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Nicholson's blowflies revisited: A fuzzy modeling approach
Fuzzy Sets and Systems 158 (2007) 1083 – 1096 www.elsevier.com/locate/fss
Nicholson’s blowflies revisited: A fuzzy modeling approach夡 Iosef Rashkovsky, Michael Margaliot∗ School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel Received 5 April 2006; received in revised form 20 August 2006; accepted 3 November 2006 Available online 30 November 2006
Abstract We apply fuzzy modeling to derive a mathematical model for a biological phenomenon: the regulation of population size in the Australian sheep-blowfly Lucilia cuprina. This behavior was described by several ethologists and fuzzy modeling allows us to transform their verbal descriptions into a well-defined mathematical model. The behavior of the resulting mathematical model, as studied using both simulations and rigorous analysis, is congruent with the behavior actually observed in nature. We believe that the fuzzy modeling approach demonstrated here may supply a suitable framework for biomimicry, that is, the design of artificial systems based on mimicking natural behavior. © 2006 Elsevier B.V. All rights reserved. Keywords: Linguistic modeling; Differential equations with time-delay; Periodic behavior; Switched systems with time delay; Population dynamics; Emergent behavior; Biomimicry
1. Introduction In many of the “soft sciences” (e.g., psychology, sociology, ethology) scientists provided verbal descriptions and explanations of various phenomena based on field observations. Obtaining a suitable mathematical model, describing the observed system or behavior, can greatly enhance our ability to understand and study it in a scientific manner. Indeed, mathematical models are very useful in summarizing and interpreting empirical data. Furthermore, once derived, such models allow us to analyze the system both qualitatively and quantitatively using mathematical tools. Tron and Margaliot [20,21] advocated fuzzy logic theory as the most suitable tool for transforming verbal descriptions of various observed phenomena into suitable mathematical models. This approach is congruent with the notion that the real power of fuzzy logic is in its ability to handle and manipulate linguistic information based on perceptions (see, e.g., [7,12,13,23]). Indeed, fuzzy modeling is routinely used to transform the knowledge of an expert, be it a physician or a process operator, into a computer algorithm. Yet, not enough attention has been paid to its possible use as a tool to assist human observers in transforming their verbal descriptions into mathematical models. The fuzzy modeling approach is based on transforming the verbal description into a set of if-then rules stated using natural language. Inferencing this rule-base yields a well-defined mathematical model. This approach has several advantages. The fuzzy rule-base represents the real system in a form that corresponds closely to the way humans 夡 An
abridged version of this paper was presented at the IEEE World Congress on Computational Intelligence (WCCI 2006).
∗ Corresponding author. Tel.: +972 3 6407768; fax: +972 3 6407095.
E-mail address: [email protected] (M. Margaliot) URL: http://www.eng.tau.ac.il/∼michaelm (M. Margaliot). 0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2006.11.001
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perceive it. Thus, the model is understandable, even by nonprofessionals, and each parameter has a readily perceivable meaning. The model is flexible and can be easily altered to incorporate new phenomena. The information is kept in three different forms: the original verbal description, the fuzzy rule-base, and the mathematical model. The interplay between these representations provides a synergistic view of the system. For example, suppose that simulations of the mathematical model indicate that its behavior is different than that of the real system. By analyzing the simulations, it is sometimes possible to find which fuzzy rule must be altered and how. Since the rules are stated in natural language, any change in the rules can be interpreted in terms of a change of the original verbal description. Thus, the modeling process can also be used to prove or refute the modeler’s ideas as to how the natural system behaves and why. Several factors make the fuzzy approach particularly suitable for developing models of animal behavior. First, ethologists usually provide detailed verbal descriptions of the phenomena they are studying. Some of these descriptions can be easily converted into fuzzy rules. For example, Fraenkel and Gunn describe the behavior of a cockroach, that becomes stationary when a large part of its body surface is in contact with a solid object, as: “A high degree of contact causes low activity. . .” [8, p. 23]. Second, mathematical models of animal behavior may also have interesting engineering applications. In fact, the development of artificial products or machines that mimic biological phenomena, sometimes referred to as biomimicry, is recently attracting considerable interest (see, e.g., [1,14,5]). The basic idea is that during their course of evolution living systems developed sophisticated mechanisms for solving various challenges. Designers of artificial products, that must function in the real world, are often required to address similar problems. Thus, information extracted from biological behavior may inspire suitable artificial designs. For example, animals must develop efficient mechanisms for orienting themselves in space. Similar problems arise in robotics, as planning and realizing oriented movements is of crucial importance in designing autonomous robots. An important component in biomimicry is the ability to perform reverse engineering of an animal’s functioning and then implement this behavior in an artificial system. We believe that the fuzzy modeling approach may be very suitable for addressing biomimicry in a systematic manner. Namely, start with a verbal description of an animal’s behavior (e.g., foraging in ants) and, using fuzzy logic theory, obtain a mathematical model of this behavior which can be implemented in artificial systems (e.g., autonomous robots). In this paper, we use the fuzzy modeling approach to derive a mathematical model for a phenomenon of animal behavior: the regulation of population size in the blowfly Lucilia cuprina. The mechanisms involved in the regulation process were described and explained in the classical works of Nicholson [16,17]. The behavior of the resulting mathematical model, as studied using both simulations and rigorous analysis, is congruent with the behavior actually observed in nature. The remainder of this paper is organized as follows. Section 2 reviews the verbal descriptions and explanations given by Nicholson. Section 3 applies fuzzy modeling to transform these descriptions into a mathematical model. Section 4 analyzes the mathematical model. Sections 5 describes the results of simulations and compares these to the behavior actually observed in nature. Section 6 discusses the quantitative aspects of the model. The final section concludes. 2. The self-adjustment of population in the Lucilia cuprina Population dynamics plays a fundamental role in the animal kingdom. Sophisticated feedback mechanisms regulate the size of the population, adjusting it to the constantly changing internal and external factors. In a classic study, Nicholson [17] analyzed the regulation of population size in laboratory colonies of the Australian sheep-blowfly Lucilia cuprina. He introduced a small number of flies into a cage and, from then on, maintained predetermined conditions leaving the population to its own device. Although the environment remained invariable, the population size showed large-amplitude quasi-periodic oscillations [4] (see Fig. 1). One of the cultures was maintained for a period of two years and the oscillations continued for this entire period of time. Many different interacting factors regulate the population’s size and lead to this nontrivial behavior. The first of these is competition between the individual members of the population for the fixed amount of food. If the population size is small, then the competition is not severe and interbreeding leads to an increase in the population size. However, at a certain point the food supply is no longer sufficient to maintain the population. This leads to increased mortality and decreased natality. The population size decreases, and the competition is relaxed again, thus completing a cycle.
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9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0
50
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day Fig. 1. Fluctuations in laboratory populations of Lucillia cuprina.
The regulating mechanisms are further complicated because of the various stages in the blowfly’s life cycle: (1) adults interbreed and lay eggs; (2) immature larva hatch from the egg; (3) the larva turn into pupa; and (4) the pupa transforms into adult. Nicholson’s experiments are interesting for several reasons. First, it is surprising that fixed conditions, in a controlled environment, lead to such extreme oscillations in the population size. Second, from a dynamic systems point of view, the quasi-periodic behavior is not trivial and raises interesting questions. For example, is the behavior chaotic [10] or, alternatively, the result of a periodic process “probed” by stochastic factors? 1 Third, Nicholson was able to separate the food supply of the adult flies and the larvae. This allowed him to experiment different types of competition. For example, by supplying a surplus quantity of food for the adult flies and limiting the larval food, it is possible to isolate the effect of larval competition. Finally, the mechanisms that regulate the population usually operate at the level of individual animals. For example, if a larva does not receive sufficient food, then it will not pupate. Yet, the combined effect is an emergent behavior regulating the total population size. This was already noted by Nicholson [17, p. 154]: “. . .each of these groups has characteristics which are more than the sum of those of the constituent individuals.” Population dynamics is a very rich field. Methods for analyzing population processes can be roughly divided into two groups. The first group includes methods based on applying statistical tools to analyze the time series. The second group includes methods that develop a mathematical model for the observed phenomena. For more information on these topics, see [11,3] and the many references therein. In the next section, we use fuzzy modeling to develop a mathematical model for the behavior described by Nicholson. 3. Mathematical modeling The fuzzy modeling approach transforms verbal descriptions into mathematical models using several stages. First, the variables in the model are determined. For example, in a description of a moving animal, the animal’s location is a variable in R3 . Second, the verbal information is stated as fuzzy rules relating the variables. These rules include fuzzy terms that must be defined mathematically using suitable membership functions. Finally, applying the inference mechanism to the rules yields a well-defined mathematical model [20]. Here we use a slightly different approach. Following the seminal paper [9], we begin by writing down a balance equation for the population of sexually mature adults at time t which we denote by x(t). The rate of change in x(t) is the difference between the rate of recruitment to the adult population r(t), and the total adult death rate d(t): x(t) ˙ = r(t) − d(t). 1 For more on the interplay between deterministic and stochastic forces in population dynamics, see [3].
(1)
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The death rate is modeled by the linear function d(t) = x(t),
(2)
where > 0 is the per capita death rate. Thus, we assume that the functional form of the model is known, except for the function r(t) [22]. We use fuzzy modeling to determine the function r(t), and thus complete the mathematical model. 3.1. Fuzzy modeling 3.1.1. The fuzzy rules Our rules are based on the following description [17, p.156]: “When the numbers of adults were high such vast numbers of eggs were laid that all of the food provided was consumed while the larvae were still too small to pupate; consequently no adult offspring resulted from eggs laid during such periods. The adult numbers therefore dwindled progressively, until a point was reached at which the intensity of larval competition became so reduced that some of the larvae attained a sufficient size to pupate. These gave rise to egg-lying adults after a developmental period of about two weeks.” Let T denote the development time, that is, the time needed for a laid egg to turn into a sexually mature adult. According to the verbal description, if the number of adult flies at time t − T is high, then the recruitment rate at time t will be zero. On the other hand, if the number of adult flies at time t − T is low, then the recruitment rate at time t will be positive. We state this as two fuzzy rules: • If x(t − T ) is high then r(t) = 0. • If x(t − T ) is low then r(t) = b, where b > 0. Note that we make no distinction between adult competition and larval competition. We assume that both forms of competition lead to the same result: a diminishing recruitment rate. Yet, this seems to be in agreement with Nicholson’s observations [17, p. 157]: “In cultures governed by larval competition only, a large number of flies laid so many eggs that larval over-crowding prevented any offspring from reaching maturity. In cultures governed by adult competition only, adult over-crowding prevented any eggs being produced.” 3.1.2. Defining the fuzzy terms The next step is to define the fuzzy terms in the rules using suitable membership functions. We use high (z) = (z; ch , wh ),
low (z) = (z; cl , wl ),
where wh , wl > 0, 0 < cl < ch , and ⎧ 0 ⎪ ⎪ ⎨ (z − c + w)/w (z; c, w) := (−z + c + w)/w ⎪ ⎪ ⎩ 0
if if if if
z c − w, c − w < z c, c < z c + w, c+w
(see Fig. 2). Note that this type of triangular membership function is quite common in fuzzy modeling. The piecewise-linear structure of will allow us to provide a rather complete analysis of the mathematical model. It is also possible of course to use other types of membership functions (e.g., trapezoidal). Yet, the triangular membership function seems more appropriate since it has a single maximum point. This is in agreement with Nicholson’s [17, p. 155] observations: “. . .there was not merely an increase in mortality with increasing larval density, but also that, after a critical point had been reached, further increase in density caused an actual decrease in the total number of flies emerging from a unit quantity of larval food.” We can now infer the fuzzy rules to obtain a mathematical formula for r(t).
Δ (z; c, w)
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1
c-w
c
c+w z
Fig. 2. The function (z; c, w).
3.1.3. Fuzzy inference We use simple additive inferencing (see, e.g., [2]). This yields r(t) = blow (x(t − T )).
(3)
It follows from this equation that the parameter b is the maximal reproduction rate that the population, as a whole, can achieve. Substituting (2) and (3) in (1) yields the complete model: x(t) ˙ = b(x(t − T ); cl , wl ) − x(t).
(4)
It is important to note that this model, or any first-order model for that matter, is an oversimplified representation of the real dynamics. A more accurate model must take into account many additional factors. First, the four distinct life stages of the flies. Second, the fact that the development time T may depend on various external factors and also vary between different flies. Third, a crowded environment cannot be characterized by the single function x(t). A crowded environment is actually a dynamic situation experienced differently by the members of the population. The larvae that develop most rapidly and emerge first are likely to experience a very different environment than those that emerge later [15]. Summarizing, we cannot expect that this model, for any function r, will reproduce the real data precisely, and we must pose a more modest aim. Namely, a model that accounts for some of the salient features apparent in the real data. In the next two sections, we study the behavior of (4) using both rigorous analysis and simulations. 4. Analysis Denoting y(t) := x(t)/b yields y(t) ˙ = (by(t − T ); cl , wl ) − y(t) = (y(t − T ); c, w) − y(t),
(5)
where c := cl /b and w := wl /b. It follows from standard arguments [6, Chapter VI] that (5) admits a unique solution y(t), t 0, for any initial condition y(t) ˙ = (t),
t ∈ [−T , 0],
where is a continuous function. Substituting the definition of in (5) yields ⎧ −y(t) ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎨ (y(t − T ) − c + w) − y(t) w y(t) ˙ = 1 ⎪ ⎪ (−y(t − T ) + c + w) − y(t) ⎪ ⎪ w ⎪ ⎩ −y(t)
if y(t − T ) c − w, if c − w < y(t−)c, if c < y(t − T ) c + w, if c + w < y(t − T ).
Note that this is a switched system where both the dynamics and the switching-law include time delay.
(6)
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4.1. Local behavior The equilibrium solutions of (5) are y(t) ≡ ye where ye satisfies (ye ; c, w) = ye . The number and location of solutions depend on the relation between the parameters. We will assume that cl − wl > 0 (see Fig. 2) so c > w. In this case • if > 1/c then (5) has a single equilibrium y(t) ≡ 0; • if = 1/c then (5) has two equilibrium solutions y(t) ≡ 0 and y(t) ≡ c; • if < 1/c then (5) has three equilibrium solutions y(t) ≡ y0 := 0, y(t) ≡ y1 := (c − w)/(1 − w) and y(t) ≡ y2 := (c + w)/(1 + w). In the first case, the death rate is so high that the population dwindles until it reaches complete extinction. The second case is “nongeneric”, as the parameters must satisfy an equality relation. Thus, we assume from here on that < 1/c. In this case, we also have < 1/w, so y0 < y1 < y2 . Proposition 1. Suppose that c > w and that < 1/c. Then, y(t) ≡ 0 is locally asymptotically stable and y(t) ≡ y1 and y(t) ≡ y2 are unstable. Proof. See the Appendix. The stability of y0 implies of course that if the initial population is too small, then it will diminish completely. The fact that y1 and y2 are unstable suggests that for other initial conditions, we may expect nontrivial behavior. 4.2. Global behavior We begin by showing that the population size is always bounded. It follows from (5) that −y(t) y(t) ˙ 1 − y(t) ∀t. This immediately yields the following result. Proposition 2. Suppose that y(0) > 0. Then, 0 < y(t)y(0) + 1/
∀t 0.
(7)
In other words, the population size always remains bounded above and below. Note that this agrees with the behavior observed in nature [17, p. 156]: “The density-governed system of reaction just described held the population in a state of stability, in the sense that it prevented both indefinite increase and indefinite decrease in numbers.” The next result shows that, under suitable conditions, the solution y(t) is periodic. For z ∈ R and times t1 < t2 , we use the notation y[t1 , t2 ) > z as a shorthand for “y(t) > z for all t ∈ [t1 , t2 )”. Theorem 1. Suppose that there exists a time such that y[ − T , ) > c + w and y() = c + w. Suppose also that the following conditions hold: c+w ln < T , (8) c−w c+w c ln < w(1 − w − c), (9) c w(c + w) exp(T ) < c ln exp(−T ) +
c2
c2 , − w2
1 1 c2 < ln 2 exp(T ). 2 w c −w (c + w)
Then the solution y(t), t , of (5) is a periodic function.
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y (t)
I1
I2
I3
I4
I5
I6
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c+w c c-w 0 1 2
T 3 4 5 T + 1
T + 2 T + 3 T + 4
T +5
η
t Fig. 3. A schematic view of one cycle of the periodic function y(t).
The proof, given in the Appendix, shows that each period has the schematic form shown in Fig. 3 (where, without loss of generality, is assumed to be zero). In particular, there are two maxima per cycle. The reason for this can be explained as follows. Recall that the fuzzy rules imply that the adult recruitment rate, at time y(t + T ), will be different than zero only if c − w < y(t) < c + w.
(12)
As we can see from Fig. 3, (12) is satisfied for t ∈ (0, 2 ). This yields a maximal value of y(t) about T seconds later. Condition (12) holds also for t ∈ (3 , 5 ). This yields another maximum point about T seconds later. Since 2 − 0 > 5 − 3 , the interval (0, 2 ) yields a stronger effect on y then the interval (3 , 5 ). This is why the first maximum value is larger than the second. Note that Fig. 3 also implies that y(t) repeatedly crosses the values of the equilibrium solutions y1 and y2 since c − w < y1 < y2 < c + w. This is in agreement with Nicholson’s observations [17, p. 156]: “. . . but, because there was a lag of about two weeks between the initiation of corrective reaction and its operation upon the population, the population change inducing the reaction continued without change in direction for this period, so causing excessive reaction and an alternating over- and under-shooting of the equilibrium position.” The periodic behavior implies, of course, that the moving time average of y(t) converges to a constant. Again, this agrees with Nicholson’s observations [17, p. 156]: “In each of the numerous experimental cultures of L. cuprina which have now been studied, it was found that, although there was commonly a great difference between mortality and natality during any given day, these two factors became equal in intensity over periods of several months. Thus, in spite of the usual great disparity between mortality and natality at any given time in L. cuprina cultures held under constant conditions, the innate tendency of the insects to multiply was found to be counteracted exactly by density induced reaction.” 5. Simulations Our simulations were motivated by one of the experiments performed by Nicholson [16,17]. To isolate the effect of larval competition, Nicholson provided surplus food for the adult flies while limiting the food supply for the larvae. In one of his experiments, Nicholson examined two cultures with identical conditions except that the daily quota of meat for the larvae was 50 g in one culture and 25 g in the second.
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3000 2500 2000 1500 1000 500 0 0
20
40
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80
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day Fig. 4. The function x(t) for m = 50 (solid line) and m = 25 (dashed line).
In order to simulate this scenario, we need to relate the daily quota of meat to the parameters in our model. If the quota of meat increases, then the competition is relaxed and more eggs will eventually turn into adults. This means that the value of the parameter b increases. Also, the recruitment rate will be greater than zero for a larger range of egg numbers. This suggests that the definition of the term low must include a larger set of values. Thus, we simulated (4) with the parameters b = 20m,
cl = 8m,
wl = 6m,
= 0.15
and
T = 14.8,
where m represents the daily quota of meat. The initial condition was x(t) = 20m for t ∈ [−T , 0]. Fig. 4 depicts x(t) as a function of time for two values of m. It may be seen that the quantitative behavior in both cases is similar. Increasing m leads to a scaling up of x(t). This is congruent with the results observed by Nicholson [17, p. 157]: “…the average numerical level of a population controlled by food supply varies directly with the quantity of this supply . . .the violence and other characteristics of the oscillation are unaltered by the difference in the rate of food supply.” 6. Quantitative behavior Although we mainly considered qualitative behavior, our results seem to provide a reasonable quantitative agreement with the results measured by Nicholson. Life history data shows that the time needed for eggs to develop into sexually mature eggs is T = 14.8 ± 0.4 days [9]. Gurney et al. [9] found that in Nicholson’s data 2.5 <
Tcycle < 2.7, T
29 <
xmax < 53, xmin
where Tcycle is the period of one cycle and xmax (xmin ) is the largest (smallest) number of adult flies during the cycle. In the simulations described above, we used T = 14.8. Analyzing the results for x(t) yields the following results. For m = 50: Tcycle = 37.75, xmax = 2881.40 and xmin = 75.84. For m = 25: Tcycle = 37.75, xmax = 1440.7 and xmin = 37.92. Hence, in both cases Tcycle = 2.55, T
xmax = 37.99, xmin
and these values agree with the actual measurements.
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7. Discussion In many of the “soft sciences” (e.g., psychology, sociology, ethology) scientists provided verbal descriptions and explanations of various phenomena based on field observations. Obtaining a suitable mathematical model, describing the observed system or behavior, can greatly enhance our ability to understand and study it in a scientific manner. Fuzzy modeling seems to be the most suitable tool for transforming this verbal information into analytical models. We demonstrated this by using fuzzy modeling to develop a mathematical model for the mechanisms that regulate population size as described by Nicholson. The fuzzy part of the model is quite simple, as it includes only two fuzzy rules. This is a natural result of the fact that the verbal description is quite simple, and the fuzzy rules are directly derived from this description. The resulting mathematical model has several features that are common in models derived using fuzzy modeling. It is very closely related to the verbal description and, although simple, it provides a reasonable qualitative and quantitative agreement with the actual results measured by Nicholson. The model is also simple enough to allow a rather complete analysis and we proved that, under some conditions on the values of the parameters, the solution is (1) periodic; and (2) produces the characteristic “double-humped peak”. To the best of our knowledge, this is the first model for which such a proof exists. Our model suggests that the quasi-periodic behavior is produced by a self-sustaining limit cycle perturbed by stochastic factors. The underlying periodic behavior is caused by delayed density dependence. This agrees with the analysis of Gurney et al. [9] who also concluded that the fluctuations observed by Nicholson, in the adult food-limited case, are of the limit cycle type. Recently, considerable research interest is devoted to the design of artificial systems inspired by the behavior of living systems. An important component in this field is the ability to perform reverse engineering of the living system’s functioning and then implement this behavior in an artificial system. We believe that the fuzzy modeling approach may be very suitable for addressing this problem in a systematic manner. Namely, start with a verbal description of an animal’s behavior (e.g., foraging in ants) and, using fuzzy logic theory, obtain a mathematical model of this behavior which can be immediately implemented in artificial systems (e.g., autonomous robots) [19]. Acknowledgments We are grateful to the anonymous reviewers for several constructive comments. Appendix. Proofs Proof of Proposition 1. To analyze the behavior near y(t) = y0 , t ∈ [−T , 0], consider a function (t) such that 0 (t)c − w for all t ∈ [−T , 0]. Then (6) yields y(t) = exp(−t)y(0), t ∈ [0, T ]. Hence, y[0, T ] c − w and (6) implies that y(t) = exp(−t)y(T ), t ∈ [T , 2T ]. Continuing in this fashion yields limt→∞ y(t) = 0. Hence, y0 is locally asymptotically stable. We now consider the local behavior in the vicinity of y1 . Denoting z(t) := y(t) − y1 yields ⎧ −(z(t) + y1 ) if z(t − T )c − w − y1 , ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎨ z(t − T ) − z(t) if c − w − y1 < z(t − T ) c − y1 , z˙ (t) = w 1 (13) 1 − c ⎪ ⎪ − < z(t − T ) c + w − y , z(t − T ) − z(t) + 2 if c − y 1 1 ⎪ ⎪ 1 − w ⎪ w ⎩ −(z(t) + y1 ) if c + w − y1 < z(t − T ). Consider the initial condition y(t) = (t), t ∈ [−T , 0] with |(t) − y1 | so |z(t)| for t ∈ [−T , 0]. For > 0 sufficiently small (13) yields z˙ (t) = (1/w)z(t − T ) − z(t). Since w < 1, standard arguments [18, Proposition 4] imply that 0 is an unstable equilibrium of z(t), so y1 is an unstable equilibrium of (5). A similar analysis in the vicinity of y2 completes the proof. Proof of Theorem 1. We will use the following result.
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Proposition 3. Consider the delay differential equation y(t) ˙ = c0 + c1 y(t) + c2 y(t − T ),
t t0 ,
with c1 = 0 and the initial condition y(t − T ) = a + exp(c1 t)
k
bi t i ,
t ∈ [t0 , t0 + T ].
i=0
Then, bi−1 c 0 + c2 a (exp(c1 (t − t0 )) − 1) + c2 exp(c1 t) (t i − t0i ) c1 i k+1
y(t) = exp(c1 (t − t0 ))y(t0 ) +
i=1
for all t t0 . Proof. By direct calculation. Without loss of generality, we assume that = 0. We divide the time axis into consecutive intervals I1 , I2 , . . . , and analyze the solution y(t) on every interval (see Fig. 3). We denote the restriction of y(t) on interval Ii by yi (t). Interval 1: I1 = [0, T ]. ˙ = −y(t). The solution along this interval is then Here, y(t − T )c + w for all t ∈ I1 , so (6) yields y(t) y1 (t) = exp(−t)(c + w).
(14)
It follows that for 1 := (1/) ln (c + w)/c and 2 := (1/) ln(c + w)/(c − w), we have y1 (1 ) = c and y1 (2 ) = c − w. It is easy to verify that (8) implies that 2 < T . Interval 2: I2 = [T , T + 1 ]. Here, c < y(t − T ) < c + w for all t ∈ I2 , so (6) yields y(t) ˙ = c0 + c1 y(t) + c2 y(t − T ),
(15)
with c0 = (c + w)/w, c1 = −, and c2 = −1/w. It follows from (14) that y(t − T ) = exp(−t)b0 , with b0 = exp(T )(c + w). Applying Proposition 3 with t0 = T and k = 0 yields 1 1 c+w y2 (t) = w exp(−t) + + T − t − exp(−(t − T )) . w
(16)
Substituting t = T + 1 and using the definition of 1 yields y2 (T + 1 ) = c exp(−T ) +
1 c c+w − ln . w c
(17)
It is easy to verify that (9) implies that y2 (T + 1 ) > c + w. We now examine y˙2 (t). It follows from (16) that y˙2 (t) = exp(−t)Q(t), where Q(t) is a linear polynomial. Using (14) and (15) yields c+w 1 y(T ˙ )= − y(T ) − y(0) w w = −y(T ) < 0.
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Similarly, (15) and the definition of 1 yield c+w 1 y(T ˙ + 1 ) = − y(T + 1 ) − y(1 ) w w = 1 − y(T + 1 ) > 0.
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(20)
It now follows from (18)–(20) that there exists tm ∈ I2 such that y˙2 (t) < 0 for all t ∈ [T , tm ), y˙2 (tm ) = 0, and y˙2 (t) > 0 for all t ∈ (tm , T + 1 ]. Hence, y2 attains a unique minimum at t = tm and is monotonically increasing for t > tm . Hence, there exist 3 , 4 , 5 , with T < 3 < 4 < 5 < T + 1 , such that y2 (3 ) = c − w, y2 (4 ) = c, and y2 (5 ) = c + w. Interval 3: I3 = [T + 1 , T + 2 ]. Here, c − w < y(t − T ) < c for all t ∈ I3 , so (6) yields y(t) ˙ = c0 + c1 y(t) + c2 y(t − T ), with c0 = (w − c)/w, c1 = −, and c2 = 1/w. It follows from (14) that y(t − T ) = exp(−t)b0 , with b0 = exp(T )(c + w). Applying Proposition 3 with t0 = T + 1 and k = 0 yields c−w y3 (t) = exp(−(t − T − 1 ))y2 (T + 1 ) + (exp(−(t − T − 1 )) − 1) w c+w + exp(−(t − T ))(t − T − 1 ) w c+w 2 c+w 1 w−c = exp(−(t − T )) w exp(−T ) − ln + +t −T + , (21) w c w where the second equation follows from the definition of 1 and (17). Differentiating this expression, and using the definition of 1 and (17), we find that y3 (t) attains a unique maximum at time tm := T − w exp(−T ) + (2/) ln ((c + w)/c). It is easy to verify that tm < T + 2 . The maximal value is y3 (tm ) =
1 (w 2 − c2 + c2 exp(w exp(−T ))). w(c + w)
Eq. (21) and the definition of 2 yield c−w 1 c2 . w exp(−T ) + ln 2 y3 (T + 2 ) = w c − w2
(22)
Interval 4: I4 = [T + 2 , T + 3 ]. ˙ = −y(t). The solution along this interval is then Here, y(t − T ) < c − w for all t ∈ I4 , so (6) yields y(t) y4 (t) = exp(−(t − T − 2 ))y3 (T + 2 ) c+w = exp(−(t − T )) y3 (T + 2 ), c−w where the second equation follows from the definition of 2 . Hence, y4 (T + 3 ) = exp(−3 )
(23)
c+w y3 (T + 2 ), c−w
and substituting (22) yields y4 (T + 3 ) = exp(−3 )
1 c2 c+w . w exp(−T ) + ln 2 w c − w2
(24)
Using the fact that 3 < T + 1 and (10) yields y4 (T + 3 ) > c + w. Interval 5: I5 = [T + 3 , T + 4 ].
(25)
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Here, c − w < y(t − T ) < c for all t ∈ I5 , so (6) yields y(t) ˙ = c0 + c1 y(t) + c2 y(t − T ),
(26)
with c0 = (w − c)/w, c1 = −, and c2 = 1/w. We know that t − T 3 and, therefore, y(t − T ) c − w. It follows that w−c 1 y(t) ˙ − y(t) + (c − w) = −y(t) w w and, therefore, y(t) exp(−(t − T − 3 ))y(T + 3 ).
(27)
In particular, using (24) yields
1 c2 c+w w exp(−T ) + ln 2 y(T + 4 ) exp(−(4 − 3 )) exp(−3 ) w c − w2 2 c 1 = exp(−4 )(c + w) exp(−T ) + . ln 2 w c − w2
Using the fact that 4 T + 1 yields
c2 1 ln 2 y(T + 4 ) exp(−(T + 1 ))(c + w) exp(−T ) + w c − w2 2 1 c = exp(−T )c exp(−T ) + , ln 2 w c − w2
and (10) implies that y(T + 4 ) > c exp(−2T ) + c + w. Hence, y5 (t) > c + w
for all t ∈ I5 .
(28)
We now derive an upper bound for y5 (t). Since t − T 4 , y(t − T ) c and this yields w−c 1 y(t) ˙ − y(t) + c w w = 1 − y(t), so y5 (t)1/ + exp(−(t − T − 3 ))(y(T + 3 ) − 1/). Using (24) and (11), we find that y5 (t) < 1/.
(29)
We now analyze y˙5 (t). It follows from (16) that c+w 1 1 y(t − T ) = w exp(−(t − T )) + + 2T − t − exp(−(t − 2T )) , w and using (26) and Proposition 3 yields y5 (t) = r + exp(−t)P (t), where r is a constant, and P (t) is a quadratic polynomial. Hence, y˙5 (t) = exp(−t)Q(t), where Q(t) is a quadratic polynomial.
(30)
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Using (26) and the definition of 3 yields y(T ˙ + 3 ) < 0.
(31)
Similarly, using (26), (29) and the definition of 4 yields y(T ˙ + 4 ) = 1 − y(T + 4 ) > 0.
(32)
It now follows from (30)–(32) that there exists tm ∈ I5 such that y˙5 (t) < 0 for all t ∈ [T + 3 , tm ), y˙5 (tm ) = 0, and y˙5 (t) > 0 for all t ∈ (tm , T + 4 ]. Hence, y5 (t) attains a unique minimum at time tm . Interval 6: I6 = [T + 4 , T + 5 ]. Here, c < y[t − T , t] < c + w so (6) yields y(t) ˙ = c0 + c1 y(t) + c2 y(t − T ), with c0 = (c + w)/w, c1 = −, and c2 = −1/w. We know that 4 t − T 5 and therefore c y(t − T ) c + w. It follows that c+w 1 y(t) ˙ − y(t) − (c + w) w w = −y(t), so y(t) exp(−(t − T − 4 ))y(T + 4 ). In particular, using (27) yields y(t) exp(−(t − T − 4 )) exp(−(4 − 3 ))y(T + 3 ) 1 c2 exp(−(t − T ))(c + w) exp(−T ) + , ln 2 w c − w2 and using the fact that t − T 5 implies that c2 1 . ln 2 y(t) exp(−5 )(c + w) exp(−T ) + w c − w2 By definition, 5 T + 1 and therefore, as in the analysis in interval I5 , we find that y6 (t) > c + w
for all t ∈ I6 .
(33)
Arguing as in the analysis in interval I5 , we find that there exists tm ∈ I6 such that y˙6 (t) > 0 for all t ∈ [T + 4 , tm ), y˙6 (tm ) = 0, and y˙6 (t) < 0 for all t ∈ (tm , T + 5 ]. Hence, y6 (t) attains a unique maximum at t = tm . Interval 7: I7 = [T + 5 , 2T + 5 ]. Using (25), (28) and (33) yields y[t − T , t]c + w, so y7 (t) = exp(−(t − T − 5 ))y6 (T + 5 ).
(34)
Thus, y7 (t) is a strictly decreasing function. Note that (33) yields y6 (T + 5 ) > c + w. We now consider two possible cases. Case 1: Suppose that there exists ∈ I7 such that y7 () = c +w. In this case, y[−T , ) > c +w, so y(t) = y(t +) for all t ∈ [0, T ]. Hence, the solution from here on is periodic with period . Case 2: Suppose that there is no ∈ I7 such that y7 () = c + w. In this case, y(t) > c + w for all t ∈ I7 . Defining I8 := [2T + 5 , 3T + 5 ], we see that y8 (t) = exp(−(t − 2T − 5 ))y7 (2T + 5 ), so y(t) continues to decrease. If there is a point ∈ I8 such that y8 () = c + w then, as in Case 1, we get that y(t) is periodic from here on. Otherwise, y(t) will continue to decrease exponentially in the next interval, and so on. Eventually, there must be a time such that y() = c + w and the solution will be periodic from this point on.
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References [1] Y. Bar-Cohen, C. Breazeal (Eds.), Biologically Inspired Intelligent Robots, SPIE Press, 2003. [2] J.M. Benitez, J.L. Castro, I. Requena, Are artificial neural networks black boxes, IEEE Trans. Neural Networks 8 (1997) 1156–1164. [3] O.N. Bjornstad, B.T. Grenfell, Noisy clockwork: time series analysis of population fluctuations in animals, Science 293 (2001) 638–643. [4] D.R. Brillinger, J. Guckenheimer, P. Guttorp, G. Oster, Empirical modelling of population time series data: the case of age and density dependent vital rates, Lectures on Mathematics in the Life Sciences, vol. 13, 1980, pp. 65–90. [5] C. Chang, P. Gaudiano, Biomimetic robotics, Robotics and Autonomous Systems 30 (1) (2000) 1–2. [6] R. D. Driver, Ordinary and Delay Differential Equations, Applied Mathematical Sciences, vol. 20, Springer, Berlin, 1977. [7] D. Dubois, H.T. Nguyen, H. Prade, M. Sugeno, Introduction: the real contribution of fuzzy systems, in: H.T. Nguyen, M. Sugeno (Eds.), Fuzzy Systems: Modeling and Control, Kluwer, Dordrecht, 1998, pp. 1–17. [8] G.S. Fraenkel, D.L. Gunn, The Orientation of Animals: Kineses, Taxes, and Compass Reactions, Dover Publications, New York, 1961. [9] W.S.C. Gurney, S.P. Blythe, R.M. Nisbet, Nicholson’s blowflies revisited, Nature 287 (1980) 17–21. [10] A. Hastings, C.L. Hom, S. Ellner, P. Turchin, H. Charles, J. Godfray, Chaos in ecology: is mother nature a strange attractor?, Annu Rev. Ecol. Systematics 24 (1993) 1–33. [11] B.E. Kendall, C.J. Briggs, W.W. Murdoch, P. Turchin, S.P. Ellner, E. McCauley, R.M. Nisbet, S.N. Wood, Why do populations cycle? a synthesis of statistical and mechanistic modeling approaches, Ecology 80 (1999) 1789–1805. [12] M. Margaliot, G. Langholz, Fuzzy Lyapunov based approach to the design of fuzzy controllers, Fuzzy Sets and Systems 106 (1) (1999) 49–59. [13] M. Margaliot, G. Langholz, New Approaches to Fuzzy Modeling and Control—Design and Analysis, World Scientific, Singapore, 2000. [14] C. Mattheck, Design in Nature: Learning from Trees, Springer, Berlin, 1998. [15] L.D. Mueller, Theoretical and empirical examination of density-dependent selection, Annu. Rev. Ecol. Syst. 28 (1997) 269–288. [16] A.J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zool. 2 (1954) 9–65. [17] A.J. Nicholson, The self-adjustment of populations to change, Cold Spring Harbor Symposium on Quantitative Biology vol. 22, 1957, pp. 153–173. [18] S.-I. Niculescu, Erik I. Verriest, L. Dugard, J.-M. Dion, Stability and robust stability of time-delay systems: a guided tour. in: L. Dugard, E.I. Verriest, (Eds), Stability and Control of Time-Delay Systems, Lecture Notes in Control and Information Sciences, vol. 228, Springer, Berlin, 1998, pp. 1–71. [19] V. Rozin, M. Margaliot, The fuzzy ant. 2006, (Online), available www.eng.tau.ac.il/∼michaelm , submitted for publication. [20] E. Tron, M. Margaliot, Mathematical modeling of observed natural behavior: a fuzzy logic approach, Fuzzy Sets and Systems 146 (2004) 437–450. [21] E. Tron, M. Margaliot, How does the Dendrocoleum lacteum orient to light? a fuzzy modeling approach, Fuzzy Sets and Systems 155 (2005) 236–251. [22] S.N. Wood, Partially specified ecological models, Ecological Monogr. 71 (2001) 1–25. [23] L.A. Zadeh, Fuzzy logic = computing with words, IEEE Trans. Fuzzy Systems 4 (2) (1996) 103–111.
Nicholson’s blowflies revisited: A fuzzy modeling approach夡 Iosef Rashkovsky, Michael Margaliot∗ School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel Received 5 April 2006; received in revised form 20 August 2006; accepted 3 November 2006 Available online 30 November 2006
Abstract We apply fuzzy modeling to derive a mathematical model for a biological phenomenon: the regulation of population size in the Australian sheep-blowfly Lucilia cuprina. This behavior was described by several ethologists and fuzzy modeling allows us to transform their verbal descriptions into a well-defined mathematical model. The behavior of the resulting mathematical model, as studied using both simulations and rigorous analysis, is congruent with the behavior actually observed in nature. We believe that the fuzzy modeling approach demonstrated here may supply a suitable framework for biomimicry, that is, the design of artificial systems based on mimicking natural behavior. © 2006 Elsevier B.V. All rights reserved. Keywords: Linguistic modeling; Differential equations with time-delay; Periodic behavior; Switched systems with time delay; Population dynamics; Emergent behavior; Biomimicry
1. Introduction In many of the “soft sciences” (e.g., psychology, sociology, ethology) scientists provided verbal descriptions and explanations of various phenomena based on field observations. Obtaining a suitable mathematical model, describing the observed system or behavior, can greatly enhance our ability to understand and study it in a scientific manner. Indeed, mathematical models are very useful in summarizing and interpreting empirical data. Furthermore, once derived, such models allow us to analyze the system both qualitatively and quantitatively using mathematical tools. Tron and Margaliot [20,21] advocated fuzzy logic theory as the most suitable tool for transforming verbal descriptions of various observed phenomena into suitable mathematical models. This approach is congruent with the notion that the real power of fuzzy logic is in its ability to handle and manipulate linguistic information based on perceptions (see, e.g., [7,12,13,23]). Indeed, fuzzy modeling is routinely used to transform the knowledge of an expert, be it a physician or a process operator, into a computer algorithm. Yet, not enough attention has been paid to its possible use as a tool to assist human observers in transforming their verbal descriptions into mathematical models. The fuzzy modeling approach is based on transforming the verbal description into a set of if-then rules stated using natural language. Inferencing this rule-base yields a well-defined mathematical model. This approach has several advantages. The fuzzy rule-base represents the real system in a form that corresponds closely to the way humans 夡 An
abridged version of this paper was presented at the IEEE World Congress on Computational Intelligence (WCCI 2006).
∗ Corresponding author. Tel.: +972 3 6407768; fax: +972 3 6407095.
E-mail address: [email protected] (M. Margaliot) URL: http://www.eng.tau.ac.il/∼michaelm (M. Margaliot). 0165-0114/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2006.11.001
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perceive it. Thus, the model is understandable, even by nonprofessionals, and each parameter has a readily perceivable meaning. The model is flexible and can be easily altered to incorporate new phenomena. The information is kept in three different forms: the original verbal description, the fuzzy rule-base, and the mathematical model. The interplay between these representations provides a synergistic view of the system. For example, suppose that simulations of the mathematical model indicate that its behavior is different than that of the real system. By analyzing the simulations, it is sometimes possible to find which fuzzy rule must be altered and how. Since the rules are stated in natural language, any change in the rules can be interpreted in terms of a change of the original verbal description. Thus, the modeling process can also be used to prove or refute the modeler’s ideas as to how the natural system behaves and why. Several factors make the fuzzy approach particularly suitable for developing models of animal behavior. First, ethologists usually provide detailed verbal descriptions of the phenomena they are studying. Some of these descriptions can be easily converted into fuzzy rules. For example, Fraenkel and Gunn describe the behavior of a cockroach, that becomes stationary when a large part of its body surface is in contact with a solid object, as: “A high degree of contact causes low activity. . .” [8, p. 23]. Second, mathematical models of animal behavior may also have interesting engineering applications. In fact, the development of artificial products or machines that mimic biological phenomena, sometimes referred to as biomimicry, is recently attracting considerable interest (see, e.g., [1,14,5]). The basic idea is that during their course of evolution living systems developed sophisticated mechanisms for solving various challenges. Designers of artificial products, that must function in the real world, are often required to address similar problems. Thus, information extracted from biological behavior may inspire suitable artificial designs. For example, animals must develop efficient mechanisms for orienting themselves in space. Similar problems arise in robotics, as planning and realizing oriented movements is of crucial importance in designing autonomous robots. An important component in biomimicry is the ability to perform reverse engineering of an animal’s functioning and then implement this behavior in an artificial system. We believe that the fuzzy modeling approach may be very suitable for addressing biomimicry in a systematic manner. Namely, start with a verbal description of an animal’s behavior (e.g., foraging in ants) and, using fuzzy logic theory, obtain a mathematical model of this behavior which can be implemented in artificial systems (e.g., autonomous robots). In this paper, we use the fuzzy modeling approach to derive a mathematical model for a phenomenon of animal behavior: the regulation of population size in the blowfly Lucilia cuprina. The mechanisms involved in the regulation process were described and explained in the classical works of Nicholson [16,17]. The behavior of the resulting mathematical model, as studied using both simulations and rigorous analysis, is congruent with the behavior actually observed in nature. The remainder of this paper is organized as follows. Section 2 reviews the verbal descriptions and explanations given by Nicholson. Section 3 applies fuzzy modeling to transform these descriptions into a mathematical model. Section 4 analyzes the mathematical model. Sections 5 describes the results of simulations and compares these to the behavior actually observed in nature. Section 6 discusses the quantitative aspects of the model. The final section concludes. 2. The self-adjustment of population in the Lucilia cuprina Population dynamics plays a fundamental role in the animal kingdom. Sophisticated feedback mechanisms regulate the size of the population, adjusting it to the constantly changing internal and external factors. In a classic study, Nicholson [17] analyzed the regulation of population size in laboratory colonies of the Australian sheep-blowfly Lucilia cuprina. He introduced a small number of flies into a cage and, from then on, maintained predetermined conditions leaving the population to its own device. Although the environment remained invariable, the population size showed large-amplitude quasi-periodic oscillations [4] (see Fig. 1). One of the cultures was maintained for a period of two years and the oscillations continued for this entire period of time. Many different interacting factors regulate the population’s size and lead to this nontrivial behavior. The first of these is competition between the individual members of the population for the fixed amount of food. If the population size is small, then the competition is not severe and interbreeding leads to an increase in the population size. However, at a certain point the food supply is no longer sufficient to maintain the population. This leads to increased mortality and decreased natality. The population size decreases, and the competition is relaxed again, thus completing a cycle.
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9000 8000 7000 6000 5000 4000 3000 2000 1000 0 0
50
100
150
200
250
300
350
day Fig. 1. Fluctuations in laboratory populations of Lucillia cuprina.
The regulating mechanisms are further complicated because of the various stages in the blowfly’s life cycle: (1) adults interbreed and lay eggs; (2) immature larva hatch from the egg; (3) the larva turn into pupa; and (4) the pupa transforms into adult. Nicholson’s experiments are interesting for several reasons. First, it is surprising that fixed conditions, in a controlled environment, lead to such extreme oscillations in the population size. Second, from a dynamic systems point of view, the quasi-periodic behavior is not trivial and raises interesting questions. For example, is the behavior chaotic [10] or, alternatively, the result of a periodic process “probed” by stochastic factors? 1 Third, Nicholson was able to separate the food supply of the adult flies and the larvae. This allowed him to experiment different types of competition. For example, by supplying a surplus quantity of food for the adult flies and limiting the larval food, it is possible to isolate the effect of larval competition. Finally, the mechanisms that regulate the population usually operate at the level of individual animals. For example, if a larva does not receive sufficient food, then it will not pupate. Yet, the combined effect is an emergent behavior regulating the total population size. This was already noted by Nicholson [17, p. 154]: “. . .each of these groups has characteristics which are more than the sum of those of the constituent individuals.” Population dynamics is a very rich field. Methods for analyzing population processes can be roughly divided into two groups. The first group includes methods based on applying statistical tools to analyze the time series. The second group includes methods that develop a mathematical model for the observed phenomena. For more information on these topics, see [11,3] and the many references therein. In the next section, we use fuzzy modeling to develop a mathematical model for the behavior described by Nicholson. 3. Mathematical modeling The fuzzy modeling approach transforms verbal descriptions into mathematical models using several stages. First, the variables in the model are determined. For example, in a description of a moving animal, the animal’s location is a variable in R3 . Second, the verbal information is stated as fuzzy rules relating the variables. These rules include fuzzy terms that must be defined mathematically using suitable membership functions. Finally, applying the inference mechanism to the rules yields a well-defined mathematical model [20]. Here we use a slightly different approach. Following the seminal paper [9], we begin by writing down a balance equation for the population of sexually mature adults at time t which we denote by x(t). The rate of change in x(t) is the difference between the rate of recruitment to the adult population r(t), and the total adult death rate d(t): x(t) ˙ = r(t) − d(t). 1 For more on the interplay between deterministic and stochastic forces in population dynamics, see [3].
(1)
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The death rate is modeled by the linear function d(t) = x(t),
(2)
where > 0 is the per capita death rate. Thus, we assume that the functional form of the model is known, except for the function r(t) [22]. We use fuzzy modeling to determine the function r(t), and thus complete the mathematical model. 3.1. Fuzzy modeling 3.1.1. The fuzzy rules Our rules are based on the following description [17, p.156]: “When the numbers of adults were high such vast numbers of eggs were laid that all of the food provided was consumed while the larvae were still too small to pupate; consequently no adult offspring resulted from eggs laid during such periods. The adult numbers therefore dwindled progressively, until a point was reached at which the intensity of larval competition became so reduced that some of the larvae attained a sufficient size to pupate. These gave rise to egg-lying adults after a developmental period of about two weeks.” Let T denote the development time, that is, the time needed for a laid egg to turn into a sexually mature adult. According to the verbal description, if the number of adult flies at time t − T is high, then the recruitment rate at time t will be zero. On the other hand, if the number of adult flies at time t − T is low, then the recruitment rate at time t will be positive. We state this as two fuzzy rules: • If x(t − T ) is high then r(t) = 0. • If x(t − T ) is low then r(t) = b, where b > 0. Note that we make no distinction between adult competition and larval competition. We assume that both forms of competition lead to the same result: a diminishing recruitment rate. Yet, this seems to be in agreement with Nicholson’s observations [17, p. 157]: “In cultures governed by larval competition only, a large number of flies laid so many eggs that larval over-crowding prevented any offspring from reaching maturity. In cultures governed by adult competition only, adult over-crowding prevented any eggs being produced.” 3.1.2. Defining the fuzzy terms The next step is to define the fuzzy terms in the rules using suitable membership functions. We use high (z) = (z; ch , wh ),
low (z) = (z; cl , wl ),
where wh , wl > 0, 0 < cl < ch , and ⎧ 0 ⎪ ⎪ ⎨ (z − c + w)/w (z; c, w) := (−z + c + w)/w ⎪ ⎪ ⎩ 0
if if if if
z c − w, c − w < z c, c < z c + w, c+w
(see Fig. 2). Note that this type of triangular membership function is quite common in fuzzy modeling. The piecewise-linear structure of will allow us to provide a rather complete analysis of the mathematical model. It is also possible of course to use other types of membership functions (e.g., trapezoidal). Yet, the triangular membership function seems more appropriate since it has a single maximum point. This is in agreement with Nicholson’s [17, p. 155] observations: “. . .there was not merely an increase in mortality with increasing larval density, but also that, after a critical point had been reached, further increase in density caused an actual decrease in the total number of flies emerging from a unit quantity of larval food.” We can now infer the fuzzy rules to obtain a mathematical formula for r(t).
Δ (z; c, w)
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1
c-w
c
c+w z
Fig. 2. The function (z; c, w).
3.1.3. Fuzzy inference We use simple additive inferencing (see, e.g., [2]). This yields r(t) = blow (x(t − T )).
(3)
It follows from this equation that the parameter b is the maximal reproduction rate that the population, as a whole, can achieve. Substituting (2) and (3) in (1) yields the complete model: x(t) ˙ = b(x(t − T ); cl , wl ) − x(t).
(4)
It is important to note that this model, or any first-order model for that matter, is an oversimplified representation of the real dynamics. A more accurate model must take into account many additional factors. First, the four distinct life stages of the flies. Second, the fact that the development time T may depend on various external factors and also vary between different flies. Third, a crowded environment cannot be characterized by the single function x(t). A crowded environment is actually a dynamic situation experienced differently by the members of the population. The larvae that develop most rapidly and emerge first are likely to experience a very different environment than those that emerge later [15]. Summarizing, we cannot expect that this model, for any function r, will reproduce the real data precisely, and we must pose a more modest aim. Namely, a model that accounts for some of the salient features apparent in the real data. In the next two sections, we study the behavior of (4) using both rigorous analysis and simulations. 4. Analysis Denoting y(t) := x(t)/b yields y(t) ˙ = (by(t − T ); cl , wl ) − y(t) = (y(t − T ); c, w) − y(t),
(5)
where c := cl /b and w := wl /b. It follows from standard arguments [6, Chapter VI] that (5) admits a unique solution y(t), t 0, for any initial condition y(t) ˙ = (t),
t ∈ [−T , 0],
where is a continuous function. Substituting the definition of in (5) yields ⎧ −y(t) ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎨ (y(t − T ) − c + w) − y(t) w y(t) ˙ = 1 ⎪ ⎪ (−y(t − T ) + c + w) − y(t) ⎪ ⎪ w ⎪ ⎩ −y(t)
if y(t − T ) c − w, if c − w < y(t−)c, if c < y(t − T ) c + w, if c + w < y(t − T ).
Note that this is a switched system where both the dynamics and the switching-law include time delay.
(6)
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4.1. Local behavior The equilibrium solutions of (5) are y(t) ≡ ye where ye satisfies (ye ; c, w) = ye . The number and location of solutions depend on the relation between the parameters. We will assume that cl − wl > 0 (see Fig. 2) so c > w. In this case • if > 1/c then (5) has a single equilibrium y(t) ≡ 0; • if = 1/c then (5) has two equilibrium solutions y(t) ≡ 0 and y(t) ≡ c; • if < 1/c then (5) has three equilibrium solutions y(t) ≡ y0 := 0, y(t) ≡ y1 := (c − w)/(1 − w) and y(t) ≡ y2 := (c + w)/(1 + w). In the first case, the death rate is so high that the population dwindles until it reaches complete extinction. The second case is “nongeneric”, as the parameters must satisfy an equality relation. Thus, we assume from here on that < 1/c. In this case, we also have < 1/w, so y0 < y1 < y2 . Proposition 1. Suppose that c > w and that < 1/c. Then, y(t) ≡ 0 is locally asymptotically stable and y(t) ≡ y1 and y(t) ≡ y2 are unstable. Proof. See the Appendix. The stability of y0 implies of course that if the initial population is too small, then it will diminish completely. The fact that y1 and y2 are unstable suggests that for other initial conditions, we may expect nontrivial behavior. 4.2. Global behavior We begin by showing that the population size is always bounded. It follows from (5) that −y(t) y(t) ˙ 1 − y(t) ∀t. This immediately yields the following result. Proposition 2. Suppose that y(0) > 0. Then, 0 < y(t)y(0) + 1/
∀t 0.
(7)
In other words, the population size always remains bounded above and below. Note that this agrees with the behavior observed in nature [17, p. 156]: “The density-governed system of reaction just described held the population in a state of stability, in the sense that it prevented both indefinite increase and indefinite decrease in numbers.” The next result shows that, under suitable conditions, the solution y(t) is periodic. For z ∈ R and times t1 < t2 , we use the notation y[t1 , t2 ) > z as a shorthand for “y(t) > z for all t ∈ [t1 , t2 )”. Theorem 1. Suppose that there exists a time such that y[ − T , ) > c + w and y() = c + w. Suppose also that the following conditions hold: c+w ln < T , (8) c−w c+w c ln < w(1 − w − c), (9) c w(c + w) exp(T ) < c ln exp(−T ) +
c2
c2 , − w2
1 1 c2 < ln 2 exp(T ). 2 w c −w (c + w)
Then the solution y(t), t , of (5) is a periodic function.
(10) (11)
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y (t)
I1
I2
I3
I4
I5
I6
1089
I7
c+w c c-w 0 1 2
T 3 4 5 T + 1
T + 2 T + 3 T + 4
T +5
η
t Fig. 3. A schematic view of one cycle of the periodic function y(t).
The proof, given in the Appendix, shows that each period has the schematic form shown in Fig. 3 (where, without loss of generality, is assumed to be zero). In particular, there are two maxima per cycle. The reason for this can be explained as follows. Recall that the fuzzy rules imply that the adult recruitment rate, at time y(t + T ), will be different than zero only if c − w < y(t) < c + w.
(12)
As we can see from Fig. 3, (12) is satisfied for t ∈ (0, 2 ). This yields a maximal value of y(t) about T seconds later. Condition (12) holds also for t ∈ (3 , 5 ). This yields another maximum point about T seconds later. Since 2 − 0 > 5 − 3 , the interval (0, 2 ) yields a stronger effect on y then the interval (3 , 5 ). This is why the first maximum value is larger than the second. Note that Fig. 3 also implies that y(t) repeatedly crosses the values of the equilibrium solutions y1 and y2 since c − w < y1 < y2 < c + w. This is in agreement with Nicholson’s observations [17, p. 156]: “. . . but, because there was a lag of about two weeks between the initiation of corrective reaction and its operation upon the population, the population change inducing the reaction continued without change in direction for this period, so causing excessive reaction and an alternating over- and under-shooting of the equilibrium position.” The periodic behavior implies, of course, that the moving time average of y(t) converges to a constant. Again, this agrees with Nicholson’s observations [17, p. 156]: “In each of the numerous experimental cultures of L. cuprina which have now been studied, it was found that, although there was commonly a great difference between mortality and natality during any given day, these two factors became equal in intensity over periods of several months. Thus, in spite of the usual great disparity between mortality and natality at any given time in L. cuprina cultures held under constant conditions, the innate tendency of the insects to multiply was found to be counteracted exactly by density induced reaction.” 5. Simulations Our simulations were motivated by one of the experiments performed by Nicholson [16,17]. To isolate the effect of larval competition, Nicholson provided surplus food for the adult flies while limiting the food supply for the larvae. In one of his experiments, Nicholson examined two cultures with identical conditions except that the daily quota of meat for the larvae was 50 g in one culture and 25 g in the second.
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3000 2500 2000 1500 1000 500 0 0
20
40
60
80
100
120
day Fig. 4. The function x(t) for m = 50 (solid line) and m = 25 (dashed line).
In order to simulate this scenario, we need to relate the daily quota of meat to the parameters in our model. If the quota of meat increases, then the competition is relaxed and more eggs will eventually turn into adults. This means that the value of the parameter b increases. Also, the recruitment rate will be greater than zero for a larger range of egg numbers. This suggests that the definition of the term low must include a larger set of values. Thus, we simulated (4) with the parameters b = 20m,
cl = 8m,
wl = 6m,
= 0.15
and
T = 14.8,
where m represents the daily quota of meat. The initial condition was x(t) = 20m for t ∈ [−T , 0]. Fig. 4 depicts x(t) as a function of time for two values of m. It may be seen that the quantitative behavior in both cases is similar. Increasing m leads to a scaling up of x(t). This is congruent with the results observed by Nicholson [17, p. 157]: “…the average numerical level of a population controlled by food supply varies directly with the quantity of this supply . . .the violence and other characteristics of the oscillation are unaltered by the difference in the rate of food supply.” 6. Quantitative behavior Although we mainly considered qualitative behavior, our results seem to provide a reasonable quantitative agreement with the results measured by Nicholson. Life history data shows that the time needed for eggs to develop into sexually mature eggs is T = 14.8 ± 0.4 days [9]. Gurney et al. [9] found that in Nicholson’s data 2.5 <
Tcycle < 2.7, T
29 <
xmax < 53, xmin
where Tcycle is the period of one cycle and xmax (xmin ) is the largest (smallest) number of adult flies during the cycle. In the simulations described above, we used T = 14.8. Analyzing the results for x(t) yields the following results. For m = 50: Tcycle = 37.75, xmax = 2881.40 and xmin = 75.84. For m = 25: Tcycle = 37.75, xmax = 1440.7 and xmin = 37.92. Hence, in both cases Tcycle = 2.55, T
xmax = 37.99, xmin
and these values agree with the actual measurements.
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7. Discussion In many of the “soft sciences” (e.g., psychology, sociology, ethology) scientists provided verbal descriptions and explanations of various phenomena based on field observations. Obtaining a suitable mathematical model, describing the observed system or behavior, can greatly enhance our ability to understand and study it in a scientific manner. Fuzzy modeling seems to be the most suitable tool for transforming this verbal information into analytical models. We demonstrated this by using fuzzy modeling to develop a mathematical model for the mechanisms that regulate population size as described by Nicholson. The fuzzy part of the model is quite simple, as it includes only two fuzzy rules. This is a natural result of the fact that the verbal description is quite simple, and the fuzzy rules are directly derived from this description. The resulting mathematical model has several features that are common in models derived using fuzzy modeling. It is very closely related to the verbal description and, although simple, it provides a reasonable qualitative and quantitative agreement with the actual results measured by Nicholson. The model is also simple enough to allow a rather complete analysis and we proved that, under some conditions on the values of the parameters, the solution is (1) periodic; and (2) produces the characteristic “double-humped peak”. To the best of our knowledge, this is the first model for which such a proof exists. Our model suggests that the quasi-periodic behavior is produced by a self-sustaining limit cycle perturbed by stochastic factors. The underlying periodic behavior is caused by delayed density dependence. This agrees with the analysis of Gurney et al. [9] who also concluded that the fluctuations observed by Nicholson, in the adult food-limited case, are of the limit cycle type. Recently, considerable research interest is devoted to the design of artificial systems inspired by the behavior of living systems. An important component in this field is the ability to perform reverse engineering of the living system’s functioning and then implement this behavior in an artificial system. We believe that the fuzzy modeling approach may be very suitable for addressing this problem in a systematic manner. Namely, start with a verbal description of an animal’s behavior (e.g., foraging in ants) and, using fuzzy logic theory, obtain a mathematical model of this behavior which can be immediately implemented in artificial systems (e.g., autonomous robots) [19]. Acknowledgments We are grateful to the anonymous reviewers for several constructive comments. Appendix. Proofs Proof of Proposition 1. To analyze the behavior near y(t) = y0 , t ∈ [−T , 0], consider a function (t) such that 0 (t)c − w for all t ∈ [−T , 0]. Then (6) yields y(t) = exp(−t)y(0), t ∈ [0, T ]. Hence, y[0, T ] c − w and (6) implies that y(t) = exp(−t)y(T ), t ∈ [T , 2T ]. Continuing in this fashion yields limt→∞ y(t) = 0. Hence, y0 is locally asymptotically stable. We now consider the local behavior in the vicinity of y1 . Denoting z(t) := y(t) − y1 yields ⎧ −(z(t) + y1 ) if z(t − T )c − w − y1 , ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎨ z(t − T ) − z(t) if c − w − y1 < z(t − T ) c − y1 , z˙ (t) = w 1 (13) 1 − c ⎪ ⎪ − < z(t − T ) c + w − y , z(t − T ) − z(t) + 2 if c − y 1 1 ⎪ ⎪ 1 − w ⎪ w ⎩ −(z(t) + y1 ) if c + w − y1 < z(t − T ). Consider the initial condition y(t) = (t), t ∈ [−T , 0] with |(t) − y1 | so |z(t)| for t ∈ [−T , 0]. For > 0 sufficiently small (13) yields z˙ (t) = (1/w)z(t − T ) − z(t). Since w < 1, standard arguments [18, Proposition 4] imply that 0 is an unstable equilibrium of z(t), so y1 is an unstable equilibrium of (5). A similar analysis in the vicinity of y2 completes the proof. Proof of Theorem 1. We will use the following result.
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Proposition 3. Consider the delay differential equation y(t) ˙ = c0 + c1 y(t) + c2 y(t − T ),
t t0 ,
with c1 = 0 and the initial condition y(t − T ) = a + exp(c1 t)
k
bi t i ,
t ∈ [t0 , t0 + T ].
i=0
Then, bi−1 c 0 + c2 a (exp(c1 (t − t0 )) − 1) + c2 exp(c1 t) (t i − t0i ) c1 i k+1
y(t) = exp(c1 (t − t0 ))y(t0 ) +
i=1
for all t t0 . Proof. By direct calculation. Without loss of generality, we assume that = 0. We divide the time axis into consecutive intervals I1 , I2 , . . . , and analyze the solution y(t) on every interval (see Fig. 3). We denote the restriction of y(t) on interval Ii by yi (t). Interval 1: I1 = [0, T ]. ˙ = −y(t). The solution along this interval is then Here, y(t − T )c + w for all t ∈ I1 , so (6) yields y(t) y1 (t) = exp(−t)(c + w).
(14)
It follows that for 1 := (1/) ln (c + w)/c and 2 := (1/) ln(c + w)/(c − w), we have y1 (1 ) = c and y1 (2 ) = c − w. It is easy to verify that (8) implies that 2 < T . Interval 2: I2 = [T , T + 1 ]. Here, c < y(t − T ) < c + w for all t ∈ I2 , so (6) yields y(t) ˙ = c0 + c1 y(t) + c2 y(t − T ),
(15)
with c0 = (c + w)/w, c1 = −, and c2 = −1/w. It follows from (14) that y(t − T ) = exp(−t)b0 , with b0 = exp(T )(c + w). Applying Proposition 3 with t0 = T and k = 0 yields 1 1 c+w y2 (t) = w exp(−t) + + T − t − exp(−(t − T )) . w
(16)
Substituting t = T + 1 and using the definition of 1 yields y2 (T + 1 ) = c exp(−T ) +
1 c c+w − ln . w c
(17)
It is easy to verify that (9) implies that y2 (T + 1 ) > c + w. We now examine y˙2 (t). It follows from (16) that y˙2 (t) = exp(−t)Q(t), where Q(t) is a linear polynomial. Using (14) and (15) yields c+w 1 y(T ˙ )= − y(T ) − y(0) w w = −y(T ) < 0.
(18)
(19)
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Similarly, (15) and the definition of 1 yield c+w 1 y(T ˙ + 1 ) = − y(T + 1 ) − y(1 ) w w = 1 − y(T + 1 ) > 0.
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(20)
It now follows from (18)–(20) that there exists tm ∈ I2 such that y˙2 (t) < 0 for all t ∈ [T , tm ), y˙2 (tm ) = 0, and y˙2 (t) > 0 for all t ∈ (tm , T + 1 ]. Hence, y2 attains a unique minimum at t = tm and is monotonically increasing for t > tm . Hence, there exist 3 , 4 , 5 , with T < 3 < 4 < 5 < T + 1 , such that y2 (3 ) = c − w, y2 (4 ) = c, and y2 (5 ) = c + w. Interval 3: I3 = [T + 1 , T + 2 ]. Here, c − w < y(t − T ) < c for all t ∈ I3 , so (6) yields y(t) ˙ = c0 + c1 y(t) + c2 y(t − T ), with c0 = (w − c)/w, c1 = −, and c2 = 1/w. It follows from (14) that y(t − T ) = exp(−t)b0 , with b0 = exp(T )(c + w). Applying Proposition 3 with t0 = T + 1 and k = 0 yields c−w y3 (t) = exp(−(t − T − 1 ))y2 (T + 1 ) + (exp(−(t − T − 1 )) − 1) w c+w + exp(−(t − T ))(t − T − 1 ) w c+w 2 c+w 1 w−c = exp(−(t − T )) w exp(−T ) − ln + +t −T + , (21) w c w where the second equation follows from the definition of 1 and (17). Differentiating this expression, and using the definition of 1 and (17), we find that y3 (t) attains a unique maximum at time tm := T − w exp(−T ) + (2/) ln ((c + w)/c). It is easy to verify that tm < T + 2 . The maximal value is y3 (tm ) =
1 (w 2 − c2 + c2 exp(w exp(−T ))). w(c + w)
Eq. (21) and the definition of 2 yield c−w 1 c2 . w exp(−T ) + ln 2 y3 (T + 2 ) = w c − w2
(22)
Interval 4: I4 = [T + 2 , T + 3 ]. ˙ = −y(t). The solution along this interval is then Here, y(t − T ) < c − w for all t ∈ I4 , so (6) yields y(t) y4 (t) = exp(−(t − T − 2 ))y3 (T + 2 ) c+w = exp(−(t − T )) y3 (T + 2 ), c−w where the second equation follows from the definition of 2 . Hence, y4 (T + 3 ) = exp(−3 )
(23)
c+w y3 (T + 2 ), c−w
and substituting (22) yields y4 (T + 3 ) = exp(−3 )
1 c2 c+w . w exp(−T ) + ln 2 w c − w2
(24)
Using the fact that 3 < T + 1 and (10) yields y4 (T + 3 ) > c + w. Interval 5: I5 = [T + 3 , T + 4 ].
(25)
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Here, c − w < y(t − T ) < c for all t ∈ I5 , so (6) yields y(t) ˙ = c0 + c1 y(t) + c2 y(t − T ),
(26)
with c0 = (w − c)/w, c1 = −, and c2 = 1/w. We know that t − T 3 and, therefore, y(t − T ) c − w. It follows that w−c 1 y(t) ˙ − y(t) + (c − w) = −y(t) w w and, therefore, y(t) exp(−(t − T − 3 ))y(T + 3 ).
(27)
In particular, using (24) yields
1 c2 c+w w exp(−T ) + ln 2 y(T + 4 ) exp(−(4 − 3 )) exp(−3 ) w c − w2 2 c 1 = exp(−4 )(c + w) exp(−T ) + . ln 2 w c − w2
Using the fact that 4 T + 1 yields
c2 1 ln 2 y(T + 4 ) exp(−(T + 1 ))(c + w) exp(−T ) + w c − w2 2 1 c = exp(−T )c exp(−T ) + , ln 2 w c − w2
and (10) implies that y(T + 4 ) > c exp(−2T ) + c + w. Hence, y5 (t) > c + w
for all t ∈ I5 .
(28)
We now derive an upper bound for y5 (t). Since t − T 4 , y(t − T ) c and this yields w−c 1 y(t) ˙ − y(t) + c w w = 1 − y(t), so y5 (t)1/ + exp(−(t − T − 3 ))(y(T + 3 ) − 1/). Using (24) and (11), we find that y5 (t) < 1/.
(29)
We now analyze y˙5 (t). It follows from (16) that c+w 1 1 y(t − T ) = w exp(−(t − T )) + + 2T − t − exp(−(t − 2T )) , w and using (26) and Proposition 3 yields y5 (t) = r + exp(−t)P (t), where r is a constant, and P (t) is a quadratic polynomial. Hence, y˙5 (t) = exp(−t)Q(t), where Q(t) is a quadratic polynomial.
(30)
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Using (26) and the definition of 3 yields y(T ˙ + 3 ) < 0.
(31)
Similarly, using (26), (29) and the definition of 4 yields y(T ˙ + 4 ) = 1 − y(T + 4 ) > 0.
(32)
It now follows from (30)–(32) that there exists tm ∈ I5 such that y˙5 (t) < 0 for all t ∈ [T + 3 , tm ), y˙5 (tm ) = 0, and y˙5 (t) > 0 for all t ∈ (tm , T + 4 ]. Hence, y5 (t) attains a unique minimum at time tm . Interval 6: I6 = [T + 4 , T + 5 ]. Here, c < y[t − T , t] < c + w so (6) yields y(t) ˙ = c0 + c1 y(t) + c2 y(t − T ), with c0 = (c + w)/w, c1 = −, and c2 = −1/w. We know that 4 t − T 5 and therefore c y(t − T ) c + w. It follows that c+w 1 y(t) ˙ − y(t) − (c + w) w w = −y(t), so y(t) exp(−(t − T − 4 ))y(T + 4 ). In particular, using (27) yields y(t) exp(−(t − T − 4 )) exp(−(4 − 3 ))y(T + 3 ) 1 c2 exp(−(t − T ))(c + w) exp(−T ) + , ln 2 w c − w2 and using the fact that t − T 5 implies that c2 1 . ln 2 y(t) exp(−5 )(c + w) exp(−T ) + w c − w2 By definition, 5 T + 1 and therefore, as in the analysis in interval I5 , we find that y6 (t) > c + w
for all t ∈ I6 .
(33)
Arguing as in the analysis in interval I5 , we find that there exists tm ∈ I6 such that y˙6 (t) > 0 for all t ∈ [T + 4 , tm ), y˙6 (tm ) = 0, and y˙6 (t) < 0 for all t ∈ (tm , T + 5 ]. Hence, y6 (t) attains a unique maximum at t = tm . Interval 7: I7 = [T + 5 , 2T + 5 ]. Using (25), (28) and (33) yields y[t − T , t]c + w, so y7 (t) = exp(−(t − T − 5 ))y6 (T + 5 ).
(34)
Thus, y7 (t) is a strictly decreasing function. Note that (33) yields y6 (T + 5 ) > c + w. We now consider two possible cases. Case 1: Suppose that there exists ∈ I7 such that y7 () = c +w. In this case, y[−T , ) > c +w, so y(t) = y(t +) for all t ∈ [0, T ]. Hence, the solution from here on is periodic with period . Case 2: Suppose that there is no ∈ I7 such that y7 () = c + w. In this case, y(t) > c + w for all t ∈ I7 . Defining I8 := [2T + 5 , 3T + 5 ], we see that y8 (t) = exp(−(t − 2T − 5 ))y7 (2T + 5 ), so y(t) continues to decrease. If there is a point ∈ I8 such that y8 () = c + w then, as in Case 1, we get that y(t) is periodic from here on. Otherwise, y(t) will continue to decrease exponentially in the next interval, and so on. Eventually, there must be a time such that y() = c + w and the solution will be periodic from this point on.
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