On the continuity of the function describing the times of meeting impulsive set and its application

Sanyi Tang; Wenhong Pang

doi:10.3934/mbe.2017072

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2017, Volume 14, Issue 5&6: 1399-1406. Doi: 10.3934/mbe.2017072

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On the continuity of the function describing the times of meeting impulsive set and its application

Sanyi Tang^, and
Wenhong Pang

School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China

* Corresponding author: Sanyi Tang
* Corresponding author: Sanyi Tang

Received: April 24, 2016

Accepted: September 18, 2016

Published: November 2017

The first author is supported by the National Natural Science Foundation of China (NSFC 11631012,11471201), and by the Fundamental Research Funds for the Central Universities (GK201701001).

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Abstract

The properties of the limit sets of orbits of planar impulsive semi-dynamic system strictly depend on the continuity of the function, which describes the times of meeting impulsive sets. In this note, we will show a more realistic counter example on the continuity of this function which has been proven and widely used in impulsive dynamical system and applied in life sciences including population dynamics and disease control. Further, what extra condition should be added to guarantee the continuity of the function has been addressed generally, and then the applications and shortcomings have been discussed when using the properties of this function.

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Mathematics Subject Classification: Primary: 34A37; Secondary: 47N60.

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Figure 1. Illustrations of impulsive set, phase set and definition of impulsive semi-dynamical system for model (2). The parameter values are fixed as: $r=1, K=52, \beta=0.19, \eta=0.45, \omega=0.19, \delta=0.36, \tau=5, ET=35, \theta=0.7$

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Figure 2. Three possible trajectories of model (2) with $x_{\Gamma_1}<(1-\theta)ET<ET<x_{\Gamma_2}$ for model (2). The parameter values are fixed as: $r=1, K=52, \beta=0.19, \eta=0.45, \omega=0.19, \delta=0.36, \tau=5, ET=35, \theta=0.7(A), 0.659(B)$ and $0.5(C)$

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Figure 3. Continuity of Poincaré map and time function without impulse of model (2) for different $\theta$. The other parameter values are fixed as: $r=1, K=52, \beta=0.19, \eta=0.45, \omega=0.19, \delta=0.36, \tau=5, ET=35$

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