A Simulation Approach to Understanding The Effect of Mimicry on Prey’s Flourishing When Predators Decline Due to Environmental Disturbance

Hongchun Qu,Kaidi Zou,Dandan Zhong,Li Yin,Xiaoming Tang

College of Automation,Chongqing University of Posts and Telecommunications,Chongqing 400065,China

Keywords Environmental disturbance Simulation-based modelling Batesian mimicry Indirect interactions Predator-prey model

Abstract Ecological interactions and their consequences to system evolution in the context of environmental disturbance are of special concern in ecological conservation since the environmental conditions have been changing so quickly in the past decades.Understanding how these interactions,particularly the indirect ones such as mimicry,could change prey variabilities in facing of predator loss is an interesting question.In this research,we incorporated Batesian mimicry into a three-species predator prey system to investigate the role of mimicry on regulating prey abundance when the system is suffering predator loss in various patterns.The Netlogo mimicry model was adopted to run the simulation experiments.We found that the timing of predator loss interacting with mimicry can induce significant difference in mimic prey’s abundance if partial predators were removed from the system.However,the variations of frequency of predator loss did not change the mimic prey’s abundance in all conditions.Our findings suggested that indirect interactions can change the final species composition on the long term evolutionary scale if environmental disturbances occur in the particular time window. This is the first report that addresses the question of how indirect interactions such as mimicry affects species abundance when environmental disturbance occurred. We expect that this finding could shed the light on conditions under which species and ecological balance can be better managed when environmental disturbances are inevitable to come.

1 Introduction

Predation,a “top-down”force in nature rules ecosystems for millions of years(Fraser,2011).However,in contrast with photosynthesis and nutrient cycling,the importance of predation had been underestimated by ecologists for a long time until a scientific consensus was emerging that predators are critical to the formation of feedbacks that control and regulate the ecosystem(Berger et al.,2001).The reality of predator loss is far worse than what we can intuitively perceive,e.g.,not just feeling how awful to never again see such a creature.Because the disappearance of predator,extinction or even just reduction in numbers,is irreversible and likely to cause cascade disasters across food webs and ecosystems(Dunne and Williams,2009).In recent years,considerable evidences of damage to ecosystem by predator reduction have been shown(Zdilla,2010),for example,over fishing of cod in the costal North Atlantic caused uncontrolled and hyper-abundant lobster and sea urchins,which has been the primary threat to the biodiversity in the Gulf of Maine(Steneck et al.,2013).

Understanding how predator disappearance would affect the population size of its preys is critical and the primary step to protect biodiversity and avoid potential threat to ecosystems.It has been widely accepted that predator disappearance is likely to flourish preys(Zeckhauser,2017),but could different types of disappearance event make difference in the growth of the prey population?Generally,the disappearance of a species occurs gradually in nature because of the large population existing as well as the large spatial scale it can reach(Graham et al.,2000).However,due to the climate change and over development of human society in recent years,the events of rapid disappearance become more common than ever,especially in the case when the environmental conditions change more rapidly than the species can adapt(Sahney and Benton,2008).Therefore,the disappearance of predator can be varied in amount and time(Rowland,2009),e.g.,the amount of predator could decline gradually in a relative long period or suddenly decrease in a time point,which is known as gradual disappearance or rapid disappearance.In addition,predator may not only be reduced in numbers,the event of disappearance could happen any time during the long-term evolution process of the ecosystem,which means the timing of disappearance might be another potential factor that affects system dynamics.

Indirect interactions between preys are potential forces shaping preys population(Lang and Benbow,2013).For instance,mimicry,the similarity of one prey(mimic species)in appearance to another(model species)plays an important role to gain protection in facing the common predator species(Maynard-Smith and Harper,2004).The variation of mimic species as well as the traits of predators would be potential factors that change the population of mimic preys(Wickler,1968).The higher the mutation rate of the mimic prey,the more likely they shift their appearance approaching the model species,which in turn enhance the mimic population density.The perceptibility of the predator might also be important to mediate the indirect interactions between preys.When a predator has a good memory,it can hold the yucky feeling much longer after eating an unpalatable prey.Then the mimic species is well protected.Therefore,we would expect that mutation and memory are two key factors regulating mimicry that poses indirect interactions between prey populations.

The primary goal of this research,was to understand whether gradually and rapidly disappearance of predators could make difference in flourishing prey population.Then,we aimed at testing the effect of timing of predator disappearance on prey population dynamics.In addition,we were interested in whether mimicry plays an important role in regulating these effects.To bridge this gap,we used simulation approach based on a spatially explicit mimicry model(Wilensky,1997)on NetLogo simulation platform(Wilensky,1997)to test the hypotheses and discussed what we found.Current models that used to investigate dynamics in predator-prey system are mostly based on the differential equation Lotka-Volterra model(Yorke and Anderson,1973).The deficits of this type of model are(Deangelis and Mooij,2005):1)they works on population level and provide no individual level traits,which is the major obstacle to model individual specific behaviour such as perception,memory and learning;2)they are very difficult to present spatial heterogeneities,which are very common in nature and probably the potential factors that affect model results;and 3)the more species in the model,the higher the complexity in modelling and analysing.These are primary reasons that we chose the spatially explicit model,which is easy to implement and able to incorporate stochastic and individualized behaviour(Devaurs and Gras,2010;Belvisi and Venturino,2013).

Although the research questions emerged from this paper have not been directly reported around the ecological and environmental community,we do have seen some related works which are focusing either on evolution and environmental disturbance or mimicry itself.Ponge(2013)developed a theoretical model to compare the adaptability of two groups of organisms under environmental changing pressure.Long term evolutionary divergence in adaptively of the two groups has been clearly shown.One group with better energic optimization ability tends to perform well in stable environment,whereas the other group without energic optimization strategy(i.e.,using energy in foraging,growth and reproduction)adapted well in unpredictable environment.A theoretical model proposed by Kokko et al.(2010)showed interesting result that alternative prey can change model mimic dynamics between parasitism and mutualism.This model revealed that both mutualistic and par-asitic relationship between model and mimic are possible and the availability of alternative prey can easily alter this relationship.In order to understand the condition for the persistence of mimicry,Seno and Kohno(2012)suggested a mathematical model of population dynamics for Batesian mimicry system.The model introduced a new concept of searching image based on predators’experience.They found that mimicry persistence mainly depends on predators’memory instead of the population size of the mimic species.These models are important to investigation of mimicry and evolution.However,they did not directly address the scientific questions that we are going to explore in this research.

In this paper,we firstly introduced the three-species predator-prey model with mimicry and how it works in a spatially explicit manner.Then,we described the simulation experiments and parameterization.Finally,the simulation results were analysed,and findings were discussed with conclusion.

2 The model

The model we used in our research is the famous mimicry simulation model developed by Wilensky(1997)on NetLogo simulation platform.We chose the NetLogo mimicry model not only because it is totally free and of its spatial explicit features,but also based on the following reasons.First,the NetLogo mimicry model established an excellent evolutionary framework under which all elements of mimicry such as individual appearance,memory and learning can be parameterized,and the system dynamics can be readily observed over time.This facilitation could be very helpful to focus our efforts on modelling logic instead of coding.Second,the NetLogo mimicry model has been widely accepted and cited by the ecological and environmental research community.The firmly grounded model could significantly reduce the uncertainty and bias in building a model from scratch.Third,all models in NetLogo simulation platform provides source code that allows us thoroughly understanding the model logic,from which we can adapt the model to our researches with necessary revisions.

The model contains one predator species,birds,and two prey species,monarchs and viceroys which are respectively butter flies and moths.Monarchs are fed with toxic milkweed thus they taste yucky and are inedible to birds.However,viceroys are harmless and palatable to birds.Technically,monarchs and viceroys are unrelated species but share with highly similar appearance probably due to long term evolution.This the so-called Batesian mimicry in which the palatable species(the mimic)take advantage of the visual similarity with the toxic species(the model)to be protected from being eaten by predators.

The model runs in a discrete time manner,i.e.,each individual does its action in each simulation step,or tick(Fig.1 A and B).At the beginning of each model run,monarchs and viceroys have different colour and thus are visually distinguishable to birds.During model runs,the three species,birds,monarchs and viceroys fly randomly in a two-dimensional world.Monarchs and viceroys are preyed by birds when they encounter.When a bird eats a monarch,it memorizes the monarch’s colour as “yucky”and avoid capturing preys in the same colour in the next several steps until its memory decays to null.A bird can remember up to three yucky colours and the previously memorized yucky colour is replaced by the newly tested yucky colour.Monarchs and viceroys regenerate through asexual reproduction to compensate the loss of population.The chance of reproduction depends on their population size.The closer their population size to the carrying capacity that is 225 individuals for each prey species,the lower the chance to reproduce.In addition,a simply random test(4%)is applied before the reproduction can be made.Once a monarch or viceroy is reproducing,the offspring is either an identical copy of the parent or having a mutation rate to be a mutant which is the same species as the parent but has a random colour between the model and mimic species(i.e.,from 15 to 105).Both monarchs and viceroys have equal chance to produce a mutant.

Fig.1 Screen shot of the NetLogo mimicry simulation model and behavior of predator(A)and prey(B)individuals.

The model has three critical parameters to be manipulated in this research to test our hypotheses.Memory duration defines the time interval that a bird can memorize a yucky colour in relation to its experience of eating a monarch.The larger the memory retention,the longer the time in which the bird avoids capturing preys of memorized yucky colour.Mutation rate controls the probability of generating a mutant offspring.No mutation means no colour overlapping between model and mimic species,thus all preys are identifiable by birds.High mutation rate leads to confusion of prey identification,but the flip side is that it introduces too much yucky colours to be memorized on the other hand.Thus,there still are considerable number of viceroys could be eaten,which means the efficiency of protection due to mimicry would be discounted.The proper combination of memory duration and mutation rate is vital to achieve Batesian mimicry in the three-species predator-prey system.In addition,there are two environmental relevant parameters,the timing of predator removal event and the frequency of each removal.The manipulation of the two parameters models the situation in which the predator-prey system is disturbed by the removal of a certain number of predators at a specific time point.

3 Simulation experiments

Before we conducted the simulation experiments that are environmental disturbance relevant,we did some pilot simulations based on the predator-prey model to thoroughly explore how the two factors,i.e.,memory duration and mutation rate,affect the mimicry between the two prey species.In these pilot simulations,we increased the preys’mutation rate and predators’memory duration gradually from 0.0 to 1.0 and from 0 to 40,respectively.Since the two factors represent traits of the different species that are respectively the prey and the predatorspecies,we assumed that the two factors are independent to each other.Therefore,when the one factor was increasing,the other was kept as constant.From these pilot simulation experiments,we could clarify in which situation the indirect interactions(mimicry)are important to protect the Viceroy,and to find when the system would go stable by observing the time series charts.

Table 1 Model parameters.

In order to simulate the effects of environmental disturbance that causes predator loss,we removed a certain number of birds from the predator-prey system during simulation.Since the environmental disturbance could vary in both magnitude and time,the different types of removal events were applied during simulation.We defined the types of removal as rapid removal and gradual removal.In rapid removal,the number of birds was cut down from 75 to 10 at one time,i.e.,65 birds disappeared suddenly.However,in gradual removal,the 65 birds were separately removed in 5 batches with a time interval of 200 ticks,in each of which only one- fifth of(13)birds were removed from the system.The timing of removal was defined as the time point when the removal events occurred.We assumed that there might be various direct and indirect interactions among predator and prey populations when the predator-prey system evolves,therefore the disturbance that occurred before and after those interactions becoming well situated could have different impact on the system dynamics.We let the removal events occurring at the time point of 200 and 1000 respectively,to model disturbance striking the unstable and stable predator-prey system.We therefore defined the disturbance combination of removal timing and type as five models,i.e.,model A,B,C,D and E(Table 2),where no disturbance was applied in model A that can be regarded as a control for comparison purpose.

To investigate the effects of mimicry,i.e.,the indirect interactions among prey species,on the disturbance of predator loss,we changed the value of mutation rate of the two prey species and the memory duration of birds for all the five models.According to the value range defined in the predator-prey model,we used three levels for the two factors,i.e.,low,intermediate(mid)and high mutation rate and memory duration.For mutation rate,the value of the three levels were 0.2,0.4 and 0.8;for memory duration,the value of the three levels were 8,16 and 32 ticks of simulation steps.

Then,we designed the total 45 simulation experiments for the combination of all interested conditions in a full factorial manner,as shown in Table 2.Each simulation experiment ran 2000 ticks before ending.In order to eliminate the effects of randomness,we replicated each simulation experiment for 100 times.In each simulation experiment,we collected colour of each prey individual and the population changes over time of each prey species for analysis.

Table 2 Parameters for simulation experiments.

Fig.2 Viceroys population changes when different mutation rate(A)and memory duration(C)were applied.Phase diagrams of population dynamics of monarch vs viceroy under different mutation rates(B)and memory durations(D).Error bars represent standard errors.

4 Results

4.1 The effects of preys’mutation rate and predators’memory duration on mimicry

Batesian mimicry enhances the protection of imitating species(i.e.,viceroy species,the mimic)through the indirect interactions with the imitated species(i.e.,monarch species,the model).The population size of viceroy is therefore naturally selected as the indicator of mimicry strength between the two preys.In other words,the more viceroys in the system,the stronger the mimicry protection.Simulation results showed that higher mutation rate leads smaller viceroy population size,but the trend stop continuing after the mutation rate approximately beyond 0.4.Lower mutation rate,e.g.,between 0.1 and 0.2,can sustain the largest number of viceroys(Figure 2A).The phase diagram also showed that the increment of mutation rate causes a nonlinear declination of mimicry strength(Figure 2B).However,the elongation of memory monotonically increased the mimicry strength.The explanation is quite straightforward.Since predators only memorize negative experience,the longer the yucky colours have been held in their memory,the more avoidance of capturing preys that have the memorized yucky colours will be.This therefore increased the survival probability of the viceroy species.

Fig.3 Color distribution of the two prey species population,monarchs and viceroys at the beginning of the simulations(A)and comparisons at the end of simulations between different mutation rates that were 0.2(B),0.4(C)and 0.8(D)respectively.

The effects of mutation rate on mimicry can be explained by looking into how offspring colour changes with the variations of preys’mutation rate(Figure 3).At the beginning of simulation,monarch and viceroy were completely visual distinguishable by their colours which were respectively 15 and 105(Figure 3A).However,at the end of simulation,varying in mutation rate changed the colour distribution of the preys.In general,viceroys are more likely to be eaten by birds due to being palatable,which means that their population is replaced by their offspring more quickly than monarchs.This is the reason why we can observe that lower mutation rate maintained the large population of monarch offspring of red colour(15),but most of the viceroy population evolved from their ancestors’blue colour(105)to red colour(15,Figure 3B)to gain more protection.In this situation,most viceroys shared the same colour with monarchs to achieve the strongest mimicry strength.As mutation rate went high,the number of monarchs that remained the colour of 15 became less and less because they reproduced more mutants that are randomly coloured from 15 to 105(Figure 3C and 3D).The mimicry favourable situation where large population of prey aggregated in the same colour became rare and rare,then the number of viceroys protected by monarchs decreased.One should note that in this model,the number of yucky colour that a bird can memorize was limited to 3,which is applied in most predator-prey system.This explained the situation in Figure 3D where more paired monarchs and viceroys were in the same colour but sustained less viceroys than the situation where mutation rate was low.

4.2 The effects of timing and type of predator removal on flourishing prey population

Surprisingly,gradual removal did not make a difference in the final viceroy population compared with rapid removal,i.e.,the frequency of removal did not change the system behaviour.Comparisons between model B and D,where part of predator was removed rapidly and gradually,showed that viceroy population eventually went back to the same level no matter how much the mutation rate was applied(Figure 4A and 4C)except for the situation where mutation rate was 0.4.The same held for model C and E(Figure 4B and 4D).This consistency remained in both situations where removal was made before stable and after stable(Figure 5A and 5D).This can be explained by that the three-species predator-prey system is a linear system in some cases.It is suggested that the number of viceroys sustained by the system through the indirect interactions(mimicry)with monarchs is approximately a linear relationship with the number of predators.Therefore,once a proportion of predators was swept out from the system,a certain number of viceroys added to the system no matter how the predators were removed.This system behaviour can be observed from the phase diagram in Figure 4,where the trajectories of monarch and viceroy population stabilized on the diagonal line.

The most interesting finding was that the timing of predator removal did make a difference in viceroys’population growth(ANOVA,df=198,F=7.433,p＜0.001,Figure 6A and 6B).But this only occurred in the situation where mutation rate was intermediate(0.4)and memory duration was high(32),as shown in Figure 5B and 5E.When predator removal was done at tick 200 where the viceroy population was still going up,the average of viceroy population went to 208 in BOTH model B and D.However,when predator removal was done at tick 1000 where the viceroy population was close to the stable state,the average of viceroy population went to 203 in BOTH model C and E.

This finding can be explained by looking into the evolution process where viceroys varied their colours and sought for protection.At the beginning of simulation,most viceroys were in blue colour which was easily identified by bird,thus their population went down quickly.When some viceroys produced mutants that were in red colour and naturally protected by monarchs of the same colour,the mutant population started to growth because they had higher probability to generate red offspring.In this way,the population of viceroy mutants was overgrowth and reached a level that beyond the mimicry protection.Because the viceroy density became higher,birds had higher chance to encounter a palatable and red viceroy,which obviously destructed the memory of yucky experience contributed to mimicry.Then the population of viceroy mutants started to go down,as shown in ticks between 0 and 500 in Figure 5.If some birds were removed from this system at the stage when viceroy mutants were overgrowth,these extra number of viceroy mutants were kept in the system because parts of their natural enemies no longer existed.On the contrary,if some birds were removed from this system when viceroy population was close to stable,i.e.,the population was balanced between being hunted and protected,thus no extra viceroy mutants were produced.This is the reason to explain why the timing of predator removal made a difference in flourishing mimic population.

However,this difference was only found in a certain situation where the preys’mutation rate was intermediate(0.4)and the predators’memory duration was high(32).As we discussed earlier,the combination of low mutation rate and high memory duration can achieve the strongest mimicry strength that protect viceroy well.But results showed that higher or lower mimicry strength(i.e.,the indirect interactions between prey species)did not make difference in viceroy growth when parts of predators were removed.It suggested that in strong indirect interaction scenario where mutation rate was intermediate and memory duration was high,more viceroys were supported due to the strong mimicry strength.In other words,the balance of viceroy population between being hunted and protected has been well kept all the way during system evolution.This explanation held when mimicry strength was low.

5 Discussion and conclusion

5.1 Why individual-based simulation approach was used to under stand the indirect interactions between prey species

In this research,we took advantage of the spatially explicit individual based model to investigate the complex interactions among prey and predator individuals in the context where environmental disturbances are considered.In this three-species predator prey system,interactions not only exist between predators and preys where predation and learning occur directly but also are shown between preys in an indirect way as Batesian mimicry(Bates,1861).To analyse these complex interactions and predict what the role they play in the system subject to environmental disturbance,we need to zoom down into the individual level to model variabilities in space,behaviour and learning rather than population level(Qu et al.,2013).Although differential equation based Lotka-Volterra models are classical and typical tools to study predator prey system and its dynamics(Llibre and Valls,2007),they are difficult to specify individual level traits such as perception,memory and learning ability which are natural elements forming complex indirect interactions between individuals,species and populations.To the best of our knowledge,no Lotka-Volterra models with inbuilt mimicry evolution have been found.By linking Batesian mimicry into predator-prey system,we were able to reveal how individual behaviours can indirectly affect system evolution and dynamics in facing environmental threats.

Fig.4 Trajectories of the monarch and viceroy population with mutation rate from 0.2 to 0.8 of the four model B,C,D and E where memory durations were all set as high(32).

5.2 Why indirect interactions are important in facing the environmental disturbance

Environmental disturbance is a major threat to ecosystems and generally reshape their functions by changing species composition(McDonald et al.,2016).A typical disturbance is species disappearing and even extinction,which poses pressures to other species that they are directly or indirectly interact with.Direct interactions such as predations,are usually obvious and not difficult to observe.However,indirect interactions such as exploitation competition and defence which are mediated by third species,are not very easy to observe and quantify(Dunn,2010).In the three-species predator prey system,preys are expected to increase due to the loss of predators that are killed by disturbance,but their population change may vary due to the indirect interactions occurred between the model and mimic preys.We have observed significant difference in prey population growth when the timing of predator removal changed during mimicry evolution.

Fig.5 Time series chart of viceroy population dynamics of model A,B,C,D,E and F with different combination of mutation rate and memory duration.

This is the first report that addresses the question,which is how indirect interactions such as mimicry affects species abundance when environmental disturbance occurred.In this system,the population of viceroy,one of the two preys,is not only a function of predator abundance but also affected by the abundance of another prey species,the monarch.Viceroys tend to evolve towards the colour of monarchs to increase their survival probability.Viceroy mutants are protected by monarchs but there is a hidden balance in which the sustainable abundance of viceroy mutant must be associated with a certain number of monarchs acting as the source of yucky experience of predator birds.If the viceroy mutants are overgrowth,the excess part is destructive power to weaken mimicry.Because the more viceroys,the higher chance a bird can encounter a palatable viceroy,thus forgets the yucky experience over time.If monarchs provide more protection than the relative small population of viceroy,viceroys tend to increase to reach that balance.The system evolves towards such hidden balance but can be disturbed by environmental changes such as the loss of predators.Our findings suggested that this indirect interaction is able to change the final species composition on the long term evolutionary scale if environmental disturbances occurs in the particular time window.

5.3 What are the implications of the simulation findings to ecological conservation

Fig.6 Viceroy population(mean of the number of viceroy individuals)comparisons between model B and C(A)as well as comparisons between model D and E(B)when different combinations of mutation rate and memory duration were applied.Difference between model B and C,model D and E in intermediate mutation rate and high memory duration were statistically tested by ANOVA,df=198,F=7.433,p＜0.001(marked as***).

Interactions and their consequences in the context of environmental disturbance are of special concern in ecological conservation since the environmental conditions have been changed so quickly in the past decades(Stephen et al.,2004).The concept of Batesian mimicry fits into our three-species predator prey system because it can help investigating and explaining how and why these indirect interactions change species composition in a long term scale.Our simulation findings suggested that species can be affected not only by direct predation,but also indirect interactions mediated by the third species,which are very common in the food web(Johnstone,2002).In addition,the timing of environmental disturbance occurred is also critical in reshaping system feedbacks and functions on the evolutionary time scale.This research extended the basic predator prey model and incorporated into complex relations between prey species,which could possibly shed the light on conditions under which species and ecological balance can be better managed when environmental disturbances are inevitable to come.

Acknowledgements

Funding was received from the National Natural Science Foundation of China(61871061)and Chongqing Research Program of Basic Research and Frontier Technology(cstc2017jcyjAX0453,cstc2015jcyjA40007).