The effectiveness of any IRM strategy is related to two kinds of parameters: the operational parameters characterizing the strategy and the biological parameters characterizing the target insect [36]. The operational parameters include the relative size of the refuge zone (v) in relation to the whole field and the selection pressure exerted by the Cry toxin on susceptible insects (sBtA, sBtB). The biological parameters are the initial frequency of the resistance allele for each locus (ArFreq, BrFreq) and its dominance (hAr, hBr), the fitness cost associated with resistance (fcostA, fcostB) and its dominance (hfcA, hfcB), the initial population size (nzero), the population intrinsic growth rate (r) and the field carrying capacity (K). Each parameter is associated with a default value taken from the literature and measured on a lepidopteran insect pest (see Table 1).

The considered system is a section of a homogeneous region of 260 ha, as proposed by Guse et al. [38]. This field is closed and composed of two adjacent zones: the Bt zone and the refuge zone. The insect population is uniformly distributed among the zones and cannot leave the area. Each generation is divided into a succession of simple stages and the generations are discrete and continuous (i.e., no diapause) (see Figure 1). The time unit is the non-overlapping generation.

At step 0, there is panmictic mating at the field scale followed by random egglaying between the two habitats (Step 1). Oviposition choice tests indicate that females are unable to discriminate between the two plants varieties and show no oviposition preferences [64,65]. There is no hatching difference between the two zones [64]. After eclosion, the larvae start feeding and are subjected to selection pressure according to their genotypes and to the zone in which they develop (Step 2). Toxin exposure occurs only for larvae. Next, the model calculates the genotype frequencies and the population sizes (Step 3). The adults emerge and the cycle starts again at Step 0. Subject to the condition of panmictic mating and random egglaying, the genotype frequency is present in Hardy-Weinberg proportions in each zone at the beginning of each generation [47]. This Hardy–Weinberg equilibrium is disturbed from the first step of selection but will be restored at Step 0 in the next generation. There is an equation for each stage, implemented in R [66].

Usually, one prerequisite of the refuge theory is that the resistance level of the pool of susceptible insects from a refuge must remain unaffected by an outward migration from the Bt zone population [67]. Our model is more realistic: the random egglaying of the resistant females allows the introduction of resistant offspring in the refuge; thus, the refuge is not 100% susceptible but can be “polluted” by resistant alleles.

The first part of this model is based on population genetics theory and allows us to follow the Ar frequency at each generation for one-toxin Bt plant or the resistance haplotype frequency (ArBr) for a two-toxin Bt plant. This model is inspired by the model of Mallet and Porter [68] and adapted from the equation of Crow and Kimura [69], which indicates the direction of change in the resistance allele frequency at each generation (t).

The second part of the model is based on population dynamics theory and allows us to follow the quantitative population changes. Here, reproduction is density-dependant and the population growth follows Ricker’s formulation of a classic discrete logistic population model (equation 2) [70]. The number of pests at time N_(t+1) depends on the population size at time N_(t), on the intrinsic rate of growth (r) and on the carrying capacity (K).

The model investigates the effect of each parameter on the evolution of both resistance and population size. Two indicators are used as standards for comparing simulations.

The most common indicator of resistance spread is the number of generations required to reach 50% frequency of the resistance allele Ar for Bt plants expressing one toxin and 50% frequency of the resistant haplotype ArBr for the pyramided plants (GF50) [36,68]. This is a convenient measure of resistance, which is independent of assumptions regarding population growth [31].

The second indicator is the percentage of population decrease between the first two generations and after ten generations. Such an indicator is faster to obtain: only ten generations are required. Secondly, it can be positive or negative, indicating that the population density is increasing or decreasing, respectively, making it more easily interpretable. We consider that the impact of the IRM strategy on population density must be assessed, because it is an important indicator of the sustainability of the strategy.

Our simulation results are consistent with the HD/R strategy assumptions, with published theoretical models and experimental data [9,23,24,25,26,27,31,71]. The spread of resistance in the population is efficiently delayed by Bt plants producing one toxin with a high concentration level. The critical threshold of 50% resistance allele frequency in the population (GF50) is reached at the 48th generation (see Figure 2A and Table 2A). According to our results, this strategy is more effective with Bt plants expressing simultaneously two toxins because resistance is reversed in the population (see Table 2B). Only double homozygotes (ArArBrBr) are resistant and resistance to two toxins is a rare event (initially ArArBrBr frequency: 5.06 × 10⁻¹² and ArAr frequency: 2.25 × 10⁻⁶). The resistance alleles are mainly present in the heterozygote insects which are eliminated by the toxins. This induces a decrease of the resistance allele frequency in time. The control of the pest population is similar for Bt plants with one or two toxins: there is a reduction of 94% of the population between the first two generations and a elimination of the population after ten generations (100% population decrease between the tenth and the first generation (see Table 2A,B).

There is no guarantee that the frequency of the resistance allele is rare. In Arizona, a field population of Pectinophora gossypiella (Lepidoptera: Gelechiidae) (Saunders) presented in 1997 an estimated mean frequency of a major resistance allele to Cry1Ac Bt cotton ca. 100 times higher than the theoretical threshold [12]. In the southeastern United States, 14 field populations of Helicoverpa zea (Lepidoptera: Noctuidae) (Boddie) had more than a hundred-fold higher than theoretical resistance to Cry1Ac Bt cotton, and showed a survival increase on leaves of Bt cotton plants [20,72,73,74]. Similarly, high resistance reports have been made for Spodoptera frugiperda (Lepidoptera: Noctuidae) (Smith) to Cry1F Bt corn in Puerto Rico and Busseola fusca (Lepidoptera: Noctuidae) (Fuller) to Cry1Ab Bt corn in South Africa [75,76].

Resistance to one-toxin Bt plants can be problematic for the pyramided Bt plant because two-toxin Bt plants can have the same Cry toxin as the previous crop of one-toxin Bt plants (BollgardII^®cotton produced Cry1Ac and Cry2Ab derives from BollgardI^® cotton protected by Cry1Ac). We consider this last situation in our model, with the complementary hypothesis that, due to previous exposure to one of the toxins, the resistance allele to this toxin is not rare.

For the one-toxin Bt plants, the initial frequency of the resistance allele has a high impact on resistance spread in the population (see Table 2A, Figure 3A). When the resistance allele is not rare, the efficiency of the resistance control drastically decreases. 48 generations are necessary to reach GF50 with an initially rare resistance allele (Ar Freq: 0.0015). This threshold is reached after eight generations for a 0.015 Ar frequency and after two generations for a 0.15 Ar frequency. Our results validate that the HD/R strategy is a preventive strategy, which is only suitable with initially rare resistance alleles [31,32]. The control of pest density is also influenced by the initial allele frequency (see Table 2A). After two generations, there is a 94% population decrease with 0.0015 and 0.015 initial Ar frequencies but a 92% population decrease with a 0.15 initial Ar frequency. However, whatever the initial resistance allele frequency, the insect population is eradicated after ten generations.

For the Bt plants expressing two toxins, resistance management remains effective when only one resistance allele is not rare in the population (see Figure 3B and Table 2B). The high toxin concentration (sBtA = sBtB: 1) eliminates all heterozygotes, whereas the homozygous resistant insects (ArArBrBr) are very rare even if one resistance allele is not infrequent (0.0015² * 0.15²). In these cases, resistance frequency decreases in the population (so, the GF50 threshold is never reached) and there is a 94% population decrease after two generations and an elimination of the pest population after ten generations. When the two resistance alleles are not rare (Ar = Br frequency: 0.15), resistance spread is rapid (GF50: 8) but the population density is still well controlled (94% and 99.9% population decrease after two and ten generations, respectively) (see Table 2B and Figure 3C). Therefore, although the initial resistance allele frequency for one- or two-toxin Bt plants does not seem to influence population density, it can nonetheless have a catastrophic impact on resistance control if the two toxins are not initially rare.

The efficiency of the high toxin concentration for one-toxin Bt plants on pest management was experimentally demonstrated in Arizona with a large-scale, long-term (ten years) study. The extensive use of Bt cotton had caused regional suppression of Pectinophora gossypiella, an ecological cotton specialist [77]. Thus, regions with substantial Bt fields imposing high mortality could have negative effects on population growth of a specific pest with a narrow host range before resistance occurs because their diet would be more affected [9,18]. According to Carrière et al. [61], the high dose limits the spectrum of resistance mutation conferring resistance in a particular species. Only few mutations could generate sufficient effects to overcome the toxin. The most widely observed mechanism in the laboratory and in the field is related to a reduced binding affinity of Cry toxins to midgut membrane receptors [78].

However, experimental studies have suggested that high Cry toxin concentration significantly increases the development duration of several insect pests [13,46,79,80,81,82,83]. The slowing down of insect development induces an asynchrony in emergence between the refuge and the Bt zone [29,84,85,86], which disrupts random mating and leads to assortative mating between adults in each of the two zones [81,86,87]. The reduction of random mating would disturb the effectiveness of the HD/R strategy and some researchers therefore recommend using Bt plants with low toxin concentration.

We test the efficiency of the refuge strategy associated with Bt plants producing a lower toxin concentration (sBtA: 0.93 or 0.50) with a resistance allele that is not completely recessive (hAr: 0.23 or 0.53, respectively) (see Table 2A and Figure 4A). Our simulations suggest that resistance is no more efficiently controlled for Bt plants expressing one toxin than for those expressing two (GF50: 7 and 20, respectively). The pest density is not as reduced after two generations (population decrease: 86% and 39%, respectively) but this difference of population control disappear after ten generations (population decrease: 99.9% and 98.7%, respectively).

Concerning the two-toxin Bt plants, we first test the association: one high toxin concentration/one low toxin concentration (see Figure 4B,C and Table 2B). For sBtA: 1-sBtB: 0.93, the GF50 threshold is never reached and for sBtA:1-sBtB: 0.50, GF50 is reached after 107 generations. A 94% population decrease is recorded for these two situations after two generations. Next, we consider Bt plants producing the two toxins at lower concentrations and we conclude that less than 25 generations are required to reach GF50 (sBtA = sBtB: 0.53, GF50: 25; sBtA: 0.53-sBtB: 0.93, GF50: 22; sBtA = sBtB: 0.93, GF50: 21). After two generations, the reduction in population density is also not so effective (sBtA = sBtB: 0.53: 67% population decrease; sBtA: 0.53–sBtB: 0.93: 90% population decrease) except for sBtA = sBtB: 0.93 (94% population decrease). However, after ten generations, the population control is efficient whatever the initial condition (100% population decrease). Thus, resistance is efficiently managed when Bt plants express at least one toxin at high concentration. We do not recommend producing Bt plants with two toxins at low concentrations unless there are no effective alternative methods (biological control, host plant resistance, etc.).

As previously suggested, HD/R strategy applied for two-toxin Bt plants is efficient even if the second toxin is produced at low concentration (sBtA: 1 and sBtB: 0.93 or 0.50) or if one resistance allele is rare in the population (0.015 or 0.15). However, what happens if two-toxin Bt plants produce one toxin at low titer when a resistance allele is not rare in the population?

Resistance is still reversed with sBtA: 1-sBtB: 0.93 associated to Ar frequency 0.0015 and Br frequency 0.15 (see Figure 5A,B and Table 2B). Resistance management strategy is also effective with sBtA: 1–sBtB: 0.53 associated to Ar frequency 0.0015 and Br frequency 0.15 (GF50: 90).

However, if the rare resistance allele (0.0015) is not associated with the high toxin concentration (sBtA: 1), resistance management is less viable (see Figure 5B and Table 2B). With sBtA: 1–sBtB: 0.93 associated with Ar frequency 0.15 and Br frequency 0.0015, the threshold GF50 is reached after only nine generations. With sBtA: 1–sBtB: 0.50 associated with Ar frequency 0.15 and Br frequency 0.0015, the threshold is reached after 20 generations. In conclusion, for Bt plants expressing one high toxin concentration combined with one low toxin concentration, it is risky to reuse a toxin already produced by a one-toxin Bt plant. The resistance allele frequency associated with this “old” toxin may not be rare. Following our results, the rare resistance allele has to be associated with the high toxin concentration for optimal resistance management. Introduction of two new toxins in the plants is recommended when low concentrations are used to limit resistance risk. Moreover, Zhao and colleagues [30] suggest that regulatory agencies consider deregulating one-toxin Bt plants as soon as two-toxin Bt plants sharing one toxin are available.

Whatever the situation, the population decrease is around 94% after two generations and 100% after ten generations, in every case (see Table 2B).

A key factor in the HD/R strategy is the refuge zone. The efficiency of the strategy is improved with high refuge zone proportion. The larger is the relative size of the refuge zone, the more insect resistance is delayed (see Figure 6 and Table 2A). However, for Bt plants with two toxins at low concentration, resistance is efficiently managed with a 30% refuge zone but the pest population is not so well controlled after two generations (population decrease: 65% for sBtA = sBtB: 0.93; 63% for sBtA: 0.93 and sBtB: 0.53; 45% for sBtA = sBtB: 0.53) (see Figure 6C, Figure 6D and Table 2B). However, after ten generations, we observe an elimination of the pest population whatever the refuge size.

Today, US EPA recommendations allow a small refuge size (5%). The critical question is whether enough susceptible insects will survive in the refuge to provide an effective source of susceptible alleles. As a resistance episode progresses, the refuge zone could be slightly “polluted” by random oviposition. The resistance level of the refuge zone may increase and make it inefficient. However, according to our results, a 5% refuge seems to be sufficient to control resistance. That could increase acceptability of this strategy by growers unwilling to sacrifice a large field area as a refuge. On the other hand, it has been shown that immigration of native susceptible insects could also restore the susceptibility of a population in the Bt zone [88]). Counting on immigration of susceptible alleles instead of using refuge zones would be feasible only if immigration patterns of susceptible insects are well known and consistent, and, usually one does not know a priori whether such events would occur. Furthermore, extremely high immigration is not compatible with overall pest control objectives (due to the damage caused) and if immigrants are not susceptible, they will not delay resistance [42].

Even though individuals with double resistance may be very rare initially, a refuge is still necessary for two-toxin Bt plants to delay resistance (our results, [31,32]). The refuge zone may be costly in terms of yield but the use of two-toxin Bt plants with high dose can reduce the amount of refuge required to delay resistance and for extended period in comparison with one-toxin Bt plants [31,89]. With a 5% refuge zone, one-toxin Bt plants delay resistance during 48 generations (GF50), whereas with two-toxin Bt plants, the delay is longer (GF50: never reached in our model). Therefore, two-toxin Bt plants have the potential to reduce the requirements for refuges in successful resistance management, which could encourage the industry to develop two-toxin Bt plants [90]. However, undersized refuges remain risky, for instance when mortalities of heterozygote insects are lower than expected. The more prudent way to deploy transgenic crops remains to keep refuges as large as economically feasible [31].