341CASTANHO ET AL. Biol Res 39, 2006, 341-352Biol Res 39: 341-352, 2006 BRFuzzy subset approach in coupled population dynamicsof blowflies

MARÍA J.P. CASTANHO,1 KARINE F. MAGNAGO,2 RODNEY C. BASSANEZI3 andWESLEY A.C. GODOY4

1 DEMAT, Unicentro, Guarapuava, Paraná, Brazil.2 Departamento de Matemática, Fundação Universidade Federal do Rio Grande, Rio Grande, Rio Grande doSul, Brazil.3 IMECC, Universidade Estadual de Campinas, Campinas, São Paulo, Brazil.4 Departamento de Parasitologia, Instituto de Biociências, Universidade Estadual Paulista, Botucatu,São Paulo, Brazil.

ABSTRACT

This paper is a study on the population dynamics of blowflies employing a density-dependent, non-linearmathematical model and a coupled population formalism. In this study, we investigated the coupledpopulation dynamics applying fuzzy subsets to model the population trajectory, analyzing demographicparameters such as fecundity, survival, and migration. The main results suggest different possibilities in termsof dynamic behavior produced by migration in coupled populations between distinct environments and therescue effect generated by the connection between populations. It was possible to conclude that environmentalheterogeneity can play an important role in blowfly metapopulation systems. The implications of these resultsfor population dynamics of blowflies are discussed.

Key terms: Fuzzy logic, blowflies, coupled populations, population biology.

Corresponding author: Wesley A.C. Godoy, Departamento de Parasitologia, Instituto de Biociências, Universidade EstadualPaulista, 18618-000 Botucatu - SP, Brazil, Tel.: (55-14) 3811-6239, Fax: (55-14) 3815-3744, E-mail: wgodoy@ibb.unesp.br

Received: December 22, 2005. Accepted: January 23, 2006

INTRODUCTION

A central goal in population biology is tounderstand temporal fluctuations inpopulation abundance (Berryman, 1999).Such fluctuations, however, often exhibitapparently cyclic population behavior ormore complex dynamic expressed by erraticand random oscillations (May, 1974;Cushing et al. 2003). Levels of variationmay range from small percentages toseveral orders of magnitude when analyzedwith deterministic or stochastic models oreven if directly investigated from empiricaldata (Renshaw, 1999; Cushing et al., 2003).

The description of deterministic complexdynamics in population models has receivedspecial attention in recent years because ithas indicated surprising effects on theunderstanding of several biologicalprocesses (May, 1974, 1975, 1995). High

growth rate values may lead theoreticalpopulations to cyclically or unpredictablyfluctuate (May, 1974; Godoy et al., 2001).Thus, fluctuations in population size can bechaotic, that is, apparently random but, infact, strictly deterministic. However,fluctuations in population size are extremelysensitive to initial conditions; minutedifferences in initial x values (xn) lead tovery large differences in future populationfluctuations, which makes predictionsimpossible (Edelstein-Keshet, 1988).

In addition to these conclusions aboutpopulation dynamics, Renshaw (1991)advocates full recognition that theenvironment has a spatial dimension, sinceindividual population members rarely mixhomogeneously over the territory available tothem but develop within separate subregions.Subsequent interaction between thesesubregions, whether in the form of migration

CASTANHO ET AL. Biol Res 39, 2006, 341-352342

of individuals or cross-infection of disease,can vary from purely local to involving theentire area studied (Turchin, 1998).

Despite the great importance of thesefindings, several models assume thatpopulations are homogeneous, comprised ofindividuals all interacting with one another,which would justify the use of average r-values for whole populations. Nevertheless,it is possible that many local populationsdiffer in their growth rates, since theenvironments are likely heterogeneous(Hanski, 1999; Harrison and Taylor, 1997;Godoy and Costa, 2005).

An understanding of the processes leadingto population fluctuations in ametapopulational context with environmentalheterogeneity, as well as persistence and/orextinction, is relevant to many questions inpopulation biology, such as life historyevolution, the success of colonizing species,and the management of endangered speciesand zoo populations (Hanski, 1999). Thecauses of extinction may be related to severalfactors: demographic processes, such asrandom fluctuations in birth and death ratesand sex ratio; seasonal and other changes inthe environment, including predation andcompetition; catastrophes, disease outbreaks,and genetic problems, including theaccumulation of deleterious mutations or theloss of adaptive variation (Lawton and May,1996).

Traditional approaches to mathematicalmodeling applied to population dynamicsand risks of extinction require advancedmathematical skills, principally becausepopulations tend to fluctuate as a function oftheir variability in demographic parameters(Gotelli, 1995; Godoy and Costa, 2005).This is why, when studying animal behavior,many mathematical models actually arepresented by mathematicians rather thanethologists or ecologists. But even when theresearchers cannot produce a mathematicalmodel directly, they might be able todescribe the system and its behaviorlinguistically (Barros et al., 2000). In thissense, the question would be: how can wetransform linguistic observations intomathematical components in order toanalyze dynamic behavior patterns and/orsusceptibility to local or global extinction?

Krivan and Colombo (1998) developed anon-stochastic methodology to deal with theuncertainty in models of populationdynamics, including a fuzzy approach.Although a stochastic approach led to anumber of useful results (May, 1974; Turelli,1986), there also is some criticismconcerning its appropriateness in models ofpopulation biology. This criticism is relatedmainly to the fact that particular noises aremathematical frameworks which, though areasonable model in physics and electronics,may not be a suitable description ofdisturbances in biological systems (Steele,1985; Halley, 1996).

The concepts of fuzzy set and fuzzy logicwere introduced by Zadeh (1965). Zadehwas working in the field of controlengineering. His intention, when introducingthis theory, was to deal with problemsinvolving knowledge expressed in vague,linguistic terms. Classically, a set is definedby its members. An object may be either amember or a non-member, the characteristicof a traditional crisp set. The connectedlogical proposition also may be true or false.This concept of crisp set may be extended tothe notion of a fuzzy set with theintroduction of the idea of partial truth. Anyobject may be a member of a set “to somedegree,” and a logical proposition may betrue “to some degree” (Bezdek, 1993).

Fuzzy modeling is the most effectiveapproach to transform linguistic data intomathematical formulas and vice versa.Indeed, Dubois and Prade (1998) state thatthe real power of fuzzy logic lies in itsability to combine modeling (constructing afunction that accurately mimics given data)and abstracting knowledge from the data.Fuzzy theory is also an alternative approachto study population changes, since it allowsfractional membership in multiple clusters,making many forms of biologicalorganization more realistic (Schaefer andWilson, 2002). A fuzzy set admits gradationbetween established boundaries andprovides a graphic description thatexpresses how the transition from one toanother takes place.

This paper is a comparative study aboutthe population dynamics of exotic and nativeblowflies, employing a non-linear

343CASTANHO ET AL. Biol Res 39, 2006, 341-352

mathematical model for population growth(Prout and McChesney, 1985) connected to acoupled population formalism (Roughgarden,1998) that incorporates fuzzy logic. Webelieve that a fuzzy approach relates to theempirical and philosophical foundations ofour principal concern, which can be expressedby a question. How can environmentalvariability affect the dynamic behavior incoupled populations of L. eximia, a nearticand neotropical blowfly species (Prado andGuimarães, 1982, Madeira et al., 1989) andC. albiceps, an exotic species introduced intothe Americas about thirty years ago(Guimarães et al. 1978, 1979)?

The objective of this paper is toinvestigate the coupled populationdynamics of the blowfly by applying fuzzysubsets to model population trajectory,considering different environmental qualitylevels, and by analyzing demographicparameters, such as fecundity, survival, andmigration.

MATERIAL AND METHODS

Laboratory populations of L. eximia and C.albiceps were used (Godoy et al. 2001, Silvaet al. 2003). The trials were carried out usingF2, the progeny of one generation that hadcompleted its life cycle in the laboratory.Exploitative intraspecific competition amongimmatures of L. eximia and C. albiceps,which is known to occur under naturalconditions, was established in the laboratory.The experimental setting was designed toevaluate at least five larval densities, 100,200, 400, 600, 800 larvae per vial (7.2 cm by13.8 cm), with two replicates for eachdensity. Fecundity (Table I) was measuredby counting the number of eggs per femaleand expressed as average daily egg output,based on the length of the gonotrophic cycleat 25ºC (Linhares, 1988). Survival wasestimated as the number of adults emergingper vial and as a function of the larvaldensities (Table I).

TABLE I

Mean daily fecundity and survival in larval densities

Density Survival Fecundity

n Mean n Mean sd

L. eximia

100 2 91.5 32 6.53 1.11

200 2 84 32 7.03 1.58

400 2 59 32 6.14 1.21

600 2 38 31 5.29 0.55

800 2 36 32 4.05 0.71

C. albiceps

100 2 54 25 26.46 4.13

200 2 34 54 21.02 2.96

400 2 12 29 19.24 2.97

600 2 7.2 17 15.91 3.15

800 2 6.9 22 13.63 2.87

1,000 2 2 10 8.57 2.43

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Mathematical models

The mathematical model developed by Proutand McChesney (1985), a standard functionof density-dependence, was used toinvestigate the dynamics of laboratorypopulations of L. eximia and C. albiceps byapplying a fuzzy coupled-populationapproach. The density dependent model isbased on a finite difference equation, and thepopulation dynamics is based on the numberof immatures, eggs or larvae, in succeedinggenerations, nt+1 and nt. It incorporates twodensity-dependent processes, namely, thevariation in fecundity (F) and survival (S), asa function of the density of immature forms,nt. The recursion is expressed by a non-linear finite difference equation, written as:

(1),

where F* and S* are the intercepts in theexponential regression analysis of fecundityand survival as a function of larval density(Table II). These parameters describe thetheoretical values for maximum fecundityand survival, respectively. The factor 1/2indicates that only half of the populationconsists of adult females that contribute eggsto the next generation. The values of f and sare regression coefficients that estimate theeffect of slope of fecundity and survival onthe density of immatures (Table II).

TABLE II

Parameters for the regression analysis offecundity and survival on larval densities

Lucilia eximia FECUNDITY SURVIVAL

Intercept in y 9.08 1Regression coefficient 0.01 0.0014t value 77.3 0.94r2 0.6517 0.92ANOVA 296.64 115.71

Chrysomya albiceps FECUNDITY SURVIVAL

Intercept in y 27.11 0.565Regression coefficient 1 x 10-3 3 x 10-3

t value 18.36* 5.48*

r2 68.5 75ANOVA 337* 30*

* P < 0.001

The exponential function was usedbecause it fitted the blowfly data as well asor even better than the linear and hyperbolicfunctions. In addition, linear regressionproduces larger slopes (in absolutemagnitude), which in turn, produce largereigenvalues that do not accurately describethe model dynamics at carrying capacity(Mueller 1985). Furthermore, the decrease infecundity as a function of the density ofimmatures can be viewed biologically as aPoisson process described by an exponentialfunction (Rodríguez, 1989).

The model for two coupled populationscan be written as

(2).

In this model, m is the probability that anorganism from subpopulation 1 disperses tosubpopulation 2 and vice versa, i.e., it is theprobability that an organism will migrate(Roughgarden, 1998). Therefore, (1-m) isthe probability that an organism will remainin its original patch and will not migrate toanother patch. The term nx,t is the numberof individuals in the population at time tand location x, where x is 1 or 2. Thegeometric growth rate at location x at time tis r. If m is zero, the equations describe twoseparate uncoupled populations, and if m is1/2, the two populations are completelymixed and are actually one population.Combining Prout and McChesney (1985)and coupled population equations yields

(3).

In this model, the geometric growth rate (r)was removed in order to introduce thedemographic parameters of fecundity (F)and survival (S), which are functions of thelarval density. The parameters F, S and m areconventionally known for their great densitydependence, associated with resource

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availability and environmental factors (Proutand McChesney, 1985; Godoy et al. 1997).Hence, in this study they were consideredlinguistic variables and therefore estimatedusing fuzzy rules. Figure 1 describes thecomponents of a system based on fuzzy rules(Pedrycz and Gomide 1998).

medium, and large; and environment ashostile, slightly unfavorable, and favorable.Fecundity and survival were classified aslow, moderate and high. These terms weremodeled mathematically by fuzzy sets intheir respective domains according to thefigures 2 A, B, C and D. Fecundity andsurvival were modeled with maximum andminimum values experimentally obtainedfor each species (Table 1). Migration wasmodeled according to Figure 3 at domain[0, 1].

The fuzzy rules, which incorporated aset of premises written as “if – then” can bedescribed as follows:

If the population is small and theenvironment favorable then the fecundityand survival are high and the migration low.

If the population is small and theenvironment slightly unfavorable then thefecundity is high, the survival medium, andthe migration low.

Inputs of the system are the linguisticvariables Populations and Environment.Then the linguistic terms were attributed tothem: populations were classified as small,

Figure 1: Framework of systems based onfuzzy rules.

Figure 2: A. Population, B. Environment, C. Fecundity and D. Survival for blowflies.

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If the population is small and theenvironment hostile then the fecundity ismedium, the survival low, and themigration high.

If the population is medium and theenvironment if favorable then the fecundityis high, the survival medium, and themigration low.

If the population is medium and theenvironment slightly unfavorable then thefecundity is medium, survival low, and themigration high.

If the population is medium and theenvironment is hostile then fecundity andsurvival are low and the migration high.

If the population is large and theenvironment is favorable then the fecundityis medium, the survival low, and themigration moderate.

If the population is large and theenvironment is slightly unfavorable thenthe fecundity and the survival are low andthe migration high.

If the population is large and theenvironment is hostile then the fecundityand the survival are low and the migrationhigh.

The Mamdani fuzzy inference combinesthe pertinence degrees associated with eachinput value by the minimum operator andaggregates the rules through the maximumoperator. This method expects the output

membership functions to be fuzzy sets andwas employed in order to evaluate fuzzyrules and produce output for each rule asfollows:

Rule 1: If x is A1 and y is B1, then z is C1.Rule 2: If x is A2 and y is B2, then z is C2.The method described by Figure 4 has a

fuzzy set as output. The process, whichconverts this set into a numeric value, isknown as defuzzification. Specifically, inthis study we employed the centre of areamethod or centre of gravity written as:

where R is the bold area at Figure 4.Simulations were run by using equation (3)with each iteration having the parameters F,S, and m estimated with Matlab toolboxFuzzy, Version 6.0.

RESULTS AND DISCUSSION

Unilateral migration (m2 = 0) in L. eximiafrom a hostile to a slightly unfavorableenvironment produced local extinction in n1,and n2 exhibited a positive monotonic stableequilibrium (Fig. 5A). However, bilateralmigration avoided local extinction, leaving

Figure 3: Migration of blowflies (high, low, moderate).

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the two populations with values above zero(Figs. 5B and D). When comparing figures5A and 5B, we can see that the bilateralmigration avoids local extinction. Whencomparing figures 5C and 5D, which showunilateral and bilateral migration respectivelyfrom a slightly unfavorable to a favorableenvironment, the positive influence of thebilateral migration on n1 is evident.

When simulating migration from hostileto slightly unfavorable environments for C.albiceps, similar results were observed.With unilateral migration, a local extinctionwas found, and with bilateral migration, noextinction was observed (Figs. 6 A – D).Specifically, when comparing figures 6Cand 6D, it is possible to observe the sameresult found in figures 5C and 5D. Fromfavorable to slightly unfavorableenvironments, C. albiceps exhibited a stableequilibrium for both coupled populationswhen the migration was unilateral (Fig.7A). However, with bilateral migration,both populations exhibited a two-point limitcycle and out-of-phase oscillations (Figs.

7B). Between favorable environments andunder unilateral migration, one populationexhibited a two-point limit cycle and theother, a monotonic stable equilibrium (Fig.7C). However, with bilateral migration,both populations exhibited a two-point limitcycle and out-of-phase oscillations.

The main results found in this study werethe different dynamic behaviors produced bymigration between coupled populations andthe rescue effect generated by the connectionbetween them. It is also possible to concludethat environmental heterogeneity can play animportant role in blowfly metapopulationsystems. Populations produced in differentenvironments in terms of quality may inducedistinct results with respect to populationdynamics, including out-of-phase oscillations(Fig. 7D), which can increase the chance oflocal or global persistence (Svensson, 1999;Godoy and Costa, 2005).

On a very interesting note, Rohde andRohde (2001) investigated how theconfusion of subpopulations withmetapopulations may affect the

Figure 4: Mandani model with composition Max-min.

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demonstration of complex behaviorfluctuations in population size. They showedthat fluctuations in population size are verysensitive to the number of subpopulationswith different growth rates (i.e., theheterogeneity of the metapopulation) and theinitial population size (Rohde and Rohde,2001). With these results, it is possible toconclude that the lack of unambiguousevidence for chaos in natural populations(Berryman, 1999; Turchin and Ellner, 2000)could be due to the fact that several studiessupposedly focusing on subpopulationsshould be focused on metapopulations(Rohde and Rohde, 2001).

Environmental heterogeneity frequentlyhas been studied in the context of source-sinkdynamics (Pulliam, 1988; Pulliam, 1996;

Figure 5: Recursion for twenty generations with N1,0 = 300 and N2,0 = 0 simulating unilateralmigration (A) and bilateral migration (B) from hostile to slightly hostile environments. Recursionhaving N1,0 = 200 and N2,0 = 0, with unilateral (C) and bilateral (D) migration from slightlyunfavorable to favorable environments.

Frouz and Kindlmann, 2001). A source is asubpopulation in which births exceed deathsand emigration exceeds immigration andwhich may be considered a net exporter ofindividuals (Pulliam, 1988). A sink, on theother hand, is a subpopulation in whichdeaths exceed births and immigration exceedsemigration (Pulliam, 1988). In the real world,some habitats are clearly more suitable forsurvival and/or reproduction than others(Pulliam, 1996). Hence, individuals migratingbetween habitats of different quality aresubjected to life-condition change, which canaffect their growth rates (Roughgarden,1998). Therefore migration and quality ofenvironment are important factors to preventextinction in sink populations (Pulliam, 1996;Frouz and Kindlmann, 2001).

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The effect of random environmentalvariation on population dynamics also hasbeen well documented (Goodman, 1987;Pimm, 1991; Ariño and Pimm, 1995). Apopulation in a variable environment withexchange of individuals betweensubpopulations will experience variation inboth time and space. At any given moment,each subpopulation cannot be correlatedperfectly with other subpopulations (Rantaet al., 1995). Hence, both the degree ofcorrelation with environmental variationand the dispersal pattern amongsubpopulations could affect local and globaldynamics. We observed that theenvironmental heterogeneity was capable of

Figure 6: Simulations in populations of Chrysomya albiceps. A and B: N1,0 = 300 and N2,0 = 0,with migration (m2 = 0) between hostile and slightly unfavorable environments (0.1 in population 1and 0.4 in population 2) with unilateral migration (m = 0) in A and bilateral migration in B. In Cand D N1,0 = 200 and N2,0 = 0, respectively, migration between slightly unfavorable and favorableenvironments (0.6 in population 1 and 0.7 in population 2), with unilateral migration for C andbilateral migration for D.

influencing both the persistence and thepopulation dynamic behavior. However, C.albiceps seems to have suffered moreinfluence from migration and quality of theenvironment than L. eximia, since these twofactors have produced changes of dynamicbehavior in C. albiceps but not in L. eximia.

In terms of persistence, the two speciesapparently suffered the same influence fromthe factors. The good performance of thespecies analyzed by the theoreticalapproach can be explained by at least twobiological reasons: the pattern of populationdynamics and larval behavior. Luciliaeximia is a species that exhibits a weakassociation between seasonal factors and

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abundance, since it is found practicallyevery month of the year (Moura et al.,1997; Carvalho et al., 2004). In addition,this species may avoid negative interactionswith other species as for example thepredator species C. albiceps – by visitingcarcasses in the initial stages ofdecomposition, when other blowfly speciesare not yet present (Moura et al., 1997).Chrysomya albiceps, on the other hand, is alarval intraguild predator species and,therefore, can exhibit an alternative way toobtain food resources under food scarcity(Faria et al., 1999; Rosa et al., 2004).

Approaches focusing on metapopulationand environmental heterogeneity have beenuncommon in insects, principally studies

combining theory and experimentation.Recently, Godoy and Costa (2005)investigated the effect of migration on thelength of persistence of coupled localpopulations of Tribolium in differentenvironments to analyze how spreading therisk works. They observed that highenvironmental heterogeneity was associatedwith the longest population persistence(Godoy and Costa, 2005). However, nosystematic study has been proposed tocompare the vulnerability of introduced andnative blowfly species to local and globalextinction in a metapopulation context,taking account different levels ofenvironmental heterogeneity, as proposedin this study.

Figure 7: Simulations in populations of Chrysomya albiceps. A and B with N1,0 = 600 and N2,0 =200 respectively, favorable environmental in population 1 (0.9) and slightly unfavorable inpopulation 2 (0.5), having unilateral migration for A and bilateral migration for B. For C and D N1,0

= 300 and N2,0 = 250 respectively, with favorable environments (1 for population 1 and 0.95 forpopulation 2), having unilateral migration in C and bilateral migration in D.

351CASTANHO ET AL. Biol Res 39, 2006, 341-352

We have introduced the idea of usingfuzzy logic to investigate the populationdynamics of blowflies. Fuzzy logiccontrollers are commonly used to controlsystems with complex and unknowndynamics, but for expositional clarity herewe have given an example of its use withthe well-studied blowfly system (Godoy etal., 2001; Silva et al., 2003).

The fuzzy theory employed here resultedin important findings since, based onscientific intuition, it was possible toincorporate the subjectivity of thebiological information into the model. Thefuzzification of the parameters of the modelallowed fluctuation inherited from thepopulation process (Godoy et al., 2001).New studies employing the fuzzy approachin dynamics of interaction are encouragedin order to contribute for the understandingof the structure of complex systems like theblowfly community.

ACKNOWLEDGEMENTS

KFM and WACG were supported by afellowship from Conselho Nacional dePesquisa CNPq.

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