Symbol | Definition | Value |
$F$ | 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 | $1.1$ |
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 [
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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 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
Symbol | Definition | Value |
$F$ | 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 | $1.1$ |
Table 2. Parameter set one
Symbol | Definition | Value |
$F$ | 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 | $1.2$ |
Table 3. Parameter set two
Symbol | Definition | Value |
$F$ | 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 | $1.3$ |
Table 4. Parameter set three
Symbol | Definition | Value |
$F$ | 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 | $1.2$ |
Table 5. Parameter set four
Symbol | Definition | Value |
$F$ | 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 | $1.2$ |
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The essential spectrum of
The contour
The image of
The image of
Plots
Plot of the Evans function given by Eq. (26). The blue and red solid lines in both plots show
Solution to the boundary value problem (6). Figure
A diagram showing the uniqueness of
Simulations corresponding to parameter set three, listed in Table 2
Simulations corresponding to parameter set two, listed in Table 3
Simulations corresponding to parameter set three, listed in Table 4
Simulations corresponding to parameter set four, listed in Table 5