The model presented here examines the probabilities of Wolbachia invasion into an isolated uninfected population. The model is unique in its individual-based representation of variation in key traits among adult females and in the resolution of larval dynamics within the host population. The model presented here predicts, as in previous modeling studies, that maternal inheritance (MI) and the relative fecundity of adult females (RF) are key parameters that determine the potential for population replacement. Specifically, population replacement occurs only at high MI or RF. In contrast, population replacement can occur at low CI or low IF. The simulation of adult females as individuals demonstrates that MI requires higher parameter values than RF for successful population replacement. The new parameter, relative larval viability (RLV), like MI and RF, requires high parameter values before population replacement can occur.
The relative larval viability between Wolbachia infected and uninfected individuals (RLV) is the most important determinant of population replacement, requiring the highest parameter values for invasion. The model predicts that reductions in infected larval survival can substantially reduce the probability of population replacement (Figure 4). While a majority of prior studies have examined for an effect in adults, recent studies have determined that, at high levels of intraspecific competition, Wolbachia infected larvae experience reduced survival . However, few theoretical studies have examined the impact of immature lifestages on the invasion of Wolbachia. Here, we demonstrate that reductions in RLV will inhibit Wolbachia invasion into an uninfected host population.
Recent work has highlighted the prevalence of Wolbachia, and its ability to invade populations [1, 20]. Studies have suggested that Wolbachia infection affects larval survival and development only when intraspecific competition is high [29, 44]. Given the predictions from our model, Wolbachia can only invade a population when RLV is very high. Therefore, the density of conspecifics in larval habitats is predicted to have significant impacts on the probability of population replacement. Similarly, the abundance and variety of larval habitats may have significant impact on the invasion of Wolbachia. The distribution, utilization and variety of larval habitats is well known for some insects, particularly mosquitoes [45–48]. Theoretical studies considering the effect of metapopulation structure and larval rearing conditions may elucidate the mechanism by which Wolbachia can invade natural populations given low initial infection frequencies.
The level of CI in insects varies widely [44, 49–51]. Our model shows that the intensity of CI has relatively little effect on the probability of population replacement when the rate of CI exceeds 60%. Furthermore, when CI = 0, the model presented here predicts population replacement can occur at low probabilities (Figure 4). Some Wolbachia infections do not cause CI, but are found at high frequencies in natural populations [44, 50, 52]. Previous theoretical studies indicate that CI or a sex-ratio distorter is not required for population replacement when endosymbionts can alter female traits [44, 53]. However, results presented here suggest that non-CI inducing Wolbachia infections can establish and persist in a population without increasing or altering host fitness, given high MI, RF, and RLV. Since the population considered by the model presented here is relatively small (N ≈ 110 adults), genetic drift could perhaps influence the probability of population replacement . To investigate the importance of genetic drift, the population size in the model was increased. In model simulations where the total adult population size is greater than approximately 200, population replacement does not occur when there is no effect of CI (i.e. CI = 0). However, when population size is increased, the general response patterns in Figure 4 are not altered.
High maternal inheritance rates have been observed consistently in natural populations [55–57]. Furthermore, theoretical studies predict the probability of population replacement declines as maternal inheritance decreases [12, 21, 22]. Similar to previous studies, results presented here suggest that maternal inheritance (MI) must be high for a Wolbachia infection to invade an uninfected population and persist. Specifically, MI must be higher than 90% to attain a realistic probability of population replacement.
The effect of Wolbachia infections on adult female fitness has been well documented empirically and theoretically [11, 15, 16, 22, 24, 58, 59]. Here, as in previous theoretical studies, the model predicts that the relative fecundity of adult females (RF) must be high to facilitate population replacement.
For all parameters, the probability of population replacement approached an absolute maximum of 90% given the conditions defined in Table 1. Here, the initially examined IF value is relatively high (0.5), analogous to artificial introductions examined in prior theoretical work . Subsequently, lower IF values have been simulated (Figure 4), including the introduction of a single, infected female (Table 2). The model predicts that Wolbachia invasion can occur at the lowest IF values and demonstrates an increasing probability of invasion with the higher introduction levels, with the probability of population replacement approaching 100%. Additional simulations determined that when IF is held constant and the total adult population size is increased, the probability of population replacement approaches one given the conditions defined in Table 1. This result suggests genetic drift can affect the probability of population replacement in small populations and may facilitate or hinder the spread of Wolbachia from low initial frequencies .
The model presented here predicted lower population replacement probabilities than those predicted by previous stochastic models (Table 2 and Figure 5) . Rasgon and Scott  noted a similar behavior where implementing population age-structure and overlapping generations increased deterministic thresholds. The inclusion of additional life stages and stage-structure in this stochastic model may explain the reduced probabilities of population replacement. However, the model presented here predicted marginally higher probabilities of population replacement when either maternal inheritance or the relative fecundity of infected females had a magnitude of 0.8. The increased probability of population replacement predicted by the model presented here is likely a result of the individual-based representation of the adult female life stage that includes stochastic survival.
The model here addresses a single, panmictic, isolated population but could be expanded to include metapopulation structure. If introduction events can be assumed to occur randomly, then the surrounding subpopulations should generally tend to inhibit population replacement, because migration between subpopulations would dilute the proportion of infected individuals. However, as demonstrated here, genetic drift may influence the invasion of Wolbachia in smaller subpopulations. The spatial spread of Wolbachia has been assessed analytically by others and defines the conditions needed for Wolbachia to spread through space [20, 24].
The majority of models that address the invasion of Wolbachia into uninfected populations have examined populations without lifestage subdivisions, suggesting that additional empirical studies focused on understanding larval dynamics are needed . Many of the parameters defined here may be difficult to determine in natural populations , but our results demonstrate the importance of understanding the role of life history parameters and their interactions, despite the difficulties. Furthermore, the sensitivity analysis of the model presented here demonstrates that the magnitudes of particular parameters strongly influence the potential for spread and establishment of Wolbachia; these (e.g., Wolbachia effects on immature fitness) should be the focus of future empirical and theoretical studies. Future theoretical studies could further address parameter sensitivity by hyper-cube sampling, but this would require information about the distribution of parameters to investigated .