1 Department of Mathematical Statistics, Agriculture University,
al.Mickiewicza 21, 31-120 Cracow, Poland
2 Department of Physics and Nuclear Techniques, AGH,
al. Mickiewicza 30, 30-059 Cracow, Poland
3 Institute of Computer Science, AGH,
al. Mickiewicza 30, 30-059 Cracow, Poland
4 Academic Computer Centre CYFRONET,
ul. Nawojki 11, 30-950 Cracow, Poland
bubak@uci.agh.edu.pl, phone: 617 39 64, fax: 633 80 54
The Penna bit-string model of life history [1], in which an individual is represented just by one chromosome, is the most commonly used for the study of dynamics of population with mutations. In the standard version, harmful mutations cause death of an individual if they exceed a threshold value $T$. The genetic mutations are inherited from parents. A strong advantage of the Penna model model is its flexibility which makes it easy to produce a many versions of the model, which allows to study the role of different factors such as influence of parental care, overfishing or hunting etc., on the population [2].
In this paper we concentrate on the role of fluctuations in $T$ on the population growth dynamics. Such a distribution of $T's$ may reflect some individual resistance to illnesses, or some non-inhereted somatic mutations, or other factors. Instead one single value, we consider a distribution of $T$ with nonzero width $\Delta$. Individuals with low threshold $T$ die earlier, even before birth, and we may adjust the $T$ distribution parameters so as to recover the claimed ratio around $0.6$ of human embryos which die before birth [3]. Therefore with introducing $\Delta>0$ we simulaneously need to readjust the birth rate $B$ so as to compensate for that losses and get same inflow of newly born individuals as in the standard model.
If such fluctuations in $T$ are switched on, we observe some structural changes in the final population. Average life time is higher and the age distribution of the population is different. That is, the number of individuals $pop$ at given age versus $age$ decreases with different slope. The Verhulst factor [4], which controls death rates caused by limited environmental capacity, makes comparable percentage of death. However death ratio due to bad mutations differs, especially among younger individuals. The age distribution of mutations seems to remain same, yet survival distribution against $age$ is again altered.
This kind of computer simulations (similar to those with Ising model) is based mainly on bit and integer operations. Genome of an individual is implemented as a computer word with 32 bits. Activation of mutation corresponds to setting up a bit at a position related to a year of life. The simulation program was written in C and parallelized on the instruction level and with pragmas and compiler directives.
The simulations were carried out on HP S2000 at the Academic Computer Center CYFRONET-KRAKOW.
Acknowledgements.
We are grateful to Dietrich Stauffer and Stanislaw Cebrat for new ideas on possible modifications and improvements in the Penna model. This research was partly supported by the Agriculture University and AGH grants.
1 Penna T. J. P., A Bit-String Model for Biological Ageing, J. Stat. Phys. 78 (1995) 1629.
2 Bernardes A. T., Monte Carlo Simulations of Biological Ageing, Ann. Rev. of Computational Physics 4 (1996) 359.
3 Copp A. J., Trends Genet. 11 (1995) 87.
4 Brown D. and Rolhery P., Models in Biology: Mathematics, Statistics and Computing, Wiley, New York, 1993.