Received 14 June 1999
Abstract
We review the results obtained using the Penna model for biological ageing (T.J.P. Penna, J. Stat. Phys. 78 (1995) 1629) when dierent strategies of parental care are introduced into evolv-ing populations. These results concern to: longevity of semelparous populations; self-organization of female menopause; the spatial distribution of the populations and nally, sexual delity.
c
1999 Elsevier Science B.V. All rights reserved.
PACS: 87.23.c; 05.10.-a
Keywords:Penna model; Biological ageing; Self-organization
1. Introduction
In the asexual version of the Penna model each individual is represented by a com-puter string of 32 bits, that can be regarded as a “chronological genome”. If theith bit is equal to 1 the individual starts to suer from the eects of a given genetic disease at his ith period of life. Each time-step of the simulation corresponds to read one bit of all the strings, and each individual can live at most for 32 periods (“years”). If at a given age the number of accumulated diseases (bits 1) reach the limit valueT, the individual dies. Lack of space and food is also taken into account through the Verhulst factor
V=N(t)=Nmax, whereN(t) is the actual size of the population andNmax is the maximum
environmental capacity. At every time-step a random number between zero and one is generated for each individual, and compared with V: if the number is smaller than V
the individual dies, independently of his age or genome. When the individual reaches
E-mail address:[email protected] (S. Moss de Oliveira)
Fig. 1. Normalized survival rates of three sexual populations, each one corresponding to a dierent period of parental care: solid line (circles) – ASPC= 0; dot–dashed line (squares) –ASPC= 1; long dashed (stars)
–ASPC= 2.
the minimum reproduction age R, it generates b ospring every year. The genome of the baby is a copy of the parent’s one, with M random deleterious mutations.
In the sexual version each “genome” has two bit-strings instead of one, recessive mutations are distinguished from dominant ones and there is crossing and recombination of the strings during reproduction. For detailed explanations see [1]; for a review of the model see [2,3] and references therein.
2. Parental care in semelparous populations
In this section we introduce parental care in the Penna model for semelparous species, i.e., for species that reproduce only once in life. The Harlequin Stink Bug (Tectocoris diophthalmus) from Australia is an example: because she lays only one batch, she defends her eggs aggressively, since they are her sole chance for reproductive success [4]. We adopted two dierent strategies of parental care: (1) babies with a living mother are protected from the Verhulst factor deaths until they reach a limit age APC;
(2) any baby younger than age ASPC is killed if its mother dies (Strong Parental Care).
The survival rate is dened asNk(t)=Nk−1(t−1), where Nk(t) is the population with
age k at time t. Unexpectedly, we obtained that the survival rates do not change with the rst strategy, independently of the parental care period considered [5].
However, when we adopted the second strategy we found that the nal survival age is pushed fromRtoR+ASPC. In Fig. 1 we show the normalized survival rates of three
sexual populations, each one corresponding to a dierent period of parental care. In all cases there is a single reproduction age R= 10 for females with males reproducing every year from age 10 until death.
3. Self-organization of female menopause age
Fig. 2. Histogram of the females menopause age for dierent periods of maternal care. When the maternal care period is too short, there is no self-organization of menopause age.
that menopause “may have arisen as a reproductive adaptation to a life-cycle already characterized by senescence, unusual hazards in pregnancy and childbirth, and a long period of juvenile dependence” [6]. In order to test this hypothesis we introduced the following ingredients into the Penna model: (1) Maternal Care: If at a time step a female (mother) dies, all her ospring which are younger than or at age ASPC
auto-matically die. (2) Reproductive Risk: At the moment of giving birth, we calculate the reproductive risk of a female, Risk =Gd=T, where is a predened factor which can
reduce or increase the whole risk function, and Gd is the number of diseases already
accumulated at the female’s current age. (3) Age of menopause Am: At the
begin-ning of the simulation males and females can reproduce until the end of their lives (Am= 32). When a female with a given value of Am gives birth to a daughter, the
daughter’s value of Am is the same as its mother with a probability Pm, or is equal to
Am±1 with probability (1−Pm=2).
In Fig. 2 we show that (after many generations) for long enough periods of strong maternal care the age of menopause self-organizes. However, if we consider reproduc-tive risk alone or strong parental care alone, no organization appears. We also obtained that 20% of the fertile female population have post-reproductive life [7].
4. Spatial distribution of the populations
Fig. 3. Final distribution of the population on the lattice considering that if the mother moves, she brings the children under maternal care (age62) with her (case a).
Fig. 4. Final distribution of the population on the lattice considering that the mother cannot move if she has any child still under maternal care (case b).
given by
Vi; j= 1−Ni; j(t)=Ni; j(max):
At every timestep each individual has a probabilitypm= 0:2 to walk to the
neighbour-ing site that presents the smallest occupation, if this occupation is also smaller or equal to that of the current individual’s site. We start the simulations randomly distributing one individual per site on a diluted lattice of 150×150 sites. The initial population
N(0) = 10 000 (asexual) individuals and the carrying capacity per site is Ni; j(max)= 34
[8]. Now our strategy of child-care consists in dening a maternal care period APC
during which no child can move alone.
a partner. The non-faithful male accepts any invitation, but his ospring still under parental care pay the price for the abandonment: they have an extra probability Pd of
dying. The delity state of the father is transmitted to the male ospring. We start the simulations with half of the males faithful, and half non-faithful. With such a strategy we obtained that for Pd60:3, after many generations there are no faithful males inside
the population. However, for Pd¿0:4 there is always a fraction of faithful males in
the population, this fraction increasing with increasing Pd. For Pd= 1 all the males
become faithful [10]. Such result shows that not always the strategy of the highest reproduction rate is the best one.
Acknowledgements
A.O. Sousa, J.S. Sa Martins, A.T. Bernardes and K.M. Fehsenfeld are acknowledged for the collaboration on the papers cited here; to CNPq, CAPES and FAPERJ for nancial support.
References
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[3] S. Moss de Oliveira, P.M.C. de Oliveira, D. Stauer, Evolution, Money, War and Computers, Teubner, Stuttgart-Leipzig, 1999.
[4] D.W. Tallamy, Sci. Am. (January 1999) 50.
[5] K.M. Fehsenfeld, J.S. Sa Martins, S. Moss de Oliveira, A.T. Bernardes, Int. J. Mod. Phys. C 9 (1998) 935.
[6] G.C. Williams, Evolution 11 (1957) 398.
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[9] V. Morell, Science 281 (1983) 1998.
