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Stability of travelling waves in a Wolbachia invasion

  • * Corresponding author: Matthew H. Chan

    * Corresponding author: Matthew H. Chan 

PSK was supported by the Australian Research Council, Discovery Project (DP160101597)

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  • 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 [4] 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 Chan & Kim [4] 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.

    Mathematics Subject Classification: Primary: 35C07, 35B35, 35K57, 62M15, 34L16, 92B05.

    Citation:

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  • Figure 1.  The essential spectrum of $\mathcal{L}$ is given by $\lambda$ in the shaded regions. The blue dashed and solid lines represent $\lambda_-^{1, 2}$ and $\lambda_+^{1, 2}$ respectively. The red line indicates the absolute spectrum given by Eq. (17) and the red dot at the origin represents an eigenvalue. Note that these are not drawn to scale for visualisation purposes

    Figure 2.  The contour $C$

    Figure 3.  The image of $C$ under $D(\lambda)$, where $r_s =0.1$ and $r_b =10$

    Figure 4.  The image of $C$ under $D(\lambda)$, where $r_s =0.001$ and $r_b =500$

    Figure 5.  Plots $(a)$ and $(b)$ show the change in argument for $D[C]$ corresponding to Figures 3 and 4 respectively

    Figure 6.  Plot of the Evans function given by Eq. (26). The blue and red solid lines in both plots show $D(\lambda)$ with $\alpha = 1.1$ and $\alpha = 1$ respectively. The dashed lines in plot $(a)$ mark the edge of the absolute spectrum corresponding to each $\alpha$. For $\alpha = 1.1$, the only roots are at $\lambda =0$ and at the edge of the absolute spectrum $\lambda =-0.002607$, whereas for $\alpha = 1$ we were unable to detect a zero at the edge of the absolute spectrum due to its proximity to the origin

    Figure 7.  Solution to the boundary value problem (6). Figure $(a)$ shows the wave profile of $\hat{u}(z)$ and $\hat{v}(z)$, represented by solid and dashed lines respectively. Figure $(b)$ shows the heteroclinic connection between equilibrium states $\mathbf{e_{-}} = (1-\alpha \mu, 0)$ and $\mathbf{e_{+}} = (0, 1-\frac{\mu}{F})$, where the solid and dashed line represent the solution in $u-u'$ and $v-v'$ space respectively

    Figure 8.  A diagram showing the uniqueness of $c_*$ by dimension counting, where $\tilde{\mathbf{e}}_-$, $\tilde{\mathbf{e}}_+$ denote the equilibria at the end points of the heteroclinic orbit in (33). In the illustration, $W^{u, s}(\tilde{\mathbf{e}}_{-, +}) \times \mathcal{C}_\varepsilon$ are shown as 2-dimensional manifolds which intersect transversally in 3-dimensional space. The one dimensional intersection corresponds to the heteroclinic connection between $\tilde{\mathbf{e}}_-$ and $\tilde{\mathbf{e}}_+$ at $c_*$

    Figure 9.  Simulations corresponding to parameter set three, listed in Table 2

    Figure 10.  Simulations corresponding to parameter set two, listed in Table 3

    Figure 11.  Simulations corresponding to parameter set three, listed in Table 4

    Figure 12.  Simulations corresponding to parameter set four, listed in Table 5

    Table 1.  Parameter values and definitions

    SymbolDefinitionValue
    $F$Relative fecundity of uninfected to infected females $1.0526$
    $s_h$Probability of embryo death due to CI $0.45$
    $\mu$Mortality rate $0.0162$
    $\alpha$Reduction in lifespan due to infection $1.1$
     | Show Table
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    Table 2.  Parameter set one

    SymbolDefinitionValue
    $F$ Relative fecundity of uninfected to infected females$1.05$
    $s_h$ Probability of embryo death due to CI$0.7$
    $\mu$ Mortality rate$0.03$
    $\alpha$Reduction in lifespan due to infection$1.2$
     | Show Table
    DownLoad: CSV

    Table 3.  Parameter set two

    SymbolDefinitionValue
    $F$ Relative fecundity of uninfected to infected females $1.1$
    $s_h$ Probability of embryo death due to CI $0.9$
    $\mu$ Mortality rate $0.02$
    $\alpha$ Reduction in lifespan due to infection $1.3$
     | Show Table
    DownLoad: CSV

    Table 4.  Parameter set three

    SymbolDefinitionValue
    $F$ Relative fecundity of uninfected to infected females $1.1$
    $s_h$ Probability of embryo death due to CI $0.8$
    $\mu$ Mortality rate $0.05$
    $\alpha$ Reduction in lifespan due to infection $1.2$
     | Show Table
    DownLoad: CSV

    Table 5.  Parameter set four

    SymbolDefinitionValue
    $F$ Relative fecundity of uninfected to infected females $1.4$
    $s_h$ Probability of embryo death due to CI $0.9$
    $\mu$ Mortality rate $0.1$
    $\alpha$ Reduction in lifespan due to infection $1.2$
     | Show Table
    DownLoad: CSV
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