• Previous Article
    The dynamics of technological change under constraints: Adopters and resources
  • DCDS-B Home
  • This Issue
  • Next Article
    Spreading speeds and traveling wave solutions in a competitive reaction-diffusion model for species persistence in a stream
December  2014, 19(10): 3283-3298. doi: 10.3934/dcdsb.2014.19.3283

A model for the biocontrol of mosquitoes using predatory fish

1. 

Department of Mathematics and Department of Zoology and Physiology, University of Wyoming, Laramie, WY, 82071, United States

2. 

Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom

Received  June 2013 Revised  March 2014 Published  October 2014

We present a mathematical model for the localised control of mosquitoes using larvivorous fish. It is supposed that the adult mosquitoes choose among a finite number of isolated ponds for oviposition and that these ponds differ in various respects including physical size, survival prospects and maturation times for mosquito larvae. We model a mosquito control effort that involves stocking some or all of these ponds with larvivorous fish such as the mosquitofish Gambusia affinis. The effect of doing so may vary from pond to pond, and the ponds are coupled via the adult mosquitoes in the air. Also, adult mosquitoes may avoid ovipositing in ponds containing the larvivorous fish. Our model enables us to predict how the larvivorous fish should be allocated between ponds, and shows in particular that only certain ponds should be stocked if there is a limited supply of the fish. We also consider oviposition pond selection by mosquitoes, and show that in some situations mosquitoes might do better to simply choose a pond at random.
Citation: Rongsong Liu, Stephen A. Gourley. A model for the biocontrol of mosquitoes using predatory fish. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3283-3298. doi: 10.3934/dcdsb.2014.19.3283
References:
[1]

M. Araujo, L. H. S. Gil and A. e-Silva, Larval food quantity affects development time, survival and adult biological traits that influence the vectorial capacity of Anopheles darlingi under laboratory conditions, Malaria Journal, 11 (2012), 261pp.

[2]

L. Blaustein and R. Karban, Indirect effects of the mosquitofish Gambusia affinis on the mosquito Culex tarsalis, Limnology and Oceanography, 35 (1990), 767-771.

[3]

, Centers for Disease Control (CDC), Anopheles Mosquitoes,, , (). 

[4]

J. D. Charlwood, T. Smith, P. F. Billingsley, W. Takken, E. O. K. Lyimo and J. H. E. T. Meuwissen, Survival and infection probabilities of anthropophagic anophelines from an area of high prevalence of Plasmodium falciparum in humans, Bull. Entomol. Res., 87 (1997), 445-453.

[5]

J. E. Deacon, C. Hubbs and B. J. Zahuranec, Some effects of introduced fishes on the native fish fauna of southern Nevada, Copeia, 1964 (1964), 384-388. doi: 10.2307/1441031.

[6]

S. A. Gourley, R. Liu and J. Wu, Some vector borne diseases with structured host populations: Extinction and spatial spread,, SIAM J. Appl. Math., 67 (): 408.  doi: 10.1137/050648717.

[7]

S. A. Gourley and S. Ruan, A delay equation model for oviposition habitat selection by mosquitoes, J. Math. Biol., 65 (2012), 1125-1148. doi: 10.1007/s00285-011-0491-8.

[8]

W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.

[9]

L. A. Krumholz, Reproduction in the western mosquitofish, Gambusia affinis affinis (Baird & Girard), and its use in mosquito control, Ecological Monographs, 18 (1948), 1-43.

[10]

G. K. Meffe, D. A. Hendrickson and W. L. Minckley, Factors resulting in decline of the endangered Sonoran topminnow Poeciliopsis occidentalis (Atheriniformes: Poeciliidae) in the United States, Biological Conservation, 25 (1983), 135-159.

[11]

G. K. Meffe, Predation and species replacement in American southwestern fishes: a case study, Southwestern Naturalist, 30 (1985), 173-187. doi: 10.2307/3670732.

[12]

A. Mokany and R. Shine, Oviposition site selection by mosquitoes is affected by cues from conspecific larvae and anuran tadpoles, Austral Ecology, 28 (2003), 33-37. doi: 10.1046/j.1442-9993.2003.01239.x.

[13]

P. B. Moyle, Inland Fishes of California, University of California Press, Berkeley, CA, 1976.

[14]

M. H. Reiskind and A. A. Zarrabi, Water surface area and depth determine oviposition choice in Aedes albopictus (Diptera: Culicidae), J. Med. Entomol., 49 (2012), 71-76.

[15]

J. B. Silver, Mosquito Ecology: Field Sampling Methods, Springer, 2008.

[16]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.

[17]

M. Spencer, L. Blaustein and J. E. Cohen, Oviposition habitat selection by mosquitoes (culiseta longiareolata) and consequences for population size, Ecology, 83 (2002), 669-679. doi: 10.2307/3071872.

[18]

M. J. Wonham, T. de-Camino-Beck and M. A. Lewis, An epidemiological model for West Nile virus: Invasion analysis and control applications, Proc. R. Soc. Lond. Ser. B., 271 (2004), 501-507. doi: 10.1098/rspb.2003.2608.

[19]

M. Yoshioka, J. Couret, F. Kim, J. McMillan, T. R. Burkot, E. M. Dotson, U. Kitron and G. M. Vazquez-Prokopec, Diet and density dependent competition affect larval performance and oviposition site selection in the mosquito species Aedes albopictus (Diptera: Culicidae), Parasit Vectors, 5 (2012), 225pp.

show all references

References:
[1]

M. Araujo, L. H. S. Gil and A. e-Silva, Larval food quantity affects development time, survival and adult biological traits that influence the vectorial capacity of Anopheles darlingi under laboratory conditions, Malaria Journal, 11 (2012), 261pp.

[2]

L. Blaustein and R. Karban, Indirect effects of the mosquitofish Gambusia affinis on the mosquito Culex tarsalis, Limnology and Oceanography, 35 (1990), 767-771.

[3]

, Centers for Disease Control (CDC), Anopheles Mosquitoes,, , (). 

[4]

J. D. Charlwood, T. Smith, P. F. Billingsley, W. Takken, E. O. K. Lyimo and J. H. E. T. Meuwissen, Survival and infection probabilities of anthropophagic anophelines from an area of high prevalence of Plasmodium falciparum in humans, Bull. Entomol. Res., 87 (1997), 445-453.

[5]

J. E. Deacon, C. Hubbs and B. J. Zahuranec, Some effects of introduced fishes on the native fish fauna of southern Nevada, Copeia, 1964 (1964), 384-388. doi: 10.2307/1441031.

[6]

S. A. Gourley, R. Liu and J. Wu, Some vector borne diseases with structured host populations: Extinction and spatial spread,, SIAM J. Appl. Math., 67 (): 408.  doi: 10.1137/050648717.

[7]

S. A. Gourley and S. Ruan, A delay equation model for oviposition habitat selection by mosquitoes, J. Math. Biol., 65 (2012), 1125-1148. doi: 10.1007/s00285-011-0491-8.

[8]

W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.

[9]

L. A. Krumholz, Reproduction in the western mosquitofish, Gambusia affinis affinis (Baird & Girard), and its use in mosquito control, Ecological Monographs, 18 (1948), 1-43.

[10]

G. K. Meffe, D. A. Hendrickson and W. L. Minckley, Factors resulting in decline of the endangered Sonoran topminnow Poeciliopsis occidentalis (Atheriniformes: Poeciliidae) in the United States, Biological Conservation, 25 (1983), 135-159.

[11]

G. K. Meffe, Predation and species replacement in American southwestern fishes: a case study, Southwestern Naturalist, 30 (1985), 173-187. doi: 10.2307/3670732.

[12]

A. Mokany and R. Shine, Oviposition site selection by mosquitoes is affected by cues from conspecific larvae and anuran tadpoles, Austral Ecology, 28 (2003), 33-37. doi: 10.1046/j.1442-9993.2003.01239.x.

[13]

P. B. Moyle, Inland Fishes of California, University of California Press, Berkeley, CA, 1976.

[14]

M. H. Reiskind and A. A. Zarrabi, Water surface area and depth determine oviposition choice in Aedes albopictus (Diptera: Culicidae), J. Med. Entomol., 49 (2012), 71-76.

[15]

J. B. Silver, Mosquito Ecology: Field Sampling Methods, Springer, 2008.

[16]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995.

[17]

M. Spencer, L. Blaustein and J. E. Cohen, Oviposition habitat selection by mosquitoes (culiseta longiareolata) and consequences for population size, Ecology, 83 (2002), 669-679. doi: 10.2307/3071872.

[18]

M. J. Wonham, T. de-Camino-Beck and M. A. Lewis, An epidemiological model for West Nile virus: Invasion analysis and control applications, Proc. R. Soc. Lond. Ser. B., 271 (2004), 501-507. doi: 10.1098/rspb.2003.2608.

[19]

M. Yoshioka, J. Couret, F. Kim, J. McMillan, T. R. Burkot, E. M. Dotson, U. Kitron and G. M. Vazquez-Prokopec, Diet and density dependent competition affect larval performance and oviposition site selection in the mosquito species Aedes albopictus (Diptera: Culicidae), Parasit Vectors, 5 (2012), 225pp.

[1]

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

[2]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042

[3]

Jan Sieber, Matthias Wolfrum, Mark Lichtner, Serhiy Yanchuk. On the stability of periodic orbits in delay equations with large delay. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3109-3134. doi: 10.3934/dcds.2013.33.3109

[4]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219

[5]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations and Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493

[6]

Anatoly Neishtadt. On stability loss delay for dynamical bifurcations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (4) : 897-909. doi: 10.3934/dcdss.2009.2.897

[7]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577

[8]

Elena Braverman, Sergey Zhukovskiy. Absolute and delay-dependent stability of equations with a distributed delay. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2041-2061. doi: 10.3934/dcds.2012.32.2041

[9]

Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095

[10]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361

[11]

István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773

[12]

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

[13]

C. Connell McCluskey. Global stability of an $SIR$ epidemic model with delay and general nonlinear incidence. Mathematical Biosciences & Engineering, 2010, 7 (4) : 837-850. doi: 10.3934/mbe.2010.7.837

[14]

Azmy S. Ackleh, Keng Deng. Stability of a delay equation arising from a juvenile-adult model. Mathematical Biosciences & Engineering, 2010, 7 (4) : 729-737. doi: 10.3934/mbe.2010.7.729

[15]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[16]

Jianghao Hao, Junna Zhang. General stability of abstract thermoelastic system with infinite memory and delay. Mathematical Control and Related Fields, 2021, 11 (2) : 353-371. doi: 10.3934/mcrf.2020040

[17]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations and Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032

[18]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295

[19]

Leonid Shaikhet. Stability of equilibriums of stochastically perturbed delay differential neoclassical growth model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1565-1573. doi: 10.3934/dcdsb.2017075

[20]

Samuel Bernard, Fabien Crauste. Optimal linear stability condition for scalar differential equations with distributed delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1855-1876. doi: 10.3934/dcdsb.2015.20.1855

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (83)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]