Oikos 121: 813–822, 2012 doi: 10.1111/j.1600-0706.2011.20181.x © 2012 The Authors. Oikos © 2012 Nordic Society Oikos Subject Editor: Justin Travis. Accepted 21 October 2011
Interplay between Allee effects and collective movement in metapopulations Aina Astudillo Fernandez, Thierry Hance and Jean Louis Deneubourg A. Astudillo Fernandez (
[email protected]) and J. L. Deneubourg, Univ. Libre de Bruxelles, Campus Plaine, CP231, Boulevard du Triomphe, BE-1050 Brussels, Belgium. – T. Hance, Univ. Catholique de Louvain, Carnoy bte L7.07.04, Croix du Sud 4-5, BE-1348 Louvain-la-Neuve, Belgium.
Population growth can be positively or negatively dependent on density. Therefore, the distribution pattern of individuals in a patchy environment can greatly affect the growth of each subpopulation and thereby of the metapopulation. When population growth presents positive density-dependence (Allee effect), the distribution pattern becomes crucial, as small populations have an increased extinction risk. The way in which individuals move between patches largely determines the distribution pattern and thereby the population dynamics. Collective movement, in particular, should be expected to increase the potential number of colonisers and therefore the probability of colonising success. Here, we use mathematical modelling (differential equations and stochastic simulations) to study how collective movement can influence metapopulation dynamics when Allee effects are at stake. The models are inspired by the two-spotted spider mite, a phytophagous pest of recognised agricultural importance. This sub-social mite displays trail laying/following behaviour that can provoke collective movement. Moreover, experimental evidence suggests that it is subject to Allee effects. In the first part of this study we present a single-species population growth model incorporating Allee effects, and study its properties. In the second part, this growth model is integrated into a larger simulation model consisting of a set of interconnected patches, in which the individuals move from one patch to the other either independently or collectively. Our results show that collective movement is more advantageous than independent dispersal only when Allee effects are present and strong enough. Furthermore they provide a theoretical framework that allows the quantification of the interplay between Allee effects and collective movement.
The way in which individuals distribute themselves in a patchy environment can greatly affect their population dynamics. This is mainly due to the fact that population growth is highly dependent on local population density, which in turn depends on the distribution of individuals between patches. The behavioural strategies that determine inter-patch movement and habitat selection play an important role in shaping distribution of the individuals among connected populations (Hanski 2001). However, the interplay between such behaviour and metapopulation dynamics is poorly studied (Bowne and Bowers 2004), mainly because ethologists and ecologists often work at different time scales (Lima and Zollner 1996). The importance of dispersal behaviour is even greater for species that are subject to the Allee effect (Allee 1931, 1949, Stephens et al. 1999, Courchamp et al. 2008). The Allee effect arises when each individual benefits from the presence of conspecifics in a way that makes its own fitness increase with the population size or density (Stephens et al. 1999, Courchamp et al. 2008). Examples are numerous across living beings and range from plants suffering from lack of pollinators when at low densities (Groom 1998),
to mongooses benefiting from the improved anti-predator behaviour of large groups (Clutton-Brock et al. 1999). In population dynamics, this can translate into a demographic Allee effect, that is when the per capita growth rate of the population is positively dependent on population size (or density) (Stephens et al. 1999, Courchamp et al. 2008). A direct consequence is that the growth rate of the population can become very low at low densities, a phenomenon known as depensation (Lidicker 2010). In the case of a strong Allee effect, the growth rate can become negative beyond what is called an Allee threshold and drive the population to extinction (Stephens et al. 1999, Courchamp et al. 2008). Several studies have recognised the importance of the Allee effect in shaping population and meta population dynamics (Hanski and Gilpin 1991, 1996, Hanski and Gaggiotti 2004). This has been shown in spatially implicit models (Amarasekare 1998a, Zhou and Wang 2004) as well as spatially explicit models (Brassil 2001, Roy et al. 2008, Sato 2009). These studies show that the Allee effect influences colonisation rates because small populations do not colonise successfully, and this can have a great impact on the resulting metapopulation dynamics. 813
Collective behaviour (aggregation, collective movement) increases the cohesion of groups during dispersal, which in turn can be expected to enhance the success of colonisation. In spite of this, relatively few metapopulation studies deal with behaviour during interpatch-movements (Etienne et al. 2002, Bowne and Bowers 2004). These studies focus mainly on dispersal rates and mortality during dispersal (Zhou and Wang 2004, 2005), and the movement pattern is usually considered a diffusion or a random process (Brassil 2001, Etienne et al. 2002). Some metapopulation studies, examine the effect of social interactions during habitat choice, focusing on aggregation (Lof et al. 2009) or conspecific attraction (Ray et al. 1991, Greene 2003). However, the social interactions that take place during dispersal itself, such as collective movement have never been studied in the context of metapopulation dynamics. During collective movement, the direction chosen by each individual is influenced by the movement of the others in such a way that all or most of the individuals move in the same direction. There are many mechanisms known for collective movement (reviewed by Sumpter 2010) and the patterns that are formed are often spectacular (e.g. bird flocks, fish schools, ant trails, caterpillar processions). Here, we focus on a particular type of collective movement: trail laying/following behaviour. This mechanism is based on a simple positive feedback: individuals mark their paths with a trail and are attracted to it. They tend to choose the paths with the highest concentration of trail. By taking those paths they lay more trail on them, reinforce their attractiveness, and so on. As a result most or all of the individuals collectively choose one of the paths. This mechanism is very typical of ants that use it to indicate food sources to nest mates (Pasteels et al. 1987). Some species, such as the Argentine ant (Deneubourg et al. 1990) lay a trail systematically, even when they are just exploring. Examples of exploratory trails are also found in other arthropods: for instance, social caterpillars (Fitzgerald 1995) and two-spotted spider mites (Saito 1977, Yano 2008). This behaviour is peculiar in the sense that it can potentially lead a part of the population to an area that has previously not been evaluated. Our hypothesis is that this behaviour is closely linked with the presence of Allee effects, and that it is only advantageous for species subject to Allee effects. The objective of this paper is to test this hypothesis using a metapopulation dynamics approach. More precisely, we model the growth dynamics of a set of interconnected populations, and evaluate their success depending on the migration strategy and on the intensity of Allee effects. We begin by presenting a single-population model including Allee effects on reproduction and/or mortality (section 2). Then, we gradually expand the model and add new elements to it, drawing conclusions at every stage of the process. First we add fragmentation of the habitat and quantify the tradeoff between Allee-effects and competition (section 3). Then, we allow diffusion between the patches and see how Allee effects can shape the spatial distribution of individuals (section 3). Next, we add a limitation on resources and finally, we formulate a complete model incorporating collective movement in a multi-patch environment (section 4). This model allows us to compare the success of populations according to their migration strategy and the intensity of the Allee effect. 814
A general model for Allee effects There are several ways to model Allee effects (reviewed by Courchamp et al. 2008). The models are most often phenomenological: an Allee threshold is explicitly included as a parameter, and the equations are such that the per capita growth rate is negative below the threshold and positive beyond the threshold. Component Allee effect models, such as the one used in Berec et al. (2007) are more concrete, in that they can include Allee effects on reproduction and/or mortality. Here, we also consider the Allee effect on mortality separately from the Allee effect on reproduction. The individual reproduction rate r (number of offspring per individual per time unit) is formulated as an increasing function of the number of individuals (r′(X ) 0) and the individual mortality rate m (number of deaths per individual per time unit) as a decreasing function of the number of individuals (m′(X ) 0). The simplest functions satisfying these criteria are Eq. 1 and 2. r( X ) r a X
(1)
m( X ) m
(2)
b 1( X / s )
In Eq. 1, r is the basic birth rate (r(0)) and a is the Allee effect coefficient, that is the slope of the increase in reproduction rate according to population size. The higher the coefficient, the steeper the increase of the reproduction rate. If a 0, there is no Allee effect on reproduction. In Eq. 2, m is the minimal death rate, and b is the increase in mortality due to an Allee effect, that is the increase in mortality when the population size approaches zero. As the population increases, the mortality rate decreases to the minimal mortality rate m. Parameter s is the number of individuals for which the increase in the mortality rate is halved. When s 0, there is no Allee effect on mortality, and the death rate is constant and equal to m. Although this model contains more parameters than phenomenological models, it presents the advantage of being easier to parameterise: experiments can determine the relation between number of offspring (or survival) and population size. In order to understand how these functions affect population dynamics, let us incorporate them in a modified version of the logistic growth function Eq. 3. For simplicity, we take m 0. dX X bX ( r a X ) X 1 dt K 1 X / s
(3)
The carrying capacity K of the population is the number of individuals that the available resources can sustain. If a s 0, there are no Allee effects and the population growth is described by the classical logistic function (Verhulst 1838). There are two steady states, that is values of X (number of individuals in the population) for which the population growth is equal to zero and the population size does not vary. The steady state X 0 is unstable, and the steady state X K is stable. An initial population different from 0 will vary until it reaches the carrying capacity K. However, with Allee effects (a 0 and/or s 0), the population dynamics are different. For some parameter
values there can be three steady states (Fig. 1a). Two of the three steady states are stable, and one is unstable. The two stable steady states (X 0 and X Xs) are separated by the unstable steady state Xu. This unstable steady sate acts as a threshold: below this critical population size, the population decreases until it reaches X 0. Beyond the threshold, the population increases until it reaches X Xs (Fig. 1b). This threshold is characteristic of strong Allee effects and is referred to as the ‘Allee threshold’. This bistability zone exists whether the Allee effect is present in reproduction, in mortality or in both. From now on, we will only consider the case of an Allee effect on mortality. However, our results (including those that follow) are also true for Allee effects on reproduction, at least qualitatively. Allee effects and spatial distribution If the growth rate of a population is dependent on its population size, the growth rate of a system of populations (a metapopulation) is dependent on each local population size, or more generally on the distribution of the individuals between different patches (Hanski and Gilpin 1996). This in turn depends among others on three features: the number of patches, the total population size, and the distribution pattern (homogeneous or heterogeneous) (Andrewartha
and Birch 1954, Brown 1984). In this section we will focus on the influence of the two first features (number of patches and population size), with a fixed distribution pattern. The ideal free distribution theory (IFD) predicts that individuals of the same species competing for resources should distribute themselves homogeneously among equal patches (Fretwell and Lucas 1969, Fretwell 1972). Obviously, this provides the best resource exploitation pattern when competition rules the population dynamics. Whether this is also true for species submitted to Allee effects is questionable. Indeed, in those species there is a tradeoff between having access to more resources by spreading into several patches, and maintaining large groups to benefit from the presence of conspecifics. This tradeoff can be quantified with our model. Let the habitat of population X be fragmented into patches of equal size and quality. Let n be the number of patches that are occupied by population X. The ideal free distribution predicts that the subpopulations that occupy the n patches are of equal size (x1 x2 ...xn). The growth dynamics of each subpopulation is given by Eq. 4. If the initial sizes of each population are identical, the total population is given by Eq. 5. dxi x b xi r xi 1 i K dt 1 xi / s
(4)
dX X bX r X 1 nK 1 X / n s dt
(5)
The tradeoff is perceptible in Eq. 5: the total carrying capacity (nK ) is proportional to the number of patches (the more patches, the larger the population at the steady state), and the number of individuals for which the mortality is halved (ns) is also proportional to n (the more patches, the higher the mortality thus the slower the population growth). For a given total population divided into n subpopulations of equal size, there is an optimal number of subpopulations such that the speed at which the growth rate increases is at its highest (Eq. 6). n
Figure 1. Bifurcation diagram for Allee effects on mortality (a). The bifurcation diagram is exactly the same in the case of Allee effects on reproduction, except that the parameters change: s is replaced by r/a. In the parameter zone corresponding to r b (K s)2/4sK, there are two stable steady states (solid lines), separated by an unstable steady state (dashed line). This unstable steady state acts as a threshold (b) below which the population decreases to extinction.
X s( K b / sr 1)
(6)
In theory, the simple process of moving from the more crowded patches to the less crowded patches (diffusion) should result in IFD if all the patches are of equal quality. This is intuitive for species with a logistic population growth, but not necessarily for species subject to Allee effects. To test whether diffusion leads to IFD in our model, movement from one patch to the others is incorporated to the model through a diffusion term: D is the diffusion coefficient, and n is the number of patches that are connected to the ith patch. For simplicity, we will consider the case of two populations: x b x1 dx1 r x1 1 1 D( x1 x2 ) K 1 x1 / s dt x dx2 b x2 r x2 1 2 D( x2 x1 ) K 1 x2 / s dt
(7) 815
In the case of normal logistic growth (s 0), the only stable steady state is a symmetrical distribution of the individuals between the two patches (x1 x2 K ). However, in the case of an Allee effect there can be up to nine steady states. As the bifurcation diagram shows (Fig. 2), the stable distribution can be asymmetrical if the diffusion coefficient is very low. The mechanism that underlies this asymmetrical stability is known as source sink dynamics (Amarasekare 1998b): the large population is constantly increasing; the small population is constantly decreasing because it is below the Allee threshold. The constant flow of individuals from the larger population (source) to the smaller population (sink) is equal to the rates of increase and decrease of the two populations, therefore the system is maintained in an asymmetrical state. Effect of collective movement on population dynamics The central question of this study is whether populations that move collectively are more successful than populations that spread randomly when Allee effects are at stake. So far, we have presented a way to model Allee effects, and we have shown the properties of the model in the equilibrium state. The objective of this paper is to use this model to compare the growth of populations with and without collective movement, and to do this for various intensities of the Allee effect. In this section we examine the metapopulation growth in its full dynamics rather than at the equilibrium state. More precisely, we look at the dynamics of invasion of one initial population into a fragmented environment. The theoretical set-up is a set of identical resource patches, connected to one another by dichotomic branches (Fig. 3) and therefore organised in a certain number of levels that are colonised successively. Although the aim of the present study is theoretical, we find it important to discuss a model that has the potential to be tested experimentally. Therefore, in order to maintain
a connection with a biological reality, we base this model on the lifestyle of the two-spotted spider mite. This phytophagous mite is one of the most documented agricultural pests, therefore there is enough information about it to make realistic estimations of the parameters. Furthermore, it is subject to Allee effects: they live in large groups sheltered by a dense silk web. The web is collectively spun and grants them protection against predators (McMurtry et al. 1970), bad weather conditions (Davis 1952) and pesticides (McMurtry et al. 1970). Also, it has been shown that increasing group size enhances oviposition, survival (Van Impe 1985, Le Goff et al. 2010), and silk production (Le Goff pers. comm.). Finally these spider mites lay silk systematically as they walk (Saito 1977), and this silk acts as an attractive trail (Yano 2008). This can lead in some situations to collective movement (Astudillo Fernandez et al. unpubl.). This involves some adjustments to the models presented in earlier sections. So far, the resources have been considered constant (they renew at the same rate at which they are consumed). In phytophagous pests, this condition is rarely met, as they consume the plants faster than the plants grow. Therefore, a variable representing a finite quantity of resources R is introduced in the model. Population growth
The daily individual consumption is R / (w R). Its value is one unit per day when there are infinite resources and zero when there are no resources left. Parameter w is the quantity of resources for which the individual consumption is halved. A proportion g of the resource consumption is allocated to reproduction. We use the general form of the mortality function (Eq. 2), with a minimal mortality rate (m) different from zero. The system of equations for a single population is: dX g XR b − m X 1 X / s dt wR dR XR dt wR
(8)
In the absence of an Allee effect the only condition that is required for the population to grow is that there are sufficient resources (R wm / (g – m)). Unsurprisingly, with Allee
Figure 2. Bifurcation diagram for the system of two connected populations with diffusion and Allee effects. There are four stable steady states (solid lines): the trivial steady state x1 x2 0, the top steady state x1 x2 xs that is symmetrical, and two asymmetrical steady states x1 xh and x2 xl or vice versa. The bifurcation diagram was calculated for the following parameter values: r 0.1, K 50, b 0.2, s 8.
816
Figure 3. Theoretical set-up used to evaluate the colonisation potential of populations. The initial population is introduced in the 1st level. Each population is connected to two populations of the next level by a bifurcation. In the present study, we use a setup with 6 levels.
effects a second condition is necessary: the population size has to exceed a threshold (Allee threshold): dX 0 if dt
b( w R ) X s 1 (g m)R mw
Collective movement
As the individuals exhaust the resources in a given patch, they move on to one of the two subsequent patches through a bifurcation. They can choose between the two patches either independently or collectively. Collective movement is incorporated into the model as follows: each individual perceives the choice of the preceding individuals and chooses preferentially the side that most of them selected. The choice is made according to a choice function designed to study trail following behaviour (Eq. 9) (Deneubourg et al. 1990). pL
( k X L )e ( k X R )e and pR e e ( k X L ) ( k X R ) ( k X L )e ( k X R )e
(9)
In this choice function, pL and pR are the probabilities of choosing the left and the right, respectively. XL and XR are the number of individuals that went to the left and to the right, respectively. Parameter k represents the inherent attractiveness of each patch and e gives an estimate of the system’s amplification potential. If e 0, the probabilities are both equal to 0.5 and the choice of each individual is independent from the choice of the others. If k is very large, the system also tends towards an independent choice. Low values of k and high values of e promote a collective choice, meaning that most or all of the individuals are likely to choose the same patch. Simulations
The simulations were run in a set-up with six levels (a total of 63 populations) The initial population (X(0) 15) is introduced in the top patch at t 0. The initial quantity of resources is equal for every patch (R(0) 1000). At each time step the births and deaths are calculated according to a discrete version of the growth model (Eq. 10). g X (t )R (t ) b − m X (t ) l X (t ) I w R (t ) 1 X (t ) / s X (t )R (t ) R (t ) R (t ) (10) w R (t ) X (t 1) X (t )
Birth and death rates were chosen so as to be realistic in the light of the literature on T. urticae life tables (Van Impe 1985) and in order to induce strong Allee effects (the condition of existence of an Allee threshold is m b l g). Experiments have been done on Tetranychus urticae to quantify Allee effects on this species (Le Goff et al. 2010). They reveal that the relation between fertility and group size is roughly linear. At this stage, we cannot use those experiments to estimate the model parameters for T. urticae because the group sizes tested (up to six individuals) are below the group sizes that we deal with in the simulations, and it would be too hazardous to extrapolate their results to larger groups.
For simplicity, births and deaths were calculated with deterministic equations that involved no noise. Movement, however was probabilistic. Stochasticity was needed in movement, because the number of individuals involved is low (at each time step, less than 5 individuals emigrate). At such low numbers, the probabilistic nature of behaviour can have an important effect. Furthermore, stochasticity is needed in the trail model to create collective movement: without the appearance of initial random asymmetries, it is impossible for amplification to take place. A constant proportion l of the population emigrates at each time step. We estimated the emigration coefficient (l 0.03) based on the percentage of preovipositing females (the migration stage) in the stable age distribution (Van Impe 1985). The number of immigrants that arrives from the previous patch (I) is calculated according to the choice function in Eq. 9. For each individual that leaves a population, a random number between 0 and 1 is generated and compared to the probability of taking left. If the number is lower than the probability, the emigrant chooses the left patch, otherwise it chooses the right patch. We ran the simulations in two conditions: with independent movement (the exponent of the choice function was set to zero e 0), and with collective movement (e 2, k 0.5). This was done for different intensities of the Allee effect (s 0,1,2...20).
Results The variation of the populations through time is shown in Fig. 4 for some example parameter sets. It is perceptible, at least qualitatively, that in the absence of an Allee effect, metapopulations that move independently have a more efficient growth than those that move collectively. The reason for this is visible in the time plots of the sub-populations: on one hand, independent dispersers colonise all of the available patches (for every patch there is a population peak), and as a result they have access to more resources. On the other hand, collective dispersers get trapped in their collective choice, leaving a vast quantity of resources unexploited (in each level, only one patch is exploited). Although with the given parameters collective choice takes place almost systematically, the stochasticity of the process can create exceptions. If by chance even a few individuals take the side that wasn’t collectively chosen, a new population can grow in the absence of Allee effects. This can be seen in the lower left metapopulation: in the 6th level, two patches are colonised instead of one. When Allee effects are at stake, the situation is inversed: metapopulations that display collective migration grow more successfully than those that migrate independently. Indeed, when the individuals disperse independently, the colonising groups are not large enough to overcome the Allee threshold. Collective migration on the other hand ensures that the number of immigrants is high enough for a successful colonisation. These qualitative observations support our hypothesis that collective movement is more advantageous for species subject to Allee effects. Nonetheless, a quantification of the success of populations is necessary in order to make an objective comparison between the two conditions. We define four indices of success: the persistence time of 817
Figure 4. Timeplots for four example populations. The sub-populations go extinct when the resources are exhausted, or when the death rate is too high. Some individuals, however, move on to the next level and colonise new patches giving rise to new sub-populations. Eventually, when all resources are depleted, all populations go extinct. The parameters of each population differ in the intensity of the Allee effect (s 0 or s 20) and in the type of movement (e 0 or e 2 and k 0.5). The rest of the parameters have the following values: g 0.15, w 10 6, m 0.04, b 0.195, l 0.03.
the metapopulation, the length of the growth period, the metapopulation peak, and the biopotential (total biomass produced) and sustained during the persistence time of the metapopulation (Fig. 5). As it is shown in Fig. 6, populations that move indepen dently are systematically more successful than those that move collectively when there is no Allee effect. As the intensity of the Allee effect increases, the advantage of the independent movers over the collective movers decreases and is eventually inversed for all of the indices. In Fig. 6, the parameters are chosen to be consistent with the literature on the two-spotted spider mite. However, our model is generic of phytophagous arthropods, and is robust against parameter variation. As we show in the appendix, the same result would be obtained for species with different basic parameters, such as dispersal rates (l) or maximal mortality rates (b). For two of the indices (persistence time and time of growth), the first part of the curve has a negative slope. This means that when the Allee effect is lower than 7 (approximately), the advantage of independent movers increases with the intensity of Allee effect. This is somehow unexpected, but it does not affect the general conclusion that if the Allee effect is intense, the collective movers perform better than the independent movers for all four indices. Beyond a certain intensity of the Allee effect, the two strategies perform equally bad. At that stage, the Allee effect is too strong, and the number of colonisers never overcomes the Allee threshold. 818
Discussion In this paper we assess the effect of collective movement on the success of metapopulations subjected to different degrees of Allee effects. In the first part, we model population growth with component Allee effects on mortality and/ or reproduction. In the second part, this population growth model is integrated into a multi-patch simulation model. Our results show that collective movement is an advantageous strategy only when Allee effects are strong enough. The properties of our population growth model are ana logous to those of phenomenological Allee effect models. Indeed, they all display the usual characteristics associated with the cubic form of the growth rate equation, namely the appearance of thresholds and bistability (Amarasekare 1998b, Courchamp et al. 2008), and the stability of asymmetrical steady states in a two-population system with diffusion (Gruntfest et al. 1997, Amarasekare 1998b, Gyllenberg et al. 1999). The first question addressed is the effect of fragmentation on a population submitted to Allee effects. The individuals are distributed between several patches and movement is not allowed between patches. The results show that for a fixed distribution pattern (ideal free distribution or IFD), there is an optimal relation between the total population size and the number of patches that the population is divided into. The situation considered is a simplified version of natural
Figure 5. Indices of success of a population in the theoretical set-up.
situations in which a group of a given size has to be divided into several groups. These situations can arise for example during the colonisation of a fragmented habitat. Should the group of colonisers remain together and invade one single patch, or should it spread over several patches? If the group spreads, how many subgroups should be made? This is an important decision for species that disperse in groups. Some species disperse by budding, (e.g. bees (Seeley 1996), social spiders (Vollrath 1982) or spider mites (Clotuche et al. 2011)), meaning that a certain number of individuals leave the initial population and settle together on a new site. The way in which the number of buds and their resulting size influences population dynamics can be formulated with our model. An example that takes place at a smaller scale, but that follows the same general principle is the question of grouping eggs. A female choosing where to lay her eggs might have to choose between putting all of the eggs in the same patch or distributing them into several patches. In a system of interconnected patches of equal quality, the distribution patterns are determined to a great extent by the type of migration: independent movement promotes IFD, whereas collective movement gives rise to heterogeneous distributions. The central question of this study is therefore how collective movement affects the population dynamics in a multi-patch system. Some studies recognise the importance of interpatch movement for metapopulation dynamics (Hanski 2001, Fowler 2009) but very few experimental studies quantify movement and its effect be it at the metapopulation level (Bowne and Bowers 2004) or at the metacommunity level (Logue et al. 2011). Some theoretical works measure the effect of migration rate and mortality during migration on metapopulation dynamics (Zhou and Wang 2005), more specifically on the Allee-like threshold of a metapopulation (Zhou and Wang 2004). Others even discuss the impact of different dispersal strategies. However, the different migration strategies proposed are most often independent of any social influence: diffusion processes,
Figure 6. Comparison of the two movement strategies (collective movement vs independent movement) according to the intensity of the Allee effect. The four indices of success were calculated in the two conditions, based on 100 simulations per parameter set. The ratio between the index with collective movement and the index with independent movement was calculated. The parameter values were the same as in Fig. 4, except for the intensity of the Allee effect. When the Allee effect exceeds an intensity of approximately s 8, the ratio is higher than 1, meaning that collective movement scores a higher index.
different types of random dispersal (Etienne et al. 2002) or directed movement along environmental gradients (Armsworth and Roughgarden 2005). A few recent papers include social influence on the decision to disperse (Meier et al. 2011, Fowler 2009). More precisely, individuals make decisions to avoid the negative consequences of the Allee effect. However, these studies look at the decisions to leave a population, rather than on the behaviours that take place during movement (Fowler 2009). The consequences of social influence on the direction of dispersal (once the individuals have decided to move) are poorly studied. Collective movement in particular has to our knowledge never been examined in this context. Some studies address a related question, namely the effect of conspecific attraction during settlement on metapopulations subjected to Allee effects (Greene 2003, Ray et al. 1991, Reed and Dobson 1993, Lof et al. 2009). The two mechanisms in question are essentially different: in conspecific attraction, the animals use information about the population density on a given patch to decide whether or not they will settle on that patch. In the trail following situation, the animals are exploring and make the directional decision without information about the patch they are going to find. However, both mechanisms rely on positive feedbacks, thus can give rise to similar heterogeneous distribution patterns. Unsurprisingly, the conclusions of Greene (2003) agree with ours in that the heterogeneous distribution is more advantageous with Allee effects. 819
That collective movement is more advantageous for species submitted to Allee effects might seem rather intuitive, or at least accessible through mere verbal reasoning. Nevertheless, studying this question with the help of a mathe matical model gives the quantitative conditions for which this is true. For example, it is only true for high intensities of the Allee effect. There are even ranges of parameters for which the advantage of independent dispersers over collective dispersers increase with Allee effects. The model is inspired by the lifestyle of T. urticae, and parameterised to be consistent with what we know of this species. However, as the sensitivity analysis shows (see appendix), our main finding holds for species that have other life parameters. The model is generic of phytophagous pests and is adaptable to several species displaying trail following and Allee effects, such as some ants (Deneubourg et al. 1990) and some social caterpillars (Fitzgerald 1995). Furthermore, the situation modelled in the simulations is analogous to an invasion of insect populations submitted to Allee effects, such as for example the gypsy moth (Contarini et al. 2009). The optimal distribution of individuals (from homo genous to heterogeneous) depends on the intensity of Allee effects, but evidently also on the total population size and the number of available patches. In dynamic systems, these two features change constantly and one would expect a truly adaptive strategy to be responsive to such changes. In nature, it is very likely that the actual behaviour displayed by trail following species is more sophisticated than is assumed in the trail following model that we use here (Deneubourg et al. 1990). In this model, collective behaviour is regulated only by the positive feedback due to trail laying/trail following. As a result, the outcome when there is collective movement (high values of e and low values of k) is always a heterogeneous distribution. However, for the strategy to be truly adaptive it should have a more flexible outcome (IFD or heterogeneous distribution) according to the available number of patches and/or the population size. We think that this is only possible if some negative feedback comes into play. For example, it is likely that as the trail concen tration increases, the attractiveness of the trail stops increasing at some point or even starts decreasing. It has been shown that if the perception of the trail concentration saturates at high trail concentrations, the stable outcome of the trail model can be either asymmetrical or symmetrical (Aron et al. 1989). The dispersal behaviour of T. urticae seems to generate variable outcomes. Indeed, binary choice experiments on isolated individuals have shown that it is very attracted to silk trails (Yano 2008). However, binary choice experiments on groups of mites show that the collective choice of one patch does not happen systematically and most often the outcome is a symmetric distribution of the individuals (Astudillo Fernandez et al. unpubl.). In theoretical terms, this corresponds to low values of both k and e (Astudillo Fernandez et al. unpubl.). It is possible that the same qualitative mechanism (trail following) obeys to different quantitative rules (different values of the parameters) in other conditions. For example, the attraction to silk is modulated by the state of satiety of the individuals (Le Goff 2011). Two-spotted spider mites combine trail following with other dispersal behaviours, such as individual ballooning 820
(Smitley and Kennedy 1985, 1988) and collective aerial dispersal in silk balls (Clotuche et al. 2011). Little is known about the conditions under which they display one or the other behaviour (Smitley and Kennedy 1988), but we suspect that similar behavioural rules might underlie both trail following and collective silk ball formation. Environmental differences would be the key factor determining which behaviour is favoured. This would be a parsimonious explanation of the flexibility exhibited by the spider mite’s dis persal behaviour. In this paper, we study two phenomena that operate at different time scales: collective movement and Allee effects. Through different processes, they can both provoke a heterogeneous distribution of the individuals in a homogeneous environment (collective choice of path in the first case, source–sink dynamics in the second case). Coupled, the two phenomena act in ‘cooperation’ to improve the population growth in a set of connected patches. Our hypothesis is that Allee effects are a likely ultimate cause for collective movement: moving together suggests that the benefit derived from the presence of conspecifics outweighs the cost of competition. This study provides theoretical support for this hypothesis, as well as quantitative tools that may be used along with experiments to validate this hypothesis in species displaying both Allee effects and trail following, such as the two-spotted spider mite. Acknowledgements – AAF is supported by a grant from the FRIA (Fonds pour la Formation à la Recherche dans Industrie et l’Agriculture). JLD is a research associate from the Belgian National Funds for Scientific Research. The authors are also indebted to the National Fund for Scientific Research (FNRS, Belgium) for funding this project through the Fund for Fundamental and Collective Ressearch (FRFC, convention 2.4622.06). We also thank D. Lalor for proofreading the manuscript.
References Allee, W. 1931. Animal aggregations, a study in general sociology. – Univ. of Chicago Press. Allee, W. 1949. Principles of animal ecology. – Saunders. Amarasekare, P. 1998a. Allee effects in metapopulation dynamics. – Am. Nat. 152: 298–302. Amarasekare, P. 1998b. Interactions between local dynamics and dispersal: insights from single species models. – Theor. Popul. Biol. 53: 44–59. Andrewartha, H. G. and Birch, L. C. 1954. The distribution and abundance of animals. – Univ. of Chicago Press. Armsworth, P. R. and Roughgarden, J. E. 2005. The impact of directed versus random movement on population dynamics and biodiversity patterns. – Am. Nat. 165: 449–465. Aron, S. et al. 1989. Trail-laying behaviour during exploratory recruitment in the Argentine ant, Iridomyrmex humilis (Mayr). – Biol. Behav. 14: 207–217. Berec, L. et al. 2007. Multiple Allee effects and population management. – Trends Ecol. Evol. 22: 185–191. Bowne, D. and Bowers, M. 2004. Interpatch movements in spatially structured populations: a literature review. – Landscape Ecol. 19: 1–20. Brassil, C. 2001. Mean time to extinction of a metapopulation with an Allee effect. – Ecol. Modell. 143: 9–16. Brown, J. H. 1984. On the relationship between abundance and distribution of species. – Am. Nat. 124: 255–279.
Clotuche, G. et al. 2011. The formation of collective silk balls in the spider mite Tetranychus urticae Koch. – PLoS One 6: e18854. Clutton-Brock, T. H. et al. 1999. Predation, group size and mor tality in a cooperative mongoose, Suricata suricatta. – J. Anim. Ecol. 68: 672–683. Contarini, M. et al. 2009. Mate-finding failure as an important cause of Allee effects along the leading edge of an invading insect population. – Entomol. Exp. Appl. 133: 307–314. Courchamp, F. et al. 2008. Allee effects in ecology and conservation. – Oxford Univ. Press. Davis, D. 1952. Influence of population density on Tetranychus multisetis. – J. Econ. Entomol. 45: 652–654. Deneubourg, J. et al. 1990. The self-organizing exploratory pattern of the Argentine ant. – J. Insect Behav. 3: 159–168. Etienne, R. et al. 2002. The interaction between dispersal, the Allee effect and scramble competition affects population dynamics. – Ecol. Modell. 148: 153–168. Fitzgerald, T. 1995. The tent caterpillars. – Cornell Univ. Press. Fowler, M. S. 2009. Density dependent dispersal decisions and the Allee effect. – Oikos 118: 604–614. Fretwell, S. D. 1972. Populations in a seasonal environment. – Princeton Univ. Press. Fretwell, S. and Lucas, H. 1969. On territorial behaviour and other factors influencing habitat distribution in birds. – Acta Biotheor. 19: 16–36. Greene, C. 2003. Habitat selection reduces extinction of populations subject to Allee effects. – Theor. Popul. Biol. 64: 1–10. Groom, M. 1998. Allee effects limit population viability of an annual plant. – Am. Nat. 151: 487–496. Gruntfest, Y. et al. 1997. A fragmented population in a varying environment. – J. Theor. Biol. 185: 539–547. Gyllenberg, M. et al. 1999. Allee effects can both conserve and create spatial heterogeneity in population densities. – Theor. Popul. Biol. 56: 231–242. Hanski, I. 2001. Population dynamic consequences of dispersal in local populations and in metapopulations. – In: Clobert, J. et al. (eds), Dispersal. – Oxford Univ. Press, pp. 283–298. Hanski, I. and Gilpin, M. 1991. Metapopulation dynamics: brief history and conceptual domain. – Biol. J. Linn. Soc. 42: 3–16. Hanski, I. and Gilpin, M. 1996. Metapopulation biology: ecology, genetics and evolution. – Academic Press. Hanski, I. A. and Gaggiotti, O. E. 2004. Ecology, genetics and evolution of metapopulations. – Academic Press. Le Goff, G. 2011. Benefits of aggregation in Tetranychus urticae. – PhD thesis, Univ. Catholique de Louvain, Belgium. Le Goff, G. et al. 2010. Group effect on fertility, survival and silk production in the web spinner Tetranychus urticae (Acari: Tetranychidae) during colony foundation. – Behaviour 147: 1169–1184. Lidicker, W. Z. 2010. The Allee effect: its history and future importance. – Open Ecol. J. 3: 71–82. Lima, S. L. and Zollner, P. A. 1996. Towards a behavioral ecology of ecological landscapes. – Trends Ecol. Evol. 11: 131–135. Lof, M. E. et al. 2009. Odor-mediated aggregation enhances the colonization ability of Drosophila melanogaster. – J. Theor. Biol. 258: 363–370.
Logue, J. B. et al. 2011. Empirical approaches to metacommunities: a review and comparison with theory. – Trends Ecol. Evol. 26: 482–491. McMurtry, J. A. et al. 1970. Ecology of tetranychid mites and their natural enemies: a review. I. tetranychid enemies: their biological characters and the impact of spray particles. – Hilgardia 40: 331–390. Meier, C. M. et al. 2011. Mate limitation causes sexes to coevolve towards more similar dispersal kernels. – Oikos 120: 1459–1468. Pasteels, J. et al. 1987. Self-organization mechanisms in ant societies: the example of food recruitment. – In: Pasteels, J. M. and Deneubourg, J. L. (eds), From individual to collective behaviour in social insects. Birkhäuser. Ray, C. et al. 1991. The effect of conspecific attraction on meta population dynamics. – Biol. J. Linn. Soc. 42: 123–134. Reed, J. M. and Dobson, A. P. 1993. Behavioural constraints and conservation biology: conspecific attraction and recruitment. – Trends Ecol. Evol. 8: 253–256. Roy, M. et al. 2008. Generalizing Levins metapopulation model in explicit space: models of intermediate complexity. – J. Theor. Biol. 255: 152–161. Saito, Y. 1977. Study on spinning behavior of spider mites (Acari, Tetraychidae). I. Method for quantitative evaluation of the mite webbing, and the relationship between webbing and walking. – Jpn J. Appl. Entomol. Zool. 21: 27–34. Sato, K. 2009. Allee threshold and extinction threshold for spatially explicit metapopulation dynamics with Allee effects. – Popul. Ecol. 51: 411–418. Seeley, T. D. 1996. The wisdom of the hive: the social physiology of honey bee colonies. – Harvard Univ. Press. Smitley, D. and Kennedy, G. 1985. Photo-oriented aerial dispersal behavior of Tetranychus urticae (Acari, Tetraychidae) enhances escape from the leaf surface. – Ann. Entomol. Soc. Am. 78: 609–614. Smitley, D. and Kennedy, G. 1988. Aerial dispersal of the twospotted spider mite (Tetranychus urticae) from field corn. – Exp. Appl. Acarol. 5: 33–46. Stephens, P. et al. 1999. What is the Allee effect? – Oikos 87: 185–190. Sumpter, D. 2010. Collective animal behaviour. – Princeton Univ. Press. Van Impe, G. 1985. Contribution à la conception de stratégies de contrôle de l’acarien tisserand commun, Tetranychus urticae Koch (Acari: Tetranychidae). – PhD thesis, Univ. Catholique de Louvain, Belgium. Verhulst, P. F. 1838. Notice sur la loi que la population suit dans son accroissement. – Corr. Math. Phys. 10: 113–121. Vollrath, F. 1982. Colony foundation in a social spider. – Z. Tierpsychol. 60: 313–324. Yano, S. 2008. Collective and solitary behaviors of twospotted spider mite (Acari: Tetranychidae) are induced by trail following. – Ann. Entomol. Soc. Am. 101: 247–252. Zhou, S.-R. and Wang, G. 2004. Allee-like effects in metapopu lation dynamics. – Math. Biosci. 189: 103–113. Zhou, S.-R. and Wang, G. 2005. One large, several medium, or many small? – Ecol. Modell. 191: 513–520.
821
Appendix 1 Sensibility analysis to parameter values In order to test the robustness of our model against para meter variation, we ran the simulations for different values of the parameters. Here, we show the results for the two parameters that are the most likely to affect the dynamics of invasion, namely the dispersal rate l and the maximal mortality rate b. We tested a large sample of parameter
values that were included in the range that provokes strong Allee effects (m b l g). As Fig. A1 shows, the collective movers perform better than independent movers beyond a certain value of the Allee effect, and this is true regardless of the value of the tested parameter.
Figure A1. Sensitivity analysis to two parameters. The green surfaces show the value of the index ratio (collective movement vs independent movement) according to the intensity of the Allee effect and according to the parameter in question (a) dispersal rate l and (b) maximal mortality rate b). The ratio between the index with collective movement and the index with independent movement was calculated based on 100 simulations for each parameter set. The parameter values were the same as in Fig. 4, except for the intensity of the Allee effect, and the parameter in question (l or b). When the Allee effect exceeds a certain intensity, the index ratio becomes higher than 1, and this is true for any value of l or b taken within the tested range.
822
