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.