By using the theory of calculus on time scales and some mathematical methods, several dynamic inequalities on time scales are established. Based on these results, we derive some sufficient conditions for permanence of predator-prey system incorporating a prey refuge on time scales. Finally, examples and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

References

1 Huang C.-Y., Zhao M., and Zhao L.-C., Permanence of periodic predator-prey system with two predators and stage structure for prey, Nonlinear Analysis. Real World Applications. (2010) 11, no. 1, 503–514, https://doi.org/10.1016/j.nonrwa.2009.01.001, 2570569, ZBL1189.34085.
10.1016/j.nonrwa.2009.01.001
Web of Science® Google Scholar
2 Chen F. and You M., Permanence, extinction and periodic solution of the predator-prey system with Beddington-DeAngelis functional response and stage structure for prey, Nonlinear Analysis. Real World Applications. (2008) 9, no. 2, 207–221, https://doi.org/10.1016/j.nonrwa.2006.09.009, 2382372, ZBL1142.34051.
10.1016/j.nonrwa.2006.09.009
Web of Science® Google Scholar
3 Cui J. and Song X., Permanence of predator-prey system with stage structure, Discrete and Continuous Dynamical Systems B. (2004) 4, no. 3, 547–554, https://doi.org/10.3934/dcdsb.2004.4.547, 2073958, ZBL1100.92062.
10.3934/dcdsb.2004.4.547
Web of Science® Google Scholar
4 Chen L. and Chen L., Permanence of a discrete periodic Volterra model with mutual inter-ference, Discrete Dynamics in Nature and Society. (2009) 2009, 11, 205481, https://doi.org/10.1155/2009/205481.
10.1155/2009/205481
Web of Science® Google Scholar
5 Fan Y.-H. and Li W.-T., Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response, Journal of Mathematical Analysis and Applications. (2004) 299, no. 2, 357–374, https://doi.org/10.1016/j.jmaa.2004.02.061, 2098248, ZBL1063.39013.
10.1016/j.jmaa.2004.02.061
Web of Science® Google Scholar
6 Yang X., Uniform persistence and periodic solutions for a discrete predator-prey system with delays, Journal of Mathematical Analysis and Applications. (2006) 316, no. 1, 161–177, https://doi.org/10.1016/j.jmaa.2005.04.036, 2201755, ZBL1107.39017.
10.1016/j.jmaa.2005.04.036
Web of Science® Google Scholar
7 Bohner M. and Peterson A., Dynamic equations on time scales, An Introduction with Applications, 2001, Birkhauser, Boston, Mass, USA, 1843232.
10.1007/978-1-4612-0201-1
Google Scholar
8 Hilger S., Analysis on measure chains—a unified approach to continuous and discrete calculus, Results in Mathematics. (1990) 18, no. 1-2, 18–56, 1066641, ZBL0722.39001.
10.1007/BF03323153
Google Scholar
9 Spedding V., Taming nature′s numbers, New Scientist. (2003) 2404, 28–31.
Google Scholar
10 McKellar R. C. and Knight K., A combined discrete-continuous model describing the lag phase of Listeria monocytogenes, International Journal of Food Microbiology. (2000) 54, no. 3, 171–180, 2-s2.0-0033997998, https://doi.org/10.1016/S0168-1605(99)00204-4.
10.1016/S0168-1605(99)00204-4
CAS PubMed Web of Science® Google Scholar
11 Tisdell C. C. and Zaidi A., Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Analysis. Theory, Methods & Applications. (2008) 68, no. 11, 3504–3524, https://doi.org/10.1016/j.na.2007.03.043, 2401364, ZBL1151.34005.
10.1016/j.na.2007.03.043
Web of Science® Google Scholar
12 Fazly M. and Hesaaraki M., Periodic solutions for predator-prey systems with Beddington-DeAngelis functional response on time scales, Nonlinear Analysis. Real World Applications. (2008) 9, no. 3, 1224–1235, https://doi.org/10.1016/j.nonrwa.2007.02.012, 2392407, ZBL1145.92035.
10.1016/j.nonrwa.2007.02.012
Web of Science® Google Scholar
13 Li Y. and Hu M., Three positive periodic solutions for a class of higher-dimensional functional differential equations with impulses on time scales, Advances in Difference Equations. (2009) 2009, 18, 698463, 2574704, ZBL1213.34117.
10.1155/2009/698463
Google Scholar
14 Bi L., Bohner M., and Fan M., Periodic solutions of functional dynamic equations with infinite delay, Nonlinear Analysis. Theory, Methods & Applications. (2008) 68, no. 5, 1226–1245, https://doi.org/10.1016/j.na.2006.12.017, 2381667, ZBL1135.34335.
10.1016/j.na.2006.12.017
Web of Science® Google Scholar
15 Li Y., Zhao L., and Liu P., Existence and exponential stability of periodic solution of high-order Hopfield neural network with delays on time scales, Discrete Dynamics in Nature and Society. (2009) 2009, 18, 573534, https://doi.org/10.1155/2009/573534, 2556328, ZBL1187.34129.
10.1155/2009/573534
Web of Science® Google Scholar
16 Adivar M. and Raffoul Y. N., Shift operators and stability in delayed dynamic equations, Rendiconti del Seminario Matematico. (2010) 68, no. 4, 369–396, 2815209, ZBL1227.34096.
Google Scholar

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