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Globally asymptotic stability in two periodic delayed competitive systems. (English) Zbl 1148.34046

The authors study the following Lotka-Volterra type competition systems with multiple time delays and periodic coefficients \[ \begin{aligned} y_1^{\prime}(t)=&y_1(t)[r_1(t)-a_1(t)y_1(t)+\sum_{i=1}^{n}b_{1i}(t)y_1(t-\tau_i(t)) -\sum_{j=1}^{m}c_{1j}(t)y_2(t-\rho_j(t))],\\ y_2^{\prime}(t)=&y_2(t)[r_2(t)-a_2(t)y_2(t)+\sum_{j=1}^{m}b_{2j}(t)y_2(t-\eta_j(t)) -\sum_{i=1}^{n}c_{2i}(t)y_1(t-\sigma_i(t))], \end{aligned}\tag{1} \] where \(a_1, a_2, c_{1j}, c_{2j}\in C(R, (0,+\infty)),\) \(b_{1i}, b_{2i}\in C(R, R)\), \(\tau_i, \sigma_i,\) \( \rho_j, \eta_j\in C^1(R, [0,+\infty))\) \((i=1,2,\dots, n; j=1,2,\dots, m)\) are \(\omega\)-periodic functions, \(r_k\in C(R, R)\) are \(\omega\)-periodic functions satisfying \(\int_{0}^{\omega}r_kdt>0, k=1,2\). Sufficient conditions are derived for the existence of positive periodic solutions to system (1) by using the continuation theorem developed by R. E. Gaines and J. L. Mawhin [Coincidence degree and nonlinear differential equations. Springer, Berlin (1977; Zbl 0339.47031)].

MSC:

34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)

Citations:

Zbl 0339.47031
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References:

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