Complex signals alter recognition accuracy and conspecific acceptance thresholds

Abstract

Many aspects of behaviour depend on recognition, but accurate recognition is difficult because the traits used for recognition often overlap. For example, brood parasitic birds mimic host eggs, so it is challenging for hosts to discriminate between their own eggs and parasitic eggs. Complex signals that occur in multiple sensory modalities or involve multiple signal components are thought to facilitate accurate recognition. However, we lack models that explore the effect of complex signals on the evolution of recognition systems. Here, we use individual-based models with a genetic algorithm to test how complex signals influence recognition thresholds, signaller phenotypes and receiver responses. The model has three main results. First, complex signals lead to more accurate recognition than simple signals. Second, when two signals provide different amounts of information, receivers will rely on the more informative signal to make recognition decisions and may ignore the less informative signal. As a result, the particular traits used for recognition change over evolutionary time as sender and receiver phenotypes evolve. Third, complex signals are more likely to evolve when recognition errors are high cost than when errors are low cost. Overall, redundant, complex signals are an evolutionarily stable mechanism to reduce recognition errors.

This article is part of the theme issue ‘Signal detection theory in recognition systems: from evolving models to experimental tests’.

1. Introduction

Recognition is important in many contexts, including identifying species, kin, mates, individuals and nest-mates [1,2]. Much recognition involves phenotype matching, where receivers assess sender phenotypes, compare sender phenotypes to a template of expected phenotypes, then make a decision about how to treat senders based on the match between the template and phenotype [3,4]. For example, during nest-mate recognition, guards smell conspecifics and compare conspecific odour to the expected odour of nest-mates. If the individual smells enough like the expected odour of nest-mates, the guard will accept the individual as a nest-mate. If the individual does not smell enough like a nest-mate, the guard will reject the individual and treat it as a non-nest-mate [5–7].

Recognition typically involves signals that coevolve between senders and receivers. Signals used during recognition usually benefit senders (i.e. the individuals displaying the trait) by influencing the behaviour of receivers (i.e. the individuals perceiving the trait). Because senders benefit from receiver responses, selection favours sender phenotypes that effectively convey information to receivers [8]. For example, both senders and receivers in a social insect colony benefit when nest-mate recognition is accurate. The benefits of accurate recognition influence the evolution of odours that signal nest-mate identity and the perceptual and cognitive processes that facilitate accurate recognition. Recognition can occur without senders benefitting or sender/receiver coevolution. For example, humans can recognize a squirrel based on its appearance. However, squirrel evolution is not influenced by how well humans recognize squirrels. Our model focuses on recognition that involves coevolution between senders and receivers.

A major challenge of recognition is that it is not possible for receivers to make perfect recognition decisions. Instead, the phenotypes used as recognition signals have broad and overlapping distributions, so errors are common. Receivers can make two types of errors (figure 1a) [9]. (i) Rejection errors occur when receivers reject individuals that should not be rejected (e.g. guards prevent nest-mates from entering their own nest). Rejection errors are similar to type I errors in statistics, when a true hypothesis is rejected. (ii) Acceptance errors occur when receivers accept individuals they should not accept (e.g. guards allow non-nest-mates to enter the nest). Acceptance errors are similar to type II errors in statistics, when a false hypothesis is accepted.

Figure 1.

The phenotypic distributions of senders and their relationship with acceptance threshold and phenotypic dissimilarity. (a) The schematic view of acceptance and rejection errors (modified from [9]). The probability distribution of a desirable sender's trait value is depicted by the solid curve. The probability distribution of an undesirable recipient's trait value is depicted by the dashed curve. The threshold of acceptance or rejection is the dot-dashed line. The dotted area above the threshold shows rejection errors (type I error). These are trait values for which desirable senders are rejected. The striped area below the threshold shows acceptance errors (type II errors). These are trait values for which undesirable senders are accepted. (b–d) Examples of the sender traits and the threshold at the end of simulations. The acceptance threshold (broken lines) is optimized according to the phenotype dissimilarity between the average trait value of desirable and undesirable senders. If the phenotypic dissimilarity is 0, the undesirable senders can have the same average trait value as the desirable senders (b). If the phenotypic dissimilarity is 1 or 2, respectively, then the traits of undesirable are always higher than desirable senders (c,d).

Conspecific acceptance threshold theory [9] makes predictions about how receivers optimize recognition behaviour in different situations. Reeve [9] proposed that receivers have an ‘acceptance threshold’ that is the maximum amount of dissimilarity between template and trait that receivers will tolerate without rejecting an individual. The probabilities of acceptance and rejection errors change based on the location of the acceptance threshold. If the receiver reduces one type of error, it is inevitable that the other type of error will increase (figure 1a). For example, if the receiver alters the acceptance threshold to reduce rejection errors, acceptance errors will increase. As a result, there is a trade-off to balance both error types. The placement of the optimal acceptance threshold that maximizes receiver fitness varies based on multiple variables [6,7,10].

Reeve [9] illustrated that the acceptance threshold is influenced by factors like the cost of making mistakes, the frequency of desirable and undesirable senders, and the phenotypes of desirable and undesirable senders. For example, if accepting undesirable senders is very low cost (e.g. non-nest-mates do not do any damage to a nest), receivers are expected to have a very permissive acceptance threshold (see electronic supplementary material, §S2 for related discussion in complex signals). They will allow many non-nest-mates on the nest and will not accidently reject nest-mates [7]. On the other hand, if accepting undesirable senders is very costly (e.g. non-nest-mates destroy the nest), receivers will have a very restrictive acceptance threshold. They will allow few non-nest-mates onto the nest, but will also reject many nest-mates.

The acceptance threshold model focused on recognition based on a single trait. However, over the last decades, there has been a growing appreciation that most animal signals are ‘complex’ [11–14], involving multiple signal components in a single sensory modality (multi-component signalling) or in multiple sensory modalities (multimodal signalling) [12]. Many complex signals are thought to be ‘redundant’, in that multiple signals convey similar information [15,16]. For example, egg recognition can be based on egg size, shape, colour and pattern [17,18]. Species recognition may be based on multiple aspects of coloration, odour, and auditory signals like songs. Female swordtail fish distinguish between heterospecific and conspecific males more accurately based on both chemical and visual traits than based on one alone [19]. Even when recognition occurs in a single sensory modality, traits often have multiple components. For example, nest-mate recognition is based on the amount and distribution of multiple chemicals [5]. Thus far, it is not clear how redundant complex signals alter the evolution of optimal acceptance thresholds or error rates.

Here, we update the acceptance threshold model to test how multi-component traits influence the evolution of acceptance thresholds. We test how the number of traits affects the evolution of the optimal acceptance threshold, acceptance errors and rejection errors. We construct individual-based models that vary in population properties, including phenotypic similarity between desirable and undesirable senders, as well as the cost of making recognition errors. We allow the phenotypes of desirable and undesirable senders to evolve, so the model also provides insight into how sender phenotypes evolve in response to receiver recognition behaviour thresholds. The goal of the models is to find the threshold that optimizes receiver fitness by allowing receivers to reject undesirable senders and accept desirable senders.

2. Methods

We use individual-based models following a genetic algorithm to understand the evolution of acceptance thresholds. All simulated data were generated using the C language. The code used for this study is available at https://github.com/mingpapilio/Codes_ComplexSignal. This method explicitly models the reproductive events, the process of acceptance or rejection, and the mutations within populations. In other words, each individual has its own trait values and acceptance thresholds. There are two categories of individuals in our models: (i) receivers make decisions about whether to accept/reject other individuals and also act as desirable senders (hosts, nest-mates), and (ii) undesirable senders (brood parasites, non-nest-mates).

The receiver makes the acceptance/rejection decision based on its acceptance threshold and the trait values of the desirable and undesirable senders. Senders with trait values above threshold are rejected, whereas senders with values below or equal to the threshold are accepted (figure 1a). Senders are accepted when both traits (traits 1 and 2) meet the receiver's threshold and are rejected when either trait 1 or trait 2 does not meet the receiver's threshold. There are no interactive effects between traits. For example, the phenotypes of trait 2 does not influence decisions about trait 1, consistent with empirical work on assessment of redundant complex signals [12,20].

In the model, traits provide information to receivers and reduce receiver uncertainty [21]. More informative traits produce a greater reduction in uncertainty. For example, traits that have a greater phenotypic dissimilarity between desirable and undesirable recipients are more informative and decrease receiver uncertainty more than traits for which the desirable and undesirable recipients have very similar phenotypes.

Only the accepted senders survive to establish future populations. Specifically, the surviving desirable senders produce the desirable senders of the next generation. During reproduction, desirable senders act as receivers because they assess the trait values of all offspring, compare the trait values with their acceptance threshold, then choose which offspring to raise. Receiver reproductive success is reduced if they reject their own offspring (termed desirable senders) or accept foreign offspring (termed undesirable senders). Undesirable senders only survive to reproduce if they are accepted by a receiver. Surviving undesirable senders die after producing undesirable senders of the next generation.

The consequences of recognition errors vary across models. For simplicity, we first assume that accepting an undesirable sender kills all desirable senders produced by the focal receiver. This is similar to a brood parasitism scenario where accepted brood parasites kill the host's offspring. In subsequent models, we relax this assumption by varying levels of cost from accepting an undesirable sender (§3c and electronic supplementary material, §S2). The genotypes, including the acceptance thresholds and the trait values of the desirable and undesirable senders, are heritable with a slight chance of mutation. The model simulates small magnitude mutations for the traits of the undesirable senders. The magnitude of the mutations is determined by drawing random numbers from a normal distribution with average of 0 and standard deviation of 0.25. The initial difference between desirable and undesirable senders is 5.

The genetic algorithm finds the optimum genotype that maximizes reproductive success. For desirable senders, successful genotypes have acceptance thresholds and trait values that reduce the probability of recognition errors. For undesirable senders, successful genotypes have trait values that ‘trick’ receivers into accepting them as offspring. Successful genotypes will increase in frequency in the population through generations. Model results reflect equilibrium trait values that have been stable for generations.

We begin with the most basic scenario, where the acceptance threshold(s) is evolvable, but the two senders have fixed phenotypic variability. That is, the average phenotypic dissimilarity (d) between desirable and undesirable senders as well as the distributions of phenotypes are constant (e.g. figure 1b–d). For simplicity, we allow the acceptance thresholds in different traits to evolve independently and do not impose a cost related to the number of traits used during recognition. Under these assumptions, we explore difference in the optimal threshold(s) between using one trait and two traits (figures 2 and 3).

Figure 2.

Threshold and error rate comparison between using a single trait or two traits for recognition when phenotypic dissimilarities between desirable and undesirable senders are fixed. (a,b) The optimal acceptance threshold when using a single trait (vertical blue dashed line) or two traits (vertical orange solid lines). (c,e) Based on the acceptance threshold and the phenotype of senders, we calculate the probability of accepting a desirable sender (c) and the probability of accepting an undesirable sender (e) when only one trait is used. (d) Similarly, when two traits are used, the probability of accepting a desirable sender can be calculated from the intersection of acceptance probabilities for each trait. (f) The probability of accepting undesirable senders is also obtained through the intersection of acceptance probabilities for each trait. (g–l) Each panel corresponds to (a–f), but the phenotypic dissimilarities are different in the two traits. Owing to limited space, we only compare here the results of using two traits with the results of using the trait with greatest dissimilarity. The error rate comparison for other phenotypic dissimilarities (d = (2, 0)) is shown in electronic supplementary material, figure S1. Finally, note that the rejection error is equal to (1 − probability of accepting desirable senders), whereas the acceptance error is equal to the probability of accepting undesirable senders. (Online version in colour.)

Figure 3.

Error comparisons of using one trait and two traits when phenotypic dissimilarities are fixed. Each set of three bars, from left to right represent (i) light grey—using trait 1 with the first phenotype dissimilarity value, (ii) grey—using trait 2 with the second phenotype dissimilarity value, and (iii) dark grey—using both trait 1 and trait 2. Each error bar shows the standard deviation of the error probabilities from 100 simulation repetitions. Each simulation lasted 3000 generations and we took the average error of the last 300 generations as one repetition. The arrows indicate places where using two traits has a lower error than using a single trait. Note that the potential probability range of rejection and acceptance errors is different, with rejection error probability (0, 0.3) and acceptance error probability (0,1).

We then examine the more realistic case, where both the acceptance threshold(s) and the trait values of the undesirable senders are changeable. Because the trait values of the undesirable senders can evolve, the average phenotypic dissimilarity (d) between desirable and undesirable senders is also evolvable. The distributions of trait values of desirable senders are fixed, which simplifies the model but does not lose much biological realism. For simplicity, we also set an additional non-evolvable threshold at the lower point of the desirable senders' trait distribution. Senders are only accepted when their values are above this additional threshold. This assumption is biologically realistic because receivers are unlikely to accept the senders with trait values much lower than the trait values of desirable senders. We analyse the probability of making errors through time (figure 4), the final trait distributions (figure 5) and the average error over repetitions (figure 6) under this scenario (see electronic supplementary material, table S1 for summary of parameter values of the model).

Figure 4.

Error comparisons of using one trait and two traits when both acceptance thresholds and phenotypic dissimilarities are allowed to evolve. The simulation was 500 repetitions, with the error bars showing the standard deviation of the final 50 repetitions. Each set of three bars, from left to right represent (i) light grey—using trait 1, (ii) grey—using trait 2, (iii) dark grey—using both trait 1 and trait 2. In the single trait scenario, the trait values of the two traits converge, presumably because they have the same initial conditions. The potential probability range of rejection and acceptance errors (y-axis scales) is different, with rejection error probability (0, 0.3) and acceptance error probability (0, 1).

Figure 5.

Threshold and error rate comparison between using a single trait and two traits when both acceptance thresholds and phenotypic dissimilarities are allowed to evolve. (a) The time series of the average acceptance threshold and average genotype of undesirable senders when a single trait is used. Acceptance threshold is shown by the blue curve. (b) The time series for using two traits, where acceptance thresholds are presented by the solid orange curves. (c–f) The error rate of using a single trait and two traits at generation 24. Similarly, the error rates at generation 36 (g–j) and 96 (k–n) are analysed. See figure 2 for more explanation of probabilities in each panel, and electronic supplementary material, figure S2 for more time series of the simulations. (Online version in colour.)

Figure 6.

Acceptance threshold comparison between using a single or two traits when both acceptance thresholds and phenotypic dissimilarities are allowed to evolve. Since the trait values of undesirable recipients are evolvable, we a use histogram to represent the average trait value of each simulation (500 repetitions for each panel). Note that the simulations using a single trait are data from two separate simulations (a,c), while the simulations using two traits are from the same sets of data (b,d). (Online version in colour.)

3. Results

(a). Acceptance threshold can evolve, but phenotypic traits cannot evolve

We explore how trait dissimilarity influences the evolution of acceptance thresholds when one versus two traits are used during recognition. We vary the trait dissimilarity and test how acceptance threshold(s) evolve and how error rates change. We first model the case where both traits have the same phenotypic dissimilarity (d = (1,1)) between the average phenotype of desirable and undesirable senders (figure 2a,b). When both traits are similarly informative, using two traits during recognition results in more accurate recognition than using one trait. Both rejection errors and acceptance errors are lower when two traits are used than when one trait is used (figure 2c–f and the arrows in figure 3).

Using two traits reduces recognition errors. One key reason for the increased accuracy is that if senders are to be accepted, both traits must meet the receivers' thresholds. However, the senders will be rejected if either trait 1 or trait 2 does not meet the thresholds of the receivers. When receivers use two traits during recognition, receivers move the threshold for each trait to be less restrictive. The less restrictive threshold reduces the probability that receivers will reject desirable senders (rejection errors). If receivers used a single trait, a less restrictive threshold would automatically increase acceptance errors. However, because acceptance is based on the intersection of acceptance in trait 1 and trait 2, acceptance errors are reduced despite the less restrictive threshold for each trait. Using two traits both increases the probability of accepting desirable senders and decreases the probability of accepting undesirable senders (figure 2d,f and electronic supplementary material, figure S2).

We also model the case where traits have different phenotypic dissimilarities between desirable and undesirable senders. In these simulations, one trait has relatively lower phenotypic dissimilarity (d = 0.5), while the other trait has higher phenotypic dissimilarity (d = 1.5) (figure 2g,h). Notably, using two traits does not significantly alter error rates when traits have different phenotypic dissimilarities between desirable and undesirable senders (figure 3). Instead, receivers make decisions using only the trait with higher phenotypic dissimilarity and ignore the trait with lower phenotypic dissimilarity (figure 2g,h). The trait with higher dissimilarity (d = 1.5) between desirable and undesirable senders allows receivers to more accurately discriminate between desirable and undesirable senders. Paying attention to the less informative trait (d = 0.5) does not increase recognition accuracy or influence probability of accepting undesirable senders (figure 2i–l).

(b). Acceptance threshold and phenotypic dissimilarity between desirable and undesirable senders can evolve

In this section, we allow the trait values of undesirable senders to evolve, which can alter the phenotypic dissimilarity between desirable and undesirable senders. One key result of this model is that using two traits during recognition is more effective than using a single trait because using two traits reduces acceptance errors (figure 4b). Receivers that use two traits during recognition are less likely to accept undesirable senders. However, using two traits during recognition does not change the probability of making rejection errors (figure 4a). This result differs from the fixed trait scenario where using two traits is only beneficial when the two traits provide similar information quality (figure 3, d = (1,1)). Using two traits reduces recognition errors because it is more difficult for senders to perfectly match two different traits simultaneously than a single trait.

The evolutionary dynamics of multiple traits are a key reason why using two traits for recognition decreases errors. When there are two traits that could be used for recognition, natural selection favours receivers that ignore the less informative trait (i.e. smaller trait dissimilarity, d, between desirable and undesirable senders) and pay attention to the more informative trait (larger trait dissimilarity, d, between desirable and undesirable senders). Receiver behaviour exerts a strong selective pressure on the trait values of undesirable senders to decrease phenotypic dissimilarity with desirable senders. When a trait becomes less informative to receivers, receivers will switch their attention to the more informative trait. These coevolutionary dynamics lead to trait values and acceptance thresholds that either reach a stable state (e.g. figure 6) or continue cycling (see electronic supplementary material, figure S3 for more examples of the evolutionary dynamics among trait values and acceptance thresholds). In summary, using two traits allows receivers to reduce acceptance error because receivers can use the more informative trait. If the focal trait becomes less informative, receivers can switch their attention to the other trait.

(c). Different costs of acceptance errors

Finally, we test how reducing the costs of an acceptance error influences the evolution of acceptance thresholds and error rates. In previous models, receivers have zero reproductive success if they accept an undesirable sender. In this new model, the cost of making an error is much lower (e.g. 0.25–0.75 reduction in offspring production). Notably, the use of two traits for recognition is beneficial regardless of the precise cost of errors, though the benefit of using two traits decreases as the cost of errors decreases.

Consistent with previous models, our results indicate that a higher cost of accepting undesirable senders favours more restrictive thresholds that reduce acceptance errors (figure 7a). However, a more restrictive threshold will inevitably cause higher rejection errors (figure 7b). In other words, the relative costs of making rejection and acceptance errors influence the evolution of acceptance threshold in a straightforward manner consistent with previous work [9].

Figure 7.

The errors and survival rates of senders in response to various costs when both acceptance thresholds and phenotypic dissimilarities are allowed to evolve. Each line shows the average error or survival at given cost values, where blue dashed lines represent recognition using a single trait and orange solid lines represent recognition using two traits. See electronic supplementary material, §S2 for more details and results about the model with variable costs. (Online version in colour.)

4. Discussion

Our simulation has three major results. First, using complex signals during recognition improves recognition accuracy. Receivers that use complex signals make fewer acceptance and rejection errors. For example, birds that use two traits to recognize their eggs are less likely to accidentally accept brood parasitic eggs and less likely to reject their own eggs than birds that use one trait for recognition. One key reason that complex signalling decreases recognition errors is that using multiple traits provides flexibility. When a particular trait becomes less informative (i.e. lower trait dissimilarity), receivers switch their attention to a more informative trait. Over time, the signalling system either reaches a stable state or cycles between multiple traits. Second, when two signals provide different amounts of information, receivers will rely on the more informative signal to make recognition decisions. When one signal is much more informative than the other, receivers will use the more informative signal alone and ignore the less informative signal. As a result, the particular signals used for recognition are not fixed over evolutionary time. Instead, it is more accurate to think of recognition as an evolutionary arms race where the particular signals used for recognition change as undesirable senders evolve more effective mimicry. Third, complex signals are most beneficial when making recognition errors is very costly. When recognition errors are not very costly, receivers may use a single trait for recognition.

(a). Complex signals improve recognition accuracy

The model illustrates that using two traits improves recognition accuracy because undesirable senders are less able to accurately match two traits simultaneously than to match a single trait. Using multiple traits for recognition creates more moving targets for undesirable senders to match. As a result, it is more difficult for undesirable senders to ‘catch up’ and match the phenotype of both traits. In the simulation, undesirable senders end up with one trait that accurately mimics the phenotypes of desirable senders and one trait that does not or recognition cycles between multiple traits (figure 5).

One limitation of the simulation is the populations reproduce asexually. Sexual reproduction could allow undesirable senders to more rapidly mimic complex signals; sexual reproduction allows genetic exchange between diverse lineages so beneficial alleles are more rapidly assembled into the same genome [22]. Future work that models a sexually reproducing population would be useful to test the effect of sexual reproduction on the evolution of complex recognition. Overall, our results illustrate that accurate mimicry in many traits is more difficult than accurate mimicry in a single trait. As a result, complex signals improve recognition accuracy.

Empirical work supports the prediction that signal receivers make more accurate recognition decisions when they use complex signals. Bees discriminate more accurately between rewarding and non-rewarding flower types when the flowers differ in both appearance and odour than when the flowers differ in only one sensory modality [23]. Complex signals may improve foraging by providing bees with additional sources of information. Mosquitofish identified predators more quickly and accurately when they were provided with both visual and olfactory traits than when either visual or olfactory traits were provided separately [24].

Comparative analyses also suggest that complex signals are more likely to evolve when accurate recognition is beneficial. For example, bird species that experience brood parasitism have eggs that are more variable across multiple parameters than bird species that do not experience brood parasitism. The high egg variation is thought to occur because complex signalling across multiple egg parameters reduces recognition errors by hosts [25,26]. Similarly, ant populations that commonly experience social parasitism have greater chemical trait diversity than ant populations that lack social parasitism. Recognition trait diversity may be favoured because it is more difficult for parasites to match complex recognition profiles of hosts [27]. Therefore, complex signals are more likely to arise in taxa with recognition challenges, likely because complex signals improve recognition accuracy.

(b). Receivers use more informative traits when traits differ in information content

The second major result of the simulation is that receivers pay attention to the more informative trait and ignore the less informative trait when traits convey different amounts of information. On the one hand, this result is not surprising. Of course, receivers will make decisions based on the most informative traits. On the other hand, this result highlights the flexibility of recognition systems. The signals used for recognition are not an evolutionary endpoint. Instead, senders and receivers may frequently change which signals are used for recognition based on the phenotypic dissimilarity between desirable and undesirable senders. Over time, some signals become more informative, while others become less informative. Less informative traits may be lost or they may be maintained through negative frequency selection based on the phenotypes of undesirable senders. Both genetic and non-genetic changes in multi-component recognition signals (e.g. learning, cultural transmission) play a role in evading mimics. Therefore, recognition is often a dynamic and flexible process akin to an evolutionary arms race between undesirable senders and receivers.

Given the evolutionary arms race between undesirable senders and receivers, the signals used for recognition may frequently change over evolutionary time. As a result, animals may often have ‘ghosts of signals past’, traits that were once used for recognition that no longer provide reliable information and are ignored by receivers [28]. Consistent with this hypothesis, there are many situations where receivers seem to use only a small fraction of the available information for recognition. For example, some birds identify eggs using multiple egg characteristics, while other species use only a subset of available signals [18]. Insect cuticular hydrocarbon profiles are composed of many different chemicals [5]; however, receivers only use a small fraction of the available chemical information for recognition [29]. Were the other chemicals used for recognition in the past, then ignored when they became uninformative? Comparative analyses of the signals used for recognition across multiple, closely related taxa provide a good opportunity to test the evolutionary flexibility of recognition. We predict that contexts where the fitness of undesirable senders depends on their ability to trick receivers (e.g. mimicry, host–parasite interactions) are especially likely to lead to rapid changes in recognition signals. The changes in recognition signals may be either plastic changes that occur over developmental time or longer-term evolutionary responses.

The simulation also predicts that the particular signals used for recognition may change rapidly based on their information content. Receivers may flexibly change how they use different aspects of a complex signal when the information content varies [30,31]. For example, squirrels living in areas with higher urban noise pay closer attention to the visual aspect of an auditory–visual alarm signal than squirrels from rural areas [32], perhaps because the auditory signal is less informative when it is masked by urban noise. Similarly, female jungle fowl change which male ornament they use for mate choice based on which signals are the most informative [33]. Psychological research indicates that the cognitive system changes how different stimuli are weighted based on the signal-to-noise ratio of each stimulus. For example, humans can use both visual and tactile information to assess object height. When researchers experimentally reduced the reliability of visual information, subjects increased their reliance on tactile information [34]. Therefore, animals may often change how they use complex signals to best match shifting sensory environments. Experimental work in additional contexts and taxa will be useful to test how the phenotypic dissimilarity between desirable and undesirable senders influences how animals use complex traits and how they evaluate different components of signals into their responses.

(c). Complex signals are more common when recognition errors are costly

The final key result of the simulation is that complex signals are most beneficial when recognition errors are very costly. When errors are low cost, receivers may rely on simple signals for recognition. Empirical work largely supports this prediction, as many examples of redundant signals occur in situations where recognition errors are costly [35]. Redundant signals are used during species recognition, during parent–offspring recognition and in taxa with brood parasitism [17]. For instance, Australian sea lion mothers use acoustic, olfactory and visual signals to accurately discriminate their own pup from the many other pups in the colony [36]. Birds often use multiple features to assess whether an egg has been laid by a brood parasite, including factors like egg size, egg shape, eggshell maculation, background colour and ultraviolet reflectance [17]. If a host accepts a brood parasitic egg, the host may lose an entire breeding season's reproduction. As a result, mechanisms that improve recognition accuracy, like complex signalling, are favoured.

There are relatively few examples of redundant signals in contexts where recognition errors are low cost. For example, although complex sexual signals are common, most do not convey redundant information. Instead, multiple signals are used in different contexts or convey different information to receivers [30,37]. For example, eland antelope have non-redundant sexual signals, as males signal body size via the frequency of knee clicks and age via dewlap size [38]. Models also suggest that redundant sexual signals are rarely evolutionarily stable, especially when producing or assessing signals is costly [15]. Although choosing a lower quality mate may reduce reproductive success, it is unlikely to result in the death of all offspring. Therefore, redundant sexual ornaments may be relatively uncommon because recognition errors are not sufficiently costly.

Additional experimental work will be important to test how the cost of errors influences whether animals use simple or complex signals for recognition. Creative experiments have shown that animals change their optimal acceptance thresholds as the cost of errors changes [6,7]. Our models predict that animals will change whether they use simple versus complex traits for recognition as the costs of making an error change. Specifically, they will be more likely to use complex signals for recognition when the cost of errors is high.

This model focuses on the recognition benefits of complex signals rather than the sender or receiver costs that may disfavour complex signalling. There is much evidence that perceiving, assessing and making decisions about signals is cognitively costly [39–41]. As a result, complex signals that occur across multiple sensory modalities or involve multiple signal components will involve greater cognitive costs for receivers than simple signals [13,42,43]. Further, producing and maintaining signals is costly for senders, with some signals imposing much greater costs than other signals (e.g. body size) [44,45]. Additional empirical research will be important to assess how the costs of perceiving and producing complex signals trade-off with the recognition benefits of complex signals in different contexts.

Overall, our model shows that redundant, complex signals are an evolutionarily stable way to reduce recognition errors. The simulation makes three testable predictions. First, redundant complex signals reduce recognition errors compared with simple signals. Second, redundant signals are most likely to occur when the costs of recognition errors are high. Third, when signals contain different amounts of information, receivers will pay attention to the more informative signal and ignore the less informative signal. Although empirical work generally supports these predictions, there is much room for additional experiments testing how complex signal use and recognition thresholds vary in different contexts. Thus far, our results indicate that signal detection theory provides a useful conceptual framework for understanding the evolution of complex signals.

Data accessibility

All simulated data were generated using the C language. The code used for this study is available at https://github.com/mingpapilio/Codes_ComplexSignal.

Authors' contributions

E.A.T. conceived of the study, coordinated the study and helped write the manuscript. M.L. helped write the model code and results and critically revised the manuscript. E.C.L. critically revised the manuscript. S.-F.S. helped write the model code and results, coordinated the study and critically revised the manuscript. All authors gave final approval for publication and agree to be held accountable for the work performed.

Competing interests

We declare we have no competing interests.

Funding

This material is based in part upon work supported by the National Science Foundation under grant no. IOS-1557564 (to E.A.T). S.-F.S. was supported by and Investigator Award (AS-IA-106-L01), Academia Sinica and the Ministry of Science and Technology of Taiwan (106-2621-B-001-005-MY3 and 108-2314-B-001-009-MY3).