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Global asymptotic stability of periodic Lotka-Volterra systems with delays. (English) Zbl 0995.34071

The authors consider the periodic Lotka-Volterra-type system with finite or infinite delays \[ \begin{split} \frac{dx_{i}(t)}{dt}=x_{i}(t)\left[b_{i}(t)-a_{i}(t)x_{i}(t)- \sum_{j=1}^{n}a_{ij}(t)x_{j}(t-\tau_{ij}(t))\right.\\ \left. -\sum_{j=1}^{n} \int_{-\sigma_{ij}}^{0} c_{ij}(t,s) x_{j}(t+s) ds\right],\;\;i=1,2,\ldots,n, \end{split} \tag{1} \] where \(b_{i}(t)\), \(a_{i}(t)\), \(a_{ij}(t)\) are \(\omega\)-periodic and continuous functions on \(\mathbb{R}\); \(c_{ij}(t,s)\) are \(\omega\)-periodic and continuous with respect to \(t\) on \(\mathbb{R}\) and integrable in \(s\) on \([-\sigma_{ij},0]\); there exists a continuous positive function \(h_{0}\) defined on \((-\infty,0]\) and \(0<l=\int_{-\infty}^{0}h_{0}(s) ds<\infty\) such that \(|c_{ij}(t,s)|\leq h_{0}(s)\) for all \((t,s)\in\mathbb{R} \times[-\sigma_{ij},0]\); \(\tau_{ij}(t)\geq 0\) are \(\omega\)-periodic, continuously differentiable functions on \(\mathbb{R}\), \(d\tau_{ij}(t)/dt<1\) and \(\sigma_{ij}\) are nonnegative constants or \(\sigma_{ij}=\infty\); \(i,j=1,\ldots,n\). There are positive constants \(c_{1},\ldots, c_{n}\) such that the functions \[ \gamma_{i}(t)=c_{i}a_{i}(t)-\sum_{j=1}^{n}c_{j} \left(\frac{|a_{ji}(\psi^{-1}_{ji}(t))|} {1-\dot{\tau}_{ji}(\psi^{-1}_{ji}(t))}+ \int_{-\sigma_{ji}}^{0}|c_{ji}(t-s,s)|ds\right), \] where \(\psi^{-1}_{ji}(t)\) is the inverse function of \(\psi_{ji}(t)=1-\tau_{ji}(t)\), are nonnegative and \(\displaystyle\sum_{j=1}^{\infty}\int_{\alpha_{j}}^{\beta_{j}} \gamma_{i}(t) dt=\infty\) for any interval sequence \(\{[\alpha_{i},\beta_{i}]\}\) such that \([\alpha_{i},\beta_{i}]\cap[\alpha_{j},\beta_{j}]=\emptyset\) and \(\beta_{i}-\alpha_{i}=\beta_{j}-\alpha_{j}>0\) for \(i,j=1,2,\ldots\), \(i\neq j\).
System (1) is said to be persistent, if for any positive solution \(x_{1}(t),\ldots,x_{n}(t)\) there exist positive constants \(m\), \(M\), \(T\) such that \(m\leq x_{i}(t)\leq M\), \(i=1,\ldots,n\), for \(t\geq T\).
Under the above assumptions the main result presented in the paper asserts that if system (1) is persistent, then it has a unique positive \(\omega\)-periodic solution which is globally asymptotically stable.
Sufficient conditions for the persistence of system (1) under some additional assumptions are also given. As a sequence, the authors obtain a concrete criterion for the existence and global asymptotic stability of a positive periodic solution to the competitive system (1).

MSC:

34K20 Stability theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
45J05 Integro-ordinary differential equations
92D25 Population dynamics (general)
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