2017, 14(5-6): 1399-1406. doi: 10.3934/mbe.2017072

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

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

* Corresponding author: Sanyi Tang

Received  April 25, 2016 Revised  September 2016 Accepted  September 19, 2016 Published  May 2017

Fund Project: 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)

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.

Citation: Sanyi Tang, Wenhong Pang. On the continuity of the function describing the times of meeting impulsive set and its application. Mathematical Biosciences & Engineering, 2017, 14 (5-6) : 1399-1406. doi: 10.3934/mbe.2017072
References:
[1]

E. M. Bonotto and M. Federson, Limit sets and the Poincare Bendixson theorem in impulsive semidynamical systems, J. Differ. Equ., 244 (2008), 2334-2349. doi: 10.1016/j.jde.2008.02.007.

[2]

K. Ciesielski, On semicontinuity in impulsive dynamical systems, Bulletin of The Polish Academy of Sciences Mathematics, 52 (2004), 71-80. doi: 10.4064/ba52-1-8.

[3]

G. B. Ermentrout and N. Kopell, Multiple pulse interactions and averaging in systems of coupled neural oscillators, J. Math. Biol., 29 (1991), 195-217. doi: 10.1007/BF00160535.

[4]

R. A. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.

[5]

G. Gabor, The existence of viable trajectories in the state-dependent impusive systems, Nonlinear Anal. TMA, 72 (2010), 3828-3836. doi: 10.1016/j.na.2010.01.019.

[6]

G. Gabor, Viable periodic solutions in state-dependent impulsive problems, Collect. Math., 66 (2015), 351-365. doi: 10.1007/s13348-015-0139-x.

[7]

P. Goel and B. Ermentrout, Synchrony, stability, and firing patterns in pulse-coupled oscillators, Physica D, 163 (2002), 191-216. doi: 10.1016/S0167-2789(01)00374-8.

[8]

M. Z. HuangJ. X. LiX. Y. Song and H. J. Guo, Modeling impulsive injections of insulin: Towards aritificial pancreas, SIAM J. Appl. Math., 72 (2012), 1524-1548. doi: 10.1137/110860306.

[9]

S. K. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128. doi: 10.1016/0022-247X(90)90199-P.

[10]

J. H. LiangS. Y. TangJ. J. Nieto and R. A. Cheke, Analytical methods for detecting pesticide switches with evolution of pesticide resistance, Math. Biosci., 245 (2013), 249-257. doi: 10.1016/j.mbs.2013.07.008.

[11]

B. Liu, Y. Tian and B. L. Kang, Dynamics on a Holling Ⅱ predator-prey model with state-dependent impulsive control International J. Biomath. 5 (2012), 1260006, 18 pp.

[12]

L. F. NieZ. D. Teng and L. Hu, The dynamics of a chemostat model with state dependent impulsive effects, Int. J. Bifurcat. Chaos, 21 (2011), 1311-1322. doi: 10.1142/S0218127411029173.

[13]

J. C. Panetta, A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competitive environment, Bull. Math. Biol., 58 (1996), 425-447. doi: 10.1007/BF02460591.

[14]

B. ShulginL. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol., 60 (1998), 1123-1148.

[15]

L. StoneB. Shulgin and Z. Agur, Theoretical examination of the pulse vaccination policy in the SIR epidemic model, Math. Comput. Model., 31 (2000), 207-215. doi: 10.1016/S0895-7177(00)00040-6.

[16]

K. B. SunY. TianL. S. Chen and A. Kasperski, Nonlinear modelling of a synchronized chemostat with impulsive state feedback control, Math. Comput. Modelling, 52 (2010), 227-240. doi: 10.1016/j.mcm.2010.02.012.

[17]

S. Y. Tang and R. A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol., 50 (2005), 257-292. doi: 10.1007/s00285-004-0290-6.

[18]

S. Y. Tang and R. A. Cheke, Models for integrated pest control and their biological implications, Math. Biosci., 215 (2008), 115-125. doi: 10.1016/j.mbs.2008.06.008.

[19]

S. Y. Tang and L. S. Chen, Modelling and analysis of integrated pest management strategy, Discrete Contin. Dyn. Syst. B, 4 (2004), 759-768. doi: 10.3934/dcdsb.2004.4.759.

[20]

S. Y. TangJ. H. LiangY. S. Tan and R. A. Cheke, Threshold conditions for interated pest management models with pesticides that have residual effects, J. Math. Biol., 66 (2013), 1-35. doi: 10.1007/s00285-011-0501-x.

[21]

S. Y. Tang, W. H. Pang, R. A. Cheke and J. H. Wu, Global dynamics of a state-dependent feedback control system Advances in Difference Equations 2015 (2015), 70pp.

[22]

S. Y. TangG. Y. Tang and R. A. Cheke, Optimum timing for integrated pest management: Modeling rates of pesticide application and natural enemy releases, J. Theor. Biol., 264 (2010), 623-638. doi: 10.1016/j.jtbi.2010.02.034.

[23]

S. Y. TangB. TangA. L. Wang and Y. N. Xiao, Holling Ⅱ predator-prey impulsive semi-dynamic model with complex Poincare map, Nonlinear Dynamics, 81 (2015), 1575-1596. doi: 10.1007/s11071-015-2092-3.

[24]

S. Y. TangY. N. XiaoL. S. Chen and R. A. Cheke, Integrated pest management models and their dynamical behaviour, Bull. Math. Biol., 67 (2005), 115-135. doi: 10.1016/j.bulm.2004.06.005.

show all references

References:
[1]

E. M. Bonotto and M. Federson, Limit sets and the Poincare Bendixson theorem in impulsive semidynamical systems, J. Differ. Equ., 244 (2008), 2334-2349. doi: 10.1016/j.jde.2008.02.007.

[2]

K. Ciesielski, On semicontinuity in impulsive dynamical systems, Bulletin of The Polish Academy of Sciences Mathematics, 52 (2004), 71-80. doi: 10.4064/ba52-1-8.

[3]

G. B. Ermentrout and N. Kopell, Multiple pulse interactions and averaging in systems of coupled neural oscillators, J. Math. Biol., 29 (1991), 195-217. doi: 10.1007/BF00160535.

[4]

R. A. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445-466.

[5]

G. Gabor, The existence of viable trajectories in the state-dependent impusive systems, Nonlinear Anal. TMA, 72 (2010), 3828-3836. doi: 10.1016/j.na.2010.01.019.

[6]

G. Gabor, Viable periodic solutions in state-dependent impulsive problems, Collect. Math., 66 (2015), 351-365. doi: 10.1007/s13348-015-0139-x.

[7]

P. Goel and B. Ermentrout, Synchrony, stability, and firing patterns in pulse-coupled oscillators, Physica D, 163 (2002), 191-216. doi: 10.1016/S0167-2789(01)00374-8.

[8]

M. Z. HuangJ. X. LiX. Y. Song and H. J. Guo, Modeling impulsive injections of insulin: Towards aritificial pancreas, SIAM J. Appl. Math., 72 (2012), 1524-1548. doi: 10.1137/110860306.

[9]

S. K. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128. doi: 10.1016/0022-247X(90)90199-P.

[10]

J. H. LiangS. Y. TangJ. J. Nieto and R. A. Cheke, Analytical methods for detecting pesticide switches with evolution of pesticide resistance, Math. Biosci., 245 (2013), 249-257. doi: 10.1016/j.mbs.2013.07.008.

[11]

B. Liu, Y. Tian and B. L. Kang, Dynamics on a Holling Ⅱ predator-prey model with state-dependent impulsive control International J. Biomath. 5 (2012), 1260006, 18 pp.

[12]

L. F. NieZ. D. Teng and L. Hu, The dynamics of a chemostat model with state dependent impulsive effects, Int. J. Bifurcat. Chaos, 21 (2011), 1311-1322. doi: 10.1142/S0218127411029173.

[13]

J. C. Panetta, A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competitive environment, Bull. Math. Biol., 58 (1996), 425-447. doi: 10.1007/BF02460591.

[14]

B. ShulginL. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol., 60 (1998), 1123-1148.

[15]

L. StoneB. Shulgin and Z. Agur, Theoretical examination of the pulse vaccination policy in the SIR epidemic model, Math. Comput. Model., 31 (2000), 207-215. doi: 10.1016/S0895-7177(00)00040-6.

[16]

K. B. SunY. TianL. S. Chen and A. Kasperski, Nonlinear modelling of a synchronized chemostat with impulsive state feedback control, Math. Comput. Modelling, 52 (2010), 227-240. doi: 10.1016/j.mcm.2010.02.012.

[17]

S. Y. Tang and R. A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol., 50 (2005), 257-292. doi: 10.1007/s00285-004-0290-6.

[18]

S. Y. Tang and R. A. Cheke, Models for integrated pest control and their biological implications, Math. Biosci., 215 (2008), 115-125. doi: 10.1016/j.mbs.2008.06.008.

[19]

S. Y. Tang and L. S. Chen, Modelling and analysis of integrated pest management strategy, Discrete Contin. Dyn. Syst. B, 4 (2004), 759-768. doi: 10.3934/dcdsb.2004.4.759.

[20]

S. Y. TangJ. H. LiangY. S. Tan and R. A. Cheke, Threshold conditions for interated pest management models with pesticides that have residual effects, J. Math. Biol., 66 (2013), 1-35. doi: 10.1007/s00285-011-0501-x.

[21]

S. Y. Tang, W. H. Pang, R. A. Cheke and J. H. Wu, Global dynamics of a state-dependent feedback control system Advances in Difference Equations 2015 (2015), 70pp.

[22]

S. Y. TangG. Y. Tang and R. A. Cheke, Optimum timing for integrated pest management: Modeling rates of pesticide application and natural enemy releases, J. Theor. Biol., 264 (2010), 623-638. doi: 10.1016/j.jtbi.2010.02.034.

[23]

S. Y. TangB. TangA. L. Wang and Y. N. Xiao, Holling Ⅱ predator-prey impulsive semi-dynamic model with complex Poincare map, Nonlinear Dynamics, 81 (2015), 1575-1596. doi: 10.1007/s11071-015-2092-3.

[24]

S. Y. TangY. N. XiaoL. S. Chen and R. A. Cheke, Integrated pest management models and their dynamical behaviour, Bull. Math. Biol., 67 (2005), 115-135. doi: 10.1016/j.bulm.2004.06.005.

Figure 1.  Illustrations of impulsive set, phase set and definition of impulsive semi-dynamical system for model (2). The parameter values are fixed as: $r=1, K=52, \beta=0.19, \eta=0.45, \omega=0.19, \delta=0.36, \tau=5, ET=35, \theta=0.7$
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 $0.5(C)$
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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