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Stability of travelling waves in a Wolbachia invasion. (English) Zbl 1403.35302
Summary: Numerous studies have examined the growth dynamics of Wolbachia within populations and the resultant rate of spatial spread. This spread is typically characterised as a travelling wave with bistable local growth dynamics due to a strong Allee effect generated from cytoplasmic incompatibility. While this rate of spread has been calculated from numerical solutions of reaction-diffusion models, none have examined the spectral stability of such travelling wave solutions. In this study we analyse the stability of a travelling wave solution generated by the reaction-diffusion model of M. H. T. Chan and P. S. Kim [Bull. Math. Biol. 75, No. 9, 1501–1523 (2013; Zbl 1311.92173)] by computing the essential and point spectrum of the linearised operator arising in the model. The point spectrum is computed via an Evans function using the compound matrix method, whereby we find that it has no roots with positive real part. Moreover, the essential spectrum lies strictly in the left half plane. Thus, we find that the travelling wave solution found by [loc. cit.] corresponding to competition between Wolbachia-infected and -uninfected mosquitoes is linearly stable. We employ a dimension counting argument to suggest that, under realistic conditions, the wavespeed corresponding to such a solution is unique.
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35C07 Traveling wave solutions
35B35 Stability in context of PDEs
35K57 Reaction-diffusion equations
62M15 Inference from stochastic processes and spectral analysis
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
92B05 General biology and biomathematics
92D30 Epidemiology
Full Text: DOI
[1] L. Allen; T. J. Bridges, Numerical exterior algebra and the compound matrix method, Numerische Mathematik, 92, 197, (2002) · Zbl 1012.65079
[2] N. H. Barton; M. Turelli, Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of allee effects, The American Naturalist, 178, E48, (2011)
[3] C. Brelsfoard; S. Dobson, Short note: an update on the utility of wolbachia for controlling insect vectors and disease transmission, Asia-Pacific Journal of Molecular Biology and Biotechnology, 19, 85, (2011)
[4] M. H. Chan; P. S. Kim, Modelling a wolbachia invasion using a slow-fast dispersal reaction-diffusion approach, Bulletin of Mathematical Biology, 75, 1501, (2013) · Zbl 1311.92173
[5] R. Dautray and J. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5. Evolution Problems I, Springer-Verlag, Berlin, 1992. · Zbl 0755.35001
[6] A. Davey, An automatic orthonormalization method for solving stiff boundary-value problems, Journal of Computational Physics, 51, 343, (1983) · Zbl 0516.65065
[7] L. O. Drury, Numerical solution of Orr-Sommerfeld-type equations, Journal of Computational Physics, 37, 133, (1980) · Zbl 0448.65057
[8] P. Hancock; S. Sinkins; H. Godfray, Population dynamic models of the spread of wolbachia, The American Naturalist, 177, 323, (2011)
[9] P. A. Hancock; H. C. J. Godfray, Modelling the spread of wolbachia in spatially heterogeneous environments, Journal of The Royal Society Interface, 9, 3045, (2012)
[10] K. Hilgenboecker; P. Hammerstein; P. Schlattmann; A. Telschow; J. H. Werren, How many species are infected with wolbachia?-a statistical analysis of current data, FEMS Microbiology Letters, 281, 215, (2008)
[11] C. K. Jones, Stability of the travelling wave solution of the Fitzhugh-Nagumo system, Transactions of the American Mathematical Society, 286, 431, (1984) · Zbl 0567.35044
[12] T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Applied Mathematical Sciences. Springer New York, 2013. · Zbl 1297.37001
[13] M. Keeling; F. Jiggins; J. Read, The invasion and coexistence of competing wolbachia strains, Heredity (Edinb), 91, 382, (2003)
[14] V. Ledoux; S. Malham; V. Thümmler, Grassmannian spectral shooting, Mathematics of Computation, 79, 1585, (2010) · Zbl 1196.65132
[15] M. A. Lewis; P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43, 141, (1993) · Zbl 0769.92025
[16] C. Mcmeniman; R. Lane; B. Cass; A. Fong; M. Sidhu; Y. Wang; S. O’Neill, Stable introduction of a life-shortening wolbachia infection into the mosquito aedes aegypti, Science, 323, 141, (2009)
[17] M. Z. Ndii; R. I. Hickson; G. N. Mercer, Modelling the introduction of wolbachia into aedes aegypti mosquitoes to reduce dengue transmission, ANZIAM Journal, 53, 213, (2012) · Zbl 1316.93104
[18] B. S. Ng; W. H. Reid, An initial value method for eigenvalue problems using compound matrices, Journal of Computational Physics, 30, 125, (1979) · Zbl 0408.65061
[19] B. S. Ng; W. H. Reid, A numerical method for linear two-point boundary-value problems using compound matrices, Journal of Computational Physics, 33, 70, (1979) · Zbl 0484.65053
[20] B. Sandstede; A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Physica D: Nonlinear Phenomena, 145, 233, (2000) · Zbl 0963.34072
[21] M. Turelli, Evolution of incompatibility-inducing microbes and their hosts, Evolution, 48, 1500, (1994)
[22] M. Turelli, Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64, 232, (2010)
[23] A. P. Turley, M. P. Zalucki, S. L. O’Neill and E. A. McGraw, Transinfected Wolbachia have minimal effects on male reproductive success in Aedes aegypti, Parasites & Vectors, 6 (2013), p36.
[24] T. Walker; P. Johnson; L. Moreira; I. Iturbe-Ormaetxe; F. Frentiu; C. McMeniman; Y. Leong; Y. Dong; J. Axford; P. Kriesner; A. Lloyd; S. Ritchie; S. O’Neill; A. Hoffmann, The wmel wolbachia strain blocks dengue and invades caged aedes aegypti populations, Nature, 476, 450, (2011)
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