The dynamics of an impulsive predator-prey model with communicable disease in the prey species only. (English) Zbl 1410.92142
Summary: In this paper, we propose an impulsive predator-prey model with communicable disease in the prey species only and investigate its interesting biological dynamics. By the Floquet theory of impulsive differential equation and small amplitude perturbation skills, we have deduced the sufficient conditions for the globally asymptotical stability of the semi-trivial periodic solution and the permanence of the proposed model. We also give the existences of the “infection-free” periodic solution and the “predator-free” solution. Finally, numerical results validate the effectiveness of theoretical analysis for the proposed model in this paper.
MSC:
| 92D30 | Epidemiology |
| 92D25 | Population dynamics (general) |
| 34A37 | Ordinary differential equations with impulses |
Keywords:
impulsive differential equations; bifurcation; global asymptotical stability; Floquet theorem; pest controlReferences:
| [1] | Nagai, K.; Yano, E., Predation by orius sauteri (poppius) (heteroptera: Anthocoridae) on thrips palmi karny(thysanoptera: Thripidae). functional response and selective predation, Appl. Entomol. Zool., 35, 565-574 (2000) |
| [2] | Port, C. M.; Scopes, N. E.A., Biological control by predatory mites (phytoseiulus persimilis athias-henriot) of red spider mite (tetranychus urticae koch)infesting strawberries grown in walk-in plastic tunnels, Plant Pathol., 30, 95-99 (1981) |
| [3] | Wang, L.; Li, X.; Zhang, Y. Q.; Zhang, Y.; Zhang, K., Evolution of scaling emergence in large-scale spatial epidemic spreading, PLoS one, 6, e21197 (2011) |
| [4] | Y. Xia, C.; Wang, Z.; Sanz, J.; Meloni, S.; Moreno, Y., Effects of delayed recovery and nonuniform transmission on the spreading of diseases in complex networks, Physica A: Stat. Mech. Appl, 392, 1577-1585 (2013) |
| [5] | Xu, J.; Liu, Q.; Yin, X. D.; Zhu, S. D., A review of recent development of bacillus thuringiensis ICP genetically engineered microbes, Entom. J. East China, 15, 53-58 (2006) |
| [6] | Dean, D. H., Biochemical genetics of the bacterial insect-control agent bacillus thuringiensis: basic principles and prospects for genetic engineering, Biotechnol. Genet. Eng. Rev., 2, 341-363 (1984) |
| [7] | Y. Xia, C.; Wang, L.; Sun, S. W.; Wang, J., An SIR model with infection delay and propagation vector in complex networks, Nonlinear Dyn., 69, 927-934 (2012) |
| [8] | Hall, I. M.; Dunn, P. H., Entomophthorous fungi parasitic on the spotted alfalfa aphid, Hilgardia, 27, 159-181 (1957) |
| [9] | Jiang, X. W.; Song, Q.; Hao, M. Y., Dynamics behaviors of a delayed stage-structured predator-prey model with impulsive effect, Appl. Math. Comput., 215, 4221-4229 (2010) · Zbl 1184.92054 |
| [10] | Gimmelli, G.; Kooi, B. W.; Venturino, E., Ecoepidemic models with prey group defense and feeding saturation, Ecol. Complex., 22, 50-58 (2015) |
| [11] | Xie, Y. X.; Yuan, Z. H.; Wang, L. J., Dynamic analysis of pest control model with population dispersal in two patches and impulsive effect, J. Comput. Sci., 5, 685-695 (2014) |
| [12] | Wang, L. J.; Xie, Y. X.; Fu, J. Q., The dynamics of natural mortality for pest control model with impulsive effect, J. Franklin Inst., 350, 1443-1461 (2013) · Zbl 1293.93093 |
| [13] | Zou, L.; Xiong, Z. L.; Shu, Z. P., The dynamics of an eco-epidemic model with distributed time delay and impulsive control strategy, J. Franklin Inst., 348, 2332-2349 (2011) · Zbl 1252.34095 |
| [14] | Yan, Y. B.; Ekaka-a, E. N., Stabilizing a mathematical model of population system, J. Franklin Inst., 348, 2744-2758 (2011) · Zbl 1254.93134 |
| [15] | Wang, L.; Chen, L.; Nieto, J., The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear Anal.: Real World Appl., 11, 1374-1386 (2010) · Zbl 1188.93038 |
| [16] | Phillips, B. L.; Brown, G. P.; Webb, J. K.; Shine, R., Invasion and the evolution of speed in toads, Nature, 439, 803 (2006) |
| [17] | Lakshmikantham, V.; Bainov, D.; Simeonov, P., Theory of Impulsive Differential Equations, 27-66 (1989), World Scientific Publisher: World Scientific Publisher Singapore · Zbl 0719.34002 |
| [18] | Orr, D., Biological control and integrated pest management, (Peshin, R.; Dhawan, A. K. (2009), Springer: Springer Netherlands), 207-239 |
| [19] | Li, Y. F.; Cui, J. A., The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage, Commun. Nonlinear Sci. Numer. Simul., 14, 2353-2365 (2009) · Zbl 1221.34034 |
| [20] | Jia, J. W.; Cao, H., Dynamic complexities of holling type II functional response predator-prey system with digest delay and impulsive harvesting on the prey, Int. J. Biomath., 2, 229-242 (2009) · Zbl 1342.92177 |
| [21] | Liu, Z. J.; Chen, L. S., Periodic solution of a two-species competitive system with toxicant and birth pulse, Chaos Solitons Fract., 32, 1703-1712 (2007) · Zbl 1137.34017 |
| [22] | Lakmeche, A.; Arino, O., Nonlinear mathematical model of pulsed therapy of heterogeneous tumors, Nonlinear Anal., 2, 455-465 (2001) · Zbl 0982.92016 |
| [23] | Tang, S. Y.; Chen, L. S., The periodic predator-prey lotka-volterra model with impulsive effect, J. Mech. Med. Biol., 2, 267-296 (2002) |
| [24] | Liu, B.; Zhang, Y. J.; Chen, L. S., The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management, Nonlinear Anal. Real World Appl., 6, 227-243 (2005) · Zbl 1082.34039 |
| [25] | Stone, L.; Shulgin, B.; Agur, Z., Theoretical examination of the pulse vaccination policy in the SIR epidemic model, Math. Comput. Model., 31, 207-215 (2000) · Zbl 1043.92527 |
| [26] | d’Onofrio, A., Pulse vaccination strategy in the SIR epidemic model: Global asymptotic stable eradication in presence of vaccine failures, Math. Comput. Model., 36, 473-489 (2002) · Zbl 1025.92011 |
| [27] | d’Onofrio, A., Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. Biosci., 179, 57-72 (2002) · Zbl 0991.92025 |
| [28] | Hadeler, K. P.; Freedman, H. I., Predator-prey population with parasite infection, J. Math. Biol., 27, 609-631 (1989) · Zbl 0716.92021 |
| [29] | Panetta, J. C., A mathematical model of periodically pulsed chemotheapy: tumor recurrence and metastasis in a competition environment, Bull. Math. Biol., 58, 3, 425-447 (1995) · Zbl 0859.92014 |
| [30] | Lakmeeh, A.; Arino, O., Bifurcation of nontrivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dynam. Contin. Discret. Impuls. Syst., 7, 265-287 (2000) · Zbl 1011.34031 |
| [31] | Liu, M. X.; Jin, Z.; Haque, M., An impulsive predator-prey model with communicable disease in the prey species only, Nonlinear Anal. Real World Appl., 10, 3098-3111 (2009) · Zbl 1162.92043 |
| [32] | Wang, X.; Guo, Z.; Song, X. Y., Dynamical behavior of a pest management model with impulsive effect and nonlinear incidence rate, J. Comput. Appl. Math., 30, 381-398 (2011) · Zbl 1237.34097 |
| [33] | Baek, H. K., Qualitative analysis of Beddington-Deangelis type impulsive predator-prey models, Nonlinear Anal. Real World Appl., 11, 312-1322 (2010) · Zbl 1204.34062 |
| [34] | Wang, L. M.; Chen, L. S.; Nieto, J. J., The dynamics of an epidemic model for pest control with impulsive effect, J. Nonlinear Anal. Real World Appl., 11, 1374-1386 (2010) · Zbl 1188.93038 |
| [35] | Hui, J.; Chen, L., Dynamic complexities in a periodically pulsed ratio-dependent predator-prey ecosystem modeled on a chemostat, Chaos Solitons Fractals, 29, 407-416 (2006) · Zbl 1095.92066 |
| [36] | Abdelkader, L.; Arino, O., Bifurcation of nontrivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment, Dyn. Contin. Discret. Impuls. Syst., 7, 265-287 (2000) · Zbl 1011.34031 |
| [37] | Panetta, J. C., A mathematical model of periodically pulsed chemotheapy: tumor recurrence and metastasis in a competition environment, Bull. Math. Biol., 58, 425-447 (1996) · Zbl 0859.92014 |
| [38] | Wang, L.; Liu, Z.; Hui, J.; Chen, L., Impulsive diffusion in single species model, Chaos Solitons Fractals, 33, 1213-1219 (2007) · Zbl 1131.92071 |
| [39] | Zhang, H.; Chen, L. S.; Nieto, J. J., A delayed epidemic model with stage-structureand pulses for pest management strategy, Nonlinear Anal. Real World Probl., 9, 1714-1726 (2008) · Zbl 1154.34394 |
| [40] | Meng, X. Z.; Li, Z. Q.; Nieto, J. J., Dynamic analysis of michaelis-menten chemostat-type competition models with time delay and pulse in a polluted environment, J. Math. Chem., 47, 123-144 (2009) · Zbl 1194.92075 |
| [41] | Jiao, J. J.; Yang, X. S.; Chen, L. S.; Cai, S. H., Effect of delayed response in growth on the dynamics of a chemostat model with impulsive input, Chaos Solitons Fractals, 42, 2280-2287 (2009) · Zbl 1198.34134 |
| [42] | Nieto, J. J.; ORegan, D., Variational approach to impulsive differential equations, Nonlinear Anal. Real World Appl., 10, 680-690 (2009) · Zbl 1167.34318 |
| [43] | Benchohra, M.; Henderson, J.; Ntouyas, S. K., Impulsive Differential Equations and Inclusions, vol. 2 (2006), Hindawi Publishing Corporation: Hindawi Publishing Corporation New York · Zbl 1130.34003 |
| [44] | Xiao, Y.; Bosch, F. V.D., The dynamics of an eco-epidemic model with bio-logical control, Ecol. Model., 168, 203-214 (2003) |
| [45] | Rhodes, C. J.; Anderson, R. M., Forest-fire as a model for the dynamics of disease epidemics, J. Franklin Inst., 335, 199-211 (1998) · Zbl 0905.92026 |
| [46] | Xie, Y. X., The Research of Several Pests Control Models with Impulsive Effect (2010), Hunan University |
| [47] | Wang, L. M.; Chen, L. S.; Nieto, J. J., The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear Anal. Real World Appl., 11, 1374-1386 (2010) · Zbl 1188.93038 |
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