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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. doi: 10.1016/j.jtbi.2005.11.007. Google Scholar

[2]

R. Bellman and K. L. Cooke, Differential-Difference Equations,, Academic Press, (1963). Google Scholar

[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. doi: 10.1016/0040-5809(82)90040-5. Google Scholar

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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. doi: 10.1016/j.bulm.2005.01.002. Google Scholar

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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. doi: 10.1137/050638941. Google Scholar

[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. doi: 10.1007/s002850050194. Google Scholar

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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. doi: 10.1016/j.bulm.2004.11.008. Google Scholar

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L. Esteva and C. Vargas, A model for dengue disease with variable human population,, J. Math. Biol., 38 (1999), 220. doi: 10.1007/s002850050147. Google Scholar

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L. Esteva and C. Vargas, Analysis of a dengue disease transmission model,, Mathematical Biosciences, 150 (1998), 131. doi: 10.1016/S0025-5564(98)10003-2. Google Scholar

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L. Esteva and C. Vargas, Influence of vertical and mechanical transmission on the dynamics of dengue disease,, Mathematical Biosciences, 167 (2000), 51. doi: 10.1016/S0025-5564(00)00024-9. Google Scholar

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Z. Feng, J. X. Velasco-Hernańdez, Competitive exclusion in a vector-host model for the dengue fever,, J. Math. Biol., 35 (1997), 523. doi: 10.1007/s002850050064. Google Scholar

[12]

H. M. Giles and D. A. Warrel, Bruce-Chwatt's Essential Malariology,, $3^{rd}$ Edition, (1993). Google Scholar

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S. A. Gourley, R. Liu and J. Wu, Eradicating vector-borne disease via agestructured culling,, Journal of Mathematical Biology, 54 (2007), 309. doi: 10.1007/s00285-006-0050-x. Google Scholar

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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. doi: 10.1137/050648717. Google Scholar

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J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations,, Springer- Verlag, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar

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J. J. Hard and W. E. Bradshaw, Reproductive allocation in the western tree-holo mosquito, Aedes Sierrensis,, OIKOS, 66 (1993), 55. Google Scholar

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G. E. Hutchinson, Circular causal systems in ecology,, Ann. NY Acad. Sci., 50 (1948), 221. doi: 10.1111/j.1749-6632.1948.tb39854.x. Google Scholar

[18]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamic,, Academic Press Inc., (1993). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/S0895-7177(00)00169-2. Google Scholar

[22]

D. J. Rodríguez, Time delays in density dependence are often not destabilizing,, J. Theor. Biol., 191 (1998), 95. Google Scholar

[23]

S. Ruan, Delay differential equations in single species dynamics,, in Delay Differential Equations and Applications (eds. O. Arino, (2006), 477. doi: 10.1007/1-4020-3647-7_11. Google Scholar

[24]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995). Google Scholar

[25]

P. F. Verhulst, Notice sur la loi que la population suit dans son accroissement,, Correspondance mathématique et physique, 10 (1838), 113. Google Scholar

[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. doi: 10.1098/rspb.2003.2608. Google Scholar

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, , (). Google Scholar

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, , (). Google Scholar

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. doi: 10.1016/j.jtbi.2005.11.007. Google Scholar

[2]

R. Bellman and K. L. Cooke, Differential-Difference Equations,, Academic Press, (1963). Google Scholar

[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. doi: 10.1016/0040-5809(82)90040-5. Google Scholar

[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. doi: 10.1016/j.bulm.2005.01.002. Google Scholar

[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. doi: 10.1137/050638941. Google Scholar

[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. doi: 10.1007/s002850050194. Google Scholar

[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. doi: 10.1016/j.bulm.2004.11.008. Google Scholar

[8]

L. Esteva and C. Vargas, A model for dengue disease with variable human population,, J. Math. Biol., 38 (1999), 220. doi: 10.1007/s002850050147. Google Scholar

[9]

L. Esteva and C. Vargas, Analysis of a dengue disease transmission model,, Mathematical Biosciences, 150 (1998), 131. doi: 10.1016/S0025-5564(98)10003-2. Google Scholar

[10]

L. Esteva and C. Vargas, Influence of vertical and mechanical transmission on the dynamics of dengue disease,, Mathematical Biosciences, 167 (2000), 51. doi: 10.1016/S0025-5564(00)00024-9. Google Scholar

[11]

Z. Feng, J. X. Velasco-Hernańdez, Competitive exclusion in a vector-host model for the dengue fever,, J. Math. Biol., 35 (1997), 523. doi: 10.1007/s002850050064. Google Scholar

[12]

H. M. Giles and D. A. Warrel, Bruce-Chwatt's Essential Malariology,, $3^{rd}$ Edition, (1993). Google Scholar

[13]

S. A. Gourley, R. Liu and J. Wu, Eradicating vector-borne disease via agestructured culling,, Journal of Mathematical Biology, 54 (2007), 309. doi: 10.1007/s00285-006-0050-x. Google Scholar

[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. doi: 10.1137/050648717. Google Scholar

[15]

J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations,, Springer- Verlag, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar

[16]

J. J. Hard and W. E. Bradshaw, Reproductive allocation in the western tree-holo mosquito, Aedes Sierrensis,, OIKOS, 66 (1993), 55. Google Scholar

[17]

G. E. Hutchinson, Circular causal systems in ecology,, Ann. NY Acad. Sci., 50 (1948), 221. doi: 10.1111/j.1749-6632.1948.tb39854.x. Google Scholar

[18]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamic,, Academic Press Inc., (1993). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/S0895-7177(00)00169-2. Google Scholar

[22]

D. J. Rodríguez, Time delays in density dependence are often not destabilizing,, J. Theor. Biol., 191 (1998), 95. Google Scholar

[23]

S. Ruan, Delay differential equations in single species dynamics,, in Delay Differential Equations and Applications (eds. O. Arino, (2006), 477. doi: 10.1007/1-4020-3647-7_11. Google Scholar

[24]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995). Google Scholar

[25]

P. F. Verhulst, Notice sur la loi que la population suit dans son accroissement,, Correspondance mathématique et physique, 10 (1838), 113. Google Scholar

[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. doi: 10.1098/rspb.2003.2608. Google Scholar

[27]

, , (). Google Scholar

[28]

, , (). Google Scholar

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