We used a previously published deterministic and discrete-time simulation model [26] implemented in R (http://www.r-project.org/) and comprising of two parts. The first part is based on population genetics theory and allows for the following of resistance frequency at each generation. Because it is necessary to limit the pest population damaging the crops, the second part is a population dynamics model simulating population density changes. Two kinds of parameters are introduced: operational parameters characterizing the strategy and biological parameters characterizing the target insect. The model and its parameters are detailed in [13]. Each parameter is associated with a default value (see Table 1).

The system considered is a section of a homogeneous, closed, 260ha area [28] composed of two adjacent zones: a Bt zone and an unsprayed refuge zone characterized by v, its relative size in relation to the global field. The insect population is uniformly distributed between the zones and cannot leave the system. Resistance is an autosomal trait controlled by a single locus with two alleles: the resistance allele Ar and the susceptible allele As. ArFreq, the initial frequency of the resistance allele is set to 1.5 × 10⁻³, with no sex linkage and no maternal effect [30,31,34]. Each generation is discrete, continuous (i.e., no diapause) and divided into a succession of simple stages. The generations do not overlap (see Figure 1).

After random oviposition and egg hatching, the larvae are subjected to the selective pressure of the plants they eat. Larval mortality, sBt, is linked in our model to toxin concentration in the plants and is dependent on larval genotype (ArAr, ArAs, AsAs) and on the dominance of the resistance allele (hAr). Insect survival is also influenced by the fitness cost associated with the resistance allele (fcost) and its dominance (hfc). The fitness cost is defined as a trade-off in which alleles conferring higher fitness in one environment (e.g., presence of Bt) reduce fitness in an alternative environment e.g., absence of Bt). In our situation, there is a Bt-resistance fitness costs in the absence of Bt toxins (fitness is lower for resistant insects than for susceptible insects) [35].

In this paper, the fitness of an insect is its ability to survive in relation to its genotype and zone of origin. It is calculated following the Lenormand equations [36] (see Equation 1, Table 2). The function g(x) equals 1 if the insect is on Bt plants and equals zero otherwise.

Depending on emergence asynchrony, three different groups of adults are identified [13]. The first two groups are isolated by emergence asynchrony: one is composed of the adults which emerge from the refuge zone before the adults of the Bt zone (%RefBef) and the other one is composed of the adults emerging from the Bt zone after the adults of the refuge zone (%BtAfter). These insects are unavailable for random mating between zones. The third group is composed by adults emerging simultaneously from the refuge zone and the Bt zone (%RefRandom, %BtRandom) and available for random mating between the two zones. Mating is panmictic inside each group and restores the Hardy–Weinberg genotype proportions [37].

Plodia interpunctella, (Lepidoptera: Pyralidae), our model insect, is a common pest of stored products biologically close to the common pest of Bt maize: Ostrinia nubilalis,and very easy to culture under controlled conditions in the laboratory. In the model parameters outlined below, the values for insect mortality and for the amount of asynchrony are derived from observed correlations with toxin concentration in Bt plants [13].

Following the HD/R initial assumptions, all of the adults simultaneously emerged from Bt plants expressing a high toxin concentration and from the non-Bt zone (see Control in Table 3; sBt = 1, hAr = 0, 0%BtAfter − 100%BtRandom, 0%RefBef − 100%RefRandom). We varied the values of the emergence asynchrony in our simulations (%BtRandom and %RefRandom: 0 to 100%) to assess its general impact on the HD/R strategy effectiveness (see Test 1 and Test 2 in Table 3).

Finally, we integrated our experimental data regarding emergence asynchrony and mortality in Plodia interpunctella when Bt plants produce 0.27, 0.25 and 0.14 μg Cry1Ab/g fwt and 57 ng Cry1Fa/mg total (see Test 3 and Test 4 in Table 3).

We considered that the insect’s reproductive capacity remains stable whatever the native zone and the age of the adults at the time of the mating.

The HD/R strategy requires Bt plants expressing high toxin concentration (sBt = 1) associated with a recessive resistance allele (hAr = 0). However, in our experiments, emergence asynchrony is significantly proportional to toxin concentration [13] (see Equations 4–7). Therefore, assortative mating could be limited by increasing toxin concentration in the Bt plants. However, toxin concentration, larval mortality and functional dominance are also linked [31,34,38,39]. Resistance dominance presents a plastic response depending on environmental parameters, with the recessivity of resistance being associated with more demanding environments [40]. For example, resistance to Cry1Ac in P.gossypiella is codominant at low concentrations (sBt 0.5; hAr = 0.53), partially recessive at intermediate concentrations (sBt = 0.93; hAr = 0.23), and completely recessive at high concentrations (sBt = 1, hAr = 0) [31].

We tested the impact of two toxin concentrations (sBt = 0.50, i.e., 50% mortality of susceptible insects- and sBt = 0.93, i.e., 93% mortality of susceptible insects-) with the associated dominances (hAr = 0.53 and hAr = 0.23, respectively) on the effectiveness of the HD/R.

First, we varied emergence asynchrony (%BtRandom and %RefRandom: 0 to 100%, see Test 1 and Test 2 in Table 3) to assess the effectiveness of the HD/R strategy in relation to theoretical random mating disruption. Next, we used simulations to characterize the impact of the observed emergence asynchronies in relation to the Cry1Ab or Cry1Fa concentrations (see Test 3 and Test 4 in Table 3 and Equations 4–7).

Emergence asynchrony potentially limits the effectiveness of the HD/R strategy. Moreover, this effectiveness is also reduced when Bt plants do not produce a high toxin concentration [26]. Here, we tested whether increasing the refuge zone (v = 5, 20 and 40%) could restore the effectiveness of the strategy.

The model investigated the effect of (i) random mating, (ii) dominance as linked to toxin concentration, and (iii) the relative size of the refuge, on resistance management in a pest population exposed to Bt plants. The most common indicator for comparing the spread of resistance in different simulations is the number of generations required to reach a 50% frequency of the resistance allele in the global population (GF50) [41,42]. This is a convenient measure of resistance, independent of any assumptions regarding population growth [43].

When emergence asynchrony occurred, the number of generations required to reach a 50% frequency of the resistance allele (GF50) in each group was also calculated: GF50Bef for the adults emerging from the refuge zone before those of the Bt zone; GF50After for the adults emerging from the Bt zone after those of the refuge zone and GF50Random for the adults simultaneously emerging from the refuge zone and the Bt zone.

The indicator for population density was the percentage of population decrease between the first and the third generation. Faster to obtain (three generations are sufficient), this indicator is also easily interpretable because its sign (positive or negative) informs about changes in population density. Because it is a ratio, this indicator is independent of the initial size of the system. Our results therefore show how the system changes regardless of the area considered.

According to our simulations, the most efficient situation corresponds to the initial assumptions of the HD/R strategy: global random mating (%RefRandom = %BtRandom = 100%), Bt plants expressing a high toxin concentration (sBt = 1) eliminating 100% of the homozygote susceptible insects (AsAs) and a recessive resistance (hAr = 0). In this ideal context, the critical threshold of 50% resistance allele frequency in the population (GF50) is reached at the 48th generation, and there is a 99.7% population decrease between the first and the third generation (see Control in Table 3A). This is really the best situation for optimal control of both resistance and pest density.

When more than 10% of the adults of the refuge zone (%RefRandom ≥ 10) are involved in random mating with all of the adults emerging from the Bt zone (%BtRandom = 100%), the strategy is still efficient but slightly reduced (48 ≥ GF50 ≥ 45) (see Figure 2A and Test 1 in Table 3A). Below this 10% limit (%RefRandom < 10), the effectiveness of the HD/R strategy is highly reduced (e.g. 2%RefRandom: GF50: 37) (see Figure 2A).

On the contrary, even minor delays in adult emergence from the Bt zone highly reduce the effectiveness of the HD/R strategy (see Figure 2B). GF50 is reached after 14 generations if 90% of the adults emerging from the Bt zone are available for random mating with all of the adults of the refuge zone (90%BtRandom, 100%RefRandom) (see Figure 2Band Test 2 inTable 3A). The threshold is reached after one generation for the adults emerging from the Bt zone after the adults of the refuge zone. However, GF50 is reached after thirteen generations in the group composed by adults emerging simultaneously from the refuge and from the Bt zone, and GF50 corresponds to 14 generations in the total population.

As recorded in our previous experiments [13], a high Cry1Ab concentration induces a complete asynchrony in emergence between the adults of the two zones and, there is thus no random mating between zones (100%RefBef and 100%BtAfter) (see Test 3 in Table 3A). In this case, only one generation is necessary to reach GF50 in the group composed of adults emerging from the Bt zone. In the refuge zone and in the global system, GF50 is reached after six generations.

When Bt plants produce a high toxin concentration, the population decrease between the first and the third generation is constant whatever the asynchrony (99.7% popdecrease).

In conclusion, temporal isolation and even minor delays could influence mating success and therefore the effectiveness of the refuge [19]. In our model, the effectiveness of the HD/R strategy is negatively influenced by emergence asynchrony, mainly by the late emergence of resistant insects from the Bt zone after emergence in the refuge zone. This temporally isolated group emerging from the Bt zone is only composed of homozygous resistant insects (ArAr) because all of the ArAs and AsAs are eliminated by ingestion of the high toxin concentration expressed by Bt plants (sBt = 1 − hAr = 0). Thus, when emergence asynchrony is complete, resistance is reached in the group after one generation. Moreover, we observe that numerous adults emerging from the refuge zone carry one or two resistance alleles. Resistance appears in this non-Bt zone, where there is no selection pressure, because of the pollution by resistant alleles during random egg laying by resistant insects selected in the Bt zone. This “pollution” decreases the effectiveness of the refuge zone by diluting the resistance.

However, with Bt plants expressing a high toxin concentration, pest density control is not influenced by emergence asynchrony (see Table 3A). The effect of emergence asynchrony on population density is probably masked by the massive population decrease induced by the Bt zone because, during the first generation, few insects are able to survive on the large Bt zone. The impact of emergence asynchrony is thus not well detected and probably insignificant in comparison to mortality in the Bt zone.

Two approaches are considered to reduce the negative impact of emergence asynchrony: reducing toxin concentration, which decreases emergence asynchrony and the recessivity of the resistance allele (see Section 3.2), or increasing the relative size of the refuge zone, which provides more susceptible alleles to dilute resistance for the next generation (see Section 3.3).

When mating is random (%RefRandom = %BtRandom = 100%), resistance is better controlled with sBt = 1 − hAr = 0 (GF50: 48) because all the ArAs are eliminated in the Bt zone (ArAs survival on Bt zone: 0% see Table 2), as recommended in the HD/R strategy. If Bt plants express lower toxin concentrations, the survival of susceptible homozygotes (AsAs) increases on the Bt zone (survival on Bt zone with sBt = 0.53: 76.5% ArAs and 50% AsAs; survival on Bt zone with sBt = 0.93: 28.39% ArAs and 0.07% AsAs). sBt = 0.53 is more efficient than sBt = 0.93 to control resistance (GF50: 20 versus GF50: 7, respectively) (see Table 2, control in Table 3B,C, respectively).

Mortality induced by Bt plants influences population density control. With sBt = 0.93 − hAr = 0.23, there is a 98% population decrease between the first and the third generation and a 63% decrease with sBt = 0.50 − hAr = 0.53 (see Table 3B,C, respectively).

In contrast to the high toxin concentration associated with a recessive resistant allele (sBt = 1 − hAr = 0), emergence asynchrony does not influence the effectiveness of resistance management at lower selection pressures (GF50: 20 sBt = 0.53 and GF50:7 for sBt = 0.93, regardless of emergence asynchrony) (see Figure 2A,B, Table 3B,C respectively). The emergence asynchrony experimentally observed with two Cry toxin concentrations (Cry1Ab and Cry1Fa) has no impact on population control (see Figure 2C,D). (sBt = 0.53: 63% population decrease and sBt = 0.93: 98% population decrease; regardless of emergence asynchrony).

To conclude, emergence asynchrony has no impact on the effectiveness of the strategy nor on pest density control when Bt plants do not produce high toxin concentrations. The susceptible alleles are not fully eliminated from the Bt zone (some ArAs and AsAs survive) and these susceptible alleles sufficiently dilute resistance even if asynchrony is complete. Resistance control and density control are thus only dependent on toxin concentration and on the associated dominance of the resistance allele. Resistance is better controlled with sBt = 0.53 than with sBt = 0.93 because survival of insect carrying one or two resistance alleles is higher. More susceptible alleles are thus available to dilute resistance. However, the pest population is better controlled with sBt = 0.93 than with sBt = 0.53, because more susceptible insects (AsAs and ArAs) are eliminated from the Bt zone. As lower toxin concentrations are more favorable to pest population development in the Bt zone than high toxin concentrations, growers might not welcome our recommendation of using lower Bt concentrations, but such a strategy will help to reduce resistance development and thus has long-term benefit.