Controlling food web structure by optimization of a community assembly model

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a v a i l a b l e a t w w w. s c i e n c e d i r e c t . c o m

w w w. e l s e v i e r. c o m / l o c a t e / e c o l i n f

Controlling food web structure by optimization of a community assembly model Pascal Côté, Lael Parrott⁎ Complex Systems Laboratory, Département de Géogrpahie, Université de Montréal, C.P. 6128, Succursale Centre-Ville, Montréal, QC, Canada H3C 3J7

AR TIC LE I N FO

ABS TR ACT

Article history:

In community assembly models, species are introduced from a pool of species according to a

Received 11 October 2005

random sequence of invasion. The present work describes a new approach based on genetic

Received in revised form

algorithms (GA) which generates non-random sequences in order to maximize the diversity

9 March 2006

of the community. The GA must also meet the constraint that the food web of the

Accepted 12 March 2006

community constructed in this fashion have a specified connectance. We show that the optimized sequences produce communities with a higher diversity than those generated

Keywords:

from random assembly sequences. In addition, the GA is able to generate sequences that

Assembly models

produce food webs from identical regional species pools that have different expected

Food webs

connectances. The results demonstrate the effectiveness of genetic algorithms for

Community assembly

optimizing parameters in ecological models.

Genetic algorithms

© 2006 Elsevier B.V. All rights reserved.

Lotka–Volterra dynamics

1.

Introduction

Ecological assembly models are computer algorithms that use a set of rules or equations to represent the interaction among species in a community. The fundamental difference between this kind of model and others is that species in the community come from a pool of species, identified as the regional species pool (RSP), and are introduced according to a specific sequence of invasion. Assembly models vary from simple Lotka– Volterra models (L–V) (Drake, 1990; Case, 1990; Lockwood et al., 1997; Hewitt and Huxel, 2002) to long-term evolution (Drossel et al., 2001; McKane, 2004; Loeuille and Loreau, 2005) or agent based models (Hraber and Milne, 1997) which are for the most part generalized L–V models. In these models, the network structure of the community's food web is generally not explicitly considered, and very few studies have verified whether the assembled communities have non-random structures similar to those observed for empirical food webs (Dunne et al., 2002).

The majority of the L–V models are extended versions of Pimm's assembly model (Pimm, 1980). In Pimm's assembly model, a random selection of species is chosen from a randomly structured RSP. These selected species are introduced sequentially into the community. At each invasion, a stability analysis is computed to evaluate the consequence of this invasion, which can be either the extinction of the invasive species, the success of invasion with secondary extinction(s) or a successful invasion with no secondary extinctions. Results from Pimm's study show that from the same RSP, different sequences lead to a branching of possibilities for community assembly, emphasizing the importance of historical events on community composition (Drake, 1990; Lockwood and Samuels, 2004; but see Morton et al., 1996). Moreover, certain sequences can lead to communities that resist invasion by other species from the regional species pool (Case, 1990; Morton and Law, 1997). The degree of resistance is subject to variations depending on factors such as the invasive species' initial density and the number of

⁎ Corresponding author. E-mail address: [email protected] (L. Parrott). 1574-9541/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ecoinf.2006.03.001

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the question: “What is the most diverse community possible and how do we find it”? To address this objective, we propose a new approach that uses genetic algorithms to find assembly sequences that satisfy an optimization function for food web diversity and connectance. We hypothesize that such optimization of assembly sequences should lead to the generation of communities that are structurally different from those assembled via random sequences. Being able to manipulate invasion sequences in real ecosystems to obtain a desired community may have a number of practical applications in ecological engineering and ecological restoration projects.

different species invading the community at the same time (Hewitt and Huxel, 2002). In a recent study, Montoya and Solé (2003) combined community assembly and food web theories by investigating the food web structures of communities generated by a Lotka– Volterra assembly model. The author showed that this assembly model is able to generate communities with food webs that reproduce the basic properties of empirical data like diversity S, number of links L and connectance C = L / S2 simply by varying the probability of connection and the interaction strengths of species in the regional species pool. Montoya's study and most literature on assembly models use random sequences of invasion (Pimm, 1980; Drake, 1990; Case, 1990; Hewitt and Huxel, 2002). The reason why random sequences are used is probably because the number of possible assembly sequences one can generate is extremely large for a high diversity regional species pool. However, random sequences represent only a small subset of the space composed of all possible sequences. Hence they do not constitute an appropriate and efficient tool to explore the entire parameter search space. Thus, the main objective of this paper is to develop a method of searching the space of all possibilities in order to find assembly sequences that generate communities having certain desired characteristics. For example, since we know that for a given regional species pool many different communities are possible, we might pose

Method

2.1.

Assembly model

We use the assembly model of Montoya and Solé (2003), slightly modified to include a carrying capacity for basal species. This modification is done in order to avoid infinite exponential growth when only one single basal species is present in the community. The assembly model proceeds first by the selection of species from a regional species pool (RSP) composed of basal, intermediate and top species. Next, the species are introduced into the community (Y) according to a

Genetic Algorithm Population : Assembly sequences

Regional Species Pool size = 3 N b 1, b 2 , …, b N Random assignment

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Apply Lotka-Volterra assembly model

b i: Basal species m i : Intermediate species t i : Top species

Apply crossover and mutation operators on selected communities to generate next population of assembly sequences

t1 , t2 , …, t N

Communities t7

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Fig. 1 – A solution of the genetic algorithm represents an assembly sequence where the species are selected from a regional species pool. The solution's fitness is based on the diversity and connectance of the community as created by the Lotka– Volterra assembly model (Eqs. (1) and (2)).

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predetermined assembly sequence. The community's dynamics is given by the set of Lotka–Volterra equations: 0 1 Y X dxi @ ¼ xi ð1−rxi Þbi þ aij xj A; dt j¼1

ð1Þ

to an assembly sequence of species from the regional species pool (Fig. 1). The equation of fitness for a solution xk is given by: f ðxk Þ ¼ S−gk

 gk ¼

1

Y X dxi ¼ xi @bi þ aij xj A; dt j¼1

ð2Þ

for intermediate and top species. The equations' parameters are defined as follows: xi is the density of species i, αij the interaction strength of species i on species j, βi N 0 the growth rate of basal species, βi b 0 is the mortality rate of other species and finally r is the carrying capacity of the environment. A basal species i is given by αij V 0 ∀ j and a top species i is given by αij z 0 ∀ j. An intermediate species i has αij N 0 when it feeds on species j while it has αij b 0 when it is preyed upon by species j. To include omnivores, αij may be set positive or negative when both i and j are intermediate species and also top species may eat basal species. The matrix of interaction strengths αij must be anti-symmetric (i.e αij = −αij).

2.2.

Regional species pool

For the purposes of this study, we fixed the number of basal, intermediate and top species to be of equal proportions in the regional species pool. The probability of connection pc of species i to j in the RSP is fixed and identical for all species. Interaction strengths were derived from a uniform distribution on the interval [0; αmax] for top species, [− αmax; 0] for basal species and [−αmax; αmax] for intermediate species. The rate of increase (decrease) βi is also chosen from a uniform distribution varying from ]0;1] for basal species and [−1;0[ for other types of species. At each invasion a species i is introduced in Y from the RSP with a density di = 1 implying that xi = 1 at t0. The system of Eqs. (1) and (2) is integrated from t0 = 1 to tf = 100 with a step of Δt = 0.05 which completes the invasion of a species. When the system is integrated, any species is considered extinct and is removed from the community if its density falls below a threshold set at the initial value of di. The assembly sequence is set as explained in the next section.

2.3.

ð3Þ

were S is the diversity of the communtiy Y and

for basal species and: 0

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Optimization technique for assembly sequences

The objective when generating an assembly sequence is to order species invasion in a manner which allows us to control the structure of the final community. The task at hand becomes an optimization problem: the objective function is to maximize the diversity of the community generated by the assembly sequence, with the additional constraint that the connectance of the community's food web fall within a specified interval [cmin; cmax]. To deal with this additional constraint we simply add a penalty cost in the objective function. This constraint on community connectance is included to assure a realistic structure to avoid creating communities that have high diversity but are only composed of basal species. Thus, for this problem a solution corresponds

ejC−Cg j 0

if jC−Cg jNDC ; otherwise:

ð4Þ

C is the connectance of the community, Cg the expected connectance, ΔC the tolerance on the expected connectance and ϵ is a penalty cost. Connectance is defined as the number of realized links divided by the total of possible links. So, as we exclude carnivore and consider webs to be directed (a link is the feeding relation like species A eat species B), the connectance is given by C = L / (S(S − 1) / 2), where L is the number of links. We use a genetic algorithm (GA) to optimize the assembly sequence. GAs are stochastic optimization methods using the so-called genetic operators (Goldberg, 1989). They belong to the family of population-based evolutionary computation strategies. In the field of ecology, evolutionary computation methods such as GA have been applied in ecological modeling (Hraber and Milne, 1997; Downing, 1997), assessment of ecological models (Reynold and Ford, 1999) and modeling time series (Whigham and Recknagel, 2001) (see Recknagel (2003) for an exhaustive review). In the present study, the population to which the GA is applied is a set of candidate solutions to the optimization problem. The candidate solution Xi is a 1 × l vector. The length of the assembly sequence l is user defined. A series of genetic operators is applied to the population resulting in a new set of candidate solutions. These solutions constitute an improved approximation of the optimization problem. The fitness of a solution is computed by the application of the assembly model to the sequence (Fig. 1) and the subsequent calculation of the diversity and connectance of the resulting community. The GA we designed uses the binary tournament procedure (Goldberg and Deb, 1991) for selecting the best candidate, a uniform crossover of probability pcx = 0.5 and a random mutation of probability pm = 1 / l (l=length of a chromosome, thus the length of an assembly sequence). The binary tournament procedure begins with a random selection of two candidate solutions from the current population by comparing their fitness. The solution with the best fitness value wins the tournament. To apply the crossover operator we need to execute the binary tournament twice to get another solution. Crossover is applied to the selected solutions to generate a new candidate solution. The elements of the new solution yi are chosen from their parents according to:  yi ¼

x1i x2i

if sNpcx ; otherwise;

ð5Þ

where τ is a uniformly distributed random number from [0;1]. Binary tournament and crossover operators are applied until the number of new candidate solutions is equal to the population size. Then the population is replaced by the set of new candidate solutions. This completes an iteration of the

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GA procedure. We iterate the GA until we reach a maximum number of iterations Tmax = 500. We ran two series of simulations. In the first series we investigate the effect of the regional species pool size and the length of the assembly sequence on the performance of the GA in meeting the objective function (i.e., maximize diversity for a fixed connectance value). To do so, we varied the RSP size from 180 to 540 by steps of 60 and the length of sequences from 50 to 200 by steps of 50. This led to 42 different sets of parameter values. In the second series of simulations we evaluated the performance of the GA in reaching feasible solutions i.e., food webs with the expected connectance. This series of simulations is composed of 6 sets of parameter values in which the expected connectance is varied from 0.005 to 0.08 by steps of 0.015. To deal with the random variability of the GA we ran 30 simulations for each set of parameter values.

3.

Results

The main motivation of this study was to investigate if assembly sequences could be rearranged in order for the community's structure to be different from the ones obtained through random sequences. The first series of simulations gives a clear answer to this question: The initial population of the GA, which is composed of random assembly sequences, gives rise to communities of lower diversity compared to the final optimized sequences (Fig. 2). Thus, it is possible to order invasion sequences to enhance diversity in the resulting community. The optimization phase is definitely crucial to the assembly process since for the all of the simulations run with random sequences (initial solution) none generated as much diversity as the final optimized solutions. Fig. 3 gives an example of a community obtained from a random sequence and an optimized sequence. Basal species

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Fitness (f) of the best solution

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Fig. 2 – Evolution of the fitness of the best solution. Population size = 40 and length of sequences l = 200. Graph shows the mean of 30 runs. Error bar is 1 standard deviation.

Fig. 3 – Example of communities obtained from random and optimized sequences. These communities were generated from a sequence of length l = 100. The random sequence is randomly selected from the initial population and the optimized sequence is the best-evolved solution.

are able to survive in isolated environments which explains the observation of disconnected communities. Both webs were generated using the same connectance. Therefore, the difference in complexity depends only on their respective diversity. The number of links is proportional to the number of species. Other than diversity, we do not find any other significant differences in community properties of food webs generated from random and optimized sequences. In fact the only distinction between random and optimized assembly processes resides in the composition of the sequences. We discuss the ecological implications of this result in the discussion. Before employing the GA to construct the community Y for different expected connectances, in the first series of simulations, we investigated the relationship between the size of the regional species pool RSP, the length of the assembly sequence and thediversityof Y,keepingthe expected connectance fixed at 0.05andtheconnectanceoftheRSPfixedat0.1.Fig.4indicatesthat the diversity of Y increases with the size of the RSP. This result is not surprising since the number of possible interacting species increases as thesizeof RSPincreases.The lengthof theassembly sequence also plays an important role. For identical RSP sizes, different sequence lengths lead to different diversity. On the other hand, beyond a certain length, the diversity of Y reaches a

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Number of species in the best solution

75 70 65

community. In the second series of simulations, using identical RSPs with l = 200, we vary the expected connectance C to examine the range under which the GA is able to generate feasible solutions (community assembly with the expected connectance range in Δc). Under these parameters, the GA performs well in reaching the desired connectance (Fig. 5). When varying the expected connectance from 0.02 to 0.08 only one of the simulation runs was unable to create a feasible solution. However when the connectance is too low (c b 0.005) no matter how we tune the GA parameters, no feasible solution can be found. Difficulties in reaching low connectance come from the RSP's connectance value which is probably too high to reproduce the expected value of connectance of the assembly community.

rsp=180 rsp=240 rsp=300 rsp=360 rsp=420 rsp=480 rsp=540

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4.

Fig. 4 – Diversity of the community obtained with the best solution in the GA population for different lengths of assembly sequences and different regional species pool sizes. Population size = 40. The graph shows the mean for 30 simulations.

Discussion

We propose a new approach to study community assembly. Rather than working on “high-level” parameters such as density of species at invasion, number of introduced species at each time step or RSP parameters, we exploit the use of a genetic algorithm to produce assembly sequences that maximize the diversity of the assembly community. Results show that it is indeed possible to order sequences of invasion in order to control basic food web properties. This outcome significantly highlights the weaknesses of traditional assembly models based on random sequences. The search space composed of all possible assembly sequences is undoubtedly extremely large. Consequently, using random sequences

maximumvaluewhichismuchlowerthanthesizeoftheRSP.This illustrates the fact that not all species can coexist in a random regional species pool. The second key part of this study was to verify the GA's ability to control the connectance of the final assembly

Expected C = 0.005 Expected C = 0.020 Expected C = 0.035 Expected C= 0.050 Expected C= 0.065 Expected C= 0.080

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Fig. 5 – Diversity of the community obtained with the best solution in the GA population for different expected connectance values. Horizontal lines present the tolerance Δc = 0.005 on the expected connectance. Population size = 40, l = 200, αmax = 0.1. We set the penalty cost ϵ = 100 which provides good results for these parameter ranges. We run 30 simulations for each setting with the same regional species pool of size = 540 and connectance = 0.1. Each marker presents the best solution of a GA run. The relationship between C and S concurs with the scale-variant power–law relation S–Cγ with γ ≈ 0.5 (dashed curve) (Montoya and Solé, 2003).

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51–100

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Expected C = 0.01 Top 0.33 Inter. 0.27 Basal 0.40

0.31 0.30 0.39

0.24 0.31 0.45

0.11 0.24 0.65

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Expected C = 0.1 Top 0.32 Inter. 0.37 Basal 0.31

0.27 0.50 0.23

0.32 0.30 0.39

0.26 0.37 0.37

0.33 0.39 0.28

leaves unexploited vast regions of the sequence's space and most likely the ones able to generate communities having desired and realistic food web properties. The use of an optimization technique also provides an interesting approach to analyze the structure of these assembly sequences. When the GA has to search for low connectance food webs it inserts in the sequence a much greater proportion of basal species than it does when generating highly connected communities (Table 1). Conversely when the GA optimizes sequences to provide high connectance food webs it adds greater intermediate and top species in the sequence. As their position in the food web logically suggests, intermediate species are the most connected ones of the RSP, thus, incorporating more intermediate species increases the food

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web's connectance. A higher proportion of intermediate species on the other hand seems to have a direct negative effect on the community's diversity (Fig. 5). This outcome suggests that the presence of basal species is essential to ensure the diversity of this ecosystem model. An additional appealing consequence of the optimized assembly model put forward in this paper is the decrease in the proportion of top species further along the invasion sequence. From the simulations executed with low expected connectance, the resulting assembly sequences clearly lacked top species. Empirical data indicate that the proportion of top predators in food webs is low compared to other species types (Dunne et al., 2002). This reduced proportion can be explained by the invasion process carried out by top predators. At the beginning of the assembly process, top predators can successfully invade the community but as this process goes on it becomes more difficult for other top predators to get established in the highly competitive community. Therefore as the GA's goal is to maximize the diversity of the community it only includes a small quantity of top predators in the final sequence (Fig. 6D). As a result, the proportion of basal species increases as the GA optimizes the assembly sequences (Fig. 6C). On the other hand it was interesting to note that the proportion of basal species in the community neither significantly decreases nor increases as the optimization proceeds (Fig. 6A). Thus, some basal species could be considered as “nexus” species, i.e. species that play a significant role in the composition of the final community but go extinct before the end of the assembly process (Drake et al., 1997). This result emphasizes the importance of history in the communities' composition.

0.2 500

% top species (assembly sequence)

Table 1 – Values in the table represent the proportion of a given type of species (basal, intermediate and top) which is present in the first, second, third or the fourth quarter of sequences of length l = 200

Fig. 6 – We report on this figure the diversity of the community of the best solution of the GA for each iteration (bold curves) and A) the proportion of basal species in the community, B) the proportion of top species in the community, C) the proportion of basal species in the assembly sequences, D) the proportion of top species in the assembly sequences. We use the same parameters as Fig. 5 except for the size of the regional species pool which is set at 300.

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Assembly models have recently received increased interest in restoration ecology (Temperton et al., 2004). The influence of historical events is reported in most literature on restoration and ecologists are now well aware that they “must recreate elements of assembly history to restore an ecosystem” (Lockwood and Samuels, 2004). The study of assembly sequences provides a promising avenue to investigate the impact of historical events in recreating ecological communities. The use of random sequences however does not offer any control on the properties of the communities that they generate. In contrast, optimized sequences present an innovative alternative able to generate communities of enhanced diversity and expected connectance. Moreover, from a practical point of view, if one is to restore a community, dealing with structured assembly sequences is certainly more tractable than operating on hard to manage parameters of the regional species pool like the species' interaction strengths. We exploit the use of a novel technique based on GA to modify sequences and generate communities with diversities far greater than those generated via random sequences. While is it obvious that communities in the wild do not assemble with any optimization purpose – nor do they assemble in a random process – the proposed method brings insights on how to manipulate an assembly process to obtain a community with particular food web properties. Therefore the procedure provides an efficient way to generate the structure of food webs with a simple dynamic model instead of with a static model (Cohen and Newman, 1990; Williams and Martinez, 2000). We are aware that L–V assembly models are far too simple to recreate the dynamics of empirical food webs; on the other hand, the suggested optimization method is a first step in demonstrating the significance of using nonrandom sequences in assembly models. Finally, one should note that whereas connectance is a meaningful food web property, other characteristics play as important a role in determining food web structure and should be incorporated in further studies. The proportion of basal, intermediate and top species as well as the proportion of omnivores could constitute additional control conditions on this optimized assembly model. The ability of the proposed GA-based assembly model in generating ecological communities of enhanced diversity is a positive indication that optimization techniques are powerful tools to study and rethink community assembly models.

Acknowledgements Simulations were run on equipment provided by the Réseau Québecois de Calcul de Haute Performance (RQCHP). The authors wish to thank Carol Gauthier and Michel Barrette, system analysts from the CCS (Center of Computational Science), Universtié de Sherbrooke, Montreal, Canada, for their technical support for the MPI C++ simulations made on the mammouth-series cluster. The authors also thank Élise Filotas for helpful discussions and comments on the draft of the paper. Financial support for this project was provided by the Canadian Foundation for Innovation and les Fonds québecois de la recherche sur la nature et les technologies.

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