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2014, 11(6): 1395-1410. doi: 10.3934/mbe.2014.11.1395

## A new model with delay for mosquito population dynamics

 1 Jiangsu Key Laboratory for NSLSCS, School of Mathematical Science, Nanjing Normal University, Nanjing, 210023, China 2 LAboratory of Mathematical Parallel Systems (LAMPS), Centre for Disease Modeling, Department of Mathematics and Statistics, York University, Toronto, Ontario, M3J 1P3, Canada

Received  November 2010 Revised  July 2014 Published  September 2014

In this paper, we formulate a new model with maturation delay for mosquito population incorporating the impact of blood meal resource for mosquito reproduction. Our results suggest that except for the usual crowded effect for adult mosquitoes, the impact of blood meal resource in a given region determines the mosquito abundance, it is also important for the population dynamics of mosquito which may induce Hopf bifurcation. The existence of a stable periodic solution is proved both analytically and numerically. The new model for mosquito also suggests that the resources for mosquito reproduction should not be ignored or mixed with the impact of blood meal resources for mosquito survival and both impacts should be considered in the model of mosquito population. The impact of maturation delay is also analyzed.
Citation: Hui Wan, Huaiping Zhu. A new model with delay for mosquito population dynamics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1395-1410. doi: 10.3934/mbe.2014.11.1395
##### References:
 [1] J. Arino, L. Wang and G. S. K. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theor. Biolo., 241 (2006), 109-119. doi: 10.1016/j.jtbi.2005.11.007. [2] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. [3] S. P. Blythe, R. M. Nisbet and W. S. C. Gurney, Instability and complex dynamic behaviour in population models with long time delays, Theor. Population Biol., 22 (1982), 147-176. doi: 10.1016/0040-5809(82)90040-5. [4] C. Bowman, A. B. Gumel, J. Wu, P. van den Driessche and H. Zhu, A mathematical model for assessing control strategies against West Nile virus, Bull. Math. Biol., 67 (2005), 1107-1133. doi: 10.1016/j.bulm.2005.01.002. [5] N. Chitnis, J. M. Cushing and J. M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission, SIAM J. Appl. Math., 67 (2006), 24-45. doi: 10.1137/050638941. [6] K. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352. doi: 10.1007/s002850050194. [7] G. Cruz-Pacheco, L. Esteva, J. Montaño-Hirose and C. Vargas, Modelling the dynamics of West Nile Virus, Bulletin of Mathematical Biology, 67 (2005), 1157-1172. doi: 10.1016/j.bulm.2004.11.008. [8] L. Esteva and C. Vargas, A model for dengue disease with variable human population, J. Math. Biol., 38 (1999), 220-240. doi: 10.1007/s002850050147. [9] L. Esteva and C. Vargas, Analysis of a dengue disease transmission model, Mathematical Biosciences, 150 (1998), 131-151. doi: 10.1016/S0025-5564(98)10003-2. [10] L. Esteva and C. Vargas, Influence of vertical and mechanical transmission on the dynamics of dengue disease, Mathematical Biosciences, 167 (2000), 51-64. doi: 10.1016/S0025-5564(00)00024-9. [11] Z. Feng, J. X. Velasco-Hernańdez, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (1997), 523-544. doi: 10.1007/s002850050064. [12] H. M. Giles and D. A. Warrel, Bruce-Chwatt's Essential Malariology, $3^{rd}$ Edition, Heinemann Medical Books, Portsmouth, NH, 1993. [13] S. A. Gourley, R. Liu and J. Wu, Eradicating vector-borne disease via agestructured culling, Journal of Mathematical Biology, 54 (2007), 309-335. doi: 10.1007/s00285-006-0050-x. [14] S. A. Gourley, R. Liu and J. Wu, Some vector borne disease with structured host populations: Extinction and spatial spread, SIAM J. Appl. Math., 67 (2007), 408-433. doi: 10.1137/050648717. [15] J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer- Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [16] J. J. Hard and W. E. Bradshaw, Reproductive allocation in the western tree-holo mosquito, Aedes Sierrensis, OIKOS, 66 (1993), 55-65. [17] G. E. Hutchinson, Circular causal systems in ecology, Ann. NY Acad. Sci., 50 (1948), 221-246. doi: 10.1111/j.1749-6632.1948.tb39854.x. [18] Y. Kuang, Delay Differential Equations with Applications in Population Dynamic, Academic Press Inc., Boston, 1993. [19] C. C. Lord and J. F. Day, Simulation studies of St. Louis encephalitis and West Nile viruses: The impact of bird mortality, Vector Borne and Zoonotic Diseases, 1 (2001), 317-329. [20] S. Munga, N. Minakawa and G. Zhou, Survivorship of immature stages of Anopheles gambiae s.l. (Diptera: Culicidae) in natural habitats in western Kenya highlands, Journal of Medical Entomology, 44 (2007), 58-764. [21] G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Mathematical and Computer Modelling, 32 (2000), 747-763. doi: 10.1016/S0895-7177(00)00169-2. [22] D. J. Rodríguez, Time delays in density dependence are often not destabilizing, J. Theor. Biol., 191 (1998), 95-101. [23] S. Ruan, Delay differential equations in single species dynamics, in Delay Differential Equations and Applications (eds. O. Arino, M. L. Hbid and E. Ait Dads), NATO Sci. Ser. II Math. Phys. Chem., 205, Springer, Dordrecht, 2006, 477-517. doi: 10.1007/1-4020-3647-7_11. [24] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995. [25] P. F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Correspondance mathématique et physique, 10 (1838), 113-121. [26] M. J. Wonham, T. de-Camino Beck and M. A. Lewis, An epidemiological model for West Nile virus: Invasion analysis and control applications, Proceedings of the Royal Society. London Ser. B, 271 (2004), 501-507. doi: 10.1098/rspb.2003.2608. [27] [28]

show all references

##### References:
 [1] J. Arino, L. Wang and G. S. K. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theor. Biolo., 241 (2006), 109-119. doi: 10.1016/j.jtbi.2005.11.007. [2] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963. [3] S. P. Blythe, R. M. Nisbet and W. S. C. Gurney, Instability and complex dynamic behaviour in population models with long time delays, Theor. Population Biol., 22 (1982), 147-176. doi: 10.1016/0040-5809(82)90040-5. [4] C. Bowman, A. B. Gumel, J. Wu, P. van den Driessche and H. Zhu, A mathematical model for assessing control strategies against West Nile virus, Bull. Math. Biol., 67 (2005), 1107-1133. doi: 10.1016/j.bulm.2005.01.002. [5] N. Chitnis, J. M. Cushing and J. M. Hyman, Bifurcation analysis of a mathematical model for malaria transmission, SIAM J. Appl. Math., 67 (2006), 24-45. doi: 10.1137/050638941. [6] K. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352. doi: 10.1007/s002850050194. [7] G. Cruz-Pacheco, L. Esteva, J. Montaño-Hirose and C. Vargas, Modelling the dynamics of West Nile Virus, Bulletin of Mathematical Biology, 67 (2005), 1157-1172. doi: 10.1016/j.bulm.2004.11.008. [8] L. Esteva and C. Vargas, A model for dengue disease with variable human population, J. Math. Biol., 38 (1999), 220-240. doi: 10.1007/s002850050147. [9] L. Esteva and C. Vargas, Analysis of a dengue disease transmission model, Mathematical Biosciences, 150 (1998), 131-151. doi: 10.1016/S0025-5564(98)10003-2. [10] L. Esteva and C. Vargas, Influence of vertical and mechanical transmission on the dynamics of dengue disease, Mathematical Biosciences, 167 (2000), 51-64. doi: 10.1016/S0025-5564(00)00024-9. [11] Z. Feng, J. X. Velasco-Hernańdez, Competitive exclusion in a vector-host model for the dengue fever, J. Math. Biol., 35 (1997), 523-544. doi: 10.1007/s002850050064. [12] H. M. Giles and D. A. Warrel, Bruce-Chwatt's Essential Malariology, $3^{rd}$ Edition, Heinemann Medical Books, Portsmouth, NH, 1993. [13] S. A. Gourley, R. Liu and J. Wu, Eradicating vector-borne disease via agestructured culling, Journal of Mathematical Biology, 54 (2007), 309-335. doi: 10.1007/s00285-006-0050-x. [14] S. A. Gourley, R. Liu and J. Wu, Some vector borne disease with structured host populations: Extinction and spatial spread, SIAM J. Appl. Math., 67 (2007), 408-433. doi: 10.1137/050648717. [15] J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations, Springer- Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [16] J. J. Hard and W. E. Bradshaw, Reproductive allocation in the western tree-holo mosquito, Aedes Sierrensis, OIKOS, 66 (1993), 55-65. [17] G. E. Hutchinson, Circular causal systems in ecology, Ann. NY Acad. Sci., 50 (1948), 221-246. doi: 10.1111/j.1749-6632.1948.tb39854.x. [18] Y. Kuang, Delay Differential Equations with Applications in Population Dynamic, Academic Press Inc., Boston, 1993. [19] C. C. Lord and J. F. Day, Simulation studies of St. Louis encephalitis and West Nile viruses: The impact of bird mortality, Vector Borne and Zoonotic Diseases, 1 (2001), 317-329. [20] S. Munga, N. Minakawa and G. Zhou, Survivorship of immature stages of Anopheles gambiae s.l. (Diptera: Culicidae) in natural habitats in western Kenya highlands, Journal of Medical Entomology, 44 (2007), 58-764. [21] G. A. Ngwa and W. S. Shu, A mathematical model for endemic malaria with variable human and mosquito populations, Mathematical and Computer Modelling, 32 (2000), 747-763. doi: 10.1016/S0895-7177(00)00169-2. [22] D. J. Rodríguez, Time delays in density dependence are often not destabilizing, J. Theor. Biol., 191 (1998), 95-101. [23] S. Ruan, Delay differential equations in single species dynamics, in Delay Differential Equations and Applications (eds. O. Arino, M. L. Hbid and E. Ait Dads), NATO Sci. Ser. II Math. Phys. Chem., 205, Springer, Dordrecht, 2006, 477-517. doi: 10.1007/1-4020-3647-7_11. [24] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1995. [25] P. F. Verhulst, Notice sur la loi que la population suit dans son accroissement, Correspondance mathématique et physique, 10 (1838), 113-121. [26] M. J. Wonham, T. de-Camino Beck and M. A. Lewis, An epidemiological model for West Nile virus: Invasion analysis and control applications, Proceedings of the Royal Society. London Ser. B, 271 (2004), 501-507. doi: 10.1098/rspb.2003.2608. [27] [28]
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