Varying the Population Size of Artificial Foraging
Swarms on Time Varying Landscapes
Carlos Fernandes1,3, Vitorino Ramos2, Agostinho C. Rosa1
1 LaSEEB-ISR-IST, Technical Univ. of Lisbon (IST),
Av. Rovisco Pais, 1, TN 6.21, 1049-001, Lisbon, PORTUGAL
{cfernandes,acrosa}@laseeb.org
2 CVRM-IST, Technical Univ. of Lisbon (IST),
Av. Rovisco Pais, 1, 1049-001, Lisbon, PORTUGAL
vitorino.ramos@alfa.ist.utl.pt
3 EST-IPS, Setúbal Polytechnic Institute (IPS),
R. Vale de Chaves - Estefanilha, 2810, Setúbal, PORTUGAL

Abstract. Swarm Intelligence (SI) is the property of a system whereby the collective behav-
iors of (unsophisticated) entities interacting locally with their environment cause coherent
functional global patterns to emerge. SI provides a basis with which it is possible to explore
collective (or distributed) problem solving without centralized control or the provision of a
global model. In this paper we present a Swarm Search Algorithm with varying population of
agents. The swarm is based on a previous model with fixed population which proved its effec-
tiveness on several computation problems. We will show that the variation of the population
size provides the swarm with mechanisms that improves its self-adaptability and causes the
emergence of a more robust self-organized behavior, resulting in a higher efficiency on search-
ing peaks and valleys over dynamic search landscapes represented here – for the purpose of
different experiments – by several three-dimensional mathematical functions that suddenly
change over time. We will also show that the present swarm, for each function, self-adapts
towards an optimal population size, thus self-regulating.
1 Introduction
Swarm Intelligence (SI) is the property of a system whereby the collective behaviors
of (unsophisticated) entities interacting locally with their environment cause coherent
functional global patterns to emerge. SI provides a basis with which it is possible to
explore collective (or distributed) problem solving without centralized control or the
provision of a global model (Stan Franklin, Coordination without Communication,
talk at Memphis Univ., USA, 1996). The well-know bio-inspired computational para-
digms know as ACO (Ant Colony Optimization algorithm [4]) based on trail forma-
tion via pheromone deposition / evaporation, and PSO (Particle Swarm Optimization
[9]) are just two among many successful examples. To tackle the formation of a co-
herent social collective intelligence from individual behaviors, in the present work,
we will address the collective adaptation of a social community to a cultural (envi-
ronmental, contextual) or informational dynamical landscape, represented here – for
the purpose of different experiments – by several three-dimensional mathematical
functions that change over time. Also, unlike past works [13], the size of our swarm
population varies over time: agents reproduce and die, according to its success on

reaching the peaks (e.g., if the swarm is requested to find the higher values of one
complex function) or deep valleys (if the goal is minimization) over the three-
dimensional landscapes. We believe that Swarms with Varying Population Size
(SVPS) provide a better model to mimic some natural features, improving not only
the population ability to evolve self-organized foraging behavior as obtained in the
past [13], while maintaining a self-regulated population adapted in real-time to differ-
ent constraints in different search landscapes. The complexity of the system, not only
provides more efficiency on the task of finding peaks/valleys, as well as supplies a
faster response to the changing environment (changing the search landscape during
the run). The progress of the population size over time also suggests that our system
may be evolving in a self-organized critical state [3], surpassing several phase transi-
tions. The present work is organized as follows; Section 2 gives an overview of re-
lated work in the area of artificial life models with varying population size. Section 3
describes the original swarm model, before the dynamic of population variation was
introduced. The proposed swarm model and its properties are described in section 4.
In Section 5 the results are shown and its implications are discussed. Finally, Section
6 concludes the paper and suggests future research, namely in what relates to Co-
Evolution.
2 Related work
In [2,3], Bak modeled an ecological system, consisting of different species repre-
sented by random fitness values, each interacting with two neighbors. After mutating,
at each time step, the least fit species, they measured the size of mutation avalanches
on the system. They concluded that the phenomenon follows a power law with slope
approximately equal to 1, suggesting that the system self-organizes to a critical state.
Genetic Algorithms (GAs) are usually implemented using populations with fixed size.
However seeking for extra performing systems, in [1], the GAVaPS - Genetic Algo-
rithm with Varying Population Size - was presented, and many works in this area
have been made since then. In this GA, the concept of “age” of an individual is intro-
duced. A chromosome remains in the population (i.e. stays “alive”) for a number of
generations proportional to its fitness (lifetime). When the age of a chromosome,
which is incremented each generation, reaches its lifetime, the individual is removed
from the population. Consequently, there is no direct selection pressure in this GA.
The individuals are randomly selected for crossover and mutation. The pressure is
assured by the fact that fitter individuals remain in the population during a larger
number of generations, thus producing more offspring than those with lower fitness.
The authors tested the algorithm with four different functions and concluded that it
outperforms the Simple GA in some cases. They also showed that the population size,
after a large initial growth, decreases, and remains stable around the initial value.
Although the results seem promising, the test functions were not very demanding, and
the GAs, simple and with varying population size, attained near-optimal solutions in
few function evaluations (between 1000 and 2000). The GAVaPS was tested again in
[7] by Fernandes et al. with the Royal Road function R4. The R4 problem has differ-
ent characteristics than those previously tested and demands a higher computation.