Bifurcation Analysis of a Mosquito Population Model with a Saturated Release Rate of Sterile Mosquitoes

Authors: Jicai Huang, Shigui Ruan http://orcid.org/0000-0002-6348-8205, Pei Yu, and Yuyue ZhangAuthors Info & Affiliations

http://doi.org/10.1137/18M1208435

Tools

Abstract

Releasing sterile mosquitoes is a method of mosquito control that uses area-wide inundative releases of sterile male mosquitoes to reduce reproduction in a field population of wild mosquitoes. In this paper, we consider a mosquito population model with a nonlinear saturated release rate of sterile mosquitoes and study the complex dynamics and bifurcations of the model. It is shown that there are a weak focus of multiplicity 3 and a nilpotent cusp of codimension 4 for various parameter values and the model exhibits Hopf bifurcation of codimension 3 and Bogdanov--Takens bifurcation of codimension 2 as the parameter values vary. Our analysis also shows that there exists a critical release rate coefficient of sterile mosquitoes, above which the mosquito population can be eliminated and below which the interacting sterile and wild mosquitoes coexist in the form of multiple periodic oscillations and steady states for some initial populations. Numerical simulations are presented to demonstrate the coexistence of a homoclinic loop and a limit cycle, the existence of two limit cycles, and the existence of three limit cycles, respectively.

Keywords

MSC codes

Get full access to this article

View all available purchase options and get full access to this article.

References

1.

P. Aguirre, E. González-Olivares, and E. Sáez, Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM J. Appl. Math., 69 (2009), pp. 1244--1262.

2.

L. Alphey, Genetic control of mosquitoes, Annu. Rev. Entomol., 59 (2014), pp. 205--224.

3.

R. Anguelov, Y. Dumont, and J. Lubuma, Mathematical modeling of sterile insect technology for control of anopheles mosquito, Comput. Math. Appl., 64 (2012), pp. 374--389.

4.

M. P. Atkinson, Z. Su, N. Alphey, L. S. Alphey, P. G. Coleman, and L. M. Wein, Analyzing the control of mosquito-borne diseases by a dominant lethal genetic system, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 9540--9545.

5.

H. J. Barclay, Models for the sterile insect release method with the concurrent release of pesticides, Ecol. Model., 11 (1980), pp. 167--177.

6.

H. J. Barclay, Models for pest control: Complementary effects of periodic releases of sterile pests and parasitoids, Theoret. Popul. Biol., 32 (1987), pp. 76--89.

7.

H. J. Barclay and M. Mackauer, The sterile insect release method for pest control: a density dependent model, Environment. Entomol., 9 (1980), pp. 810--817.

8.

M. Q. Benedict and A. S. Robinson, The first releases of transgenic mosquitoes: An argument for the sterile insect technique, Trends Parasitol., 19 (2003), pp. 349--356.

9.

A. A. Berryman, Mathematical description of the sterile male principle, Can. Entomol., 99 (1967), pp. 858--865.

10.

K. W. Blayneha and J. Mohammed-Awel, Insecticide-resistant mosquitoes and malaria control, Math. Biosci., 252 (2014), pp. 14--26.

11.

C. Boete, F. B. Agusto, and R. G. Reeves, Impact of mating behaviour on the success of malaria control through a single inundative release of transgenic mosquitoes, J. Theoret. Biol., 347 (2014), pp. 33--43.

12.

R. I. Bogdanov, The versal deformations of a singular point on the plane in the case of zero eigenvalues, Trudy Sem. Petrovsk. Vyp., 2 (1976), pp. 37--65.

13.

L. Cai, S. Ai, and J. Li, Dynamics of mosquitoes populations with different strategies of releasing sterile mosquitoes, SIAM, J. Appl. Math., 75 (2014), pp. 1223--1237.

14.

D. O. Carvalho, A. L. Costa-da-Silva, and R. S. Lees, Two step male release strategy using transgenic mosquito lines to control transmission of vector-borne diseases, Acta Tropica, 132 (2014), pp. S170--S177.

15.

S.-N. Chow, C. Li, and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, UK, 1994.

16.

C. S. Coleman, Hilbert's $16$th problem: How many cycles? in Differential Equations Model, M. Braun, C. S. Coleman, D. Drew, eds., Springer, New York, 1983, pp. 279--297.

17.

C. Dufourd and Y. Dumont, Impact of environmental factors on mosquito dispersal in the prospect of sterile insect technique control, Comput. Math. Appl., 66 (2013), pp. 1695--1715.

18.

Y. Dumont and J. M. Tchuenche, Mathematical studies on the sterile insect technique for the chikungunya disease and Aedes albopictus, J. Math. Biol., 65 (2012), pp. 809--854.

19.

F. Dumortier, R. Roussarie, and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic Theory Dynam. Systems, 7 (1987) (3), pp. 375--413.

20.

F. Dumortier, R. Roussarie, J. Sotomayor, and K. Zoladek, Bifurcation of Planar Vector Fields, Nilpotent Singularities and Abelian Integrals, Lecture Notes in Math. 1480, Springer, New York, 1991.

21.

V. A. Dyck, J. Hendrichs, and A. S. Robinson, eds., Sterile Insect Technique---Principles and Practice in Area-Wide Integrated Pest Management, Springer, New York, 2005.

22.

L. Esteva and M. H. Yang, Mathematical model to assess the control of Aedes aegypti mosquitoes by the sterile insect technique, Math. Biosci., 198 (2005), pp. 132--147.

23.

K. R. Fister and M. McCarthy, Optimal control of insects through sterile insect release and habitat modification, Math. Biosci., 244 (2013), pp. 201--212.

24.

G. Fu, R. S. Lees, D. Nimmo, D. Aw, L. Jin, P. Gray, T. U. Berendonk, H. White-Cooper, S. Scaife, H. Kim Phuc, O. Marinotti, N. Jasinskiene, A. A. James, and L. Alphey, Female-specific flightless phenotype for mosquito control, Proc. Natl. Acad. Sci. USA, 107 (2010), pp. 4550--4554.

25.

M. Gazor and P. Yu, Formal decomposition method and parametric normal form, Internat. J. Bifur. Chaos, 20 (2010), pp. 3487--3415.

26.

M. Gazor and P. Yu, Spectral sequences and parametric normal forms, J. Differential Equations, 252 (2013), pp. 1003--1031.

27.

M. Gazor and M. Moazeni, Parametric normal forms for Bogdanov-Takens singularity; the generalized saddle-node case, Discrete Contin. Dynam. Syst., 35 (2015), pp. 205--224.

28.

E. González-Olivares, B. González-Yan͂ez, J. Mena Lorca, A. Rojas-Palma, and J. D. Flores, Consequences of double Allee effect on the number of limit cycles in a predator-prey model, Comput. Math. Appl., 62 (2011), pp. 3449--3463

29.

J. Huang, Y. Gong, and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), pp. 2101--2121.

30.

J. Huang, S. Liu, S. Ruan, and X. Zhang, Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting, Commun. Pure Appl. Anal., 15 (2016), pp. 1041--1055.

31.

J. Huang, S. Ruan, and J. Song, Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response, J. Differential Equations, 257 (2014), pp. 1721--1752.

32.

E. F. Knipling, Possibilities of insect control or eradication through the use of sexually sterile males, J. Econ. Entomol., 48 (1955), pp. 459--462.

33.

E. S. Krafsur, Sterile insect technique for suppressing and eradicating insect populations: 55 years and counting, J. Agr. Entomol., 15 (1998), pp. 303--317.

34.

R. S. Lees, J. R. L. Gilles, J. Hendrichs, M. J. B. Vreysen, and K. Bourtzis, Back to the future: The sterile insect technique against mosquito disease vectors, Curr. Opin. Insect Sci., 10 (2015), pp. 156--162.

35.

M. A. Lewis and P. van den Driessche, Waves of extinction from sterile insect release, Math. Biosci., 116 (1993), pp. 221--247.

36.

C. Li, J. Li, and Z. Ma, Codimension 3 B-T bifurcation in an epidemic model with a nonlinear incidence, Discrete Contin. Dynam. Syst. Ser. B., 20 (2015), pp. 1107--1116.

37.

C. Li and C. Rousseau, A system with three limit cycles appearing in a Hopf bifurcation and dying in a homoclinic bifurcation: The cusp of order 4, J. Differential Equations, 79 (1989), pp. 132--167.

38.

J. Li, Simple mathematical models for interacting wild and transgenic mosquito populations, Math. Biosci., 189 (2004), pp. 39--59.

39.

J. Li, L. Cai, and Y. Li, Stage-structured wild and sterile mosquito population models and their dynamics, J. Biol. Dynam., 11 (2017), pp. 79--101.

40.

P. N Okorie, J. M Marshall, O. M Akpa, and O. G Ademowo, Perceptions and recommendations by scientists for a potential release of genetically modified mosquitoes in Nigeria, Malaria J., 13 (2014), 154.

41.

L. Perko, Differential Equations and Dynamical Systems, 3rd ed., Springer, New York, 2001.

42.

J. H. Werren, L. Baldo, and M. E. Clark, Wolbachia: Master manipulators of invertebrate biology, Nat. Rev. Microbiol., 6 (2008), pp. 741--751.

43.

S. M. White, P. Rohani, and S. M. Sait, Modelling pulsed releases for sterile insect techniques: Fitness costs of sterile and transgenic males and the effects on mosquito dynamics, J. Appl. Ecol., 47 (2010), pp. 1329--1339.

44.

F. Takens, Forced oscillations and bifurcations, in Applications of Global Analysis I, Comm. Math. Inst. Rijksuniv. Utrecht 3, Mathematisch Instituut der Rijksuniversiteit Utrecht, the Netherlands, 1974, pp. 1--59.

45.

D. D. Thomas, C. A. Donnelly, R. J. Wood, and L. S. Alphey, Insect population control using a dominant, repressible lethal, genetic system, Science, 287 (2000), pp. 2474--2476.

46.

D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Inst. Commun., 21 (1999), pp. 493--506.

47.

P. Yu, Computation of normal forms via a perturbation technique, J. Sound Vib., 211 (1998), pp. 19--38.

48.

P. Yu, Simplest normal forms of Hopf and generalized Hopf bifurcations, Internat. J. Bifur. Chaos, 9 (1999), pp. 1917--1939.

49.

P. Yu and A. Y. L. Leung, The simplest normal form of Hopf bifurcation, Nonlinearity, 16 (2003), pp. 277--300.

50.

Z. Zhang, T. Ding, W. Huang, and Z. Dong, Qualitative Theory of Differential Equation, Transl. Math. Monogr. 101, AMS, Providence, RI, 1992.

51.

H. Zhu, S. A. Campbell, and G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 63 (2002), pp. 636--682.