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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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