Abstract
In this paper, a predator-prey system with Michaelis-Menten functional response on time scales is investigated. First of all, we generalize the semi-cycle concept to time scales. Second, we obtain the uniformly ultimate boundedness of solutions of this system. Our results demonstrate that when the death rate of the predator is rather small without prey, whereas the intrinsic growth rate of the prey is relatively large, the species could coexist in the long run. In particular, if $\mathbb{T}=\mathbb{R}$ or $\mathbb{T}=\mathbb{Z}$ , some well-known results have been generalized. In addition, for the continuous case, we provide a new idea to prove its permanence. Finally, a numerical simulation is given to support our main results.
Highlights
The permanence is based on a global criterion for the coexistence of species, which describes a numerical technique for assembly of ecological communities of Lotka-Volterra form [ ]
In [ ], by using the semi-cycle and related concepts, they discussed the permanence of a discrete biological system
The existence of periodic solutions of predator-prey systems on time scales has been obtained by coincidence degree theory in many articles [ – ], since the existence result could be obtained by coincidence degree theory both in the continuous case and the discrete case
Summary
The permanence is based on a global criterion for the coexistence of species, which describes a numerical technique for assembly of ecological communities of Lotka-Volterra form [ ]. The existence of periodic solutions of predator-prey systems on time scales has been obtained by coincidence degree theory in many articles [ – ], since the existence result could be obtained by coincidence degree theory both in the continuous case and the discrete case. Since the permanence of this system has been obtained by the comparison theorem in the continuous case, while it has been obtained by semi-cycle concept in discrete case. Our aim is to prove the uniform ultimate boundedness of solutions of system ( ) by using the semi-cycle concept on time scales instead of comparison theorems. From Part and Part , x (t) is uniformly ultimate bounded, we can assume m ≤ exp{x (t)} ≤ M for any t ∈ [T , ∞) ∩ T, where T is sufficiently large.
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