a v a i l a b l e a t w w w . s c i e n c e d i r e c t . c o m
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e c o l m o d e l
Predator–prey fuzzy model
Magda da Silva Peixoto
a,∗, La´ecio Carvalho de Barros
b, Rodney Carlos Bassanezi
baUniversidade Federal de S ˜ao Carlos, Campus Sorocaba, PO box 3031, 18043-970, Sorocaba, SP, Brazil
bDepartamento de Matem ´atica Aplicada, IMECC, Universidade Estadual de Campinas, PO box 6065, 13083-859, Campinas,SP, Brazil
a r t i c l e
i n f o
Article history:
Published on line 11 March 2008
Keywords:
Fuzzy sets
Fuzzy rule-based systems Predator
Prey Aphids Ladybugs
Citrus Sudden Death
a b s t r a c t
In this work we have used fuzzy rule-based systems to elaborate a predator–prey type of model to study the interaction between aphids (preys) and ladybugs (predators) in citricul-ture, where the aphids are considered as transmitter agents of the Citrus Sudden Death (CSD). Simulations were performed and a graph was drawn to show the prey population, the potentiality of the predators, and a phase-plane. From this phase-plane, a classic model of the Holling–Tanner type is fitted and its parameters were found. Finally, we have studied the stability of the critical points of the Holling–Tanner model.
© 2008 Elsevier B.V. All rights reserved.
1.
Introduction
Citrus Sudden Death (CSD) is a disease that has affected sweet orange trees grafted on Rangpur lime in the south of the state of Minas Gerais and in the north of the state of S ˜ao Paulo
(Bassanezi et al., 2003). Researchers believe this disease has been caused by a virus transmitted by insects known as aphids (vector). Among the most known predators of aphids in citrus in Brazil, we point out the ladybugs(Morales and Buranr, 1985). In this paper we suggest using Zadeh (1965) fuzzy set theory to create a model that studies the interaction between a prey (aphid) and its predator (ladybug), instead of using the usual differential equations which characterize the classic deterministic models. Since we do not have sufficient infor-mation about our phenomena, it is difficult to express the variations as functions of the states. On the other hand, qual-itative information from specialists allows us to propose rules that relate (at least partially), to the state variables, with their own variations. In particular, our interest here is to elaborate a predator–prey model that represents the interaction between
∗Corresponding author.
E-mail addresses:[email protected](M. da Silva Peixoto),[email protected](L.C. de Barros),[email protected]
(R.C. Bassanezi).
aphids (preys) and ladybugs (predators) in citriculture, by using a fuzzy rule-based system. Next, we propose a classic deterministic model given by a system of ordinary differential equations, supposing there is a predator–prey system whose solutions coincide with those of the fuzzy model. In order to achieve this purpose, we intend to fit the differential equations parameters from the fuzzy model obtained. The great advantage of the obtained parameters for differential equations is the fact that we can do the stability analysis of the system.
2.
Preliminary concepts and definitions
A fuzzy subsetAof the a universal setXis defined by a mem-bership function A that assigns to each elementx ofX a
number A(x), between zero and one, which gives by degree
of membership fromxtoA. Thus, A:X→[0,1]. It interesting
to note that a classic subsetAofXis a particular fuzzy set for which the membership function is the its characteristic function ofA,A:X→ {0,1}.
Fig. 1 – Structure of fuzzy rule-based systems. Source:Jafelice et al. (2003).
2.1. Fuzzy rule-based systems
Basically, fuzzy rule-based systems have four components: an input processor; a collection of linguistic rules, called rule base; a fuzzy inference method and an output proces-sor. These components process real-valued inputs to provide real-valued outputs. Fig. 1 illustrates a fuzzy rule-based system.
The fuzzification is the process by which the input values of the system are translated into fuzzy sets of their respective universes. It is a mapping of the dominion of the real numbers led to the fuzzy dominion.
The rule base characterize the objectives and strategies used by specialists in the area, by means of a linguist rule set. It is composed by a collection of fuzzy conditional proposi-tions in the form if-then rules. An expert, interviewed to help formulate the fuzzy rules set, can articulate associations of linguistic inputs/outputs.
The fuzzy inference machine performs approximate rea-soning using the compositional rule of inference. A particular form of fuzzy inference of interest here is the Mamdani method (Pedrycz and Gomide, 1998). In this case, it aggre-gates the rules through the logical operator OR, modelled by the maximum operator and, in each rule, the logical operators AND and THEN are modelled by the minimum operator (Klir and Yuan, 1995). The fuzzy rule-based sys-tem considered in this work, consists of two inputs, two outputs, and 30 fuzzy rules of the following form: “IF the number of preys is large AND the potential of predation is very small, THEN the variation of preys increases a little AND the variation of the potential of predation increases a lot”, where large, very small, increases a little, increases a lot are fuzzy sets (see Section 4). The logic of decisions to be made, incorporated to the structure of inference of the rule base, uses fuzzy implications(Pedrycz and Gomide, 1998)to simulate the wanted decisions. It generates actions – consequents – inferred from a set of input conditions – antecedents.
Finally, in defuzzification, the value of the output linguis-tic variable inferred by the fuzzy rule is translated to a real value. The purpose is to obtain a real number that better rep-resents the fuzzy values of the output linguistic variable. A typical defuzzification scheme, the same as adopted in this paper, is the center-of-gravity method defined as follows. Let
Cbe the membership function of the output variablez, then
the real-valued output ¯zis chosen as follows:
¯
z=
zC(z) dz
C(z) dz.
We refer the reader toPedrycz and Gomide (1998)andKlir and Yuan (1995)for a detailed coverage of the fundamentals of fuzzy set and systems theory and applications. In the fol-lowing section we proceed to review the interaction between preys and predators.
3.
Predator–prey model description
The mathematical models that describe prey and predator relationship are used to study interactions between two pop-ulations, when one of them depends on the other for food and for survival. Such dynamic relationship between preys and predators are prominent subjects in Ecology (Edelstein-Keshet, 1987; Murray, 1990).
In short, we present hypotheses that characterize a predator–prey model, whose trajectories show the following features:
(1) the number of prey population and the number of predator population have an oscillatory character;
(2) an increase in the prey population is followed (with a delay) by an increase in the predator population;
(3) a decrease in the prey population is followed (with a delay) by a decrease in the predator population;
(4) if the number of predators is small, the number of preys increases;
(5) if the number of predators is large, the number of preys decreases;
(6) if the number of preys is large, the number of predators increases;
(7) if the number of preys is small, the number of predators decreases.
So this dynamics is characterized by enchained oscillations in both populations: predator and prey. What is most interest-ing is that these oscillations have the followinterest-ing property: the peak of the prey population will always occur some time before the natural enemy’s population peak.
equa-tions, which characterize the classic deterministic models that are used to model the dynamics between preys and preda-tors. In fact, our main interest in this paper is to elaborate a predator–prey model that represents the interaction between aphids (preys) and ladybugs (predators) in citriculture by using a fuzzy rule-based system.
Next, a short review of the Citrus Sudden Death and the interaction between aphids and ladybugs is presented.
3.1. Aphids×ladybugs
Citrus Sudden Death is a disease that has caused serious harm to citriculturists, up to the point of destroying big plantations in the north of the state of S ˜ao Paulo and in the south of Minas Gerais. CSD is a disease combining canopy/rootstock and it can lead plants on intolerant rootstock to death. Researches have shown that the ducts, which lead nutrients generated by the photosynthesis to the roots, become obstructed and degenerated. Without food, the roots putrefy, the tree decays and dies.(Bassanezi et al., 2003).
Researchers believe this disease has been caused by a virus transmitted by insects known as aphids (vector). Among the most known predators of aphids in citrus in Brazil, we point out the ladybugsCycloneda sanguineabelonging to the Coleoptera Order and the Coccinelidae Family . The constant occurrence of larvae and adults of ladybugs is important to control of the aphids(Hodek, 1973).
Thus, our main interest here is to take into account the particularities of our data and reports of specialists, since the quality of the predators is most importance.
In the following section, we proceed to introduce the fuzzy model, the main aim of this paper.
4.
Formulation of the predator–prey fuzzy
model
In the predator–prey systems, the structure of the predator population changes in time and there are phases where there is no predation at all. As we have observed, the ladybugs only prey upon aphids in the larva and adult stages. On the other hand, the aphids are captured by their enemies independently of their life phase(Braga and Sousa-Silva, 1999). Therefore incorporating this new characteristic into the model will help us understand predator–prey interactions. Moreover, it can be applied to the predation theory for biological control. Facts like these are not considered in the simple model of predation, for example, Lotka-Volterra. According toHsin and Yang (2003), simple models are not adequate to study the predator–prey relationship, when the populations involved present different dynamics according to their ages. And each of these preda-tors’ larva can consume up to 200 aphids a day, and the adult predators prey, on average, upon 20 aphids a day(Gravena, 2003). Hence the population of predators will consist of lar-vae and adults. So we should distinguish these subpopulations and their particularities in the predator–prey model(Peixoto et al., 2005).
From the information above, we can consider that preda-tors are differentiated in accordance with their potential of predation, through a membership function of predator class
that we will adopt here:
Pyi=
1, if larvae;0.1, if adults
and the potential of predation of a predators population as beingPy=p1+0.1×p2, wherep1is the number of larvae
pop-ulation andp2is the population of adults.
As the aphids are captured by their enemies independently of their life phase, the population of preys will not be subdi-vided, since the quality of being a prey does not depend on its time of life to be classified according to its readiness to escape from their predators.
The variables of the system are the number of preys, the potentiality of the predators (inputs) and their variations (out-puts). However, accurate knowledge about the input variables and theirs variations is not available. On the other hand, qualitative information from specialists, in particular by ento-mologists, allows us to propose rules that that relate to the variables of state, with their own variations. The fuzzy rule base is given by 30 rules of the type: “If the number of preys is large and the potential of predation is very small, then the varia-tion of preys increases a little and the variavaria-tion of the potential of predation increases a lot”.
From Mamdani Inference Method and defuzzification of the center-of-gravity, we have obtained the variation rates of the preys and the potential of predation. In each momentt, the number of preys and the potential of predation are given by the formulas:
In the performed numerical simulations we have aimed to observe the variation in the number of preys and the potential of predation. In order to achieve this, we have considered an initial number of aphids,x0, and an initial number of potential
of predation,Py0, in a branch of a tree, chosen randomly. From
the initial conditions, the fuzzy system producesx′andP′
yas
outputs. From these two last values, to getxandPyin each
iteration we have the following:
⎧
Finally, to solve the integral above we have adopted the Trapezoidal Numerical Integration, since the fuzzy system providesx′andP′
yin each iterationti. Thus the system(2)turns
to be:
Simulations of the trajectories produced by the fuzzy model follow the steps below:
• given the initial population of the preys (x0) and the initial
potential of predation (Py0) as inputs data of the fuzzy
rule-based systems;
• the fuzzy rule-based systems gives the values of the output data:x′
1andP′y1;
• from(3), we findx1andPy1;
• x1andPy1are the inputs variables of the fuzzy rule-based
systems and so forth.
The evolution of the population contingents of the preys and potential of predation given by(3) by the fuzzy model over time, and its respective phase-plane, are illustrated in
Fig. 2.
We would like to point out that even without any equations, we have managed to obtain a phase-plane where the trajecto-ries appear to converge to a cycle with certain regularity.
5.
Fitting the Holling–Tanner model
Next, we propose a classic deterministic model, given by a sys-tem of ordinary differential equations, supposing there is a predator–prey system whose solutions coincide with those of the fuzzy model described above(Peixoto, 2005). This way, we suppose that, in the classic model, there is no heterogene-ity in the class of predators. Therefore it is possible to find parameters of the new model, utilizing the phase-plane of the fuzzy model illustrated inFig. 2. We would like to com-pare the Predator–prey Fuzzy Model with the Holling–Tanner Model. In order to achieve this purpose, we intend to fit the parameters of the system given by differential equations from the new fuzzy model. We consider a predator–prey system of Holling–Tanner:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
dx
dt =rx 1− x K
− mxyD+x
dy
dt =sy 1−h y x
x(0)>0, y(0)>0
(4)
wherex(t) andy(t) denote prey and predator densities, respec-tively, as functions of time, andr, m, s, h, D, K >0.
In system(4)we assume
• the prey population grows logistically with carrying capacity
Kand intrinsic growth raterin the absence of predation;
• the predator consumes the prey according to the func-tional response p(x)=(mx/(D+x)) and grows logistically with intrinsic growth ratesand carrying capacity propor-tional to the population size of prey. In(4), the functional responsep(x) is classified into type II(Svirezhev and Logofet, 1983);
• the parameterhis the number of prey required to support one predator at equilibrium whenyequalsx/h;
• mis the maximal predator per capita consumption rate, i.e., the maximum number of preys that can be captured by a predator in each time unit;
• itemhis a measure of the food quality that the prey provides for conversion into predator births.
This choice is justifiable, because:
• The prey population grows logistically and a branch of an orange tree has carrying capacity in the absence of preda-tion.
• For the population of ladybugs, the branch of an orange tree has carrying capacity proportional to the population of preys.
• According to Morales and Buranr (1985), the number of aphids captured per day by Cycloneda sanguinea (adults, male and female), corresponds to the functional response of Holling’s type II.
Some parameters may be obtained:
• since the adult ladybug consumes, on average, 20 aphids a day, thenD=10;
• a population of 200 aphids per branch is considered large, that is why we consideredK=200 as the carrying capacity of the prey population;
• considering that an adult aphid generates up to 5 new nymphs a day, we shall taker=2;
• if the ladybug population duplicates in 1.03 weeks and only the females reproduce, we supposes=0.3.
The other parameters,mandh, were fitted according to the data (x, y), generated by the fuzzy model, substituting into Eq.
Fig. 2 – (a) The evolution of the population contingents in time, and (b) Phase-plane of the fuzzy model, withx0=110 and
Fig. 3 – (a) Phase-plane of the fuzzy model and (b) phase-plane of the determinist model withx0=110,y0=3.2 and x0=100,y0=2.3.
(4)we obtain the following system of equations:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
dx
dt =2x 1− x
200
−30.625xy
10+x
dy
dt =0.3y 1−22.142857 y x
x(0)>0, y(0)>0
(5)
wherexis the number of preys andyis the number of preda-tors.
InFig. 3we present a example of fitting for the two models. Note that it is possible to fit the curve in order to find suit-able parameters for deterministic models. This is achieved by using the phase-plane curve from the fuzzy model. The great advantage of obtaining parameters for differential equations given by(4)is the fact that we can do the stability analysis of the system.
We analyzed system(5)in order to find its critical points, that is, the pair of valuesxand ythat turn the derivatives null and kept the system in equilibrium, without changing the values ofxandy. One can note that the system above has a possible pair: (77.5, 3.5).
It is important to study what happens when the initial pop-ulationsx0andy0are very near to critic populations, i.e., (x, y)
is near (77.5, 3.5).
The system(4)is the same ofTanner (1975)for which he has written a stability analysis. Following Tanner’s procedure, we may consider isoclines equations of the prey and predator populations. The peak of the prey critical line of(4)is at (K−
D)/2. His result is:
(1) If Kis small, and the critical point to right of the peak of the prey critical line is a stable focus for all values of
s/r.
(2) If K is large, critical point is left of peak of the prey critical line, ands/ris larger than a boundary value deter-mined byrh/m, in this case, the critical point is an stable focus.
(3) If K is large, critical point is left of peak of the prey critical line, ands/ris less than a boundary value deter-mined byrh/m, then the critical point is focus of a limit cycle.
(4) IfKis infinite ands/ris smaller thanrh/m, the critical point is an unstable focus.
To analyze the stability of the critical point (77.5, 3.5), we had used the result above. The isocline equations are given by:
⎧
⎨
⎩
y= 2
30.625(10+x) 1−
x
200
y= x
22.142857
(6)
isoclines of the prey and predation population, respectively, referring to the system of Eq.(5).
The maximum of prey isocline is
K−D
2 =95> x
∗=77.5, (7)
and thus, the critical point is on the right side of the maximum. Still,
rh
m =1.446> s
r=0.15 (8)
From(7)and(8), the critical point (77.5, 3.5) is the focus of a cycle.
6.
Conclusion
This paper has suggested applying Fuzzy Logic to Ecol-ogy/Epidemiology. Primarily, we have been able to model the ladybug–aphids dynamics without using explicit differ-ential equations. We only used intuitive hypotheses of the predator–prey interaction and data from experts.
We consider the Fuzzy Sets Theory as a great contribution to the construction of mathematical models, mainly when some parameters of the differential equations are not avail-able.
The great advantage of obtaining the parameters of dif-ferential equations lies in the fact that we can analyze the stability of the system.
We would like to highlight the advantages of using fuzzy rule-based models as opposed to deterministic models:
• several differential equations parameters of the predator–prey type systems are not available;
obtained, if needed, through curve fitting procedure from the solutions obtained by the fuzzy rule-based models;
• the input and output sets of fuzzy rule-based systems can be easily constructed with the help of specialists in the field, that is, a specialist will know when the population of a par-ticular species is small, large, and so forth.
Acknowledgments
The authors are thankful to Dr. Renato Beozzo Bassanezi (Fundecitrus-Araraquara/SP, Brazil), Dr. Ar´ıcio Xavier Linhares (IB/UNICAMP-Campinas/SP-Brazil) and Dr. Carlos Roberto Sousa e Silva (DEBE/UFSCar-S ˜ao Carlos/SP-Brazil).
They also acknowledge the anonymous referees for the suggestions on improving the results.
The first author acknowledges Brazilian Research Council (CNPq) for the financial support.
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