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Physica A 273 (1999) 182–189
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Penna model in migrating population – eect of
environmental factor and genetics
Maria S. Magdon a; ∗ , Andrzej Z. Maksymowicz b
a Department

of Mathematical Statistics, Agriculture University, al. Mickiewicza 21,
31-120 Cracow, Poland
b Department of Physics and Nuclear Techniques, AGH, al. Mickiewicza 30, 30-059 Cracow, Poland
Received 31 May 1999

Abstract
We consider eect of possible migration between dierent locations on population evolution.
Examples of dierent rules based on preferences to live in bigger or smaller populations, (or
environmental capacity, or living space available), are discussed. In the limiting case of small
migration intensity, each location evolves independently according to its local rules and conditions, as expected. With increasing migration, the population distribution between locations
changes, including some critical behavior of extinction of population in some locations for specic set of the rules. Then the deserted location may become populated again if the migration is
still on increase as a result of a pressure to move. Presented version is devoted to the migration
controlled exclusively by environmental factor, yet the model is primarily designed to describe

in
uence of other factors which may control migrating processes such as inherited mutation load,
age, or other parameters associated either with individuals or the specic location, races mixing,
c 1999 Elsevier Science B.V. All rights reserved.
or recovery of environmental capacity.
PACS: 89.60.+x; 05.45.-a
Keywords: Penna model; Population evolution; Logistic equation; Migration

1. Introduction
In the simplest logistic model, population evolution may be described in terms of
the normalized to the environmental capacity N population n(t) ¡ N at time (t). Then
x (t + 1) = (1 + B)x (1 − x) ;
Corresponding author. Tel.: +48-632-16-20 extn. 261.
E-mail address: rrmagdon@cyf-kr.edu.pl (M.S. Magdon)

∗

c 1999 Elsevier Science B.V. All rights reserved.
0378-4371/99/$ - see front matter
PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 3 5 3 - 2

(1)

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183

where the right-hand side x =n=N is at time t. This equation predicts single value equilibrium x =B=(1+B) at growth rates 0 ¡ B ¡ 2, followed by cyclic solutions for larger
B and then a chaotic regime. However, it must be noted that in computer simulation
we apply probabilistic rules for the system evolution, and so the deterministic version
of the logistic equation must be replaced by its version with some noise admixture
; x(t + 1) = (1 + B) x (1 − x) + . This mimics draws for the elimination (known as the
Verhulst factor [1]) of an individual due to the limited environmental capacity instead
of purely deterministic recipe. As a result, we may not recover in the simulation cyclic
behavior which may be suppressed by the noise.
The Penna model [2,3], may be considered as a generalization of the logistic picture
for the case when the evolution rules include actual age a of an individual. Then
population x(t; a) may be considered as dierent groups of dierent ages, and after one
time step we get x(t + 1; a + 1). In the Penna model each individual is characterized
by an inherited genome which already possesses and anticipates all the future life

of the individual. The genome is a computer word which contains information of
‘bad’ mutations prescribed for the individual’s whole future life. At the beginning, the
mutations are idle and become activated by
ow of time, one piece of information
read from genome per each time step, and adding up to the already active number of
mutations. On entry to the next era we scan over all members and eliminate the ones
which are too old, or have too many active mutations (above threshold T ) or are hunted
out, or perhaps for limited environmental capacity (the Verhulst factor). However, if
the individual survives, it may give birth with probability B if the reproduction age
R is reached. The baby’s genome is inherited from the parent, and gets M additional
mutations randomly picked over its whole lifespan.
Thus dynamics of population n(t; a) is governed by a genetic load, environmental
capacity N , the growth rate B and the minimum reproduction age R, bad mutation
threshold T , the extra mutations M for the newly born and other parameters which
make the rules on how the system evolves. A strong advantage of the Penna model is
its
exibility which makes it easy to produce specic versions devoted to some picked
out scenario. This allows to study the role of dierent factors such as in
uence of
sexual selection, parental care, overshing or hunting etc., on the population [4,5].

In this paper we intend to account for migration in the population split into several
locations and with possible transfer of individuals between these locations.

2. Model
We consider a number of locations labeled by i, each with its own set of evolution
parameters. One iteration step leading from normalized population xi (t) = ni =Ni at time
t to xi (t + 1) is then a result of a scan over the whole population. For each item in
the population we consider
• : : : start from xi (t) : : :
• virtual elimination due to the Verhulst factor or other reasons, then

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M.S. Magdon, A.Z. Maksymowicz / Physica A 273 (1999) 182–189

• giving birth
• : : : nish at xi (t + 1).
This is the Penna model which we apply for s iteration steps. This yields a new population yi = xi (t + s). Then after a given number s of evolution steps, we also allow
each individual to migrate from location i to another location j with probability p(i; j).
The probability p(i; j) is negotiable according to variety of scenarios which may be

claimed or justied. After all migration moves are carried out, the full iteration cycle
is complete:
• : : : from xi (t), by elimination followed by growth, s-times, to yi = xi (t + s) : : :
• : : : from yi = xi (t + s), by migration, to xi (t + s):
Obviously, for p(i; j) = 0 the game describes independent sub-populations and the standard Penna model applies to each location. However, when migration is allowed, a new
equilibrium is established which may be dierent from the isolated islands case. In the
limiting case of logistic equation R = 0; M = 0 and large threshold T , the population
yi after elimination and growth is expected to be
yi = ki xi (1 − xi ) ;

(2)

where ki = 1 + Bi . The migration process alters yi to the new value xi (t + s),
xi (t + s) = yi +

N X
Tij ;
Ni j

(3)

where
N=

X

Ni

(4)

i

and the antisymmetrical transfer matrix Tij = −Tji is
Tij =

Nj
Ni
yj p(j; i) − yi p(i; j) :
N
N

(5)

The rst term represents the in
ow of population into ith location from any location
j, while the other term is the opposite. As we said, p(i; j) stands for the probability
of an individual to migrate from location i to location j. In the following we conned
ourselves to the uncorrelated version when
p(ij) = q(i)p(j)

(6)

and so
Tij =

Nj
Ni
yj q(j)p(i) − yi q(i)p(j) :
N
N

(7)

It is promptly seen that for the case q(i) = q and p(j) ˙ yj Nj =N , the actual population
at time (t + s) just before migration, the transfer matrix vanishes and so we get the
claimed random migration for which xi (t + s) = yi , according to the Penna scheme
(we may choose dierent q(i) and p(j) also leading to the neutral migration). For a

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185

random migration
q(i) = q;

(8)

p(j) ˙ yj Nj =N ;

where yj Nj is the population at time (t + s) after executing the s evolution steps with

no migration, and just before the nal step of virtual migration which closes the full
cycle of evolution,
• : : : x(t) → (s-steps of elimination and growth) → y → (migration) → x(t + s):
The neutral or random migration means x(t + s) = y. For non-random migration we
choose probabilities of the migration dierent from the ones leading to the random
moves. For example, for
q(i) = q=(yi Ni =N ) ;

(9)

where q is a proportionality constant, then the total number of the ‘move-out-of’ individuals is the same for each location, and independent of the actual population there.
This is a resemblance of a ‘quota’ limit policy for migrating people, and the same
for all locations, which may lead to faster escape of individuals from already deserted
areas. As a result we expect a new equilibrium between locations with a tendency to
clustering. Perhaps it is illustrative to consider the stability of the coupled (by migration
process) two identical locations in the simple logistic case. Obviously yi = yj is the
solution. However, if a small
uctuation (t) is allowed, so that one of the locations
has a surplus of population, xi + , at the cost of the other location population, xj − ,
then the system response after one full cycle with (t + s) = r(t). The ratio r may be

obtained for the pure logistic equation as
r = (1 − B) + (1 − B)(q=qmax );

qmax = 2B=(1 + B) ;

(10)

with maximum q coecient so that the probability is less that one. The system becomes
unstable for the mobility parameter q larger then a minimum qmin value when r exceeds
one. This yields the instability regime for q,
0 ¡ B2 =(1 − B2 ) ¡ 2q ¡ B=(1 + B);

0 ¡ B ¡ 0:5 :

(11)

Let us summarize the main concept. In the above example we claimed modication of
the migration probability, as compared against the reference random transfers, according
to some environmental factors such as the actual population yi Ni .
This may serve as indication of what may be expected from computer simulation,

also for the non-logistic version based on the Penna model. Actually, the dierences
from the simulations show that positions of critical qc for the phase transitions are
similar for the Penna and the logistic case.

3. Results and conclusions
Let us discuss some typical examples of population evolution for (l) logistic case
with growth rate B=0:2, and=or for (p) Penna rules with mutation rate M =1, threshold
T =3, minimum reproduction age R=4 and higher growth rate B=0:5 to balance higher

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Fig. 1. Normalized population xi = ni =Ni at 2 locations versus mobility q, logistic case for set of the model
parameters (ni ; ei ; vi ; nj ; ej , vj ) = (1; 0; 0; 0; 0; 0).

Fig. 2. Normalized population xi = ni =Ni at 2 locations versus mobility q, Penna case for set of the model
parameters (ni ; ei ; vi ; nj ; ej ; vj ) = (1; 0; 0; 0; 0; 0).

death rate due to the bad mutations. The logistic case may serve as a test for which
some analytical results were obtained. In each case we use population of order of 106
or so on a 32 bit machine. The number of iterations necessary to get an equilibrium
is around 100 iteration steps. In examples with two locations, the environmental capacities Ni were assumed the same, as we are interested in stability of the system.
Examples with 3 locations were with capacities ratio 1 : 2 : 3.
First we consider two identical locations with the mentioned ‘quota’ rule when total
number of outgoing individuals is the same for each location. This causes the preference towards clustering. Analytical results for the logistic case show instability above

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187

Fig. 3. Normalized population xi = ni =Ni at 3 locations versus mobility q, logistic case for set of the model
parameters (ni ; ei ; vi ; nj ; ej , vj ) = (1; 0; 0; 0; 0; 0): The normalized population at 2 sites is nearly the same,
see text.

Fig. 4. Normalized population xi = ni =Ni at 3 locations versus mobility q, Penna case for set of the model
parameters (ni ; ei ; vi ; nj ; ej ; vj ) = (1; 0; 0; 0; 0; 0): The normalized population at 2 sites is nearly the same,
see text.

critical qc = 0:5B2 =(1 − B2 ), and only for B ¡ 0:5, which yields qc = 0:021. So we
expect population x(q) = n=N to stay stable at same level for both locations at the
value x = B=(1 + B) = 0:17 predicted by the logistic equation until about (remember
non-deterministic implementation of deterministic logistic rules) q = 0:021, and then
followed by perhaps sharp decrease in population in one of the location as a result of
instability. This is illustrated in Fig. 1.

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M.S. Magdon, A.Z. Maksymowicz / Physica A 273 (1999) 182–189

Fig. 5. Normalized population xi = ni =Ni at 3 locations versus mobility q, Penna case for set of the model
parameters (ni ; ei ; vi ; nj ; ej ; vj ) = (0; −1; 0; 0; 0; 0):

Fig. 6. Normalized population xi = ni =Ni at 3 locations versus mobility q, Penna case for set of the model
parameters (ni ; ei ; vi ; nj ; ej ; vj ) = (0; 0; −1; 0; 0; 0):

It should be noted that for still larger q, when migration is forced to become very
intensive, the deserted location may become re-occupied again. It may be seen that
both logistic and Penna cases (see Fig. 2) are rather similar.
Similar behavior is seen in Figs. 3 and 4 for 3 locations with relative capacities
Ni = 1 : 2 : 3. Here, however, the rst two locations (1 : 2) provide nearly the same
normalized population xi at any q and xi tends to zero for intermediate range of q.
Larger q restores x1 = x2 = x3 at equilibrium.
An opposite tendency towards avoiding overcrowded locations is demonstrated in
Figs. 5 and 6 for the Penna model and caused by dierent mechanisms: avoiding large

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189

territories and avoiding densely populated spaces, respectively. This time locations of
larger capacities are less densely packed.
We conclude that migration is important and may signicantly alter population
distribution.

Acknowledgements
The work was partly supported by grant of Agriculture University in Krakow
and also by University of Mining and Metallurgy. Main computer simulations were carried out on HP Exemplar S2000 machine at the Academic Computer Center

CYFRONET-KRAKOW.
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[3] T.J.P. Penna, D. Stauer, Ecient Monte Carlo simulations for biological ageing, Int. J. Mod. Phys. C
6 (1995) 233.
[4] A.T. Bernardes, Monte Carlo simulations of biological ageing, Ann. Rev. Comput. Phys. 4 (1996) 359.
[5] S.G.F. Martins, T.J.P. Penna, Computer simulation of sexual selection on age-structured population, Int.
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