Biomimicry and Fuzzy Modeling:

A Match Made in Heaven

Michael Margaliot∗

November 18, 2007

Abstract

Biomimicry, the design of artificial systems that mimic naturalbehavior, is recently attracting considerable interest. Biomimicry re-quires a reverse engineering process; the behavior of a biological agentis analyzed in order to mimic this behavior in an artificial system. Inmany cases, biologists have already studied the relevant behavior andprovided a detailed verbal description of it. Mimicking the natural be-havior is then reduced to the following problem: how can we convertthe given verbal description into a well-defined mathematical formulaor algorithm that can be implemented by an artificial system?

Fuzzy modeling (FM), with its ability to handle and manipulateverbal information, constitutes a natural approach for addressing thisproblem. The application of FM in this context may lead to a sys-tematic approach for biomimcry, namely, given a verbal descriptionof an animal’s behavior (e.g., the foraging behavior of ants), applyFM to obtain a mathematical model of this behavior which can beimplemented by artificial systems (e.g., autonomous robots).

The purpose of this position paper is to highlight these issues andto alert the attention of the computational intelligence community tothis emerging application of FM.

Keywords: Linguistic modeling, vagueness, animal behavior, bio-inspiredsystems, mathematical modeling in biology.

∗Correspondence: Dr. Michael Margaliot, School of Elec. Eng.-Systems, TelAviv University, Tel Aviv 69978, Israel. Tel: +972 3 640 7768. Homepage:www.eng.tau.ac.il/∼michaelm Email: michaelm@eng.tau.ac.il

1

1 Introduction

Biomimicry1 is a new discipline that studies designs and processes in natureand then mimics them in order to solve human problems. Studying howplants absorb and utilize solar energy in order to develop better solar cells isan example of such engineering inspired by nature.

In order to implement a natural behavior in an artificial system, it is firstnecessary to understand the behavior. In many cases, the relevant behav-ior has already been studied by biologists. Biological theories are generallydescriptive in nature, and are stated using natural language. For example,Darwin’s original presentation of his evolutionary theory did not include asingle equation [9]. The problem of mimicking the natural behavior is thenreduced to the following problem.

Problem 1 Given a verbal description of some relevant behavior, design anartificial system that mimics this behavior.

Fuzzy logic theory has been associated with human linguistics ever sinceit was first suggested by Zadeh [66, 67]. Decades of successful applications(see, e.g., [12, 54, 55, 64, 65, 44]) suggest that the real power of fuzzy logicis in its ability to handle and manipulate verbally-stated information basedon perceptions rather than equations [11, 34, 35, 68]. In particular, fuzzymodeling (FM) is routinely used to transform the knowledge of a humanexpert, stated in natural language, into a fuzzy expert system that imitatesthe human experts’ functioning [51, 26].

This suggests that FM may be suitable for addressing Problem 1, andthus provide a systematic approach for addressing biomimicry. Given a ver-bal description of the behavior of a biological agent, apply FM to obtaina mathematical model of this behavior which can be implemented by anartificial system.

The purpose of this position paper is to summarize some of the lessonslearned from using FM to develop mathematical models of animal behavior,and to suggest the next natural step, that is, the use of FM in biomimcry.Special attention is devoted to presenting the potential advantages of FM inthis specific context.

The remainder of this paper is organized as follows. Section 2 brieflyreviews the motivation for imitating processes and designs found in nature.

1from the Greek bio, life, and mimesis, to imitate.

2

Section 3 describes several approaches for transforming verbal descriptionsinto mathematical equations or algorithms. Section 4 describes two recent ap-plications of FM for transforming verbal descriptions of animal behavior intowell-defined mathematical models. Sections 5 and 6 summarize some lessonslearned from using FM in modeling animal behavior, with an emphasis onthe unique advantages and disadvantages of the FM approach. Section 7 dis-cusses the suitability of FM in the specific context of biomimcry. The finalsection concludes.

2 Biomimicry

Over the course of evolution, living systems have developed efficient androbust solutions to various problems they encountered in their natural habi-tats. Plants have developed means for absorbing and utilizing solar energy.Foraging animals have developed ways of navigating and searching in an un-known environment. More generally, many natural beings have developedthe capabilities to reason, learn, evolve, adapt, and heal.

Scientists and engineers are interested in implementing such capabilitiesin artificial systems. For example, designing efficient schemes for navigat-ing and searching in an unknown environment is a fundamental problem inthe field of autonomous robots. It is thus natural that considerable researchhas recently been devoted to the development of artificial products or ma-chines that mimic (or are inspired by) various biological phenomena [52, 41].For example, walking robots offer tremendous potential in replacing humanswhere it is unsafe, inaccessible, or too expensive to operate. Currently thereis still a lack of a strong scientific basis for designing such robots. Yet, natureprovides us with an abundance of highly efficient walking agents. In partic-ular, many insects are capable of traversing very rough terrain with sparsefootholds. This has motivated considerable research on the design of artificialwalking robots by emulating the structure and functioning of insects [46, 4].

Other examples from the field Biomimicry or Bio-inspired systems, in-clude: evolutionary computation that addresses optimization problems byimitating some of the ideas in Darwin’s theory of evolution [30, 18]; the designof artificial structures that mimic shape laws in nature and, in particular, thestructural properties of trees [36]; autonomous robots that mimic the behav-ior of various animals [3, 8]; the design of computer security systems inspiredby the natural immune system [16, 22]; the design of optimization, search,

3

and clustering algorithms based on the behavior of social insects [6, 10]; andthe design of new materials inspired by nature, such as a self-cleaning glassbased on the drift-repellent surface of the lotus plant [15]. More informationand a suggested reading list can be found at http://www.biomimicry.net/

An important component in biomimicry is a reverse engineering processof the relevant natural behavior. In some cases, biologists have already stud-ied the relevant behavior and provided verbal descriptions of the underlyingmechanisms. For example, as noted in [50], it is customary to describe thebehavior of simple animals (e.g., ants) using simple rules of thumb [24]. Inthis case, the problem of imitating the relevant behavior is reduced to Prob-lem 1. In the next section, we review several approaches that are suitable foraddressing this problem.

3 From words to equations

During the 1950s, Jay W. Forrester and his colleagues developed system dy-namics2 as a method for modeling the dynamic behavior of complex systems.The basic idea is to represent the causal structure of the system using elemen-tary structures that include positive, negative, or combined positive and neg-ative feedback loops. These are depicted graphically, and then transformedinto a set of differential equations. The method was applied successfullyto numerous real-world applications in the social, economic, and industrialsciences. However, the inherent vagueness of the linguistic description is ig-nored, and exact terms and phrases (e.g., a temperature of 30◦ Celsius) aretreated in precisely the same way as vague verbal terms (e.g., warm weather).

A more systematic approach to modeling physical systems is qualitativereasoning (QR) [31] which transforms qualitative descriptions of a systeminto qualitative differential equations. These extend ordinary differentialequations in two ways: (1) functional relationships between variables canbe represented by functions that are monotonous, but do not have to becompletely specified; and (2) the variable values are described using a set oflandmark values rather than exact numerical values. QR was applied to sim-ulate and analyze many real world systems. However, its applications seemto suggest that it is suitable when modeling with accurate, yet incomplete,knowledge rather than with a verbal description of the system.

2To date, more than 30 books on system dynamics have appeared. A very readableone is [48]. The journal Systems Dynamics Review contains up-to-date papers.

4

3.1 Natural language and fuzzy logic

As noted in [39], the world around us is plagued by various forms of uncer-tainty, ambiguity, and vagueness. The latter property arises in the groupingof elements. Suppose that D is some universe of discourse, P is some prop-erty, and consider the set S := {x ∈ D : x has property P}. In many cases,it is impossible to provide a precise binary definition of S. For example,consider the case where D is the set of all the man-made items, and P is theproperty: x is a chair. Any attempt to provide a clear cut definition of Sis doomed to fail. This is evident from some dictionary definitions for chair,e.g., “any of various devices that hold up or support.”

A typical feature of vagueness is continuity : a small difference betweentwo objects should not lead to an abrupt change in the decision of whetherthey belong to S or not. In other words, the transition from having prop-erty P to not having it (or vice versa) is smooth.

One approach for modeling such continuity is based on characterizing therelation between an object and its properties by means of a scale. Conse-quently, the membership in the set S is no longer binary, but becomes amatter of degree. The concept of a fuzzy set provides a useful mathematicalmodel for characterizing sets that allow graded membership.

The property of vagueness is most striking in the semantics of natu-ral language. This suggests that an algorithmic or mathematical treatmentof natural language should include mechanisms that deal with the inherentvagueness.

Zadeh laid down the foundations of fuzzy sets and fuzzy logic and linkedthem to human linguistics [66, 67]. The pioneering work of Wenstop [61, 62]was aimed at building verbal models (VMs) capable of representing and pro-cessing information stated in natural language. A VM consists of three basiccomponents: (1) generative grammar–used for defining the semantics of theverbal statements; (2) fuzzy logic-based inferencing–used in the deductiveprocess; and (3) linguistic approximation–used for attaching suitable linguis-tic labels to the outputs. The entire process was implemented using the APLcomputer language. Wenstop and Kickert developed VMs for several inter-esting systems from the social sciences [62, 28][27, Ch. 7]. However, VMs arenot standard mathematical models, as their input and output are linguis-tic values rather than numerical values. Consequently, they suffer from twodrawbacks: (1) there are no methods for analyzing the behavior of VMs, sothey can only be used for simulations; and (2) when modeling dynamic sys-

5

tems the degree of fuzziness (or uncertainty) increases with every iteration,so the system can be simulated effectively only over relatively short timespans.

Kosko [29] suggested fuzzy cognitive maps (FCMs) as a tool for represent-ing causal relationships between various linguistic concepts. FCMs were usedto model several interesting real world phenomena (see [1] and the referencestherein). However, the inferencing process used in FCMs yields a discrete-time linear system. Such a simple mathematical model is inadequate formany real world systems.

Mamdani [33] designed a fuzzy algorithm for regulating a laboratory-built steam engine. The design was based on transforming the linguisticcontrol protocol, used by a skilled human operator, into a fuzzy controller.The transformation consisted of the following steps. The verbal informa-tion was stated as a set of If-Then fuzzy rules. Vague terms such as smallor large3 were modeled using suitable fuzzy sets. Inferring the fuzzy rule-base (FRB) led to a well-defined controller. This approach, although ratherad-hoc, proved to be very successful and led the way to numerous applica-tions (see, e.g., [12, 54, 55, 64]). However, most of these applications arebased not on modeling natural phenomena, but rather on transforming theknowledge of a human expert into a fuzzy expert system.

In the last 15 years or so considerable research interest has been devotedto establishing a solid mathematical foundation for fuzzy logic theory andfuzzy modeling (see, e.g., [40, 21]). Yet, the success of these more theoreticalapproaches in practical applications has yet to be demonstrated.

4 Fuzzy modeling of animal behavior

Recently, FM was applied successfully to transform verbal descriptions ofanimal behavior into well-defined mathematical models. The method consistsof four steps: (1) identifying the state-variables; (2) restating the given verbaldescriptions as an FRB relating these variables; (3) defining the fuzzy termsusing suitable membership functions; and (4) inferring the FRB to obtain awell-defined mathematical model.

This approach was used to derive mathematical models for: (1) the ter-ritorial behavior of fish [57]; (2) the orientation of a planarian to light [58];(3) the foraging behavior of ants [49]; and (4) the mechanisms regulating

3Dvorak and Novak [13] refer to such terms as evaluating linguistic expressions.

6

the population size in flies [47]. In all these examples, the starting pointwas a detailed verbal description of the natural phenomenon. Using FM thiswas transformed into a well-defined mathematical model. Simulations andrigorous analysis demonstrated the suitability of the mathematical model.

Independently, Lebar Bajec et al. [2] used fuzzy modeling to develop aninteresting algorithm for simulating the flocking behavior of birds.

We now briefly review two examples of deriving mathematical models foranimal behavior using FM. More details can be found in [57] and [58].

4.1 Territorial behavior in the stickleback

Territory plays a major role in social animal behavior [63] and results in arich set of phenomena, but how is territory created? Nobel Laureate KonradLorenz describes a specific example [32, p. 47]:

“. . . a real stickleback fight can be seen only when two males arekept together in a large tank where they are both building theirnests. The fighting inclinations of a stickleback, at any givenmoment, are in direct proportion to his proximity to his nest.. . . The vanquished fish invariably flees homeward and the victor. . . chases the other furiously, far into its domain. The farther thevictor goes from home, the more his courage ebbs, while that ofthe vanquished rises in proportion. Arrived in the precincts ofhis nest, the fugitive gains new strength, turns right about anddashes with gathering fury at his pursuer . . . ”

Note that Lorenz provided us with a complete verbal description of themechanisms underlying the dynamics. Furthermore, he also described theresulting global patterns as observed in nature:

“The pursuit is repeated a few times in alternating directions,swinging back and forth like a pendulum which at last reaches astate of equilibrium at a certain point.”

4.1.1 Fuzzy modeling

The state of the system at time t is determined by the location and fightinginclination of the fish. For fish i, let xi(t) ∈ R

n, wi(t) ∈ R, and ci ∈ Rn denote

7

the location, fighting inclination, and the location of the nest, respectively.Here n is the dimension of the state-space.

Lorenz’ description of the change in fighting inclination is transformedinto the following fuzzy rules:

• if neari(xi, ci) then wi = p

• if fari(xi, ci) then wi = −p,

where p > 0, that is, the fighting inclination increases (decreases) when thefish is near (far) its nest. Similarly, the description of the movement of fish iis transformed into:

• if neari(xi,xj) and highi(wi) then xi = xj − xi

• if neari(xi,xj) and lowi(wi) then xi = ci − xi

where xj is the location of the other fish. That is, when the other fish isnear, and the fighting inclination is high (low), then move in the direction ofthe other fish (nest).

Membership functions are used to model the fuzzy terms.4 The termneari(x,y) is modeled using ni(x,y) = exp(−||x − y||2/(k2

i )), with ki > 0.Note that ni(x,y) ≤ 1 with equality only when y = x. The term highi

is defined using hi(w) = (1 + tanh(w/ai))/2, where ai > 0 determines theslope of hi. Note that lim

w→−∞

hi(w) = 0 and limw→+∞

hi(w) = 1. The opposite

terms fari and lowi are defined using fi(x, y) = 1 − ni(x, y), and li(w) =1− hi(w), respectively.

Inferring the rules, using multiplication for the “and” operator, and centerof gravity defuzzification, yields:

wi = 2p exp(−||xi − ci||2/(k2

i ))− p,

xi = ci − xi + hi(wi)(xj − ci), i = 1, 2. (1)

Summarizing, the FM approach transformed the verbal description intothe well-defined mathematical model (1).

4For a discussion on the degrees of freedom in the FM approach, see Section 6 below.

8

0 5 10 15 20 25−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t

Figure 1: Trajectories x1(t) (solid line) and x2(t) (dashed line) for initialpositions x1(0) = −0.4 and x2(0) = 0.8. The nests are located at c1 = −1and c2 = 1.

4.1.2 Simulations

Fig. 1 depicts the results of simulations of (1) for the one-dimensional case(n = 1), with: c1 = −1, c2 = 1, p = a1 = a2 = k1 = k2 = 1, and theinitial conditions: x1(0) = −0.4, x2(0) = 0.8, w1(0) = w2(0) = 1. It may beseen that the fish follow an oscillatory movement, with one fish advancing,the other retreating until a point is reached where they switch roles. Finally,they converge to a steady state point at x1 = −0.1674 and x2 = −x1. Onemay then say that the territory of fish i is the region between its nest ci

and the point xi. Note that the model reproduces the oscillatory movementdescribed by Lorenz.

Once the verbal description is converted into a mathematical model, it isof course also possible to rigorously analyze this model, use it for predictionsof new scenarios, etc. The interested reader is referred to [57] for more detailson this specific model.

4.2 Orientation to light in the Dendrocoleum lacteum

An animals life depends on oriented movements. Such movements guidethe animal into its normal habitat or into other situations which are of im-portance to it. Various stimuli, such as light or smell, activate the living

9

mechanisms and lead to orientation.Taxis is a mechanism of orientation by means of which an animal moves

in a direction related to a source of stimulation [19]. For example, positive(negative) photo-taxis is the directed movement towards (away from) a sourceof light. Taxes require sensory organs that can accurately detect the directionof the stimulus, and a brain sophisticated enough to process the sensory dataand consequently determine the appropriate direction of movement.

Simple organisms, such as wood lice or flat-worms, do not necessarilyhave the physiological equipment needed to perform taxes. Their eyes, forexample, do little more than indicate the general intensity of light, but notits direction. In such organisms the locomotory action can be affected bythe intensity, but not the direction, of the stimulus. This type of response isreferred to as kinesis.5

Klino-kinesis6 is defined as a movement where the rate of turning, butnot the direction of turning, depends on the intensity of the stimulus. Thistype of movement appears in many flat-worms: in regions with higher lightintensity their rate of turning increases. As a result, the animals eventuallyaggregate in shadier parts of the available habitat.

In a classic paper, Philip Ullyott studied this type of behavior in theplanarian Dendrocoleum lacteum [59]. In order to determine whether thereaction is simply to the intensity of the light falling on the animal, or toits direction as well, he designed an apparatus insuring that the only objectvisible to the animal is the single patch of light directly above it. Ullyottanalyzed the animals behavior, and found that the stimulus did not affect theanimals linear velocity. Instead, increased light intensity yielded an increasein the rate of change of direction (RCD) (measured in angular degrees perminute) in which the animal moved. Ullyott defined the RCD as the sumof all the deviations in the animals path during one time unit, summing upboth right-hand and left-hand deflections as positive changes. As the lightwas switched on, the RCD immediately increased but with time it fell off,converging to a constant level, which Ullyott designated the basal RCD. Thisdecrease of response under constant stimulation suggested the existence ofan adaptation mechanism. Ullyott [59, p. 274] summarized his findings asfollows:“(1) An increase in stimulating intensity produces an increase in RCD.

5from the Greek kinesis, movement.6from the Greek klino, incline.

10

light lightbrightdim

y

C

x

A BE

D

Figure 2: Path of the “average” animal. Adapted from [17, p. 49].

(2) This initial increase in RCD falls off under constant stimulation owing toadaptation.(3) There is a basal RCD, which is an expression of the fact that turningmovements occur even in absolute darkness or at complete adaptation.”

Fraenkel and Gunn [17, Ch. V] reviewed and refined Ullyott’s work. Theydeveloped a simplified and deterministic model for the “averaged” animalsmovement. The most important simplification is the assumption that theanimal always turns to the right and always through exactly 90◦. As theRCD increases, the time between these right-hand turns decreases.

Following Ullyott, Fraenkel and Gunn furnished a heuristic explanation ofhow this behavior drives the animal to the darker regions. Suppose that theanimal is placed in a plane (described by two coordinates (x, y)), where thelight intensity gradually increases along the positive direction of the x-axis(see Fig. 2). Beginning at a point A, (and assuming that the animal is fullyadapted to the light at A), the animal continues in the positive x directionuntil making a right-hand turn at point B, and so on. Along the segment AB,the light intensity increases and the adaptation level lags behind, so theRCD increases. Along the BC segment the light intensity is constant andthe RCD decreases back to its basal level. Along the CD segment the lightintensity decreases and the RCD remains constant (note that the RCD isaffected only by an increase in the light intensity). Finally, the behavioralong the DE segment is as in the BC segment. Since the RCD increases

11

along the AB segment and, due to the adaptation process, decreases alongthe other segments (until it converges to the basal RCD), the segment ABwill be shorter than the segment CD. Hence, after a set of consecutive turnsthe animal ends up at a point E, which might be closer to the darker partof the plane than the initial point A. Note that the adaptation process playsan essential role in this explanation, since it allows the animal to (indirectly)compare a certain light intensity in the present with a previous light intensity.

Ullyott’s results were quite surprising at the time of their publication, ashe postulated that although the direction of the animals movement is notoriented to the direction of the light at all, it still yields a result similar tonegative photo-taxis, (i.e., the animal eventually finds its way to the shadierregions of its habitat): “This alternate stimulation and adaptation has aneffect on the RCD of such a kind that the animal is led automatically to theplace of minimal intensity.” [59, p. 277]

Both Ullyott and Fraenkel and Gunn provided only a verbal descriptionof the animals behavior and its outcome. Patlak [43] studied the behaviorof random particles affected by a force field. He derived a suitable Fokker-Planck-type equation characterizing the particles movement, and used thisstochastic model to analyze klino-kinesis [42]. However, his model ignoresthe adaptation process which plays a vital role in the description given byUllyott and Fraenkel and Gunn.

4.2.1 Fuzzy modeling

The verbal description given above can be transformed into a mathematicalmodel using the FM approach. The state-variables are: the animals locationin the plane (x(t), y(t)); the light intensity at each point in the plane l(x, y);the light intensity that the animal is currently adapted to la(t); the RCD r(t);the direction of movement θ(t); and the set of turning times: t1, t2, . . . . Themodel also includes two constants: the animals linear velocity v, and thebasal RCD rb.

The verbal description is restated as an FRB relating the state-variables.The adaptation process changes the level of adaptation to light in accordancewith intensity of light. We state this as two fuzzy rules:

• If l − la is positive, then la = c1

• If l − la is negative, then la = −c1

12

where c1 > 0. The RCD decreases when it is above the basal RCD, andincreases when the light stimulus is above the adaptation level. We statethis as:

• If r − rb is large, then r = −c2

• If l − la is high, then r = c3

where c2, c3 > 0. Finally, following Fraenkel and Gunn’s model, we add thecrisp rule:

• If∫ t

tlr(τ)dτ = q, then θ ← θ − π/2

Here, tl < t denotes the time when the last turn took place, and q > 0. Inother words, the RCD is constantly accumulated and whenever it reaches thethreshold q, the animal makes a right-hand turn.

The next step is to define the fuzzy membership functions. For our firstset of rules, we use µpositive(x) = ek1x/(ek1x + e−k1x), with k1 > 0, andµnegative(x) = 1− µpositive(x).

Let Sk be the piecewise linear function defined by: Sk(z) = 0 for z ≤ 0;Sk(z) = z/k for 0 < z < k, and Sk(z) = 1 for z ≥ k. For the second set ofrules, we use µlarge(x) = Sk2

(x), µhigh(x) = Sk3(x), with k2, k3 > 0.

Using additive inferencing (see, e.g., [5]), the first set of rules yields

la = c1µpositive(l − la)− c1µnegative(l − la)

= c1 tanh(k1(l − la)). (2)

Similarly, the second set of rules yields r = −c2Sk2(r−rb)+c3Sk3

(l− la). Theactual movement of the animal is given by x = v cos(θ), and y = v sin(θ).The crisp rule implies that the value θ “jumps” at the discrete times t1, t2, . . .satisfying

∫ ti+1

tir(τ)dτ = q, i = 0, 1, . . . (with t0 = 0). Thus, the model is

a hybrid system [60] combining continuous-time dynamics and discrete-timeevents.

Fig. 3 summarizes the mathematical model. The upper arrow in thisfigure corresponds to the conditional transition described by the crisp rule.When the condition holds, θ is updated, and then the evolution in timeproceeds using the continuous-time dynamics.

The FM approach allowed us to transform the verbal description of thebehavior into a mathematical model. One criteria for evaluating the suit-ability of this model is how well it mimics the patterns that were actuallyobserved in the natural system.

13

x = v cos(θ)

r = −c2Sk2(r − rb) + c3Sk3

(l − la)

la = c1 tanh(k1(l − la)) θ← θ − π/2

i← i + 1

ti ← t

∫ t

ti

r(τ)dτ = q

y = v sin(θ)

Figure 3: The hybrid model.

4.2.2 Simulations

In one of his experiments, Ullyott [59] placed the animals in total darknessfor three hours (to make them completely dark adapted). He then exposedthe animals to a sudden increase in light intensity, and measured the RCDat different times. Fig. 4 depicts his results.

We simulated a similar scenario in our mathematical model with

c1 = 5, c2 = 1, c3 = 2, k1 = k2 = 1, k3 = 2, r(0) = rb = 2, la(0) = 0, (3)

and light intensity: l(t) = 0 for t ∈ [0, 1), and l(t) = 1 for t ≥ 1. Fig. 5depicts la(t) and r(t) as a function of time. It may be seen that the adaptationlevel increases once the light is switched on, and then converges to la(t) =1 (= l(t)). Note the qualitative resemblance between the behavior of r(t)and the RCD as actually measured by Ullyott (Fig. 4).

4.2.3 Analysis

One of the principle advantages of mathematical models is that they areamenable to mathematical analysis. The next result analyzes the modelsbehavior for a step function in the light intensity.

Proposition 1 [58] Consider the hybrid model with initial conditions la(0) =0, r(0) = r0, and light intensity l(t) = 0 for t < 0, and l(t) = 1 for t ≥ 0. De-note w := c1k1k2−c2, p1 := sinh(k1), and assume that the models parameterssatisfy:

k3 ≥ 1, w > 0, and k2 ≥ r0 − rb + (p1k2c3)/(k1k3w). (4)

Thenla(t) = 1− sinh-1(p1 exp(−c1k1t))/k1, (5)

14

0

400

600

800

0

200

10 20 30 40 50

CBAA B

RC

D (

angu

lar

degr

ees

per

min

ute)

Time (minutes)

Figure 4: The relationship between RCD and duration of stimulus. AB,RCD of the animal in darkness (basal RCD). At B a light intensity of 2500ergs/(cm2·sec) was switched on. BC, adaptation to the stimulus. Each pointon the curve represents the average of fourteen experiments. (Reproducedfrom [59] with permission.)

where sinh-1 denotes the inverse hyperbolic sine function, and

r(t) = rb+exp(− c2

k2

t)(r0−rb)+c3

k1k3

∫ t

0

exp(c2

k2

(τ−t)) sinh-1(p1 exp(−c1k1τ))dτ.

(6)

It is easy to verify that the specific parameter values used in the simula-tion (3) indeed satisfy (4). Hence, (5) and (6) actually provide an analyticaldescription of the responses depicted in Fig. 5.

It is possible to show that (6) implies that r(t) decays exponentially. Thisagrees with the observations of Ullyott, who states: “. . . it is possible to seethat the falling off of RCD with time is exponential...” [59, p. 270].

Extensive literature exists on developing mathematical models in biology(see, e.g., [37, 38, 14]), using classic modeling techniques that have nothing

15

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

t0 1 2 3 4 5 6 7 8

1.95

2

2.05

2.1

2.15

t

Figure 5: la(t) (left) and r(t) (right) as a function of time. The light isswitched on at t = 1.

to do with FM. A natural question is: why use FM at all? We try to addressthis question by describing some of the advantages of FM.

5 Advantages of the fuzzy modeling approach

The most important advantages of the FM approach stem from the close con-nection between the initial verbal description and the resulting mathematicalmodel. In the FM approach the knowledge about the system is represented inthree different forms in parallel: (1) the initial verbal description and expla-nation; (2) the FRB; and (3) the mathematical model obtained by inferringthe rules. These three representations provide a synergistic overview of thesystem that may provide new insights on the studied phenomenon.

Fig. 6 summarizes the design cycle in the FM approach. The initial verbaldescription is stated as an FRB. Defining the membership functions, andinferencing yields the mathematical model. Simulations and analysis can beused to check whether the models behavior is congruent with that actuallyobserved in nature. When this is not the case, it is sometimes possible,due to the If-Then structure of the rules, to determine which fuzzy rule

16

fuzzyverbal

descriptionmodeling fuzzy

rule base

modification simulation and analysis

modelmathematicalinferencing

Figure 6: Synergy between three different forms of knowledge representation.

should be changed and how. Inferring the modified FRB yields a modifiedmathematical model, and so on. Furthermore, any change in the FRB canalso be interpreted as a change in the initial verbal description, suggestingdirections for further research of the original natural phenomenon. In thissense, the behavior of the mathematical model can be used, to some extent,to prove or refute the modeler’s ideas as to how the natural system behaves.We now demonstrate these issues using the mathematical models describedin Section 4.

5.1 Interpretability

A fuzzy model represents the real system in a form that corresponds closelyto the way humans perceive it. Thus, the model is understandable, even bynon-professionals, and each parameter has a readily perceivable meaning [20].The next example demonstrates this.

Example 1 Consider the parameters k1 and k2 in the mathematical model (1).To gain some intuitive understanding of these parameters, recall that the kisoriginate from the membership function ni(x,y) = exp(−||x−y||2/(k2

i )) usedto define the term neari(x,y). As ki decreases, the Gaussian becomes morepeaked, so fish i will sense the other fish as “near” only when it is indeedvery close to it. In other words, a decrease in ki corresponds, in some sense,to making fish i “less aggressive.” This provides a clear interpretation of ki

by relating it to the original verbal description.This interpretation can be verified by analyzing (1). Assume, without

17

loss of generality, that c2 = c and c1 = −c, for some c > 0. In this case,

x1 = −c + k1

√ln 2, x2 = c− k2

√ln 2,

w1 = a1 tanh−1(2x1 + c

x2 + c− 1), w2 = a2 tanh−1(2

x2 − c

x1 − c− 1),

where tanh−1 denotes the inverse hyperbolic tangent function, is a (locallyasymptotically stable) equilibrium point of (1). In other words, there existinitial conditions for which fish i will come to rest at a distance ki

√ln 2 from

its nest, i = 1, 2.If k1 > k2 then the equilibrium position is no longer symmetric with

respect to the nests and, eventually, fish 1 will guard a larger territory thanfish 2. This leads to a clear link between the relative aggressiveness of thefish, as manifested by the values of the parameters, and the resulting sizesof their territories. 2

5.2 Verifying the verbal description

Once the mathematical model is derived, its suitability can be examinedusing both simulations and rigorous analysis. The close connection betweenthis mathematical model and the original verbal description can be used, tosome extent, to verify the correctness of the verbal description. The nextexamples demonstrate this.

Example 2 The hybrid model for the planarian’s orientation to light in-cludes the fuzzy rule: “If l − la is high, then r = c3.” If c3 = 0, then theRCD will not increase when l− la is high. Since the increase in RCD plays acrucial role in Ullyott’s explanation, we may expect that the models behaviorwill change substantially.

To verify this, we substitute c3 = 0 in the mathematical model. Then

r(t) = −c2Sk2(r − rb),

and assuming that r(0) = rb yields r(t) = rb for all t ≥ 0. Recall that theright-hand turns take place at times t1, t2, . . . such that

∫ ti+1

ti

r(τ)dτ = q,

18

so ti+1 − ti = q/rb for all i. In other words, the right-hand turns takeplace at regular intervals. Since the linear velocity v is constant, it is clearthat x(ti) = x(ti+4) and y(ti) = y(ti+4) for all i. In other words, the planarianwill periodically return to its initial position, and the process of graduallymoving toward the shadier parts of the habitat will not take place. 2

Example 1 (continued) Consider again the stickleback model. A closerlook shows that there is actually a discrepancy between Lorenz’ verbal de-scription and the FRB. Indeed, Lorenz claimed that: “The fighting inclina-tions of a stickleback, at any given moment, are in direct proportion to hisproximity to his nest,” whereas the FRB relates the proximity to the nestwith wi, that is, the derivative of the fighting inclination.

There is a good reason for this discrepancy. Initially, we actually did usefuzzy rules with wi, and not wi, in the Then-part (e.g., if neari(x

i, ci) thenwi = p). Simulating the resulting mathematical model showed that the fishmove toward each other and then stop at an equilibrium point. There wereno periodic motions. This does not agree, of course, with the actual behaviorobserved in nature.

A moment of reflection suggested that the problem was that the fishhave no “memory,” so they stop once they reach the edge of their territory.To correct this, we changed the Then-part of the rules using wi insteadof wi. Then, when a fish approaches the border of its territory, the fightinginclination starts decreasing, but it may still be sufficient to lead it across tothe other territory, and thus initiate the oscillatory behavior.

The change in the FRB can be naturally interpreted as a change in theinitial verbal description. The modified FRB actually corresponds to themodified statement: The change in the fighting inclinations of a stickleback,at any given moment, are in direct proportion to its proximity to its nest.This is a new assertion that can be checked only by means of suitable ex-periments on the real world system. Nevertheless, this demonstrates thesynergistic view provided by the interconnection of the verbal description,the FRB, and the mathematical model. 2

Example 2 (continued) Consider again the hybrid model. Fig. 7 depictsa simulation of this model for initial conditions: r(0) = rb, x(0) = y(0) =la(0) = 0, and l(x, y) = x, that is, the light intensity increases linearly alongthe x-axis.

Fig. 7 depicts the trajectory (x(t), y(t)). It may be seen that x(t0) >

19

−2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

x

y

Figure 7: Trajectory (x(t), y(t)) for initial values x(0) = y(0) = 0, and lightintensity l(x, y) = x.

x(t4) > x(t8) > . . . , where ti are the turning times (with t0 = 0). Thus, thetrajectory indeed moves toward the darker region of the plane. This fact is inagreement with Ullyott’s observations [59, p. 272]: “The shift of the positionof the animal towards the end of the gradient was rather a gradual process,but in each case with the steep gradient, the animal was to be found at thedarker end within 2 hours from the beginning of the experiment.”

However, it may also be seen that y(t0) < y(t4) < y(t8) < . . . . Thisfeature is not described in Fraenkel and Gunn’s explanations (see Fig. 2).A moments reflection suggests that this behavior is indeed what we shouldexpect. Indeed, reconsidering the discussion preceding Fig. 2 suggests that,due to the adaptation process, the segment DE should be longer than thesegment BC. Although this is evident in retrospect, it is not necessarilyimmediate when reading the verbal explanation. The simulations of thecorresponding mathematical model, derived based on this explanation, iswhat makes this aspect of the verbal explanation apparent. 2

The examples above describe the case where the verbal description wastransformed into a mathematical model using FM. In the specific approach

20

used here, the first two stages in this transformation are [57]: (1) identifythe variables needed to define the systems state; and (2) re-state the givenverbal information as an FRB relating these variables.

It has been the author’s experience, in many examples, that by followingthese stages, it is possible to detect deficiencies in the given verbal descrip-tion. For example, it is sometimes possible to see that although some state-variable, say x(t), is needed, the verbal description does not include sufficientinformation on the relation of x(t) to the other variables.

The interesting thing is that detecting such a problem in the verbal de-scription is not necessarily immediate. It is revealed only when one tries tofollow the FM approach. This is similar to discovering that an algorithm,stated in pseudo-code, is not really complete while actually trying to encodeit in some computer language.

Scientists who provide verbal descriptions for various phenomena maychoose to use FM not only to derive a mathematical model, but simplyto verify that there are no unapparent missing parts or gaps in their ex-planations. Verifying the verbal description using this approach forces thescientist to provide a mechanistic and complete description of the relevantphenomenon. This is congruent with the methodological aspect of logic asa theory for analyzing the consistency and completeness of knowledge-basedsystems [45].

The discussion will not be complete without describing some of the down-sides of the FM approach.

6 Disadvantages of the fuzzy modeling ap-

proach

In the FM approach the fuzzy rules follow more or less directly from theverbal description. However, there are many degrees of freedom, and a greatdeal of subjectivity, in specifying the other components of the model: themembership functions, logical operators, and inferencing method. A changein any of these may yield a mathematical model with a rather different be-havior. This is usually regarded as a disadvantage, as it suggests that we caneasily miss the “correct” mathematical model.

It should be noted however that this “fuzzines” is not really a feature ofthe FM approach, rather it is a direct consequence of the inherent vague-

21

ness of verbal descriptions. Indeed, imagine a scenario in which two appliedmathematicians are given the same verbal description, and asked to developa suitable mathematical model for it. It seems reasonable to expect that thetwo may end up with different models.

The conclusion is that one should not be tempted to believe that thereexists a single “correct” mathematical model. Any reasonable approach fortransforming words into equations must contain some degrees of freedom.These should be addressed by a trial and error approach, hopefully convergingto a useful model.

The inherent vagueness of natural language is not a weakness, rather itis this vagueness that allows us to communicate valuable information parsi-moniously. This is congruent with Zadeh’s principle of incompatibility [67]:“As the complexity of a system increases, our ability to make precise andyet significant statements about its behavior diminishes until a threshold isreached beyond which precision and significance (or relevance) become al-most mutually exclusive characteristics.”

Note also that in the specific context of biomimcry, the goal is not neces-sarily to obtain a mathematical model that reproduces the relevant naturalbehavior precisely. Rather, the goal is to design an artificial system that isinspired by the natural behavior and solves a given engineering problem.

In the examples described above the degrees of freedom in the FM ap-proach were determined in attempt to reach a model that satisfies two re-quirements. The first was, of course, that the model reproduces the describedbehavior. The second requirement was that it be amenable to mathematicalanalysis.

Extensive experience and information have accumulated over decades ofsuccessful implementations of the FM approach. Consequently, there is con-siderable literature on how the various elements in the fuzzy model influenceits behavior (see, e.g., [53, 20]). If the verbal description of the system isaccompanied by quantitative data, then it is possible to apply well-knownmethods, such as fuzzy clustering, neural learning, and least squares approx-imation [20, 7, 25], for learning the parameters of the fuzzy model.

7 Fuzzy modeling and biomimcry

FM may be suitable for addressing biomimicry in a systematic manner.Namely, start with a verbal description of an animals behavior (e.g., for-

22

aging in ants), and apply FM to obtain a mathematical model of this behav-ior that can be implemented by artificial systems (e.g., autonomous robots).Although it is a natural idea, it seems that the computational intelligencecommunity is not aware of this possible application of FM in biomimcry.

It should be noted that in addition to the general advantages of FMdescribed above, there are at least two considerations that make FM par-ticularly suitable for modeling biological behavior and, in particular, animalbehavior.

First, many animal (and human) actions are non-binary, that is, thebehavior itself is “fuzzy.” For example, the response to a (low intensity)stimulus might be what Oskar Heinroth [23] called intention movements, thatis, a slight indication of what the animal is intending to do. Tinbergen [56,Ch. IV] states: “As a rule, no sharp distinction is possible between intentionmovements and more complete responses; they form a continuum.” It isinteresting to recall that Zadeh [66] defined a fuzzy set as “a class of objectswith a continuum of grades of membership.” Hence, FM seems the mostappropriate tool for studying such behaviors.

The second consideration is that studies of animal behavior often pro-vide a verbal and, therefore, an inherently vague, description of both fieldobservations and interpretations. Darwin [9, Ch. 2] states: “Nor shall I herediscuss the various definitions which have been given of the term species. Noone definition has as yet satisfied all naturalists; yet every naturalist knowsvaguely what he means when he speaks of a species.” As another example,Fraenkel and Gunn [17, p. 23] describe the behavior of a cockroach, thattends to become stationary when its body surface is in contact with a solidobject, as: “A high degree of contact causes low activity . . . .” FM is asuitable approach for addressing this vagueness. Indeed, note that the laststatement can be immediately stated in the form of a fuzzy If-Then rule: “Ifdegree of contact is high then activity is low.”

8 Concluding Remarks

Recently, the design of artificial algorithms and machines that imitate bi-ological behavior, is attracting considerable interest. In some cases, thereexist verbal descriptions of the natural behavior we aim to mimic. The prob-lem of mimicking the natural behavior is then reduced to transforming thegiven verbal information into a well-defined mathematical formula or algo-

23

rithm that can be implemented in an artificial system. A natural approachfor addressing this is FM.

In this paper, we tried to summarize some of the lessons learned fromapplying FM to develop mathematical models of animal behavior, and tosuggest the next natural step, that is, the use of FM in biomimcry. Specialattention was devoted to presenting the potential advantages of FM in thisspecific context.

We believe that FM can and should be utilized as one of the main toolsin the field of biomimicry. We hope that this paper will stimulate more workin this direction.

Acknowledgments

The author is grateful to the anonymous reviewers and the editor for theirdetailed and constructive comments. The author thanks The Company ofBiologists Ltd. for permission to reproduce one of the figures from [59].

References

[1] J. Aguilar, “Adaptive random fuzzy cognitive maps,” in Lecture Notes in

Artificial Intelligence 2527, F. J. Garijo, J. C. Riquelme, and M. Toro, Eds.Springer-Verlag, 2002, pp. 402–410.

[2] I. L. Bajec, N. Zimic, and M. Mraz, “Simulating flocks on the wing: the fuzzyapproach,” J. Theoretical Biology, vol. 233, pp. 199–220, 2005.

[3] Y. Bar-Cohen and C. Breazeal, Eds., Biologically Inspired Intelligent Robots.SPIE Press, 2003.

[4] R. D. Beer, R. D. Quinn, H. J. Chiel, and R. E. Ritzmann, “Biologicallyinspired approaches to robotics: what can we learn from insects?” Comm.

ACM, vol. 40, pp. 30–38, 1997.

[5] J. M. Benitez, J. L. Castro, and I. Requena, “Are artificial neural networksblack boxes?” IEEE Trans. Neural Networks, vol. 8, pp. 1156–1164, 1997.

[6] E. Bonabeau, M. Dorigo, and G. Theraulaz, Swarm Intelligence: from Natural

to Artificial Systems. Oxford University Press, 1999.

24

[7] G. Bontempi, H. Bersini, and M. Birattari, “The local paradigm for modelingand control: from neuro-fuzzy to lazy learning,” Fuzzy Sets Systems, vol. 121,pp. 59–72, 2001.

[8] C. Chang and P. Gaudiano, Eds., Robotics and Autonomous Systems, Special

Issue on Biomimetic Robotics, vol. 30, 2000.

[9] C. Darwin, On the Origin of Species: By Means of Natural Selection. Dover,2006, Dover Giant Thrift Ed.

[10] M. Dorigo and T. Stutzle, Ant Colony Optimization. MIT Press, 2004.

[11] D. Dubois, H. T. Nguyen, H. Prade, and M. Sugeno, “Introduction: thereal contribution of fuzzy systems,” in Fuzzy Systems: Modeling and Control,H. T. Nguyen and M. Sugeno, Eds. Kluwer, 1998, pp. 1–17.

[12] D. Dubois, H. Prade, and R. R. Yager, Eds., Fuzzy Information Engineering.Wiley, 1997.

[13] A. Dvorak and V. Novak, “Formal theories and linguistic descriptions,” Fuzzy

Sets Systems, vol. 143, pp. 169–188, 2004.

[14] L. Edelstein-Keshet, Mathematical Models in Biology. SIAM, 2005.

[15] P. Forbes, The Gecko’s Foot: Bio Inspiration-Engineering New Materials from

Nature. W. W. Norton, 2006.

[16] S. Forrest, S. A. Hofmeyr, and A. Somayaji, “Computer immunology,” Comm.

ACM, vol. 40, pp. 88–96, 1997.

[17] G. S. Fraenkel and D. L. Gunn, The Orientation of Animals: Kineses, Taxes,

and Compass Reactions. Dover Publications, 1961.

[18] D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine

Learning. Addison-Wesley Professional, 1989.

[19] G. Goodenough, B. McGuire, and R. A. Wallace, Perspectives on Animal

Behavior. Second edition, John Wiley & Sons, 2001.

[20] S. Guillaume, “Designing fuzzy inference systems from data: aninterpretability-oriented review,” IEEE Trans. Fuzzy Systems, vol. 9, pp. 426–443, 2001.

[21] P. Hajek, Metamathematics of Fuzzy Logic. Springer, 2002.

25

[22] E. Hart and J. Timmis, “Application areas of AIS: the past, the present andthe future,” in Artificial Immune Systems, ser. Lecture Notes in ComputerScience, C. Jacob, M. L. Pilat, P. J. Bentley, and J. Timmis, Eds. Springer-Verlag, 2005, vol. 3627, pp. 483–497.

[23] O. Heinroth, “Beitrage zur bilogie, insbiesonder psychologie und ethologieder anatiden,” in Proc. 5th Int. Ornithological Congress, Berlin, 1910, pp.589–702.

[24] B. Holldobler and E. O. Wilson, The Ants. Belknap Press, 1990.

[25] J. S. R. Jang, C. T. Sun, and E. Mizutani, Neuro-Fuzzy and Soft Computing:

A Computational Approach to Learning and Machine Intelligence. Prentice-Hall, 1997.

[26] A. Kandel, Ed., Fuzzy Expert Systems. CRC Press, 1992.

[27] W. J. M. Kickert, Fuzzy Theories on Decision-Making. Martinus Nijhoff,1978.

[28] W. J. M. Kickert, “An example of linguistic modeling: The case of Mulder’stheory of power,” in Advances in Fuzzy Set Theory and Applications, M. M.Gupta, R. K. Ragade, and R. R. Yager, Eds. North-Holland, 1979, pp.519–540.

[29] B. Kosko, Neural Networks and Fuzzy Systems: A Dynamical Systems Ap-

proach to Machine Intelligence. Prentice-Hall, 1992.

[30] J. R. Koza, Genetic Programming: on the Programming of Computers by

Means of Natural Selection. MIT Press, 1992.

[31] B. Kuipers, Qualitative Reasoning: Modeling and Simulation with Incomplete

Knowledge. The MIT Press, 1994.

[32] K. Z. Lorenz, King Solomon’s Ring: New Light on Animal Ways. Methuen& Co., 1957.

[33] E. H. Mamdani, “Applications of fuzzy algorithms for simple dynamic plant,”Proc. IEE, vol. 121, pp. 1585–1588, 1974.

[34] M. Margaliot and G. Langholz, “Fuzzy Lyapunov based approach to the de-sign of fuzzy controllers,” Fuzzy Sets Systems, vol. 106, pp. 49–59, 1999.

[35] M. Margaliot and G. Langholz, New Approaches to Fuzzy Modeling and Con-

trol - Design and Analysis. World Scientific, 2000.

26

[36] C. Mattheck, Design in Nature: Learning from Trees. Springer-Verlag, 1998.

[37] J. D. Murray, Mathematical Biology I, 3rd ed. Springer, 2002.

[38] J. D. Murray, Mathematical Biology II, 3rd ed. Springer, 2003.

[39] V. Novak, “Are fuzzy sets a reasonable tool for modeling vague phenomena?”Fuzzy Sets Systems, vol. 156, pp. 341–348, 2005.

[40] V. Novak, I. Perfilieva, and J. Mockor, Mathematical Principles of Fuzzy

Logic. Kluwer, 1999.

[41] K. M. Passino, Biomimicry for Optimization, Control, and Automation.Springer, 2004.

[42] C. S. Patlak, “A mathematical contribution to the study of orientation oforganisms,” Bulletin of Mathematical Biophysics, vol. 15, pp. 431–476, 1953.

[43] C. S. Patlak, “Random walk with persistence and external bias,” Bulletin of

Mathematical Biophysics, vol. 15, pp. 311–338, 1953.

[44] W. Pedrycz, Ed., Fuzzy Sets Engineering. CRC Press, 1995.

[45] I. Perfilieva, “Logical foundations of rule-based systems,” Fuzzy Sets Systems,vol. 157, pp. 615–621, 2006.

[46] M. J. Randall, Adaptive Neural Control of Walking Robots. ProfessionalEngineering Publishing, 2001.

[47] I. Rashkovsky and M. Margaliot, “Nicholson’s blowflies revisited: A fuzzymodeling approach,” Fuzzy Sets Systems, vol. 158, pp. 1083–1096, 2007.

[48] G. P. Richardson and A. L. Pugh, Introduction to System Dynamics Modeling

with Dynamo. MIT Press, 1981.

[49] V. Rozin and M. Margaliot, “The fuzzy ant,” IEEE Computational Intelli-

gence Magazine, vol. 2, pp. 18–28, 2007.

[50] S. Schockaert, M. De Cock, C. Cornelis, and E. E. Kerre, “Fuzzy ant basedclustering,” in Ant Colony, Optimization and Swarm Intelligence, ser. Lec-ture Notes in Computer Science, M. Dorigo, M. Birattari, C. Blum, L. M.Gambardella, F. Mondada, and T. Stutzle, Eds. Springer-Verlag, 2004, vol.3172, pp. 342–349.

[51] W. Siler and J. J. Buckley, Fuzzy Expert Systems and Fuzzy Reasoning.Wiley-Interscience, 2004.

27

[52] M. Sipper, Machine Nature: The Coming Age of Bio-Inspired Computing.McGraw-Hill, 2002.

[53] J. M. C. Sousa and U. Kaymak, Fuzzy Decision Making in Modeling and

Control. World Scientific, 2002.

[54] K. Tanaka and M. Sugeno, “Introduction to fuzzy modeling,” in Fuzzy Sys-

tems: Modeling and Control, H. T. Nguyen and M. Sugeno, Eds. Kluwer,1998, pp. 63–89.

[55] T. Terano, K. Asai, and M. Sugeno, Applied Fuzzy Systems. AP Professional,1994.

[56] N. Tinbergen, The Study of Instinct. Oxford University Press, 1969.

[57] E. Tron and M. Margaliot, “Mathematical modeling of observed natural be-havior: a fuzzy logic approach,” Fuzzy Sets Systems, vol. 146, pp. 437–450,2004.

[58] E. Tron and M. Margaliot, “How does the Dendrocoleum lacteum orient tolight? a fuzzy modeling approach,” Fuzzy Sets Systems, vol. 155, pp. 236–251,2005.

[59] P. Ullyott, “The behavior of Dendrocoelum lacteum II: responses in non-directional gradients,” J. Experimental Biology, vol. 13, pp. 253–278, 1936.

[60] A. van der Schaft and H. Schumacher, An Introduction to Hybrid Dynamical

Systems. LNCIS 251, Springer, 2000.

[61] F. Wenstop, “Quantitative analysis with linguistic rules,” Fuzzy Sets Systems,vol. 4, pp. 99–115, 1980.

[62] F. Wenstop, “Deductive verbal models of organizations,” in Fuzzy Reasoning

and its Applications, E. H. Mamdani and B. R. Gaines, Eds. Academic Press,1981, pp. 149–167.

[63] E. O. Wilson, Sociobiology: The New Synthesis. Harvard University Press,1975.

[64] R. R. Yager and D. P. Filev, Essentials of Fuzzy Modeling and Control. JohnWiley & Sons, 1994.

[65] J. Yen, R. Langari, and L. Zadeh, Eds., Industrial Applications of Fuzzy Logic

and Intelligent Systems. IEEE Press, 1995.

28

[66] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, pp. 338–353, 1965.

[67] L. A. Zadeh, “Outline of a new approach to the analysis of complex systemsand decision processes,” IEEE Trans. Systems, Man, Cybernetics, vol. 3, pp.28–44, 1973.

[68] L. A. Zadeh, “Fuzzy logic = computing with words,” IEEE Trans. Fuzzy

Systems, vol. 4, pp. 103–111, 1996.

29