## 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
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### 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
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