Predator prey fuzzy model 2

e c o l o g i c a l m o d e l l i n g 2 1 4 ( 2 0 0 8 ) 39–44

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journal homepage: www.elsevier.com/locate/ecolmodel

Predator–prey fuzzy model
Magda da Silva Peixoto a,∗ , La´ecio Carvalho de Barros b , Rodney Carlos Bassanezi b
a
b

˜ Carlos, Campus Sorocaba, PO box 3031, 18043-970, Sorocaba, SP, Brazil
Universidade Federal de Sao
´
Departamento de Matematica
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

a b s t r a c t

Article history:

In this work we have used fuzzy rule-based systems to elaborate a predator–prey type of

Published on line 11 March 2008

model to study the interaction between aphids (preys) and ladybugs (predators) in citriculture, where the aphids are considered as transmitter agents of the Citrus Sudden Death

Keywords:

(CSD). Simulations were performed and a graph was drawn to show the prey population,

Fuzzy sets

the potentiality of the predators, and a phase-plane. From this phase-plane, a classic model

Fuzzy rule-based systems

of the Holling–Tanner type is fitted and its parameters were found. Finally, we have studied

Predator

the stability of the critical points of the Holling–Tanner model.
© 2008 Elsevier B.V. All rights reserved.

Prey
Aphids
Ladybugs
Citrus Sudden Death

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
˜ Paulo
of Minas Gerais and in the north of the state of Sao

(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 information about our phenomena, it is difficult to express the
variations as functions of the states. On the other hand, qualitative 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

∗

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 subset A of the a universal set X is defined by a membership function A that assigns to each element x of X a
number A (x), between zero and one, which gives by degree
of membership from x to A. Thus, A : X → [0, 1]. It interesting
to note that a classic subset A of X is a particular fuzzy set
for which the membership function is the its characteristic
function of A, A : X → {0, 1}.

Corresponding author.
E-mail addresses: magda@power.ufscar.br (M. da Silva Peixoto), laeciocb@ime.unicamp.br (L.C. de Barros), rodney@ime.unicamp.br

(R.C. Bassanezi).
0304-3800/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.ecolmodel.2008.01.009

40

e c o l o g i c a l m o d e l l i n g 2 1 4 ( 2 0 0 8 ) 39–44

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 processor. 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 propositions 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 reasoning 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 aggregates 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 system 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 linguistic variable inferred by the fuzzy rule is translated to a real
value. The purpose is to obtain a real number that better represents 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
C be the membership function of the output variable z, then

the real-valued output z¯ is chosen as follows:
z¯ =


zC(z) dz

.
C(z) dz

We refer the reader to Pedrycz and Gomide (1998) and Klir
and Yuan (1995) for a detailed coverage of the fundamentals
of fuzzy set and systems theory and applications. In the following 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 populations, 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 (EdelsteinKeshet, 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 interesting is that these oscillations have the following property: the
peak of the prey population will always occur some time before
the natural enemy’s population peak.
According to the information above, our purpose is to
elaborate a fuzzy rule base that replaces differential equa-

41

e c o l o g i c a l m o d e l l i n g 2 1 4 ( 2 0 0 8 ) 39–44

tions, which characterize the classic deterministic models that
are used to model the dynamics between preys and predators. 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
˜ Paulo and in the south of Minas
in the north of the state of Sao
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 ladybugs Cycloneda sanguinea belonging 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.

that we will adopt here:
Pyi =



1,

if larvae;

0.1,

if adults

and the potential of predation of a predators population as
being Py = p1 + 0.1 × p2 , where p1 is the number of larvae population and p2 is the population of adults.
As the aphids are captured by their enemies independently
of their life phase, the population of preys will not be subdivided, 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 (outputs). However, accurate knowledge about the input variables
and theirs variations is not available. On the other hand,
qualitative information from specialists, in particular by entomologists, 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 variation of preys increases a little and the variation 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 moment t,
the number of preys and the potential of predation are given
by the formulas:

⎧
 t
⎪
⎪
x′ (s) ds
⎪
⎨ x(t) = x(t0 ) +
t0
 t
⎪
⎪
⎪
P′y (s) ds
⎩ Py (t) = Py (t0 ) +

(1)

t0

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 to Hsin 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 predators’ 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 larvae 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 predators are differentiated in accordance with their potential of
predation, through a membership function of predator class

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, Py 0 , in a branch of a tree, chosen randomly. From
the initial conditions, the fuzzy system produces x′ and P′y as
outputs. From these two last values, to get x and Py in each
iteration we have the following:


⎧
⎪
x(t
)
=
x(t
)
+
⎪
i
⎨ i+1

ti+1

x′ (s) ds

ti

⎪
⎪
⎩ Py (ti+1 ) = Py (ti ) +



(2)

ti+1

P′y (s) ds

ti

Finally, to solve the integral above we have adopted the
Trapezoidal Numerical Integration, since the fuzzy system
provides x′ and P′y in each iteration ti . Thus the system (2) turns
to be:

⎧
⎨ x(ti+1 ) = x(ti ) + 1 [x′ (ti+1 ) + x′ (ti )]
2

⎩ Py (t

i+1 )

= Py (ti ) +

1 ′
[P (t ) + P′y (ti )]
2 y i+1

(3)

Using (3) now and being ti = t0 + i and t0 = 0, we get the
values of x and Py , and thus, successively.

42

e c o l o g i c a l m o d e l l i n g 2 1 4 ( 2 0 0 8 ) 39–44

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 rulebased systems;
• the fuzzy rule-based systems gives the values of the output
data: x1′ and P′y1 ;
• from (3), we find x1 and Py1 ;
• x1 and Py1 are 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 trajectories 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 system 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 heterogeneity 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 in Fig. 2. We would like to compare 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:

⎧


mxy
dx
x
⎪
=
rx
1
−
−
⎪
⎪
K
D+x
⎨ dt



y
dy
= sy 1 − h
⎪
dt
x
⎪
⎪
⎩
x(0) > 0,

(4)

y(0) > 0

where x(t) and y(t) denote prey and predator densities, respectively, as functions of time, and r, m, s, h, D, K > 0.

In system (4) we assume
• the prey population grows logistically with carrying capacity
K and intrinsic growth rate r in the absence of predation;
• the predator consumes the prey according to the functional response p(x) = (mx/(D + x)) and grows logistically
with intrinsic growth rate s and carrying capacity proportional to the population size of prey. In (4), the functional
response p(x) is classified into type II (Svirezhev and Logofet,
1983);
• the parameter h is the number of prey required to support
one predator at equilibrium when y equals x/h;
• m is 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;
• item h is 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 predation.
• 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, then D = 10;
• a population of 200 aphids per branch is considered large,
that is why we considered K = 200 as the carrying capacity
of the prey population;
• considering that an adult aphid generates up to 5 new
nymphs a day, we shall take r = 2;
• if the ladybug population duplicates in 1.03 weeks and only
the females reproduce, we suppose s = 0.3.
The other parameters, m and h, 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, with x0 = 110 and
Py0 = 3.2.

43

e c o l o g i c a l m o d e l l i n g 2 1 4 ( 2 0 0 8 ) 39–44

Fig. 3 – (a) Phase-plane of the fuzzy model and (b) phase-plane of the determinist model with x0 = 110, y0 = 3.2 and
x0 = 100, y0 = 2.3.

To analyze the stability of the critical point (77.5, 3.5), we
had used the result above. The isocline equations are given by:

(4) we obtain the following system of equations:

⎧

30.625xy

x
dx
⎪
−
=
2x
1
−
⎪
⎪ dt
200
10 + x
⎨



dy
y
= 0.3y 1 − 22.142857
⎪
x
⎪ dt
⎪
⎩
x(0) > 0,

(5)

y(0) > 0

where x is the number of preys and y is the number of predators.
In Fig. 3 we present a example of fitting for the two models.
Note that it is possible to fit the curve in order to find suitable 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 values x and y that turn the derivatives
null and kept the system in equilibrium, without changing the
values of x and y. 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 populations x0 and y0 are very near to critic populations, i.e., (x, y)
is near (77.5, 3.5).
The system (4) is the same of Tanner (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 K is 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, and s/r is larger than a boundary value determined by rh/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, and s/r is less than a boundary value determined by rh/m, then the critical point is focus of a limit
cycle.
(4) If K is infinite and s/r is smaller than rh/m, the critical point
is an unstable focus.

2
x
(10 + x) 1 −
30.625
200
x
⎩y =
22.142857

⎧
⎨y =






(6)

isoclines of the prey and predation population, respectively,
referring to the system of Eq. (5).
The maximum of prey isocline is
K−D
= 95 > x∗ = 77.5,
2

(7)

and thus, the critical point is on the right side of the maximum.
Still,
rh
s
= 1.446 > = 0.15
m
r

(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 Ecology/Epidemiology. Primarily, we have been able to model
the ladybug–aphids dynamics without using explicit differential 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 available.
The great advantage of obtaining the parameters of differential 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;
• in the fuzzy model, we have used a rule base, instead
of systems given by equations, eliminating the difficulty
of obtaining the parameters; these parameters can be

44

e c o l o g i c a l m o d e l l i n g 2 1 4 ( 2 0 0 8 ) 39–44

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 particular 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
˜ Carlos/SP-Brazil).
Sousa e Silva (DEBE/UFSCar-Sao
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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