Constructing a new individual-based model of phosphine resistance in lesser grain borer (Rhyzopertha dominica): do we need to include two loci rather than one?

Abstract

In this article, we describe and compare two individual-based models constructed to investigate how genetic factors influence the development of phosphine resistance in lesser grain borer (R. dominica). One model is based on the simplifying assumption that resistance is conferred by alleles at a single locus, while the other is based on the more realistic assumption that resistance is conferred by alleles at two separate loci. We simulated the population dynamic of R. dominica in the absence of phosphine fumigation, and under high and low dose phosphine treatments, and found important differences between the predictions of the two models in all three cases. In the absence of fumigation, starting from the same initial frequencies of genotypes, the two models tended to different stable frequencies, although both reached Hardy–Weinberg equilibrium. The one-locus model exaggerated the equilibrium proportion of strongly resistant beetles by 3.6 times, compared to the aggregated predictions of the two-locus model. Under a low dose treatment the one-locus model overestimated the proportion of strongly resistant individuals within the population and underestimated the total population numbers compared to the two-locus model. These results show the importance of basing resistance evolution models on realistic genetics and that using oversimplified one-locus models to develop pest control strategies runs the risk of not correctly identifying tactics to minimise the incidence of pest infestation.

Appendices

Appendix 1

Figures for the original two-locus model

See Figs. 11 and 12.

Fig. 11

The population of beetles for each genotype under 0.2 mg/l × 8 days fumigation treatment for the original two-locus model. a The weekly average proportions and b daily numbers

Fig. 12

The population of beetles for each genotype under 0.01 mg/l × 14 days fumigation treatment for the original two-locus model. a The weekly average proportions and b daily numbers

Table for resistance factors

See Table 8.

Table 8 The resistance factors for the nine genotypes of the two-locus model (LC₅₀ at exposure time 48 h)

Appendix 2

Log-normal distribution as a model of time within a life stage

Log-normal distribution is a continuous distribution in which the logarithm of a variable has a normal distribution (Limpert et al. 2001). It is appropriate for modelling the time an individual spends within a given life stage because it is bounded below at zero, and because it can be parameterized to have a non-zero median and to simultaneously give a restricted range of likely values that does not necessarily include values close to zero (unlike the Weibull distribution for example).

The probability density function (pdf) and cumulative distribution function (cdf) for the log-normal distribution are, respectively,

$$ \begin{aligned} {\text{pdf:}} \\ &\quad f(t > 0) = \frac{1}{{\left( {\sigma \sqrt {2\pi } } \right)t}}\exp \left( { - (\ln t - \mu )^{2} /2\sigma^{2} } \right) \end{aligned} $$

(14)

$$ \begin{aligned} &{\text{cdf:}} \\ & \quad F(t) = \frac{1}{2}\left[ {1 + {\text{erf}}\left( {\frac{\ln \, t - \mu }{\sigma \sqrt 2 }} \right)} \right],\quad {\text{erf}}(x) = \frac{1}{\sqrt \pi }\int\limits_{0}^{x} {\exp ( - u^{2} )\,{\text{d}}u} \end{aligned} $$

(15)

where μ and σ are the mean and std of the corresponding normal distribution, respectively, and erf(x) is the complementary error function.

Given the mean and std values of a sample L from a random variable T with a log-normal distribution, M _L and S _L, the parameters μ and σ can be estimated as follows. The expectation E(T) and variation Var(T) are the estimates of M _L and (S _L)², respectively, and we know E(T) and Var(T) are

$$ M_{\text{L}} \approx E(T) = \exp (\mu + \sigma^{2} /2), $$

(16)

$$ S_{\text{L}}^{2} \approx {\text{Var}}(T) = (\exp (\sigma^{2} ) - 1)[E(T)]^{2}. $$

(17)

Substituting (16) into (17) we have

$$ \exp (\sigma^{2} ) - 1 = {\text{Var}}(T)/[E(T)]^{2} \quad {\text{or}}\quad \sigma^{2} = \ln [{\text{Var}}(T)/E(T)^{2} + 1] $$

(18)

Solving Eqs. 16 and 18 yields

$$ \sigma = \sqrt {\ln [{\text{Var}}(T)/E(T)^{2} + 1]} \quad {\text{and}}\quad \mu = \ln [E(T)/\exp (\sigma^{2} /2)] $$

(19)

Thus, given the expected mean and standard deviation of the times spent within different life stages by different individuals, we have a simple method to calculate the parameters of the log-normal distribution used to stochastically generate the actual time spent within a given life stage by a given individual.

For model verification, we used the Python built-in function lognormvariate(μ,σ) to generate 10,000 random numbers from log-normal distribution with mean M _L and std S _L for the life duration of each stage of the beetle. The results are listed in Table 9.

Table 9 A generated sample of random numbers from log-normal distribution with mean M _L ≈ E(T) and (std S _L)² ≈ Var(T) for the life duration of each stage of the beetle and the sample mean and std with minimum and maximum values of the sample (sample size = 10,000)