Abstract
With the help of a continuation theorem based on Gaines and Mawhin′s coincidence degree, easily verifiable criteria are established for the global existence of positive periodic solutions of a delayed ratio‐dependent predator‐prey system with stage structure for predator. The approach involves some new technique of priori estimate. For the system without delay, by constructing a suitable Lyapunov function, some sufficient conditions which guarantee the existence of a unique global attractive positive periodic solution are obtained. Those results have further applications in population dynamics.
Highlights
Some biologists have argued that the ratio-dependent predator-prey model dx dt = x(a − bx) cx y my + x, dy dt y d + f my x + (1.1)
In [14], they further considered system (1.2) with time-varying delays, by using the coincidence degree theory, they obtained sufficient conditions which guarantee the existence of positive periodic solution
In order to obtain the existence of positive periodic solutions of (1.4), for the reader’s convenience, we will summarize in the following a few concepts and results from [16] that will be basic for this section
Summary
Some biologists have argued that the ratio-dependent predator-prey model dx dt. In [14], they further considered system (1.2) with time-varying delays, by using the coincidence degree theory, they obtained sufficient conditions which guarantee the existence of positive periodic solution. Stimulated by the works of [22, 23], in [26], Xiao considered the ratio-dependent predator-prey system with stage structure, that is dx dt x(r ax) −. 156 A stage-structured ratio-dependent predator-prey model periodic solution of ecosystem, one could refer to [6, 7, 8, 9, 10] and the references cited therein
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