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Some remarks on the homogenization of immiscible incompressible twophase flow in double porosity media
Stability of travelling waves in a Wolbachia invasion
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia 
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 reactiondiffusion 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 reactiondiffusion model of Chan & Kim [
References:
[1] 
L. Allen, T. J. Bridges, Numerical exterior algebra and the compound matrix method, Numerische Mathematik, 92 (2002), 197232. doi: 10.1007/s002110100365. 
[2] 
N. H. Barton, M. Turelli, Spatial waves of advance with bistable dynamics: Cytoplasmic and genetic analogues of allee effects, The American Naturalist, 178 (2011), E48E75. 
[3] 
C. Brelsfoard, S. Dobson, Short note: an update on the utility of Wolbachia for controlling insect vectors and disease transmission, AsiaPacific Journal of Molecular Biology and Biotechnology, 19 (2011), 8592. 
[4] 
M. H. Chan, P. S. Kim, Modelling a Wolbachia invasion using a slowfast dispersal reactiondiffusion approach, Bulletin of Mathematical Biology, 75 (2013), 15011523. doi: 10.1007/s115380139857y. 
[5] 
R. Dautray and J. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5. Evolution Problems I, SpringerVerlag, Berlin, 1992. 
[6] 
A. Davey, An automatic orthonormalization method for solving stiff boundaryvalue problems, Journal of Computational Physics, 51 (1983), 343356. doi: 10.1016/00219991(83)900980. 
[7] 
L. O. Drury, Numerical solution of OrrSommerfeldtype equations, Journal of Computational Physics, 37 (1980), 133139. doi: 10.1016/00219991(80)90008X. 
[8] 
P. Hancock, S. Sinkins, H. Godfray, Population dynamic models of the spread of wolbachia, The American Naturalist, 177 (2011), 323333. 
[9] 
P. A. Hancock, H. C. J. Godfray, Modelling the spread of wolbachia in spatially heterogeneous environments, Journal of The Royal Society Interface, 9 (2012), 30453054. 
[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 (2008), 215220. 
[11] 
C. K. Jones, Stability of the travelling wave solution of the fitzhughnagumo system, Transactions of the American Mathematical Society, 286 (1984), 431469. doi: 10.1090/S00029947198407609716. 
[12] 
T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Applied Mathematical Sciences. Springer New York, 2013. 
[13] 
M. Keeling, F. Jiggins, J. Read, The invasion and coexistence of competing wolbachia strains, Heredity (Edinb), 91 (2003), 382388. 
[14] 
V. Ledoux, S. Malham, V. Thümmler, Grassmannian spectral shooting, Mathematics of Computation, 79 (2010), 15851619. doi: 10.1090/S0025571810023239. 
[15] 
M. A. Lewis, P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141158. 
[16] 
C. Mcmeniman, R. Lane, B. Cass, A. Fong, M. Sidhu, Y. Wang, S. O'Neill, Stable introduction of a lifeshortening Wolbachia infection into the mosquito Aedes aegypti, Science, 323 (2009), 141144. 
[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 (2012), 213227. 
[18] 
B. S. Ng, W. H. Reid, An initial value method for eigenvalue problems using compound matrices, Journal of Computational Physics, 30 (1979), 125136. doi: 10.1016/00219991(79)900913. 
[19] 
B. S. Ng, W. H. Reid, A numerical method for linear twopoint boundaryvalue problems using compound matrices, Journal of Computational Physics, 33 (1979), 7085. doi: 10.1016/00219991(79)900287. 
[20] 
B. Sandstede, A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Physica D: Nonlinear Phenomena, 145 (2000), 233277. doi: 10.1016/S01672789(00)001147. 
[21] 
M. Turelli, Evolution of incompatibilityinducing microbes and their hosts, Evolution, 48 (1994), 15001513. 
[22] 
M. Turelli, Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64 (2010), 232241. 
[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. IturbeOrmaetxe, 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 (2011), 450453. 
show all references
References:
[1] 
L. Allen, T. J. Bridges, Numerical exterior algebra and the compound matrix method, Numerische Mathematik, 92 (2002), 197232. doi: 10.1007/s002110100365. 
[2] 
N. H. Barton, M. Turelli, Spatial waves of advance with bistable dynamics: Cytoplasmic and genetic analogues of allee effects, The American Naturalist, 178 (2011), E48E75. 
[3] 
C. Brelsfoard, S. Dobson, Short note: an update on the utility of Wolbachia for controlling insect vectors and disease transmission, AsiaPacific Journal of Molecular Biology and Biotechnology, 19 (2011), 8592. 
[4] 
M. H. Chan, P. S. Kim, Modelling a Wolbachia invasion using a slowfast dispersal reactiondiffusion approach, Bulletin of Mathematical Biology, 75 (2013), 15011523. doi: 10.1007/s115380139857y. 
[5] 
R. Dautray and J. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5. Evolution Problems I, SpringerVerlag, Berlin, 1992. 
[6] 
A. Davey, An automatic orthonormalization method for solving stiff boundaryvalue problems, Journal of Computational Physics, 51 (1983), 343356. doi: 10.1016/00219991(83)900980. 
[7] 
L. O. Drury, Numerical solution of OrrSommerfeldtype equations, Journal of Computational Physics, 37 (1980), 133139. doi: 10.1016/00219991(80)90008X. 
[8] 
P. Hancock, S. Sinkins, H. Godfray, Population dynamic models of the spread of wolbachia, The American Naturalist, 177 (2011), 323333. 
[9] 
P. A. Hancock, H. C. J. Godfray, Modelling the spread of wolbachia in spatially heterogeneous environments, Journal of The Royal Society Interface, 9 (2012), 30453054. 
[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 (2008), 215220. 
[11] 
C. K. Jones, Stability of the travelling wave solution of the fitzhughnagumo system, Transactions of the American Mathematical Society, 286 (1984), 431469. doi: 10.1090/S00029947198407609716. 
[12] 
T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Applied Mathematical Sciences. Springer New York, 2013. 
[13] 
M. Keeling, F. Jiggins, J. Read, The invasion and coexistence of competing wolbachia strains, Heredity (Edinb), 91 (2003), 382388. 
[14] 
V. Ledoux, S. Malham, V. Thümmler, Grassmannian spectral shooting, Mathematics of Computation, 79 (2010), 15851619. doi: 10.1090/S0025571810023239. 
[15] 
M. A. Lewis, P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141158. 
[16] 
C. Mcmeniman, R. Lane, B. Cass, A. Fong, M. Sidhu, Y. Wang, S. O'Neill, Stable introduction of a lifeshortening Wolbachia infection into the mosquito Aedes aegypti, Science, 323 (2009), 141144. 
[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 (2012), 213227. 
[18] 
B. S. Ng, W. H. Reid, An initial value method for eigenvalue problems using compound matrices, Journal of Computational Physics, 30 (1979), 125136. doi: 10.1016/00219991(79)900913. 
[19] 
B. S. Ng, W. H. Reid, A numerical method for linear twopoint boundaryvalue problems using compound matrices, Journal of Computational Physics, 33 (1979), 7085. doi: 10.1016/00219991(79)900287. 
[20] 
B. Sandstede, A. Scheel, Absolute and convective instabilities of waves on unbounded and large bounded domains, Physica D: Nonlinear Phenomena, 145 (2000), 233277. doi: 10.1016/S01672789(00)001147. 
[21] 
M. Turelli, Evolution of incompatibilityinducing microbes and their hosts, Evolution, 48 (1994), 15001513. 
[22] 
M. Turelli, Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64 (2010), 232241. 
[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. IturbeOrmaetxe, 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 (2011), 450453. 
Symbol  Definition  Value 
Relative fecundity of uninfected to infected females   
 Probability of embryo death due to CI  
 Mortality rate  
 Reduction in lifespan due to infection  
Symbol  Definition  Value 
Relative fecundity of uninfected to infected females   
 Probability of embryo death due to CI  
 Mortality rate  
 Reduction in lifespan due to infection  
Symbol  Definition  Value 
 Relative fecundity of uninfected to infected females  
 Probability of embryo death due to CI  
 Mortality rate  
 Reduction in lifespan due to infection 
Symbol  Definition  Value 
 Relative fecundity of uninfected to infected females  
 Probability of embryo death due to CI  
 Mortality rate  
 Reduction in lifespan due to infection 
Symbol  Definition  Value 
Relative fecundity of uninfected to infected females   
 Probability of embryo death due to CI  
 Mortality rate  
 Reduction in lifespan due to infection  
Symbol  Definition  Value 
Relative fecundity of uninfected to infected females   
 Probability of embryo death due to CI  
 Mortality rate  
 Reduction in lifespan due to infection  
Symbol  Definition  Value 
Relative fecundity of uninfected to infected females   
 Probability of embryo death due to CI  
 Mortality rate  
 Reduction in lifespan due to infection  
Symbol  Definition  Value 
Relative fecundity of uninfected to infected females   
 Probability of embryo death due to CI  
 Mortality rate  
 Reduction in lifespan due to infection  
Symbol  Definition  Value 
Relative fecundity of uninfected to infected females   
 Probability of embryo death due to CI  
 Mortality rate  
 Reduction in lifespan due to infection  
Symbol  Definition  Value 
Relative fecundity of uninfected to infected females   
 Probability of embryo death due to CI  
 Mortality rate  
 Reduction in lifespan due to infection  
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