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Consequences of parental care on population dynamics
Physica A 273 (1999) 140–144
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Consequences of parental care on population dynamics S. Moss de Oliveira Instituto de Fsica, Universidade Federal Fluminense Av. Litoranea ˆ s/n, Boa Viagem, Niteroi 24210-340, RJ, Brazil 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 di erent strategies of parental care are introduced into evolving 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 computer string of 32 bits, that can be regarded as a “chronological genome”. If the ith bit is equal to 1 the individual starts to su er from the e ects 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 value T , the individual dies. Lack of space and food is also taken into account through the Verhulst factor V =N (t)=Nmax , where N (t) is the actual size of the population and Nmax 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] .br (S. Moss de Oliveira) c 0378-4371/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII : S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 3 4 8 - 9
S. Moss de Oliveira / Physica A 273 (1999) 140–144
141
Fig. 1. Normalized survival rates of three sexual populations, each one corresponding to a di erent 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 o spring 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 di erent 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 de ned as Nk (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 from R to R + ASPC . In Fig. 1 we show the normalized survival rates of three sexual populations, each one corresponding to a di erent 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 The existence of post-reproductive periods observed in several species of mammals is one of the most challenging mysteries of Biology. Williams pointed out 40 years ago
142
S. Moss de Oliveira / Physica A 273 (1999) 140–144
Fig. 2. Histogram of the females menopause age for di erent 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 o spring which are younger than or at age ASPC automatically die. (2) Reproductive Risk: At the moment of giving birth, we calculate the reproductive risk of a female, Risk = Gd =T , where is a prede ned 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 beginning 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 reproductive 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 In this section we study the spatial distribution of populations subjected to di erent conditions of child-care. Each individual lives on a given site (i; j) of a square lattice, and there is a maximum allowed occupation per site. The Verhulst factor is now
S. Moss de Oliveira / Physica A 273 (1999) 140–144
143
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 probability pm = 0:2 to walk to the neighbouring 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 de ning a maternal care period APC during which no child can move alone. We considered the following conditions: (a) if the mother moves, she brings the young children with her; (b) the mother cannot move if she has any child still under maternal care. In Figs. 3 and 4 we show the con gurations of the lattice after 800 000 steps for cases (a) and (b), respectively. From these gures it can be noticed that the spatial distribution is strongly dependent on the child-care condition.
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S. Moss de Oliveira / Physica A 273 (1999) 140–144
5. Sexual delity versus high reproduction rate One of the rare examples of true monogamy in Nature is the California mouse. In this species a female is not able to sustain one to three pups alone. The pups are born at the coldest time of the year and depend on the parents body heat to survive [9]. To simulate this rare sexual behaviour we start by assuming that if a female reproduces this year, she spends the next two following years without reproducing. So we consider two time steps as the parental care period. Since in our simulations the female randomly chooses a male to mate, we impose that if the male is a faithful one, he will refuse, during this period, to mate any female that eventually chooses him as a partner. The non-faithful male accepts any invitation, but his o spring 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 o spring. We start the simulations with half of the males faithful, and half non-faithful. With such a strategy we obtained that for Pd 60:3, 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 [1] J.S. Sa Martins, S. Moss de Oliveira, Int. J. Mod. Phys. C 9 (1998) 421. [2] S. Moss de Oliveira, Physica A 257 (1998) 465. [3] S. Moss de Oliveira, P.M.C. de Oliveira, D. Stau er, 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. [7] S. Moss de Oliveira, A.T. Bernardes, J.S. Sa Martins, Eur. Phys. J. B 7 (1999) 501. [8] A.O. Sousa, S. Moss de Oliveira, Eur. Phys. J. B 9 (1999) 365. [9] V. Morell, Science 281 (1983) 1998. [10] A.O. Sousa, S. Moss de Oliveira, Eur. Phys. J. B 10 (1999) 781.
www.elsevier.com/locate/physa
Consequences of parental care on population dynamics S. Moss de Oliveira Instituto de Fsica, Universidade Federal Fluminense Av. Litoranea ˆ s/n, Boa Viagem, Niteroi 24210-340, RJ, Brazil 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 di erent strategies of parental care are introduced into evolving 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 computer string of 32 bits, that can be regarded as a “chronological genome”. If the ith bit is equal to 1 the individual starts to su er from the e ects 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 value T , the individual dies. Lack of space and food is also taken into account through the Verhulst factor V =N (t)=Nmax , where N (t) is the actual size of the population and Nmax 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] .br (S. Moss de Oliveira) c 0378-4371/99/$ - see front matter 1999 Elsevier Science B.V. All rights reserved. PII : S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 3 4 8 - 9
S. Moss de Oliveira / Physica A 273 (1999) 140–144
141
Fig. 1. Normalized survival rates of three sexual populations, each one corresponding to a di erent 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 o spring 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 di erent 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 de ned as Nk (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 from R to R + ASPC . In Fig. 1 we show the normalized survival rates of three sexual populations, each one corresponding to a di erent 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 The existence of post-reproductive periods observed in several species of mammals is one of the most challenging mysteries of Biology. Williams pointed out 40 years ago
142
S. Moss de Oliveira / Physica A 273 (1999) 140–144
Fig. 2. Histogram of the females menopause age for di erent 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 o spring which are younger than or at age ASPC automatically die. (2) Reproductive Risk: At the moment of giving birth, we calculate the reproductive risk of a female, Risk = Gd =T , where is a prede ned 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 beginning 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 reproductive 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 In this section we study the spatial distribution of populations subjected to di erent conditions of child-care. Each individual lives on a given site (i; j) of a square lattice, and there is a maximum allowed occupation per site. The Verhulst factor is now
S. Moss de Oliveira / Physica A 273 (1999) 140–144
143
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 probability pm = 0:2 to walk to the neighbouring 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 de ning a maternal care period APC during which no child can move alone. We considered the following conditions: (a) if the mother moves, she brings the young children with her; (b) the mother cannot move if she has any child still under maternal care. In Figs. 3 and 4 we show the con gurations of the lattice after 800 000 steps for cases (a) and (b), respectively. From these gures it can be noticed that the spatial distribution is strongly dependent on the child-care condition.
144
S. Moss de Oliveira / Physica A 273 (1999) 140–144
5. Sexual delity versus high reproduction rate One of the rare examples of true monogamy in Nature is the California mouse. In this species a female is not able to sustain one to three pups alone. The pups are born at the coldest time of the year and depend on the parents body heat to survive [9]. To simulate this rare sexual behaviour we start by assuming that if a female reproduces this year, she spends the next two following years without reproducing. So we consider two time steps as the parental care period. Since in our simulations the female randomly chooses a male to mate, we impose that if the male is a faithful one, he will refuse, during this period, to mate any female that eventually chooses him as a partner. The non-faithful male accepts any invitation, but his o spring 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 o spring. We start the simulations with half of the males faithful, and half non-faithful. With such a strategy we obtained that for Pd 60:3, 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 [1] J.S. Sa Martins, S. Moss de Oliveira, Int. J. Mod. Phys. C 9 (1998) 421. [2] S. Moss de Oliveira, Physica A 257 (1998) 465. [3] S. Moss de Oliveira, P.M.C. de Oliveira, D. Stau er, 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. [7] S. Moss de Oliveira, A.T. Bernardes, J.S. Sa Martins, Eur. Phys. J. B 7 (1999) 501. [8] A.O. Sousa, S. Moss de Oliveira, Eur. Phys. J. B 9 (1999) 365. [9] V. Morell, Science 281 (1983) 1998. [10] A.O. Sousa, S. Moss de Oliveira, Eur. Phys. J. B 10 (1999) 781.