Permanence in delayed ratio-dependent predator-prey models with monotonic functional responses. (English) Zbl 1152.34368
Summary: In this paper, sufficient conditions for permanence of the general delayed ratio-dependent predator-prey model
\[
\begin{cases} x^{\prime}(t)=x(t)[a(t)-b(t)x(t)]-c(t)g\left(\frac{x(t)}{y(t)}\right)y(t),\\ y^{\prime}(t)=y(t)\left[e(t)g\left(\frac{x(t-\tau)}{y(t-\tau)}\right)-d(t)\right],\end{cases}
\]
are obtained when functional response \(g\) is monotonic, where \(a(t), b(t), c(t), d(t)\) and \(e(t)\) are all positive periodic continuous functions with period \(\omega >0, \tau\) is a positive constant. We find that the conditions on existence of a positive periodic solution imply the permanence of the above system. As applications, some examples are given.
are obtained when functional response \(g\) is monotonic, where \(a(t), b(t), c(t), d(t)\) and \(e(t)\) are all positive periodic continuous functions with period \(\omega >0, \tau\) is a positive constant. We find that the conditions on existence of a positive periodic solution imply the permanence of the above system. As applications, some examples are given.
MSC:
| 34K13 | Periodic solutions to functional-differential equations |
| 92D25 | Population dynamics (general) |
| 34C25 | Periodic solutions to ordinary differential equations |
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