=Paper=
{{Paper
|id=None
|storemode=property
|title=BDI Agents with Fuzzy Perception for Simulating Decision Making in Environments with Imperfect Information
|pdfUrl=https://ceur-ws.org/Vol-627/mass_4.pdf
|volume=Vol-627
|dblpUrl=https://dblp.org/rec/conf/mallow/FariasDC10
}}
==BDI Agents with Fuzzy Perception for Simulating Decision Making in Environments with Imperfect Information==
https://ceur-ws.org/Vol-627/mass_4.pdf
BDI Agents with Fuzzy Perception
for Simulating Decision Making
in Environments with Imperfect Information
Giovani P. Farias, Graçaliz P. Dimuro and Antônio C. Rocha Costa
PPGMC - C3, Universidade Federal do Rio Grande
96201-900 Rio Grande, RS, Brazil, Email: {giovanifarias, gracaliz, ac.rocha.costa}@gmail.com
Abstract—This work introduces a model of fuzzy perception A well-known intentional model is the BDI (Beliefs, De-
for BDI agents, to support the simulation of decision making sires and Intentions) architecture, introduced by Rao and
processes in environments with imperfect information. An appli- Georgeff [8]. This model is based on the representation of
cation to a fuzzy prey-predator environment was developed, as
an example, where the process of deciding which prey a predator the agent’s beliefs about the states of the world and a set of
should attack is based on its fuzzy perception of the strength of desires, which identify those states that the agent has as goals.
the prey, and on the comparison of the preys’ strengths with By a process of deliberation, the agent formulates one or more
its own strength. Different simulations were realized for the intentions (the states which the agent is committed to bringing
comparative evaluation of different types of predator agents, about). The agent then builds a plan to achieve those intentions
in contexts with and without competition between predators.
The quantitative analysis of the simulations shows that the fuzzy (through some form of means-ends reasoning), and executes it.
predator agent presents the best scores. However, the important After that, the agent uses its perception about the environment
result is that the fuzzy predator seems to behave more adequately (which may include itself) in order to have its beliefs updated.
in the environment, in the sense that it presents an apparently Although Rao and Georgeff explicitly acknowledge that an
more natural, coherent and realistic behavior.
agent’s model of the world is incomplete, the BDI model
does not take into account the influence of the imperfect
I. I NTRODUCTION information (in the sense discussed above) acquired from the
world in beliefs, desires and intensions. In particular, it does
Fuzzy sets and Fuzzy Logic (FL) [1] may be viewed as an not consider that the agent could have a “fuzzy” perception of
attempt to formalize/mechanize two kinds of human capabili- the world. Then, in this paper, we experiment with a BDI agent
ties. The first one is the capability to reason and make rational with fuzzy perception operating in a task environment with
decisions in an environment of imperfect information (i.e, imperfect information, namely, a fuzzy prey-predator system.
of imprecision, uncertainty, incompleteness of information,
Prey-Predator systems are an important theme in the area
conflicting information, partiality of truth and partiality of
of Population Dynamics, their modeling having achieved a
possibility). And second, the capability to perform a wide
classical status through the formulation of the so-called Lotka-
variety of physical and mental tasks without any measurements
Volterra equations [9]. The particular type of systems that
and any computations [2].
we simulate was inspired by the Fuzzy Prey-Predator Model
Zadeh [3] pointed out the Incompatibility Principle, which introduced by Peixoto et al. [10].
states that “complexity and precision are incompatible prop- The paper is organized as follows. Section II discusses
erties”, arguing that the conventional numerical-based ap- related work. Section III presents some concepts on the fuzzy
proaches are inadequate to model human-like complex pro- inference system used in this work. The environment with
cesses. Therefore, “the closer one looks at a real-world prob- imperfect information inspired by the Fuzzy Prey-Predator
lem, the fuzzier becomes its solution”. Model is introduced in Sect. IV, including our approach for the
In the context of Social Simulation (SS), Grüne-Yanoff [4] fuzzy perception module to be included in the BDI architecture
and Rossiter et al. [5] remarked that one often has to deal of the predator agent. The results on simulations are discussed
with “fuzzy” social concepts, which are difficult to formalize in Sect. V. Section VI is the Conclusion.
and observe in the real-world system. For that reason, FL has
been used in SS for representing vagueness, uncertainty and II. R ELATED W ORK
subjectiveness in everyday life.
Among the agent models commonly used in agent-based In the context of social simulation, FL has been playing
simulation of decision processes in complex environments, an important role, and it is possible to find many interesting
there are the ones of an intentional nature, whose behaviors works using FL to deal with different problems that can not
can be explained by attributing certain mental attitudes to the be solved with classical simulation models and tools. In this
agents, such as knowledge, belief, desire, intention, obligation, section, we briefly present some of those works, according to
commitment (see, e.g., [6], [7]). the different issues covered by them.
Hassan et al. [11] observed that simple agent models, as agent with a fuzzy neural network was also used by Hai-bo
those normally used with exiting tools, are neither sufficient et al. [26] for application in autonomous underwater vehicles.
nor adequate to deal with the uncertainty and subjectiveness Shen et al. [27] have explored a hybrid BDI model based on
that have to be considered in the analysis of values (e.g., deliberative and fuzzy reasoning, and in [28] the model was
trust) in human societies. In their agent-based social modeling improved within the context of wireless sensor networks.
and simulation, FL was used to naturally specify attributes However, neither the nice formalization by Casali et al.
of the agents representing individuals, the evolution of the nor the other analyzed works have considered the influence
agent minds, the inheritance, the relationship and similarity of fuzzy perception on the operation of a BDI agent and its
between individuals, etc. In the same direction, in [12], fuzzy decision making.
filters were used for modeling trust in social modeling using III. O N F UZZY I NFERENCE S YSTEMS
multiagent systems. Fuzzy set theory [1], [2], [3] is based on the idea that several
In Ghasem-Aghaee and Ören [13], human personality facets elements in human thinking are not exact data, but can be
and traits (according to the Big Five and OCEAN models) approximated as classes of objects in which the transition from
were specified as conditional rules in fuzzy agents, in order to membership to nonmembership is gradual rather than abrupt,
perform human behavior simulation. With related objectives, represented by membership grades in the interval [0; 1]. Since
Dimuro et al. [14] introduced an approach based on FL for human reasoning sometimes does not follow the two-valued
the evaluation of the social exchange values generated in or multivalued logic, FL is a logic with fuzzy truths, fuzzy
the simulation of social exchanges between personality-based connectives, and fuzzy rules of inference.
agents, with the analysis of the fuzzy equilibrium equation. Fuzzy inference systems are non-linear models that aim to
Sabeur and Denis [15] presented an application of FL describe the input-output relationship of a real system using a
in the simulation of human behavior and social networks, family of linguistic If-then constructions and the inference
representing behavioral elements, such as stress, motivation mechanisms of FL. Among the several methods available
or fatigue, and sociological aspects. Hassan et al. [16] use a for fuzzy inference, we adopt in this work the Kang-Takagi-
fuzzy system to model friendship dynamics with an agent- Sugeno (KTS) method [29], where each fuzzy rule represents
based model that could manage social relationships, together a local model of the real system under consideration1 . The kth
with demographics and evolutionary crossover. rule of a KTS system with input vector X = (x1 , . . . , xN ) and
Fort and Pérez [17] used FL to model the adaptive behaviour output z presents the general form:
of the agents playing the Iterated Prisoner’s Dilemma, gov- If (x1 is A1,k ) and . . . and (xN is AN,k )
erned by Pavlovian strategies, to analyze the evolution of co-
operation. Sabater et al. [18] proposed a fuzzy representation then z = fk (X), (1)
of evaluations for the system Repage, which adopts a cognitive where the linguistic terms An,k (n = 1, . . . , N ) in the rule
theory of reputation. antecedents represent fuzzy sets with membership functions
Concerning fuzzy perception in robots, Cuesta and µn,k , which are used to partition the domains of the input
Ollero [19] used it to improve robot’s navigation, and Mobahi variables into overlapping regions. The functions fk in the rule
and Ansari [20] applied fuzzy perception to improve the consequents are usually first-order polynomials of the form:
credibility in robot’s emotions. fk (x1 , . . . , xN ) = b0,k + b1,k x1 + . . . , bN,k xN . (2)
Notice that the agent architectures proposed so far mostly
For a given input X = (x1 , . . . , xN ), the degree of fulfil-
deal with two-valued information. Casali et al. [21], however,
ment of the kth rule evaluates the compatibility of the input
incorporated a formal model to represent and reason under
X with the rule antecedent and determines the contribution of
uncertainty, introducing a general model for graded BDI
the rule’s response z = fk (x1 , . . . , xN ) to the overall model’s
agents, and an architecture, based on multi-context systems,
output. The degree of firing of the kth rule is expressed as
able to model these graded mental attitudes. In [22], the model
wk (x1 , . . . , xN ) = T1 (µA1 ,k (x1 ), . . . , µAN ,k (xN )), (3)
was used to specify an architecture for a travel assistant agent
that helps a tourist to choose holiday packages, and in [23] it where T1 is a t-norm (triangular norm). In this work, T1 is
was applied to build a recommender system for tourism. the Minimum t-norm (called Gödel t-norm), and then Eq. 3
Hybrid models can be found in the literature, introducing becomes :
some kind of fuzziness to BDI architecture. Long and Ester- wk (x1 , . . . , xN ) = min{µA1 ,k (x1 ), . . . , µAN ,k (xN )}. (4)
line [24] introduced a BDI agent, which uses fuzzy inference The overall output of a normalized first-order TSK fuzzy
engines, fuzzy controllers and classifiers, for the modeling model with K rules is given by
of co-operative societies of artificial agents, outlining some PK
social conditions necessary for agents to form joint intentions T2 (wk (x1 , . . . , xN ), fk (x1 , x2 , . . . , xN ))
and actions. Lokuge and D. Alahakoon [25] introduced a BDI z = k=1 , (5)
PK
agent coupled with a neural network and an adaptive neuro wk (x1 , . . . , xN )
fuzzy inference system for application in container terminal k=1
operations, allowing the improvement the decision making 1 The adoption of the KTS method is due to its better performance in some
process in such a complex, dynamic environment. A BDI applications, since it avoids the defuzzification step. See [29] for details.
where T2 is also a t-norm. In this work, T2 is the Product Environment
Actions
t-norm, so that Eq. 5 results in: Perceptions
P
K
wk (x1 , . . . , xN ) · fk (x1 , x2 , . . . , xN ) Sensors Actuators
z = k=1 . (6)
PK
wk (x1 , . . . , xN ) Deliber
k=1 BRF Beliefs ations
IV. A F UZZY P REY-P REDATOR E NVIRONMENT
In [10], Peixoto et al. proposed a fuzzy rule-based system Fig. 1. Part of the BDI model with a fuzzy perception module.
to elaborate a predator-prey model to study the interaction
between aphids (preys) and ladybugs (predators) in citricul-
ture. Due to the lack of available information about the phe- (i) at each movement it loses a fixed amount of weight (weight
nomenon, instead of using the usual differential equations that loss rate), and has its age incremented by a fixed value (aging
characterize the classic deterministic models, they introduced rate); (ii) at each successful attack, it gains a fixed amount of
a fuzzy approach for analyzing the problem. weight (attack reward); otherwise, it loses a fixed amount of
In this paper, we informally build on the fuzzy prey-predator weight (attack punishment); (iii) there is a minimum weight
approach for an agent-based simulation in order to analyze the that a predator can support; if it achieves a weight less than the
ability of a predator with fuzzy perception in surviving in an minimum then it dies by weakness; (iv) there is a maximum
environment of imperfect information.2 age that a predator can achieve; after that it dies by ageing.
In this environment, the age and the weight of a prey (and
A. Characterizing the Fuzzy Predator (FP) Agent
of a fuzzy predator itself) are vague information for the fuzzy
predator. However, such information is crucial for a predator The Fuzzy Predator (FP) has a perception mechanism
to evaluate the strength level of a certain prey in comparison directly connected to the BRF (Belief Revision Function) of
with its own strength level, and, therefore, to estimate the its BDI architecture, partially depicted in Fig. 1. This means
probability of the success of its attack to such prey, which that the fuzzy perception mechanism receives as input data
is given by: the prey’s age and weight, as well as the predator’s own age
RAP − RP P and weight, all of which are perceived through the predator’s
P rob = 50 + , (7)
200 non precise sensors. Then, using the KTS inference system
where RAP and RP P are the predator’s and the prey’s (Sect. III), the predator infers the prey’s strength level, and also
strength levels, respectively. its own strength level, updating its beliefs with the inferred
We assume that (i) predators and preys are initially ran- information, in order to let this information be used in the
domly distributed in a grid; (ii) the food is always available decision process.
for the different preys, and (iii) a predator loses weight for The linguistic variables age, weight and strength
being looking around for preys and much more for each level are modeled as fuzzy sets with trapezoidal member-
unsuccessful attack (on the contrary, it gains weight if its ship functions (Fig. 2). The analysis of those linguistic vari-
attack is successful). Then, the predator survival during the ables allowed the construction of a knowledge base composed
evolution of the time depends on its decision about attacking by the linguistic rules presented in part in Table I. Table II
or not any prey it finds during its life. This decision is based on shows part of the rule base for the KTS inference system of
the imperfect information that the agent can perceive through the perception model of the FP agent, each one with 2 inputs
its fuzzy perception mechanism, which uses a fuzzy inference (age, weight) ∈ R2 and the output z ∈ R, where “young”,
system to determine the prey’s strength level and its own. “adult”, “old”, “very light”, “light”, “average”, “heavy” e
The predator is a BDI agent with beliefs3 on the following “very heavy” represent fuzzy subsets of R.
parameters: age, weight and strength level. The age and weight Example 1: In order to see how the inference system of
the agent can perceive through its perception mechanism. the fuzzy perception mechanism operates, let us consider the
The strength level can be estimated considering perceived following crisp input data: age = 16 and weight = 84.
ages and weights. The abilities of the predator are: random Those values are fuzzified, considering the membership grades
movement looking for preys, perception of preys’s age and in relation to the fuzzy subsets that define those linguistic
weight, estimation of prey’s strength level in comparison with variables, given in Fig. 2. Then, the age value age = 16 is
its own strength level at the current time, and decision on considered “young” with grade µyoung (16) = 0, 4 and “adult”
attacks to preys, which considers if the probability of success with grade µadult (16) = 0, 6. The weight value weight = 84
satisfies P rob > 0.25 (Eq. 7). The constraints of its life are: is evaluated as “heavy” with grade µheavy (84) = 0, 6 and
“very heavy” with grade µvery−heavy (84) = 0, 4.
2 Notice that we did not study population dynamics, as it was done in [10],
For each combination of those sets achieved by the input
although this can be considered in future work. data, some of the rules of the knowledge base are activated.
3 In this paper, we do not refer to the agent’s desires or intentions, only to
its beliefs, since this is the component of the BDI model that is connected to In this case, four rules are fired, namely, the rules R4 , R5 , R9
the fuzzy perception mechanism. and R10 of the Tables I and II. Using Eq. 4, it is possible to
TABLE I
L INGUISTIC RULE BASE .
If age and weight then strength level
R1 young very light very weak
R2 young light very weak
R3 young average weak
R4 young heavy average
R5 young very heavy average
R6 adult very light average
R7 adult light average
R8 adult average strong
R9 adult heavy very strong
R10 adult very heavy very strong
R11 old very light very weak
R12 old light very weak
R13 old average weak
R15 old very heavy average
knowledge base is activated. The characteristic functions of
the sets related
to the linguistic variable age
are:
1 if x ≤ 15; 1 if 15 < x < 35;
µyoung (x) = µadult (x) =
0 otherwise 0 otherwise
1 if x ≥ 35;
µold (x) = (8)
0 otherwise
The characteristic functions of the sets related to the lin-
guistic variable weight
are:
1 if x ≤ 15; 1 if 15 < x ≤ 35;
µvery−light (x) = µlight (x) =
0 otherwise 0 otherwise
1 if 35 < x ≤ 65;
µaverage (x) = (9)
0 otherwise
1 if65 < x ≤ 85 1 if x > 85
µheavy (x) = µvery−heavy (x) =
0 otherwise 0 otherwise
Example 2: Considering the same input data (age,
weight) of Ex. 1 and the characteristic functions given in
Equations 8 and 9, one has that weight = 84 and age = 16
Fig. 2. Membership function for the considered linguistic variables. are definitely evaluated as “heavy” (µheavy (84) = 1) and
“adult” (µadult (16) = 1), respectively. In this case, only the
rule R9 of the rule base of Tables I and II is activated.
find the degrees of firing of each one of those rules, as, e.g.,
Obviously, the firing degree of this rule is w9 = 1. The general
w4 = min {µyoung (16), µheavy (84)} = 0, 4. Then, one has
output, given by Eq. 6, results in the value of the straight level:
that w5 = 0, 4, w9 = 0, 6 and w10 = 0, 4. Using Eq. 6, we w9 f9 (16, 84) 1 · 88
obtain the overall output of the process, where f4 , f5 , f9 and z = = = 88
w9 1
f10 are calculated using Table II:
w4 f4 (16, 84)+w5 f5 (16, 84)+w9 f9 (16, 84)+w10 f10 (16, 84) To enrich the possible comparisons, we have implemented
z= =74,
w4 + w5 + w9 + w10 a Greedy Predator (GP), which always attacks the preys it
which represents the predator’s strength level. encounters, without considering any reasoning on strength
levels and the probability of success of its attacks to preys.
B. The Crisp Predator (CP)
V. A NALYSIS O F THE S IMULATION R ESULTS
For the comparative analysis of simulations, we imple-
mented a Crisp Predator (CP), which is a BDI agent that The simulations were realized to obtain a general view
does not consider that the information about itself and the of the behaviors of the different predators4 in two kinds
one perceived from the environment are vague or incomplete. of the Fuzzy Prey-Predator Environment: competitive (Sect-
Its perception mechanism is inspired on the perception mecha- V-A) and non-competitive environments (Sect. V-B). The
nism of the fuzzy predator, but, instead of using fuzzy subsets implementation was done in the Jason platform [30].
for the modeling of the input linguistic variables, we use 4 Since we are not analyzing population behavior, in the simulations we
classical sets with the usual characteristic functions into the only consider either 2 or 3 predators, in order to be able to compare directly
set {0, 1}. For each set of input date, only one rule of the their surviving abilities.
TABLE II
RULE BASE FOR THE KTS INFERENCE SYSTEM .
If age and weight then strength level = fk (age, weight)
x+y
R1 young very light f1 (x, y) =
2
y
x+( )
R2 young light f2 (x, y) = 2
2
(x + y) − 10
R3 young average f3 (x, y) =
2
y
R4 young heavy f4 (x, y) = (x − 1) +
2
y
R5 young very heavy f5 (x, y) = x +
2
x+y
R6 adult very light f6 (x, y) = + 25
2
x+y
R7 adult light f7 (x, y) = + 30
2
y
x + + 100
R8 adult average f8 (x, y) = 2
2
x+y
R9 adult heavy f9 (x, y) = + 63
4
x
+y
R10 adult very heavy f10 (x, y) = 2 + 40
2
(50 − x) + y
R11 old very light f11 (x, y) =
2
y
(50 − x) +
R12 old light f12 (x, y) = 2
2
(50 − x) + (y − 10)
R13 old average f13 (x, y) =
2
y
R15 old very heavy f15 (x, y) = (50 − x) +
2
The results were obtained from a total of 100 simulation
runs. In each run, the time grows in discrete units (1 time
unit = one predator movement). In the beginning of each run,
the predators present the following initial parameters: age =
1 and weight = 50. Those parameters change at each time
instant according to the following fixed rates5 : the weight loss
rate (-0,1 kg for each movement/time), the aging rate (-0.05
year for each movement/time), the attach reward (+2 kg for
each successful attack), and the attack punishment (-1 kg for
each non successful attack). The simulation run ends when all
the predators have died, either for weakness (weight less than
1 kg) or for ageing (age equal to 50 years).
A. The Competitive Environment
Fig. 3. The average of attacks (top), victories (middle) and defeats (bottom)
The competitive environment consists of 2 kinds of preda- at an age i, with 1 ≤ i ≤ 50, in a competitive environment.
tors (FP e CP) and different 250 preys. At each successful
predator attack, the corresponding defeated prey dies. Consid-
ering that there is no prey reproduction, the prey population
tends to decrease, increasing the probability of the predator Figure 3 (top) shows the average number of predators’
not finding a prey as it moves in the environment, which may attacks at each year. Observe that the number of the CP’s
cause increasing weight losses. In this sense, both predators attacks surges around the age of 15. This is so because, before
compete for the preys remaining in the environment. 15, the agent thinks that it is young (with too low strength
level), but, suddenly, as it achieves 15 years old, it concludes
5 Variations of the initial parameters and rates are not considered here, since that it is already an adult (with too high strength level). The
they affect only the agent’s deliberations, not its perceptions. increase in the number of the FP’s attacks is more gradual,
showing more coherence in its decisions. On the other hand,
one might have expected that the high number of attacks would
have lasted until around the age of 35, since it is only after
this age that the CP considers itself old. However, due the
prey population decreasing, the number of the attacks of both
predators also decreases, even before the age 35. Around the
age 35, the decrease in the number of the attacks of the CP is
much more abrupt than the smooth decreasing of the number
of the attacks of the FP, as it passes from young to adult/old.
Figure 3 (middle) presents the average of predators’ victo-
ries at each year. There is a significant increase in the number
of victories when the CP is around 15, which is an expected
result, since this is the period that, as it considers itself an
adult by this age, it increases a lot the number of attacks until
around the age of 35, when it considers itself old, as discussed
in the previous paragraph. Also, due to the decrease in the prey
population, and consequently, the decrease in the number of
attacks, the number of victories also decreases, even before
the age 35. Again, it is possible to observe that the graph
corresponding to the FP increases and decreases smoothly, as
the agent becomes old, whereas the one of the CP increases
abruptly around 15 and decreases around 35, also drastically.
Analogous analysis can be done concerning the average of
number of the predators’ defeats at each year, which is shown
in Fig. 3 (bottom).
B. The Non-competitive Environment
The non-competitive environment consists of 3 kinds of
predators (FP, CP and GP), and 250 different preys. For each
prey that dies in consequence of a predator attack, another
prey with similar characteristics appears in the environment,
at a random position. This means that the predators always
have the same chance to find a prey to attack.
Figure 4 (top) shows the average number of predators’
attacks at each year. For the same reasons discussed in
Sect. V-A, the number of the CP’s attacks surges around the
age of 15. However, since the prey population is constant
along the time, the high number of attacks of the CP lasts
until around the age of 35, and then it follows drastically. The
behavior of the FP is much more natural and coherent, since
it presents a gradual increase in the number of attacks as it
becomes an adult, and a also a smooth decrease as it becomes
old. The high number of attacks of the GP during its life was
as expected. During adulthood the numbers of attacks of all Fig. 4. The average of attacks (top), victories (middle) and defeats (bottom)
at an age i, with 1 ≤ i ≤ 50, in a non-competitive environment.
predators are similar.
Figure 4 (middle) presents the average number of predators’
victories at each year. There is an abrupt increase in the
number of victories when the CP is around 15, due to the high predators are similar. The highest numbers of victories, for
increase in the number of its attacks by this age. However, Crisp and Fuzzy predators, appear between the ages of 20 and
since the prey population does not decrease, the number 33. Analogous analysis can be done concerning the average of
of victories stays high until around the age 35. After that, number of predators’ defeats at each year (Fig. 4 (bottom)).
it decreases radically. Again, it is possible to observe that Figure 5 (top) shows the average number of accumulated
the graph corresponding to the FP increases and decreases attacks during the predators’ lives, until they reach a certain
smoothly, as the agent becomes old. The higher number of age i, with 1 ≤ i ≤ 50. As expected, the GP had an average
victories of the GP is due to its attack strategy. During number of accumulated attacks much higher than the other
adulthood, the numbers of victories of the three kinds of two predators, which had a similar attack behavior.
Fig. 6. Average lifetime (top), average weight at the end of the life (middle)
Fig. 5. Average accumulated number of attacks (top), victories (middle) and and average of number of attacks/victories/defeats (bottom) of predators.
defeats (bottom) during the predator life until the age i, with 1 ≤ i ≤ 50.
expected, the Greedy and Fuzzy predators present the lowest
Figure 5 (middle) presents the average number of accumu- and the highest average lifetimes, respectively.
lated victories during the predators’ lives, until they reach an Figure 6 (middle) presents the average weight of predators
specific age. As expected, the GP had an average number at the end of their lives. The average weight of the FP at the
of accumulated victories much higher than the other two end of its life is the highest one.
predators. However, this number for the FP is higher than that Figure 6 (bottom) shows the average number of attacks,
of the CP, as they become older. victories and defeats of predators during its whole life. The GP
Analogous analysis can be done concerning the average is the one that presents the highest averages in all categories,
number of accumulated defeats during the predators’ lives, and it is the one that has the average number of defeats
until they reach an specific age, shown in Fig. 5 (bottom). higher than that of victories. Its average numbers of attacks
Figure 6 (top) shows the average lifetime of the predators. As and victories are higher than the ones of the CP, whereas the
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