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Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal
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 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 Chan & Kim [
References:
[1] |
L. Allen and T. J. Bridges,
Numerical exterior algebra and the compound matrix method, Numerische Mathematik, 92 (2002), 197-232.
doi: 10.1007/s002110100365. |
[2] |
N. H. Barton and M. Turelli,
Spatial waves of advance with bistable dynamics: Cytoplasmic and genetic analogues of allee effects, The American Naturalist, 178 (2011), E48-E75.
|
[3] |
C. Brelsfoard and 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 (2011), 85-92.
|
[4] |
M. H. Chan and P. S. Kim,
Modelling a Wolbachia invasion using a slow-fast dispersal reaction-diffusion approach, Bulletin of Mathematical Biology, 75 (2013), 1501-1523.
doi: 10.1007/s11538-013-9857-y. |
[5] |
R. Dautray and J. Lions,
Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5. Evolution Problems I, Springer-Verlag, Berlin, 1992. |
[6] |
A. Davey,
An automatic orthonormalization method for solving stiff boundary-value problems, Journal of Computational Physics, 51 (1983), 343-356.
doi: 10.1016/0021-9991(83)90098-0. |
[7] |
L. O. Drury,
Numerical solution of Orr-Sommerfeld-type equations, Journal of Computational Physics, 37 (1980), 133-139.
doi: 10.1016/0021-9991(80)90008-X. |
[8] |
P. Hancock, S. Sinkins and H. Godfray,
Population dynamic models of the spread of wolbachia, The American Naturalist, 177 (2011), 323-333.
|
[9] |
P. A. Hancock and H. C. J. Godfray,
Modelling the spread of wolbachia in spatially heterogeneous environments, Journal of The Royal Society Interface, 9 (2012), 3045-3054.
|
[10] |
K. Hilgenboecker, P. Hammerstein, P. Schlattmann, A. Telschow and J. H. Werren,
How many species are infected with wolbachia?-a statistical analysis of current data, FEMS Microbiology Letters, 281 (2008), 215-220.
|
[11] |
C. K. Jones,
Stability of the travelling wave solution of the fitzhugh-nagumo system, Transactions of the American Mathematical Society, 286 (1984), 431-469.
doi: 10.1090/S0002-9947-1984-0760971-6. |
[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 and J. Read,
The invasion and coexistence of competing wolbachia strains, Heredity (Edinb), 91 (2003), 382-388.
|
[14] |
V. Ledoux, S. Malham and V. Thümmler,
Grassmannian spectral shooting, Mathematics of Computation, 79 (2010), 1585-1619.
doi: 10.1090/S0025-5718-10-02323-9. |
[15] |
M. A. Lewis and P. Kareiva,
Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141-158.
|
[16] |
C. Mcmeniman, R. Lane, B. Cass, A. Fong, M. Sidhu, Y. Wang and S. O'Neill,
Stable introduction of a life-shortening Wolbachia infection into the mosquito Aedes aegypti, Science, 323 (2009), 141-144.
|
[17] |
M. Z. Ndii, R. I. Hickson and G. N. Mercer,
Modelling the introduction of wolbachia into aedes aegypti mosquitoes to reduce dengue transmission, ANZIAM Journal, 53 (2012), 213-227.
|
[18] |
B. S. Ng and W. H. Reid,
An initial value method for eigenvalue problems using compound matrices, Journal of Computational Physics, 30 (1979), 125-136.
doi: 10.1016/0021-9991(79)90091-3. |
[19] |
B. S. Ng and W. H. Reid,
A numerical method for linear two-point boundary-value problems using compound matrices, Journal of Computational Physics, 33 (1979), 70-85.
doi: 10.1016/0021-9991(79)90028-7. |
[20] |
B. Sandstede and A. Scheel,
Absolute and convective instabilities of waves on unbounded and large bounded domains, Physica D: Nonlinear Phenomena, 145 (2000), 233-277.
doi: 10.1016/S0167-2789(00)00114-7. |
[21] |
M. Turelli,
Evolution of incompatibility-inducing microbes and their hosts, Evolution, 48 (1994), 1500-1513.
|
[22] |
M. Turelli,
Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64 (2010), 232-241.
|
[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 and A. Hoffmann,
The wmel wolbachia strain blocks dengue and invades caged aedes aegypti populations, Nature, 476 (2011), 450-453.
|
show all references
References:
[1] |
L. Allen and T. J. Bridges,
Numerical exterior algebra and the compound matrix method, Numerische Mathematik, 92 (2002), 197-232.
doi: 10.1007/s002110100365. |
[2] |
N. H. Barton and M. Turelli,
Spatial waves of advance with bistable dynamics: Cytoplasmic and genetic analogues of allee effects, The American Naturalist, 178 (2011), E48-E75.
|
[3] |
C. Brelsfoard and 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 (2011), 85-92.
|
[4] |
M. H. Chan and P. S. Kim,
Modelling a Wolbachia invasion using a slow-fast dispersal reaction-diffusion approach, Bulletin of Mathematical Biology, 75 (2013), 1501-1523.
doi: 10.1007/s11538-013-9857-y. |
[5] |
R. Dautray and J. Lions,
Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5. Evolution Problems I, Springer-Verlag, Berlin, 1992. |
[6] |
A. Davey,
An automatic orthonormalization method for solving stiff boundary-value problems, Journal of Computational Physics, 51 (1983), 343-356.
doi: 10.1016/0021-9991(83)90098-0. |
[7] |
L. O. Drury,
Numerical solution of Orr-Sommerfeld-type equations, Journal of Computational Physics, 37 (1980), 133-139.
doi: 10.1016/0021-9991(80)90008-X. |
[8] |
P. Hancock, S. Sinkins and H. Godfray,
Population dynamic models of the spread of wolbachia, The American Naturalist, 177 (2011), 323-333.
|
[9] |
P. A. Hancock and H. C. J. Godfray,
Modelling the spread of wolbachia in spatially heterogeneous environments, Journal of The Royal Society Interface, 9 (2012), 3045-3054.
|
[10] |
K. Hilgenboecker, P. Hammerstein, P. Schlattmann, A. Telschow and J. H. Werren,
How many species are infected with wolbachia?-a statistical analysis of current data, FEMS Microbiology Letters, 281 (2008), 215-220.
|
[11] |
C. K. Jones,
Stability of the travelling wave solution of the fitzhugh-nagumo system, Transactions of the American Mathematical Society, 286 (1984), 431-469.
doi: 10.1090/S0002-9947-1984-0760971-6. |
[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 and J. Read,
The invasion and coexistence of competing wolbachia strains, Heredity (Edinb), 91 (2003), 382-388.
|
[14] |
V. Ledoux, S. Malham and V. Thümmler,
Grassmannian spectral shooting, Mathematics of Computation, 79 (2010), 1585-1619.
doi: 10.1090/S0025-5718-10-02323-9. |
[15] |
M. A. Lewis and P. Kareiva,
Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141-158.
|
[16] |
C. Mcmeniman, R. Lane, B. Cass, A. Fong, M. Sidhu, Y. Wang and S. O'Neill,
Stable introduction of a life-shortening Wolbachia infection into the mosquito Aedes aegypti, Science, 323 (2009), 141-144.
|
[17] |
M. Z. Ndii, R. I. Hickson and G. N. Mercer,
Modelling the introduction of wolbachia into aedes aegypti mosquitoes to reduce dengue transmission, ANZIAM Journal, 53 (2012), 213-227.
|
[18] |
B. S. Ng and W. H. Reid,
An initial value method for eigenvalue problems using compound matrices, Journal of Computational Physics, 30 (1979), 125-136.
doi: 10.1016/0021-9991(79)90091-3. |
[19] |
B. S. Ng and W. H. Reid,
A numerical method for linear two-point boundary-value problems using compound matrices, Journal of Computational Physics, 33 (1979), 70-85.
doi: 10.1016/0021-9991(79)90028-7. |
[20] |
B. Sandstede and A. Scheel,
Absolute and convective instabilities of waves on unbounded and large bounded domains, Physica D: Nonlinear Phenomena, 145 (2000), 233-277.
doi: 10.1016/S0167-2789(00)00114-7. |
[21] |
M. Turelli,
Evolution of incompatibility-inducing microbes and their hosts, Evolution, 48 (1994), 1500-1513.
|
[22] |
M. Turelli,
Cytoplasmic incompatibility in populations with overlapping generations, Evolution, 64 (2010), 232-241.
|
[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 and A. Hoffmann,
The wmel wolbachia strain blocks dengue and invades caged aedes aegypti populations, Nature, 476 (2011), 450-453.
|








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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