The concomitant effects of self-limiting insect releases and behavioural interference on patterns of coexistence and exclusion of competing mosquitoes

Aedes aegypti is the dominant vector of dengue, a potentially fatal virus whose incidence has increased eightfold in the last two decades. As dengue has no widely available vaccine, vector control is key to reducing the global public health burden. A promising method is the release of self-limiting Ae. aegypti, which mate with wild Ae. aegypti and produce non-viable offspring. The resultant decrease in Ae. aegypti population size may impact coexistence with Ae. albopictus, another vector of dengue. A behavioural mechanism influencing coexistence between these species is reproductive interference, where incomplete species recognition results in heterospecifics engaging in mating activities. We develop a theoretical framework to investigate the interaction between self-limiting Ae. aegypti releases and reproductive interference between Ae. aegypti and Ae. albopictus on patterns of coexistence. In the absence of self-limiting Ae. aegypti release, coexistence can occur when the strength of reproductive interference experienced by both species is low. Results show that substantial overflooding with self-limiting Ae. aegypti prevents coexistence. For lower release ratios, as the release ratio increases, coexistence can occur when the strength of reproductive interference is increasingly high for Ae. albopictus and increasingly low for Ae. aegypti. This emphasizes the importance of including behavioural ecological processes into population models to evaluate the efficacy of vector control.


Introduction
Vector-borne diseases account for 17% of all infectious diseases and cause more than 700 000 deaths annually [1]. For example, dengue, a potentially fatal virus spread by Aedes mosquitoes, has increased eightfold in incidence over the last two decades [2]. Coupled with the rise in average global temperatures, the total burden on public health caused by vector-borne diseases is likely to further increase: Messina et al. [3] predicted 2.25 billion more people will be at risk of dengue by 2080 compared to 2015. In areas where vector-borne diseases are already present, pathogen replication, vector survival, reproduction, biting rate and the length of transmission seasons are set to increase with environmental change [3][4][5]. Furthermore, the global distribution of vectors is likely to widen: for instance, the global abundance of Aedes aegypti is predicted to increase by 20% or 30% by the end of the century, for a low and high carbon dioxide emission future, respectively [6]. Thus, it is increasingly important to have robust strategies to manage and control vector-borne diseases. As many vector-borne diseases have no widely available vaccine or disease-specific drugs (e.g. [6][7][8]), the key is employing methods to control vector population size.
However, the environment and human health can be negatively impacted by conventional, chemical-based vector control methods, such as the mass spraying of insecticide. Even pyrethroids, which have a low toxicity compared to many other insecticides [9], decrease the diversity of non-target small-bodied arthropods [10,11], cause aquatic toxicity [12,13] and negatively impact the human male reproductive system [14,15]. Additionally, the efficacy of chemical control is declining as insecticide resistance spreads [16]. Therefore, there is a growing imperative to develop novel approaches for controlling disease vectors. As such there has been a continued focus on the use of self-limiting insects for the control of mosquito-borne diseases [17][18][19]; a technique with substantial benefits [18], which is less costly than alternative control methods [19].
Self-limiting insects decrease the number of offspring contributing to the next generation by competing with wild insects for mates and subsequently producing non-viable offspring. Self-limiting systems include the sterile insect technique (SIT), where insects are irradiated so that they cannot produce viable offspring [20], and the release of insects carrying a dominant lethal (RIDL), where insects are genetically engineered to be homozygous for a dominant lethal genetic construct [17,18]. These methods target single species that are detrimental to human well-being, and do not require the direct or indirect release of harmful chemicals, and thus are considered relatively environmentally benign [19]. Previous use of SIT has been successful-resistance to the technique is infrequent, and it has successfully eliminated and controlled multiple insect pests (see Alphey et al. [20]).
However, by altering the population size of the target species, the release of self-limiting insects may have indirect environmental impacts on wider biodiversity-for instance, by affecting the interspecific interactions of the target species [21]. These indirect effects could be substantial when the population size of the heterospecific species is highly coupled to the population size of the target species. This is true for species that have a similar ecological niche to the target species, and are therefore likely to compete strongly with it for resources.
Aedes aegypti and Aedes albopictus, the two principal vectors of dengue, have overlapping realized ecological niches: they are both anthropophilic, have similar diurnal peak activity periods, use hosts to find mates, exist in urban and suburban settings and deposit their eggs above the water line in (often ephemeral) natural and artificial pools of water [22][23][24][25][26][27][28]. However, the two species cannot reproduce to form viable offspring [29]. Despite these species being native to different continents, their range expansion has resulted in frequent overlap of spatial distribution, causing either exclusion, or coexistence [30].
Two key mechanisms influencing coexistence patterns between Ae. aegypti and Ae. albopictus are interspecific resource competition and reproductive interference (reviewed by Lounibos & Juliano,[31]). Reproductive interference is where incomplete species recognition results in heterospecifics engaging in mating activities which do not produce viable offspring and cause a fitness cost to one or both of the species involved [31,32]. Thus, reproductive interference is distinct from competition, as the fitness costs incurred are not due to shared limited resources [33]. Instead, they can occur directly by female harassment and reduction in fertility, and indirectly through wasted courtship/handling time and energy [31,33]. Reproductive interference occurs across a wide range of taxa, as summarized by Gröning & Hochkirch [31]. Where two species coexist, theoretical studies have shown reproductive interference is more likely to cause exclusion than interspecific resource competition [34]. When both are considered together, reproductive interference acts synergistically with resource competition to promote exclusion; their combined effect is greater than the sum of their independent effects [35].
Reproductive interference occurs between Ae. aegypti and Ae. albopictus both in the field [36], and in the laboratory [37,38]. As discussed by Paton & Bonsall [33], at least four types of reproductive interference [31] occur between Aedes species: misdirected courtship, heterospecific mating attempts, erroneous female choice and heterospecific mating [36][37][38]. While the impact of reproductive interference upon the coupled population dynamics of Ae. aegypti and Ae. albopoictus has previously been analysed [33,35], the consequences of this mating disruption and behavioural interference on patterns of coexistence have not been explored in combination with the release of self-limiting Ae. aegypti.
Ae. aegypti is a key target for self-limiting techniques, as the dominant vector of dengue, and a vector of chikungunya, yellow fever and Zika viruses [1]. In regions where Ae. aegypti and Ae. albopictus coexist, their population dynamics are strongly coupled. As Ae. albopictus is a secondary vector to dengue, and carrier of chikungunya, yellow fever and Zika viruses, it is epidemiologically relevant: assessments of the impact of the release of self-limiting Ae. aegypti need to consider the impact upon Ae. albopictus populations.
Here, we form a theoretical framework (based on modified ecological competition equations) to investigate the interaction between the release of self-limiting Ae. aegypti and the impact of reproductive interference between Ae. aegypti and Ae. albopictus populations on patterns of coexistence and exclusion. In order to disentangle the effects of reproductive interference caused by wild mosquitoes and self-limiting Ae. aegypti, we examine scenarios where the self-limiting Ae. aegypti reproductively interfere with Ae. albopictus, and where they do not. Our results inform on the developing pragmatic applications and policy developments around the ecological (and epidemiological) consequences of the release of self-limiting Ae. aegypti to manage the dengue disease burden.

Methods
We used a set of simple differential equation models to assess how the release of self-limiting Ae. aegypti impacts its coexistence with Ae. albopictus when there is reproductive interference. Models are summarized in figure 1.
We use two baseline models: Kuno's model of reproductive interference ( [34], equations (2.1a) and (2.1b)) and a basic model of self-limiting Ae. aegypti release (equations (2.2a) and (2.2b)). The former examines the impact of reproductive interference in the absence of self-limiting Ae. aegypti release, while the latter explores the effect of self-limiting Ae. aegypti release without reproductive interference. The further two models build on these baseline models, to assess the combined impact of self-limiting Ae. aegypti release and reproductive interference: first when the self-limiting Ae. aegypti do not reproductively interfere with Ae. albopictus (equations (2.3a) and (2.3b)) and, second, when they do (equations (2.4a) and (2.4b)). Comparisons between these models allowed the impact of self-limiting Ae. aegypti release, and reproductive interference on coexistence to be examined separately and in combination. For all models with self-limiting Ae. aegypti release, we used a proportional release policy [39]: at each time point, the number self-limiting Ae. aegypti released is proportional to the number to wild Ae. aegypti.
royalsocietypublishing.org/journal/rspb Proc. R. Soc. B 288: 20210714 As with our previous work [33] other demographic processes such as birth and death rate [40,41] were assumed to be equal for both Aedes species and parameters were varied to allow the results to be investigated for a variety of environmental contexts [42,43], table 1.
(a) Baseline models (i) Kuno's model of reproductive interference Kuno [34] developed a two species interaction model that described resource competition and reproductive interference. In this model, for the ith species an increase in the heterospecific population causes a rise in density-dependent resource competition, increasing mortality and causes a rise in reproductive interference, decreasing recruitment. This model can be described by the following set of differential equations: À d A A(t) ð2:1aÞ where A represents the density of Ae. aegypti and B represents the density of Ae. albopictus. Parameters subscript A correspond to Ae. aegypti, and subscript B to Ae. albopictus. Recruitment is determined by r i , the reproductive rate, scaled by δ i , the decrease in reproductive success caused by reproductive interference. Density-dependent adult mortality is determined using α i , the strength of intraspecific competition, and β i , the strength of interspecific competition (relative to the intraspecific competition), while d i is the density-independent adult mosquito mortality rate.
(ii) Basic self-limiting Ae. aegypti release This model describes the reduction in recruitment of wild Ae. aegypti (denoted A) by the release of self-limiting Ae. aegypti. A proportional release policy [39] is used-thus, it is assumed that there is a stable proportion of self-limiting Ae. aegypti to wild Ae. aegypti. Furthermore, it is assumed that the self-limiting males always mate to produce non-viable offspring and are fully competitive with wild mosquitoes. This model can be described by À d A A(t) ð2:2aÞ   (i) Self-limiting Ae. aegypti do not cause reproductive interference Here, the released self-limiting Ae. aegypti only act to decrease the recruitment of Ae. aegypti: ð2:3aÞ (ii) Self-limiting Ae. aegypti cause reproductive interference This model extends equations (2.3a) and (2.3b). Here, the released self-limiting Ae. aegypti reproductively interfere with Ae. albopictus: ð2:4aÞ ð2:4bÞ This model assumes that self-limiting and wild Ae. aegypti equally interfere with the reproduction of Ae. albopictus.

(c) Model analysis (i) Zero-net-growth isoclines
Zero-net-growth isoclines were used to compare the outcomes of interspecific interaction across models. By definition, the zeronet-growth isoclines for each model were determined by solving dA(t)/dt = 0 and dB(t)/dt = 0 and taking positive solutions. This resulted in a linear (basic self-limiting Ae. aegypti release, equations (2.2a) and (2.2b)) or quadratic (all other models) equation for each species. Equilibria occur when the population growth rates of both species are equal zero (i.e. dA(t)/dt = 0 and dB(t)/dt = 0); where the zero-net-growth isoclines for each species cross. Equilibria were determined numerically with a multiroot function [44,45] which uses the Newton-Raphson method. Two types of equilibria are possible: exclusion of either species, or coexistence of both species. A Jacobian matrix approach was used to determine the stability of equilibrium points, where stable equilibria produce negative dominant eigenvalues and the magnitude of the eigenvalue corresponds to the stability [46].

(ii) Sensitivity analysis
Population size. We examined the impact of the strength of different parameters on the population size of each species at stable equilibria. While keeping all other parameters constant, the strength of the reproductive interference and interspecific competition experienced by each species, together with the selflimiting Ae. aegypti release ratio, were varied. This was conducted separately for exclusion and coexistence equilibria.
As with the isocline analysis, equilibria were determined numerically [44,45]. Under certain circumstances, the Newton-Raphson method fails to converge upon a root. For instance, when there is an inflection point at the root that is being approximated, the Newton-Raphson method often fails to converge but instead forms an oscillating sequence. Thus, for each sensitivity analysis, parameter values were selected to prevent these failures occurring. For this reason, the constant parameter values can vary between sensitivity analyses and comparisons were not made between sensitivity analyses.
Coexistence and exclusion boundaries. For each model with a reproductive interference term (equations (2.1a) and (2.1b), equations (2.3a) and (2.3b) and equations (2.4a) and (2.4b)), further analysis was conducted to determine the parameter space that results in stable coexistence. We examined how the strength of reproductive interference and interspecific competition influence the potential for coexistence for different self-limiting Ae. aegypti release ratios (no self-limiting Ae. aegypti release, θ = 0; weak underflooding, θ = 0.4; moderate underflooding, θ = 0.8; weak overflooding, θ = 1.2; substantial overflooding, θ = 8.0). We varied both the strength of reproductive interference and interspecific competition parameters from 0 (where they have no effect) to 1 (where heterospecifics are equivalent to conspecifics). In order to determine parameter values that result in stable coexistence, cubic expressions (cubic formula, equations (2.5)) were derived by substituting the solution for Ae. aegypti (denoted A) into the equation for Ae. albopictus (denoted B) and vice versa. When both cubic equations in a model have three positive solutions, there is stable coexistence-two unstable coexistence points surrounding one stable coexistence point [35]. For the coupled cubic equations to have at least three solutions, the discriminants of both equations must be greater than zero (equations (2.6)) and for all solutions to be positive, and thus biologically relevant, the coefficients have satisfy certain inequalities [35], see equations (2.7).
All analyses (simulations, mathematical derivations and graphical analysis) were completed in R (v. 4.0.4) and Mathematica (v. 12). Code is available at (OSF) and for further details of these analyses, see electronic supplementary material [47].

Results (a) Isocline analysis
The addition of self-limiting Ae. aegypti or reproductive interference to the baseline models can alter the shape or gradient of the zero-net-growth isocline of one or both species. This changes where the two isoclines intersect and influences the number of coexistence points, their stability and the population size of each species at those points.
By comparing the isoclines for Kuno's model of reproductive interference (equations (2.1a) and (2.1b); figure 2a) to models with reproductive interference and self-limiting Ae. aegypti release, we can determine the impact of releasing self-limiting Ae. aegypti. Where there is reproductive interference, but no release of self-limiting Ae. aegypti (Kuno's model, royalsocietypublishing.org/journal/rspb Proc. R. Soc. B 288: 20210714 equations (2.1a) and (2.1b); figure 2a), Ae. aegypti and Ae. albopictus isoclines cross at three points, one stable coexistence point flanked by two unstable coexistence points (as seen in [31]). However, underflooding (figure 2c,d) and overflooding (figure 2f,g) with self-limiting Ae. aegypti causes vertical and horizontal compression of the Ae. aegypti isocline, and minor horizontal compression of the Ae. albopictus isocline, causing the isoclines to only cross at a single point. Therefore, following underflooding and overflooding with self-limiting Ae. aegypti, coexistence is destabilized and exclusion are the only stable equilibria.
The influence of reproductive interference on the stable equilibria is assessed by comparing the isoclines of the model of self-limiting Ae. aegypti release with no reproductive interference (equations (2.2a) and (2.2b); figure 2b,e) to the models with both self-limiting Ae. aegypti release and reproductive interference (equations (2.3a) and (2.3b); figure 2c,f and equations (2.4a) and (2.4b); figure 2d,g, respectively). In the absence of reproductive interference, the dynamics depend upon the release ratio of self-limiting Ae. aegypti; in an underflooding scenario, there is only stable coexistence There is minimal difference between the results of the model where self-limiting Ae. aegypti cause reproductive interference (equations (2.4a) and (2.4b); figure 2d,g), and the model where only wild mosquitoes cause reproductive interference (equations (2.3a) and (2.3b); figure 2c,f ). This suggests that the reproductive interference caused by wild Ae. aegypti and Ae. albopictus has a much greater impact on dynamics than the reproductive interference caused by selflimiting Ae. aegypti. This is further explored in the sensitivity analysis (see coexistence and exclusion boundaries, 3b(ii)).

(b)Sensitivity analysis
Isocline analyses are limited in that they only show stable equilibria for a certain set of parameter values. Thus, we conducted further analyses to examine the sensitivity of the population sizes of Ae. aegypti and Ae. albopictus to the ratio of self-limiting Ae. aegypti (θ; electronic supplementary material, figure S1), the strength of reproductive interference (δ i ; electronic supplementary material, figure S2), and the strength of interspecific competition (β i ; electronic supplementary material, figure S3). We then assessed the sensitivity of the stable coexistence of Ae. aegypti and Ae. albopictus to the strength of reproductive interference (δ i ; figure 3), and interspecific competition (β i ; figure 4), for different ratios of self-limiting Ae. aegypti (θ).

(i) Population size
We explored the impact of different parameter values on the population size of both species at all stable equilibria. The magnitude of the ratio of self-limiting Ae. aegypti (θ), reproductive interference (δ i ) and interspecific competition (β i ) all impact the population size of both species at coexistence. In all models with the release of self-limiting Ae. aegypti, as the release ratio increases, the population size of Ae. aegypti at coexistence decreases, and Ae. albopictus increases   Although the ratio of self-limiting Ae. aegypti (θ), the strength of reproductive interference (δ i ) and strength of interspecific competition (β i ) influence the population size of both species at coexistence, only the release ratio of selflimiting Ae. aegypti (θ) has any impact on population size when there is exclusion (electronic supplementary material figure S1d-f ). When there is reproductive interference, as the release ratio of self-limiting Ae. aegypti (θ) increases, there is a decrease in the population size of Ae. aegypti when it excludes Ae. albopictus (electronic supplementary material figure S1e,f ). These results are sensical: following exclusion of a species, interspecific interactions (through resource competition and reproductive interference) will not occur, and thus will not influence population density. However, following the exclusion of Ae. albopictus, self-limiting Ae. aegypti will still suppress the population size of Ae. aegypti. In the absence of reproductive interference, Ae. aegypti cannot stably exclude Ae. albopictus (electronic supplementary material figure S1d). For all self-limiting insect release models, the population size of Ae. albopictus, when it excludes Ae. aegypti, is unaffected by the self-limiting Ae. aegypti release ratio (electronic supplementary material figure S1d-f ). Again, this is sensical, as there will be no selflimiting Ae. aegypti present following the exclusion of Ae. aegypti.     Figure 4. The effects of the strength of interspecific competition (β i ) on stable coexistence in all models with a reproductive interference term (equations (2.1a) and (2.1b); equations (2.3a) and (2.3b) and equations (2.4a) and (2.4b)). Parameter space where stable coexistence can occur is shaded in black. As in figure 3, the ratio of self-limiting Ae. aegypti release (θ) is varied across plots, from no self-limiting Ae. aegypti release (θ = 0, (a)), to weak underflooding (θ = 0.4, (b,f )), moderate underflooding (θ = 0.8, (c,g)), weak overflooding (θ = 1.2, (d,h)) and substantial overflooding (θ = 8, (e,i)). In the first row, self-limiting Ae. aegypti do not reproductively interfere with Ae. albopictus (equations (2.3a) and (2.3b)), while in the second row, they do (equations (2.4a) and (2.4b)). Constant parameter values are given in

(ii) Coexistence and exclusion boundaries
For all models with the reproductive interference term, analyses explored the ranges of strengths of reproductive interference (δ i , figure 3) and interspecific competition (β i , figure 4) that resulted in stable coexistence, for different ratios of self-limiting Ae. aegypti (θ). Similar patterns are observed for both parameters, in Kuno's model of reproductive interference (equations (2.1a) and (2.1b)) and the models including self-limiting Ae. aegypti release (equations (2.3a) and (2.3b) and equations (2.4a) and (2.4b)).
In Kuno's model of reproductive interference model (equations (2.1a) and (2.1b)), coexistence regions are biased towards the left corner, where there is low reproductive interference (figure 3a), or low interspecific competition (figure 4a) experienced by both species. In models with selflimiting Ae. aegypti release, an increase in the release ratio (θ) causes the stable coexistence parameter space to decrease in area, and change shape. As the release ratio (θ) increases, coexistence can occur when Ae. albopictus suffers increasingly high reproductive interference (figure 3) or interspecific competition (figure 4), and Ae. aegypti suffers increasingly low reproductive interference (figure 3) or interspecific competition ( figure 4).
When self-limiting Ae. aegypti do not reproductively interfere with Ae. albopictus (figures 3b-e and 4b-e), coexistence can occur when Ae. albopictus suffers greater reproductive interference and interspecific competition, than when Ae. aegypti do cause reproductive interference (figures 3f-i and 4f-i). In the substantial overflooding situation, the self-limiting insect release ratio is high enough (θ = 8) that coexistence cannot occur for any values for reproductive interference (figure 3e,i) or interspecific competition (figure 4e,i)-thus, there is only exclusion.

Discussion
This work is the first to investigate the combined role of reproductive interference and self-limiting insect releases on the coexistence of closely related disease vectors, Ae. aegypti and Ae. albopictus. It is well established, both theoretically [47][48][49][50] and through a limited set of field trials [51,52], that the release of self-limiting mosquitoes can be used to suppress the population size of Ae. aegypti, and therefore, potentially, reduce the substantial associated public health burden.
Ae. aegypti shares a similar ecological niche with Ae. albopictus, they compete for resources [24][25][26] and interfere with each other's mating attempts [36][37][38]. Previous studies have separately modelled the impact of reproductive interference [33] and self-limiting Ae. aegypti release [21] upon the coexistence of Ae. aegypti and Ae. albopictus. However, as highlighted [35], the combined effect has not been investigated. We addressed this by examining the interactions between Ae. aegypti and Ae. albopictus where there is reproductive interference, and self-limiting Ae. aegypti release. We explored the potential outcomes following the release of self-limiting Ae. aegypti, where Ae. aegypti and Ae. albopictus coexist. This work remains highly relevant as Ae. albopictus and Ae. aegypti are undergoing range expansion, making it increasingly likely that these important vectors of disease will come into contact [30]. Our results show that the ratio of self-limiting Ae. aegypti and the strength of reproductive interference can act concomitantly to determine whether coexistence is maintained, or exclusion occurs. Self-limiting Ae. aegypti releases and reproductive interference also affect the population size of one or both species. Therefore, in locations where the distributions of Ae. aegypti and Ae. albopictus overlap, both the behavioural ecological and population ecological effects of self-limiting releases have important consequences for the efficacy of vector control programmes.
To investigate the release of self-limiting Ae. aegypti, we used a proportional release policy [40], where, at each time point, the number of self-limiting Ae. aegypti is proportional to the number of wild Ae. aegypti [40]. To conduct proportional releases in the field requires constant monitoring of mosquito populations. There are well established techniques to monitor Aedes populations, and entomological surveys are necessary following self-limiting insect release in order to monitor efficacy [19]. The proportional release policy has been investigated in previous theoretical studies on self-limiting control [39,53,54]; under this release policy, fewer self-limiting mosquitoes need to be released to eradicate the target organism than in a constant release scenario (where the same number of self-limiting mosquitoes are released at each time step) [39]. As such, there could be the economic benefit of using proportional release policies, especially in areas where public health monitoring of mosquitoes is sufficient to estimate mosquito density, so no additional monitoring costs accrue. Economic benefits are an important consideration: cost-effectiveness has been highlighted as a technical aspect of self-limiting insect releases that requires further investigation [19]. One approach could be to compare the costs of different release policies to understand the implications of self-limiting releases on wider aspects of biodiversity.

(a) Population size
We examined the impact of increasing the ratio of self-limiting Ae. aegypti on the population sizes of Ae. aegypti and Ae. albopictus. Self-limiting Ae. aegypti mate with their wild counterparts to produce non-viable offspring: thus, these self-limiting insects suppress the population density of Ae. aegypti by decreasing recruitment. This is shown from our analyses, where increasing the ratio of self-limiting Ae. aegypti decreases the population density of Ae. aegypti when there is coexistence, or where Ae. aegypti excludes Ae. albopictus. As there are fewer Ae. aegypti to compete with Ae. albopictus for resources or interfere with their mating, the population size of Ae. albopictus increases with the self-limiting Ae. aegypti release ratio when there is coexistence. This result holds when self-limiting Ae. aegypti reproductively interfere with Ae. albopictus, and when they do not cause reproductive interference: thus, this outcome is governed by the decrease in the wild Ae. aegypti population size, rather than any additional increase in reproductive interference by self-limiting Ae. aegypti.
Following the release of self-limiting Ae. aegypti, where Ae. albopictus persists, it will increase in population size; either by excluding Ae. aegypti and reaching carrying capacity, or by increasing its population size in coexistence with Ae. aegypti. This is an important consideration: as highlighted by Bargielowski et al., Ae. albopictus is the principal vector of dengue in regions where Ae. aegypti is rare or uncommon (e.g. in China [55][56][57], Bangladesh [58] and South India [59,60]), and in regions of Africa native to Ae. aegypti that Ae. albopictus has recently colonized [61]. Therefore, in some regions, the decrease in public health burden caused by royalsocietypublishing.org/journal/rspb Proc. R. Soc. B 288: 20210714 lowering the Ae. aegypti population may be lessened or compensated for by the associated increase in the Ae. albopictus population size. This highlights a limitation of using species specific pest control techniques [18]. In situations where both species coexist and are significant disease vectors, the release of both self-limiting Ae. aegypti and Ae. albopictus may be a more appropriate course of action [19].
Furthermore, our results show that the strength of reproductive interference and interspecific competition affects the population densities of both species at coexistence. An increase in the strength of reproductive interference or interspecific competition reduces the population size of the species experiencing the behavioural (reproductive interference) or ecological (interspecific competition) effects. For the former, this is due to fewer successful matings, reducing recruitment, and for the latter, more competition with heterospecifics for resources, increasing the number of deaths. The subsequent reduction in population size of the focal species allows the other species to increase in population density, as there are fewer heterospecifics to compete with for resources, or to interfere with their mating. However, previous work has shown Ae. aegypti may develop some resistance to reproductive interference. Bargielowski et al. [37] showed that female Ae. aegypti from populations in allopatry with Ae. albopictus mis-mate more frequently than those from populations with a history of sympatry; suggesting that upon contact between Ae. aegypti and Ae. albopictus, Ae. aegypti are selected for their ability to evade reproductive interference by Ae. albopictus. Thus, where Ae. aegypti and Ae. albopictus have come into contact more recently, Ae. aegypti may experience stronger reproductive interference. Our results suggest this will cause a lower population size of Ae. aegypti-thus, recent contact with Ae. albopictus means fewer self-limiting Ae. aegypti need to be released to have the equivalent impact upon the Ae. aegypti population size. Therefore, these areas could be targeted at a lower economic burden. This is particularly relevant in regions where Ae. albopictus is not a disease vector.

(b) Coexistence and exclusion boundaries
Our results show that the ratio of self-limiting Ae. aegypti released determines whether coexistence is possible. A high enough release ratio of Ae. aegypti destabilizes coexistence across all strengths of reproductive interference and interspecific competition-thus, the only stable outcome is exclusion. However, at lower release ratios, coexistence is still possible. Here, the ratio of self-limiting Ae. aegypti, and whether the self-limiting Ae. aegypti cause reproductive interference, determines the strength of the reproductive interference and resource competition that can be experienced by Ae. aegypti and Ae. albopictus, and coexistence maintained.
When self-limiting Ae. aegypti reproductively interfere with Ae. albopictus, coexistence can occur when each individual Ae. aegypti causes weaker reproductive interference and interspecific competition, than when only wild Ae. aegypti reproductively interfere with Ae. albopictus. This is because, there are additional individuals (the self-limiting Ae. aegypti) that reproductively interfere with Ae. albopictus. This increases the total reproductive interference experienced by the Ae. albopictus population, for the same strength of reproductive interference imposed by each individual Ae. aegypti.
As discussed, an increase in the ratio of self-limiting Ae. aegypti causes the population size of Ae. aegypti to decrease, meaning there are fewer Ae aegypti to interfere with the mating of Ae. albopictus or compete with them for resources. Thus, as the release ratio increases, each individual Ae. aegypti can reproductively interfere and compete for resources increasingly strongly with Ae. albopictus, without changing the total ecological effects experienced by the Ae. albopictus population. As the population size of Ae. albopictus increases with the release ratio of self-limiting Ae. aegypti, the reverse occurs to the per capita reproductive interference and competition for resources caused by Ae. albopictus. For these reasons, up to a moderate release ratio of Ae. aegypti, our results show that as the ratio of selflimiting Ae. aegypti increases, coexistence can occur when the reproductive interference and resource competition suffered by Ae. albopictus is increasingly high and that suffered by Ae. aegypti is increasingly low.
However, meta-analyses suggest that Ae. albopictus is more likely to experience weaker reproductive interference, and interspecific resource competition than Ae. aegypti [30,42]. A meta-analysis by Juliano [42] showed Ae. albopictus has a competitive advantage over Ae. aegypti when there is low food quality, and high food quality results in competitive equivalence. Furthermore, Ae. aegypti is more likely to mismate than Ae. albopictus [37,38] and following mis-mating, female Ae. aegypti can become refractory to further mating [36]. This asymmetry is likely to be due to the greater species recognition abilities of Ae. albopictus [31,38]. This suggests that in the wild, as the release ratio of self-limiting Ae. aeypti increases, coexistence will only occur where both species experience an increasingly low strength of reproductive interference and interspecific resource competition, although coexistence is theoretically possible when Ae. albopictus suffers more than Ae. aegypti.
It is plausible that there are regions where the reproductive interference and resource competition experienced by Ae. aegypti is sufficiently weak for coexistence to occur, even when self-limiting Ae. aegypti are released. For instance, coexistence could be facilitated in areas with high food quality, where the competitive advantage of Ae. albopictus is reduced, resulting in lower interspecific competition [42]. Furthermore, Ae. aegypti may experience reproductive interference that is weak enough to allow coexistence following the release of self-limiting Ae. aegypti-for instance in a coexistence region in Brazil, Ae. albopictus only weakly reproductively interferes with Ae. aegypti [43]. Equally, as discussed, in regions where Ae. aegypti initially experiences strong reproductive interference, this species could develop resistance to reproductive interference prior to exclusion [37]. Additionally, in the wild, habitat partitioning in time or space could allow reproductive interference to be avoided and coexistence to be maintained [62]. Thus, the release of self-limiting Ae. aegypti could initially destabilize coexistence, but subsequent development of resistance to mis-mating in Ae. aegypti, or habitat partitioning, could recover stable coexistence.

Conclusion
Previous models have examined the impact of reproductive interference [33][34][35] and self-limiting insect release [21] on the royalsocietypublishing.org/journal/rspb Proc. R. Soc. B 288: 20210714 population dynamics between the vectors Ae. aegypti and Ae. albopictus. However, we are the first to addresses the combined impact of self-limiting insect release and reproductive interference upon Ae. aegypti and Ae. albopictus coexistence. Our results show that the strength of reproductive interference and the ratio of self-limiting Ae. aegypti are important factors that act together to determine the population size of both Ae. aegypti and Ae. albopictus and whether coexistence can occur. This highlights the importance of including behavioural ecological processes, such as reproductive interference, into population dynamic frameworks to evaluate the efficacy of vector control. Future work could focus on stochastic and spatial aspects of coexistence following self-limiting releases. Furthermore, linking these outcomes to vector-borne disease epidemiology would allow the effects of behavioural traits (such as reproductive interference) on disease spread and the public health implications to be evaluated.