Three species competition in periodic environment (Q1385106)

Consider the Lotka-Volterra system for three-competing species \[ \begin{aligned} x_1'(t) & =x_1(t) \bigl(1-x_1(t) -\alpha (t)x_2(t) -\beta (t)x_3 (t)\bigr),\\ x_2'(t) & =x_2(t) \bigl(1-\beta (t)x_1 (t)-x_2(t)-\alpha (t)x_3(t)\bigr), \tag{1}\\ x_3'(t) & =x_3(t) \bigl(1-\alpha (t)x_1(t)- \beta (t)x_2 (t)-x_3 (t)\bigr), \end{aligned} \] where \(\alpha, \beta: \mathbb{R}^1 \to\mathbb{R}^1\) are continuous, nonnegative and \(T\)-periodic for some common period \(T>0\). Set \[ \mathbb{R}^3_+ =\bigl\{x = (x_1,x_2, x_3)\in \mathbb{R}^3: x_i\geq 0; \quad i=1,2,3 \bigr\}, \] \[ L= \{x\in\mathbb{R}^3_+: x_1=x_2 = x_3\}. \] We shall show that system (1) has at least one \(T\)-periodic solution with strictly positive components. If \(\max_{0\leq t\leq T} (\alpha(t) +\beta(t))<2\), then such a solution is unique and globally asymptotically stable (or attractive) in \(\text{int} (\mathbb{R}^3_+): =\{x\in \mathbb{R}^3_+: x_i>0\), \(i=1,2,3\}\). Furthermore, if \(\min_{0\leq t\leq T} \{\alpha (t)+ \beta(t)\} >2\) then a positive (componentwise) \(T\)-periodic solution of (1) is also unique but \(\text{dist} (x(t),\;\partial \mathbb{R}^3_+) \to 0\) as \(t\to+\infty\) for every solution \(x(t)\) of (1) with \(x(t_0) \in\text{int} (\mathbb{R}^n_+) \setminus L\) for some \(t_0 \in\mathbb{R}\), where \(\text{dist} (x(t), \partial \mathbb{R}^3_+)\) is the distance from \(x(t)\) to \(\partial \mathbb{R}^3_+\).