# American Institute of Mathematical Sciences

• Previous Article
Threshold dynamics of an SIR epidemic model with hybrid of multigroup and patch structures
• MBE Home
• This Issue
• Next Article
A stochastic simulation model for Anelosimus studiosus during prey capture: A case study for determination of optimal spacing
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. [2] R. Bellman and K. L. Cooke, Differential-Difference Equations,, Academic Press, (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. 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. 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. 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. 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. 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. doi: 10.1007/s002850050147. [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. [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. [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. [12] H. M. Giles and D. A. Warrel, Bruce-Chwatt's Essential Malariology,, $3^{rd}$ Edition, (1993). [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. [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. [15] J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations,, Springer- Verlag, (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. [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. [18] Y. Kuang, Delay Differential Equations with Applications in Population Dynamic,, Academic Press Inc., (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. [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. [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. [22] D. J. Rodríguez, Time delays in density dependence are often not destabilizing,, J. Theor. Biol., 191 (1998), 95. [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. [24] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995). [25] P. F. Verhulst, Notice sur la loi que la population suit dans son accroissement,, Correspondance mathématique et physique, 10 (1838), 113. [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. [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. doi: 10.1016/j.jtbi.2005.11.007. [2] R. Bellman and K. L. Cooke, Differential-Difference Equations,, Academic Press, (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. 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. 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. 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. 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. 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. doi: 10.1007/s002850050147. [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. [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. [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. [12] H. M. Giles and D. A. Warrel, Bruce-Chwatt's Essential Malariology,, $3^{rd}$ Edition, (1993). [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. [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. [15] J. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations,, Springer- Verlag, (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. [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. [18] Y. Kuang, Delay Differential Equations with Applications in Population Dynamic,, Academic Press Inc., (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. [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. [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. [22] D. J. Rodríguez, Time delays in density dependence are often not destabilizing,, J. Theor. Biol., 191 (1998), 95. [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. [24] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems,, Mathematical Surveys and Monographs, (1995). [25] P. F. Verhulst, Notice sur la loi que la population suit dans son accroissement,, Correspondance mathématique et physique, 10 (1838), 113. [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. [27] [28]
 [1] Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325 [2] Wonlyul Ko, Inkyung Ahn, Shengqiang Liu. Asymptotical behaviors of a general diffusive consumer-resource model with maturation delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1715-1733. doi: 10.3934/dcdsb.2015.20.1715 [3] Suqi Ma, Qishao Lu, Shuli Mei. Dynamics of a logistic population model with maturation delay and nonlinear birth rate. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 735-752. doi: 10.3934/dcdsb.2005.5.735 [4] Tomas Alarcon, Philipp Getto, Anna Marciniak-Czochra, Maria dM Vivanco. A model for stem cell population dynamics with regulated maturation delay. Conference Publications, 2011, 2011 (Special) : 32-43. doi: 10.3934/proc.2011.2011.32 [5] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [6] Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051 [7] Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657 [8] Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026 [9] Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367 [10] Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121 [11] Edoardo Beretta, Dimitri Breda. Discrete or distributed delay? Effects on stability of population growth. Mathematical Biosciences & Engineering, 2016, 13 (1) : 19-41. doi: 10.3934/mbe.2016.13.19 [12] Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503 [13] Fang Han, Bin Zhen, Ying Du, Yanhong Zheng, Marian Wiercigroch. Global Hopf bifurcation analysis of a six-dimensional FitzHugh-Nagumo neural network with delay by a synchronized scheme. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 457-474. doi: 10.3934/dcdsb.2011.16.457 [14] Zuolin Shen, Junjie Wei. Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. Mathematical Biosciences & Engineering, 2018, 15 (3) : 693-715. doi: 10.3934/mbe.2018031 [15] Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 [16] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [17] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 [18] Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451 [19] Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979 [20] Kousuke Kuto. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 489-509. doi: 10.3934/dcds.2009.24.489

2017 Impact Factor: 1.23

## Metrics

• HTML views (0)
• Cited by (0)

• on AIMS