Fuzzy model of Drosophila mediopunctata population dynamics

Ecological Modelling 287 (2014) 9–15

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Fuzzy model of Drosophila mediopunctata population dynamics M.J.P. Castanho a,∗ , R.P. Mateus b , K.D. Hein a a b

Department of Mathematics, Universidade Estadual do Centro-Oeste, Guarapuava, PR, Brazil Department of Biological Sciences, Universidade Estadual do Centro-Oeste, Guarapuava, PR, Brazil

a r t i c l e

i n f o

Article history: Received 3 December 2013 Received in revised form 28 April 2014 Accepted 29 April 2014 Available online 2 June 2014 Keywords: p-Fuzzy system Population size Seasonality

a b s t r a c t Drosophila mediopunctata belongs to the tripunctata group, which is the second largest Neotropical group of the Drosophila genus, with 64 species described. This paper presents a p-fuzzy system to describe its population dynamics. An approach using fuzzy set theory was chosen to handle some problems of uncertainty in knowledge-based modeling. This model is based on population density, climatic variation and extractive activity. Simulations allow us to visualize the behavior of the population in different environments and seasons of the year. The results are consistent with previous studies showing that in preserved area the population exhibits oscillations with return to equilibrium while in altered area it can be extinguished after a certain time. © 2014 Elsevier B.V. All rights reserved.

1. Introdution The process of habitat fragmentation naturally occurs, but has been intensified by human action. Extractive, agriculture and livestock activities are responsible for the remaining fragments to become smaller faster. Population dynamics of species that live in forest fragments are under the direct consequence of such activities being carried out in the area. These consequences include population isolation with a significant reduction in the population size, restriction of gene flow, increased probability of endogamy as well as the loss of genetic variability. These consequences can restrict the ability of a population to fit in with new environmental conditions (Brown and Kodric-Brown, 1977; Keller and Largiader, 2003; Grady et al., 2006). Considering their biology, insects are highly susceptible to the adverse effects of forest fragmentation (Didham et al., 1996). Drosophila mediopunctata, a member of the tripunctata group, subgenus Drosophila is a Neotropical species and can be found from El Salvador to southern South America (Val et al., 1981). It is considered an excellent model in biology due to several factors: small size, relative abundance, particularly in Southern Brazil, wide distribution, long life cycle (around 90 days), relatively low fecundity, easy collection, identification and manipulation and high sensitivity to climate change (Tidon, 2006). It has been considered a good bioindicator of human disturbance in the environment quality because it

∗ Corresponding author at: Caixa Postal 3010, CEP 85015-430, Guarapuava, PR, Brazil. Tel.: +55 04236298348; fax: +55 04236211090. E-mail address: [email protected] (M.J.P. Castanho). http://dx.doi.org/10.1016/j.ecolmodel.2014.04.025 0304-3800/© 2014 Elsevier B.V. All rights reserved.

is a strict forest-dwelling species (Mata et al., 2008). However, due to its high sensitivity to climate change, some studies have shown that its frequency changes along the year, increasing during cold months (Saavedra et al., 1995; Bitner-Mathé and Klaczko, 1999; Cavasini et al., 2008). For the biologists who work with conservation it is important to detect endangered species and to have a mathematical model that describes population dynamics. Typically, these models are devised in a deterministic form. For example, Jansen and Sevenster (1997) built a discrete model for Drosophila population dynamics based on larval competition. The outcomes of larval competition were supported by data of populations reared in the laboratory. When working in the field, environmental data present uncertainty and heterogeneity. Heterogeneity results from different sources, structures and types of data. The uncertainty of this information results from presence of random variables, incomplete or inaccurate data, qualitative information incomplete or vague expert knowledge, among others (Salski, 1999). To include the uncertainty in the model, some authors used stochastic methodology (Godoy and Costa, 2005; Day and Possingham, 1995; Calder et al., 2003). Krivan and Colombo (1998) used models based on differential inclusions in modeling uncertainty. Fuzzy set theory (Zadeh, 1965) provides another means for dealing with uncertainty. Several authors have used this theory because of similar problems, such as insufficient data for statistical analysis (Salski, 1999), uncertainties in the available data and expert knowledge (Bock and Salski, 1998) and subjectivity in the state variables or in the parameters (Barros et al., 2000). Interestingly, Salski (1999) concluded that heterogeneous and imprecise data and vague expert knowledge can be integrated effectively using a fuzzy

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approach. Furthermore, fuzzy set theory has been successfully used in several other works to describe aspects of population dynamics of different organisms and in different ecological situations (see Leal-Ramírez et al., 2011; Schaefer et al., 2001; Cecconello, 2006; Mastorakis and Avramenko, 2007; Cheung et al., 2005; Mocq et al., 2013; Mackinson, 2000; Jafelice et al., 2011 as examples). Fuzzy rule-based systems are very useful tools of this theory. In these systems the explicit knowledge is represented in the form of if-then rules and linguistic and/or numerical information are described by membership functions. An inference method together with fuzzy rules provides the output of the model. A fuzzy rulebased system is an essential part of an iterative process to model the dynamics of a system. This system is called p-fuzzy system (Silva et al., 2013). A p-fuzzy system can be used to model population dynamics where the relationship between the state variables and the population variation is uncertain. This paper adopt a p-fuzzy system to describe the population dynamics of D. mediopunctata, a forest dwelling species that shows frequency variation along the year (related to climate change) and which can be related to conservation aspects of Atlantic Forest in South America. For a better understanding, in the next section, we introduce some concepts utilized in this work. 2. Preliminary concepts and definitions Fuzzy sets are classes with not sharply defined boundaries in which the transition from membership to non-membership is gradual. The fuzzy set theory (Zadeh, 1965) provides a natural way of dealing with problems in which are present uncertainty. It is an important tool to model population dynamics (Barros et al., 2000) where it is often necessary to combine both subjective information obtained from an expert and objective observations and measurements (Bock and Salski, 1998). Definition 1.

A fuzzy subset A of a universal set X is defined by

A = {(x, A (x))|x ∈ X and A : X → [0, 1]}

Fig. 1. Structure of a p-fuzzy system.

2 Knowledge base: contains both an ensemble of fuzzy rules and a set of membership functions, known as a database. Each rule r of the form “If X is A then Y is B” describes a relation between the fuzzy variables X and Y. This relation, R on X × Y can be determined by the relational assignment equation R (x, y) = g(A (x), B (y)),

(3)

∀(x, y) ∈ X × Y, where g is a function that generalizes the fuzzy Cartesian product. 3 Inference method: the inference in approximate reasoning is mainly the generalization of the classical inference (modus ponens). Available data are supplied to the expert system, which uses them to evaluate relevant production rules and draw all possible conclusions. A particular form of fuzzy inference of interest here is the Mamdani Method (Pedrycz and Gomide, 1998). In this method the function g is a fuzzy conjunction: g(A (x), B (y)) = A (x) ∧ B (y),

(4)

∀(x, y) ∈ X × Y, where ∧ represents the minimum operator. 4 Output processor: translates the fuzzy set resultant from the inference method into a real number. A typical defuzzification scheme, the same adopted in this work, is the center-of-gravity method.

(1)

where the membership function, A (x), is interpreted as a grade in which an element x ∈ X has a property A.

Definition 4. In the center-of-gravity method, the defuzzified value xˆ is equal to the center of gravity of fuzzy set A:

Definition 2. The support of a fuzzy set A, denoted by Supp(A), are all elements of X that belong to A to a nonzero degree,

xˆ =

Supp(A) = {x ∈ X|A (x) > 0}.

(2)

Definition 3. A fuzzy set A is normal if its membership function attains 1, that is, supA (x) = 1. x

We can employ fuzzy sets to represent linguistic variables. A linguistic variable is a variable whose values are words or sentences in a natural or artificial language. A particularly important area of application for the concept of linguistic variable is that of approximate reasoning that is the mechanism of generation of inference in knowledge based systems. Also known as fuzzy rule-based systems, these systems have the ability to linguistically specify relationships that are too complex or not well enough understood to be directly described by precise mathematical models (Sudkamp and Hammell, 1996). The basic structure of a fuzzy rule-based system comprises four main components: an input processor, a knowledge base, an inference method and an output processor (Pedrycz and Gomide, 1998). These components process real-valued inputs to provide real-valued output as follows. 1 Input processor: crisp inputs are translated into fuzzy sets of their respective universes. Fuzzy sets describe linguistic variables.

 x · A (x) x dx. x

A (x)

(5)

Definition 5. A p-fuzzy (pure fuzzy) system in Rn is a discrete dynamic system:



xk+1 = f (xk ) x0 ∈ Rn

(6)

where f(xk ) is partly known, however, the system solution is crisp because in each instant k, a value is obtained after a defuzzification process (Barros and Bassanezi, 2006). The function f is given by f(xk ) = xk + xk · xk where xk ∈ Rn is obtained by a fuzzy rule-based system. The initial condition xo is known. The structure of a p-fuzzy system is represented in Fig. 1. In p-fuzzy systems, the result obtained by rule-based system xk in changes the input variable and it collaborates to develop the iterative process. This system is in equilibrium when xk = xk+1 ↔ xk = 0. Definition 6. Let {Ai }1≤i≤k a finite family of normal fuzzy subsets, associated with linguistic variable z. The subsets are called successive if they satisfy the following requirements (Silva et al., 2013):

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Fig. 2. Structure of the fuzzy rule-based system constructed.

1 suppAi ∩ suppAi+1 = /  for 1 ≤ i < k. 2  suppAi ∩ suppAj =  if |i − j| ≥ 2. 3 i=1,k suppAi = U, where U is the domain of linguistic variable z. 4 given z1 ∈ suppAi and z2 ∈ suppAi+1 , if Ai (z1 ) = Ai+1 (z2 ) = 1 then z1 < z2 for all i ≤ 1 < k. Definition 7. Let a p-fuzzy system and a family of successive subsets {Ai }1≤i≤k . If for each x1 , x2 ∈ supp(Ai ∪ Ai+1 ), 1 ≤ i < k, the variations x1 and x2 have opposite signs, then the fuzzy subset A * = Ai ∩ Ai+1 is called a feasible set of equilibrium and suppA* is a feasible region of equilibrium (Cecconello, 2006). Theorem 1. If a p-fuzzy system S admits a feasible region of equilibrium, suppA * = / , then S has at least one state of equilibrium in suppA*, this is, ∃x* ∈ suppA * such that x * =0. Proof.

See Silva et al. (2013).

3. Material and methods Data of the D. mediopunctata population were collected during 2006 and 2008 in the two remaining forest areas, in Guarapuava, state of Paraná, southern Brazil: Parque das Araucárias and Fazenda Brandalise. Parque das Araucárias (25◦ 23 36 S, 51◦ 27 19 W) is a conservation unit, with a total area of 1.04 km2 where 0.43 km2 are Araucaria Forest (Mixed Ombrophylous Forest). Fazenda Brandalise

is a private property located approximately 6 km from Parque das Araucárias. In this fragment (approximately 4 km2 ) there is a large extractive activity (mate and pine nut) in addition to the replacement of natural habitats with cultivated areas. The regional climate is classified as mesothermal humid subtropical, according to the methodology of Köeppen. In the winter, the average temperature is 8.4 ◦ C and in the summer, 26.7 ◦ C. Some characteristics of this fly species are sensitivity to extreme temperatures (Klaczko, 2006) and the reduction of its population in altered environments (Döge et al., 2008). As temperature changes gradually, the fragment of forest is an uncertain measure and available data are insufficient requiring expert knowledge, the fuzzy set theory is appropriate to develop the model. Population density, seasonality and extractive activity were selected as the most important factors affecting the population dynamics of the D. mediopunctata. Consequently population density, seasonality are chosen as the input variables of the model and extractive activity is considered to elaborate rules. The fuzzy rule-based system is represented in Fig. 2. The domain of population density variable (x) was defined in the range [0, 700] regarding the flies collected using traps. The traps were closed with bait consisting of banana, orange and yeast. In each fragment were traced two parallel, horizontal transects, 200 m long, 15 m distant from each other. Six traps were installed in each transect, 40 m distant from each other and 1.5 m above-ground. The total capture area was 21,000 m2 (Cavasini et al., in press). It was

Fig. 3. Input and output variables.

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Fig. 4. Simulation with respect to favorable weather (March) in Fazenda Brandalise and Parque das Araucárias respectively.

reproduction in the first 30 days. Using this information the linguistic labels were determined: High Negative (HN), Medium Negative (MN), Low Negative (LN), Low Positive (LP), Medium Positive (MP) and High Positive (HP). The membership functions of fuzzy sets assumed by input and output variables of the system are represented in Fig. 3. The rule-base was composed of 36 if–then rules. It was considered little extractive activity in Parque das Araucárias and very extractive activity in Fazenda Brandalise. This was formulated with information supplied by specialists and it is described in Tables 1 and 2. The inference was made using the Mamdani method and the defuzzified result was obtained by the center-of-gravity method. The fuzzy rule-based system is a component of the iterative system represented in Fig. 1, constructed to describe the population dynamics of D. mediopunctata. As the female flies oviposit each 10 days approximately, each iteration k is equivalent to a period of 10 days.

700

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Population

considered that the flies were attracted to the traps in a radius of approximately 40 m and the traps captured less than 1/5 of the flies existing in the area. The linguistic labels used were: Very Low (VL), Low (L), Medium (M), Upper Medium (UM), High (H) and Very High (VH). The variable seasonality had the days of the year as domain, [0, 365], in which zero corresponds to January 1st. D. mediopunctata is more abundant in the cold months, but the winter in this region is very severe, so the favorable climate for its survival occurs in spring and fall. Then, the linguistic labels used were: Favorable (F), Less Favorable (LF) and Unfavorable (U). Unfavorable corresponds to the peak of the summer and winter. The output variable was variation. To establish the domain [− 0.5, 1.72] of this variable, data from the basic biology of D. mediopunctata were considered: deviation of sex ratio in 50%, 10 being the number of eggs which the female lays in the range of 10 days, 50% the approximate percentage of eggs that hatch and whose flies reach adulthood, a life cycle of 90 days without

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Fig. 5. Simulation with respect to less favorable weather (May) in Fazenda Brandalise and Parque das Araucárias respectively.

M.J.P. Castanho et al. / Ecological Modelling 287 (2014) 9–15 Table 1 Rule-base for little extractive activity. Density

Table 2 Rule-base for very extractive activity.

Seasonality

Very Low Low Medium Upper Medium High Very High

Density

Favorable

Less Favorable

Unfavorable

Low Positive Medium Positive High Positive Medium Positive Low Positive Low Negative

Low Positive Low Positive Low Positive Low Positive Low Negative Medium Negative

Low Negative Low Negative Medium Negative Medium Negative High Negative High Negative

4. Simulation results The D. mediopunctata population behavior varies with the season and the environment. Simulations were made with various initial conditions to observe the population evolution. Figs. 4–6

Favorable

Less Favorable

Unfavorable

Low Positive Low Positive Medium Positive Low Positive Low Negative Medium Negative

Low Negative Low Positive Low Positive Low Negative Medium Negative High Negative

Medium Negative Medium Negative Low Negative Medium Negative High Negative High Negative

show the simulation of this behavior during a year, with regard to an initial population equal to 100 individuals. From the simulations performed we were able to observe that the behavior was the same for both forest fragments. Starting with unfavorable climate, the population first decreased and then

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Fig. 6. Simulation with respect to unfavorable weather (January) in Fazenda Brandalise and Parque das Araucárias respectively.

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Fig. 7. Simulation with initial population equal 200 flies and less favorable weather (May) in Fazenda Brandalise and Parque das Araucárias respectively.

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Fig. 8. Simulation with respect to unfavorable weather (June) in Fazenda Brandalise and Parque das Araucárias respectively.

resumed the cyclical behavior. When the weather was less favorable or favorable, it immediately grew but in the second case took longer to reach the minimum value. When there is little extractive activity, regardless of the initial number of individuals and climate, the population reaches a maximum around 700 individuals. If there is a lot of extractive activity the maximum is around 400 individuals (see Fig. 7). Depending on the initial condition, it is possible to observe that while in the Parque das Araucárias (environmental conservation unit) the maximum population remained approximately constant in each cycle, in Fazenda Brandalise the population declined, evidencing the influence of human activity on the environment. If the initial population is below a threshold in area with very extractive activities it will be extinguished after a certain time. This threshold depends on the climate. If it is favorable, 10 individuals are enough to maintain the population; if it is unfavorable, it needs more than 50. This does not occur in the conservation area. Fig. 8 shows the simulation of population behavior over a period of three years, starting with a population of 50 flies and unfavorable weather.

With the simulations performed we were able to observe that whilst in preserved areas the D. mediopunctata population exhibits oscillations with return to equilibrium, in areas of high human activity the population can be extinguished after a certain time. 5. Discussion The model presented here is strongly based on the expert knowledge. As D. mediopunctata is a forest-dwelling species and has its population reduced in altered environments (Döge et al., 2008), we consider areas with different extractive activities. Population density and seasonality were selected as input variables because this species is influenced by the temperature of the environment (Klaczko, 2006). The results are in agreement with previous studies of population seasonal samples of this species, including its relation to the conservation status of the areas of collection. For example, in Bizzo et al. (2010), the frequency of D. mediopunctata was always low, not exceeding 12 individuals (winter 2002) per collection in each season. This study was performed in a more opened area, an arboreal

Fig. 9. Simulation considering conservation area (Parque das Araucárias) and real data (Bitner-Mathé and Klaczko, 1999).

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strand forest fragment (named restinga) in a shrub to grassland strand forest matrix in southern Brazilian coast. Therefore, this is a greater environmental impact to a forest dwelling species as D. mediopunctata. In spite of that, this species depicted a similar frequency variation during the year. Bitner-Mathé and Klaczko (1999) collected D. mediopunctata samples in a conservation area where the climate is mesothermal with average annual temperature of 27◦ C. Fig. 9 shows a simulation considering the initial population equal to 145 individuals (5 times the number of flies caught in the traps), in September, and real data. The average annual temperature in region studied in this work is 16.7◦ C with very low temperatures in winter. So, while in the area studied by Bitner-Mathé and Klaczko the winter is a favorable season because it has mild temperatures, in our study very low temperatures make it unfavorable season and the population decreases. The real data are actually like a snapshot of the population size in a specific time evaluated in the nature. Therefore, we do not have all the variation that might have occurred in the population size in the real population, just these pictures. However, they are mostly in agreement with the time frame of the simulation obtained using p-fuzzy model. 6. Conclusion It is possible to observe that fuzzy set theory is a good option to model population dynamics because of the way it deals with imprecision. Using a p-fuzzy system we were able to describe the variation in D. mediopunctata population size with regard to subjective variables such as environmental and climate in a simple way. Also, it was possible to incorporate in the model the qualitative information available. Another positive feature was the ease of transition of collected data into mathematical language. Simulation results clarified the differences between the forest fragments with extractive activity and environmental preservation policies, as well as the interference of climatic variation in such populations. Therefore, we suggest that this type of model to be used as a predictor in programs of conservation of areas where D. mediopunctata occurs. It is easily adaptable to the conditions of the area of interest. Also, the methodology presented here is an option to construct systems to describe the dynamics of several other populations. References Barros, L., Bassanezi, R., 2006. Tópicos de Lógica Fuzzy e Biomatemática, 1st ed. Colec¸ão IMECC (in Portuguese). Barros, L., Bassanezi, R., Tonelli, P., 2000. Fuzzy modelling in population dynamics. Ecol. Model. 128, 27–33. Bitner-Mathé, B.C., Klaczko, L.B., 1999. Size and shape heritability in natural populations of Drosophila mediopunctata: temporal and microgeographical variation. Genetica 105, 35–42. Bizzo, L., Gottschalk, M., De Toni, D., Hofmann, P., 2010. Seasonal dynamics of a drosophilid (Diptera) assemblage and its potential as bioindicator in open environments. Iheringia 100 (3), 185–191. Bock, W., Salski, A., 1998. A fuzzy knowledge-based model of population dynamics of the yellow-necked mouse (Apodemus flavicollis) in a beech forest. Ecol. Model. 108, 155–161. Brown, J., Kodric-Brown, A., 1977. Turnover rates in insular biogeography: effect of imigration on extinction. Ecology 58, 445–449.

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