Estimation of dispersal ability responding to environmental conditions: larval dispersal of the flightless firefly, Luciola parvula (Coleoptera: Lampyridae)

Abstract

We provided an extensional method for diffusion equation models so far presented to cover cases where diffusion coefficients temporally change. We applied this method to data sampled from mark recapture surveys to estimate the natural mean dispersal distance and diffusion coefficients of terrestrial firefly larvae, Luciola parvula Kiesenwetter (Coleoptera: Lampyridae: Luciolinae). The surveys were conducted twice (December 2009 and March–April 2010) on a Cryptomeria plantation where 100 traps were placed in a lattice pattern at 30 cm intervals (10 × 10). Marked larvae were released at the center of the lattice, and the number of recaptures was recorded. Larval dispersal became remarkably active on the first night with rainfall as well as on the following day. The estimated natural mean dispersal distances (±SE) were 100.7 (±18.4) cm (December) and 245.4 (±700.0) cm (March–April). The diffusion coefficients just after rainfall were estimated to increase by 14.2 (±6.1) times (December) and 106.0 (±55.9) times (March–April) (±SE). Larvae were expected to disperse no further from where their eggs were laid. Most of their dispersing activity took place just after rainfall. Our extensional method was able to effectively illustrate that larval dispersal was affected by rainfall. This method can be usefully applied to any other species that disperses according to environmental conditions.

Appendix: estimating parameters in Eq. (4)

In this appendix, we briefly explain the procedure of estimating parameters in Eq. (4) by the maximum likelihood method.

The mean number of surviving individuals up to time t, S(t), is expressed by Eq. (4) as given again below:

$$S(t) = N_{0} [\delta_{D} /(\lambda_{D} + \delta_{D} )]\left \{ 1 - exp \left [ { - \left( {\lambda_{D} + \delta_{D} } \right)\mathop \int \limits_{0}^{t} D\left( \tau \right)d\tau } \right ] \right \} .$$

(4)

Hence, the number of individuals captured in the ith day, i.e., captured number of individuals between t _i−1 and t _i , is calculated as follows:

$$\begin{gathered} \Delta S(t_{i} ) = S(t_{i} ) - S(t_{i - 1} ) \hfill \\ \qquad \quad= N_{0} [\delta_{D} /(\lambda_{D} + \delta_{D} )]\left \{ \exp \left [ - \left( {\lambda_{D} + \delta_{D} } \right)\mathop \int \limits_{0}^{{t_{i} }} D\left( \tau \right)d\tau \right ] - \exp \left [ - \left( {\lambda_{D} + \delta_{D} } \right)\mathop \int \limits_{0}^{{t_{i - 1} }} D\left( \tau \right)d\tau \right ] \right \} . \hfill \\ \end{gathered}$$

We assume here that the observed number of captured individuals on each day follow a Poisson distribution with the mean value described by the above Eq. If the observed number is n _i in day i, the probability that this takes place is described as follows:

$$\begin{gathered} P\left( {n_{i} , t_{i} |\lambda_{D} , \delta_{D} , d_{2} } \right) = \frac{{\Delta S\left( {t_{i} } \right)^{{n_{i} }} }}{{n_{i} !}}\exp [ - \Delta S(t_{i} )] \hfill \\ \qquad \qquad \qquad \quad \quad\,\,\, = \text{ }\frac{1}{{n_{i} !}} \times \left\{ {N_{0} [\delta_{D} /(\lambda_{D} + \delta_{D} )]\{ { \exp }[ - \left( {\lambda_{D} + \delta_{D} } \right)\mathop \int \limits_{0}^{{t_{i} }} D\left( \tau \right)d\tau ] - { \exp }[ - \left( {\lambda_{D} + \delta_{D} } \right)\mathop \int \limits_{0}^{{t_{i - 1} }} D\left( \tau \right)d\tau ] \} } \right\}^{{n_{i} }} \hfill \\ \qquad \qquad \qquad \quad \quad\,\,\, \times \left[ {{ \exp }\left\{ {N_{0} [\delta_{D} /(\lambda_{D} + \delta_{D} )]\{ { \exp }[ - \left( {\lambda_{D} + \delta_{D} } \right)\mathop \int \limits_{0}^{{t_{i} }} D\left( \tau \right)d\tau ] - { \exp }[ - \left( {\lambda_{D} + \delta_{D} } \right)\mathop \int \limits_{0}^{{t_{i - 1} }} D\left( \tau \right)d\tau ] \} } \right\}} \right] \hfill \\ \end{gathered}$$

Please note here that we have parameters λ _D , δ _D , and d ₂. Then we obtain the log likelihood function to be maximized as follows:

$$\begin{gathered} LL_{i} (\lambda_{D} , \delta_{D} , d_{2} ) = \mathop \sum \limits_{i = 1}^{i max} \log [P\left( {n_{i} , t_{i} |\lambda_{D} , \delta_{D} , d_{2} } \right)] = \mathop \sum \limits_{i = 1}^{i max} \log \left\{ {\frac{{\Delta S\left( {t_{i} } \right)^{{n_{i} }} }}{{n_{i} !}}\exp [ - \Delta S(t_{i} )]} \right\} \hfill \\ \qquad \qquad \qquad \quad = \mathop \sum \limits_{i = 1}^{i max} \bigg[{ \log }\frac{1}{{n_{i} !}} \hfill \\ \qquad \qquad \qquad \quad + n_{i} { \log }\left\{ {N_{0} [\delta_{D} /(\lambda_{D} + \delta_{D} )]\{ { \exp }[ - \left( {\lambda_{D} + \delta_{D} } \right)\mathop \int \limits_{0}^{{t_{i} }} D\left( \tau \right)d\tau ] - { \exp }[ - \left( {\lambda_{D} + \delta_{D} } \right)\mathop \int \limits_{0}^{{t_{i - 1} }} D\left( \tau \right)d\tau ] \} } \right\} \hfill \\ \qquad \qquad \qquad \quad - N_{0} [\delta_{D} /(\lambda_{D} + \delta_{D} )]\{ { \exp }[ - \left( {\lambda_{D} + \delta_{D} } \right)\mathop \int \limits_{0}^{{t_{i} }} D\left( \tau \right)d\tau ] - { \exp }[ - \left( {\lambda_{D} + \delta_{D} } \right)\mathop \int \limits_{0}^{{t_{i - 1} }} D\left( \tau \right)d\tau ] \} \bigg] \hfill \\ \end{gathered}$$

The maximum likelihood estimates of parameter values, λ _D , δ _D and d ₂, are obtained as such. Please note that we put d ₁ = 1 and d ₂ = D ₂/D ₁ to adjust the redundancy of parameters.