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The two populations’ cellular automata model with predation based on the Penna model
Physica A 312 (2002) 243 – 250
www.elsevier.com/locate/physa
The two populations’ cellular automata model with predation based on the Penna model Mingfeng He, Jing Lin ∗ , Heng Jiang, Xin Liu Department of Applied Mathematics, Dalian University of Technology, 116024 Dalian, China Received 7 May 2001
Abstract In Penna’s single-species asexual bit-string model of biological ageing, the Verhulst factor has too strong a restraining e1ect on the development of the population. Danuta Makowiec gave an improved model based on the lattice, where the restraining factor of the four neighbours take the place of the Verhulst factor. Here, we discuss the two populations’ Penna model with predation on the planar lattice of two dimensions. A cellular automata model containing movable wolves and sheep has been built. The results show that both the quantity of the wolves and the sheep 5uctuate in accordance with the law that one quantity increases while the other one decreases. c 2002 Elsevier Science B.V. All rights reserved. PACS: 05.45.−a; 89.60.+x Keywords: Penna model; Cellular automata; Lotka–Volterra model; Predation; Population
1. Introduction The colony’s evolutive law has been considered in Penna’s single-species asexual model of biological ageing [1]. It is considered under the e1ect of the Verhulst factor, the genetic diseases, the mutations and the minimum reproduction age. The model has been applied to study the cod Ash [2], and a one-species cellular automata model based on the Penna model was found on a lattice of two dimensions [3]. In the model local dying rules take the place of the Verhulst factor. The authors discovered the single-species law of spatial distribution and mortality. We develop a cellular automata model aiming to study the wolves’ and sheep’s populations with predation on a lattice ∗
Corresponding author. Fax: +(86)-0411-4708599. E-mail address: [email protected] (J. Lin).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 0 8 5 0 - 6
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M. He et al. / Physica A 312 (2002) 243 – 250
of two dimensions. And the moving, predation and the wolves’ dying of hunger are all considered in the model. In Section 2, we introduce the Penna biological ageing model and our assumptive term. The model is described in Section 3 and the results are given in Section 4. 2. The Penna model and our assumptive term Each individual has a bit-string of 32 bits. Each bit is 1 or 0. An individual stays alive if its life programme read up to its present age is not overloaded by inherited diseases and the external environment, where the individual lives are not over occupied. Getting older each individual, if survives, obtains the right to produce an o1spring. Its life programme, modiAed by random mutations, is transmitted to the o1spring. In our model we assume that the sheeps are the exclusive foods of the wolves while the sheep’s foods are always enough. Each wolf or sheep is characterized by a bit-string of 32 bits. After the indraught of moving and hungry rules we get the following model. 3. The model Consider a square lattice N × N . Each site of the lattice can only have three states: empty, occupied by a single wolf (Typhlodromus occidentalis) or occupied by a single sheep (Eotetranychus sexmaculatus). First, we deAne the following parameters: • • • • • • • • • •
the remarks of the species: i = 1 represents wolf while i = 2 represents sheep; location j on the lattice, where j = 1; 2; : : : ; N 2 ; the individual’s age Aij , where i = 1; 2; bit-string Bij (k) of length of 32, k = 1; : : : ; 32, containing life history; Hjt : the hungry time of the wolf on location j on the iteration t; Tij : the current Atness of an individual on location j; Ti : the threshold for the accepted Atness of each population in this model; Hold: the longest time of a wolf’s being hungry that is accepted in the model mi : the mutation rate of population i; Ri : the minimum reproduction age of a wolf or a sheep; Then we give the rules of the model for the wolves and sheep, respectively: We search to handle the wolves preferentially: For the wolf at the location j: First A1j is increased by 1. Then calculate the number of diseases su1ered until the present age of a wolf as T1j =
A1j
B1j (k) :
k=1
If T1j ¿ T1 or T1j ¿ 32 (the maximum age of an individual) or Hjt ¿ Hold, the wolf dies. Otherwise, Hjt is increased by 1. Then when there is one or more sheep in its four Von Neumann neighbours, it will choose the one of the maximum T2 to eat and is
M. He et al. / Physica A 312 (2002) 243 – 250
245
located at the chosen site on the next iteration lattice. Hjt becomes 0. (If there are more than one sheep having the maximum T2 , the choice will be random.) When its Von Neumann neighbours are all wolves, it will remain they on the next iteration lattice. But when they are not all wolves and not any sheep, it will choose one empty sites in the four neighbours to move onto the next iteration. At last we consider the reproduction: compare A1j with R1 , if A1j ¿ R1 , the wolf generates b1 o1spring. And since then, it will generates b1 o1spring every r1 years. Each o1spring inherits a bit-string which is identical to its mother’s bit-string except for m randomly chosen additional deleterious mutations. The age for a new born baby is set to 1. The o1spring will be located randomly at one of the empty ones of the eight more neighbours of its mother on the next iteration lattice. So if there are no empty ones, it will die. If there has been a new-born wolf at the chosen site then an extra random toss decides which baby is alive. Then we start to handle the sheep whose rules are similar to those of wolves but a little more simple. For the sheep at the location j: First A2j is increased by 1. Then calculate the number of su1ered diseases until the present age of a sheep as T2j =
A2j
B2j (k) :
k=1
If T2j ¿ T2 or T2j ¿ 32 (the maximum age of an individual), the sheep dies. When its Von Neumann neighbours are all wolves and sheep, it will remain there on the next iteration lattice. But when there are not all wolves or sheep, it will choose one empty site in the four neighbours to move onto the next iteration. Then we consider the reproduction: compare A2j with R2 , if A2j ¿ R2 , the wolf generates b2 o1spring. And since then, it will generates b2 o1spring every r2 years. Each o1spring inherits a bit-string which is identical to its mother’s bit-string except for m randomly chosen additional deleterious mutations. The age for a new-born baby is set to 1. The o1spring will be located randomly at one of the empty ones of the four Von Neumann neighbours of its mother on the next iteration lattice. And so if there are no empty ones, it will die. If there has been a new born wolf at the chosen site, it will die. But if there has been a new born sheep at the chosen site then an extra random toss decides which baby is alive. With respect to the rating of the wolves being higher than that of the sheep in the food chain, we have assumed some priorities for the wolves such as the new-born wolf has some advantages in the competition with the new-born sheep and the birthing range of the wolves is larger than that of the sheep. Meanwhile, choosing the sheep with that of the maximum T2 to eat is based on such hypothesis: the more diseases one has experienced the easier it is to be caught. 4. Results In our simulation, L = 50, N0 (w) = 100, N0 (s) = 1000. r2 is Axed as 1 and let r1 change. As in the Penna model, a 1 in an individual’s bit string which
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M. He et al. / Physica A 312 (2002) 243 – 250
Fig. 1. (a) Hold = 3; R1 = 4; r1 = 4; R2 = 3; r2 = 1; T1 = 7; T2 = 3; m1 = m2 = 1; b1 = 2; b2 = 1; The fuscous spots represent the wolves while the light-coloured ones represent the sheep. The abscissa represents the time (iteration) while the ordinate represents the quantity. The wolves’ quantity and the sheep’s quantity raise and reduce alternately. The average quantity of the sheep is about 639. (b) increase the T2 by 1 and keep the other parameters unchangeable. It means that the sheep will live two more years on average. This time the 5uctuation also shows distinct periodicity, but the quantity of the sheep raise obviously (above 700).
corresponds to a genetic disease that a1ects this individual’s life at the age corresponding to the position of this 1 in the bit-string. Initially, the individual distribution of bit-strings is random. The evolution is a synchronized iteration. It means that, according to the evolution in cellular automata, the results of application of a local rule do not in5uence the current population state, but are collected in a copy of a lattice to be used in the subsequent iteration. And also we use the periodic boundary. First, we make the Hold = 3; R1 = 4; r1 = 4; R2 = 3; r2 = 1; T1 = 7; T2 = 3; m1 = m2 = 1; b1 = 2; b2 = 1. Then we get Fig. 1(a) of the quantity’s 5uctuation of the
M. He et al. / Physica A 312 (2002) 243 – 250
247
Fig. 2. (a) and (b) correspond to Fig. 1(a) and (b), respectively. All the parameters are the same except for the expansion of the range of the wolves’ predation from the four Von Neumann neighbours to the eight more neighbours.This time the quantity’s 5uctuation is obviously larger and more similar to the results of the Lotka–Volterra model.
wolves and sheep. Second, we increase the T2 by 1 and keep the other parameters unchanged. It means that the sheep will live two more years on average. Then we get Fig. 1(b). This time the 5uctuation also shows distinct periodicity, but the quantity of the sheep raise obviously. This shows that the T2 is sensitive in our model. And so is T1 . Further we expand the preying range of the wolves. Let it change from the four Von Neumann neighbours to the eight more neighbours. Using the same parameters as before, we get Fig. 2(a) and (b) corresponding to Fig. 1(a) and (b) in which the quantity’s 5uctuation is obviously larger. It is more similar to the result of Lotka– Volterra model [4]. In fact, according to the idiographic situation, the preying range can be adjusted then.
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M. He et al. / Physica A 312 (2002) 243 – 250
Finally, we illustrate the two populations’ 5uctuation and the process of their evolution through several snapshots. Using the same parameters as in Fig. 2(b), we get seven pictures in Fig. 3. Although the beginning is random, after some steps they come to be in order.
Fig. 3. (a) – (g), Hold = 3; R1 = 4; r1 = 4; R2 = 3; r2 = 1; T1 = 7; T2 = 4; m1 = m2 = 1; b1 = 2; b2 = 1, and the wolves’ preying range is the 8 more neighbours. The empty lattice represents the empty land. (a) is the initial state in which there are 100 wolves and 1000 sheep locating on the lattice randomly. After 20 steps (b), there are few wolves and only a few sheeps on the lattice. But after 10 steps (c), the number of sheep increased and the number of wolves become a little more ready for the increase in the following steps. So in step 60 (d), the wolves have expanded their dimensions. The ones on the edges are chasing sheep while the ones at the centre will die in the subsequent steps because of lacking in sheep. When it comes to step 70 (e), the dimensions of wolves has become quite large. And in step 90 (f) it even forms an isolated belt in which only few sheep and wolves exist. But just the few sheep occupy the belt after 20 steps (g) newly. And at this time the wolves start to besiege the sheep again in the top part of the lattice. So they keep multiplying just like this.
M. He et al. / Physica A 312 (2002) 243 – 250
249
Fig. 3. Continued.
5. Expectation Recently, more and more methods which use the computers to simulate the evolution of the biological populations have been introduced. Professor J. McGlade in Warwick College once introduced one of his research: simulating a small ecological system which consists of foxes, rabbits, grass and stones. The results are exciting. In our opinion, the most advantage of using the computers to simulate is to save large amounts of the money for constructing the real environment. And it can also prove some hypotheses and get some new ideas. We only simulate two species, but as long as the memory of the computer permits, more species can be simulated.
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M. He et al. / Physica A 312 (2002) 243 – 250
References [1] [2] [3] [4]
T.J.P. Penna, J. Stat. Phys. 78 (1995) 1629. S. Moss de Oliveira, T.J.P. Penna, D. Stau1er, Physica 215 (1995) 298–304. D. Makowiec, Physica A 289 (2001) 208–222. Zhenji Li, Xiaolin Chen, Hailei Zheng, Yuwu Lian, Ecology, Science Press, Beijing, 2000 (in Chinese).
www.elsevier.com/locate/physa
The two populations’ cellular automata model with predation based on the Penna model Mingfeng He, Jing Lin ∗ , Heng Jiang, Xin Liu Department of Applied Mathematics, Dalian University of Technology, 116024 Dalian, China Received 7 May 2001
Abstract In Penna’s single-species asexual bit-string model of biological ageing, the Verhulst factor has too strong a restraining e1ect on the development of the population. Danuta Makowiec gave an improved model based on the lattice, where the restraining factor of the four neighbours take the place of the Verhulst factor. Here, we discuss the two populations’ Penna model with predation on the planar lattice of two dimensions. A cellular automata model containing movable wolves and sheep has been built. The results show that both the quantity of the wolves and the sheep 5uctuate in accordance with the law that one quantity increases while the other one decreases. c 2002 Elsevier Science B.V. All rights reserved. PACS: 05.45.−a; 89.60.+x Keywords: Penna model; Cellular automata; Lotka–Volterra model; Predation; Population
1. Introduction The colony’s evolutive law has been considered in Penna’s single-species asexual model of biological ageing [1]. It is considered under the e1ect of the Verhulst factor, the genetic diseases, the mutations and the minimum reproduction age. The model has been applied to study the cod Ash [2], and a one-species cellular automata model based on the Penna model was found on a lattice of two dimensions [3]. In the model local dying rules take the place of the Verhulst factor. The authors discovered the single-species law of spatial distribution and mortality. We develop a cellular automata model aiming to study the wolves’ and sheep’s populations with predation on a lattice ∗
Corresponding author. Fax: +(86)-0411-4708599. E-mail address: [email protected] (J. Lin).
c 2002 Elsevier Science B.V. All rights reserved. 0378-4371/02/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 2 ) 0 0 8 5 0 - 6
244
M. He et al. / Physica A 312 (2002) 243 – 250
of two dimensions. And the moving, predation and the wolves’ dying of hunger are all considered in the model. In Section 2, we introduce the Penna biological ageing model and our assumptive term. The model is described in Section 3 and the results are given in Section 4. 2. The Penna model and our assumptive term Each individual has a bit-string of 32 bits. Each bit is 1 or 0. An individual stays alive if its life programme read up to its present age is not overloaded by inherited diseases and the external environment, where the individual lives are not over occupied. Getting older each individual, if survives, obtains the right to produce an o1spring. Its life programme, modiAed by random mutations, is transmitted to the o1spring. In our model we assume that the sheeps are the exclusive foods of the wolves while the sheep’s foods are always enough. Each wolf or sheep is characterized by a bit-string of 32 bits. After the indraught of moving and hungry rules we get the following model. 3. The model Consider a square lattice N × N . Each site of the lattice can only have three states: empty, occupied by a single wolf (Typhlodromus occidentalis) or occupied by a single sheep (Eotetranychus sexmaculatus). First, we deAne the following parameters: • • • • • • • • • •
the remarks of the species: i = 1 represents wolf while i = 2 represents sheep; location j on the lattice, where j = 1; 2; : : : ; N 2 ; the individual’s age Aij , where i = 1; 2; bit-string Bij (k) of length of 32, k = 1; : : : ; 32, containing life history; Hjt : the hungry time of the wolf on location j on the iteration t; Tij : the current Atness of an individual on location j; Ti : the threshold for the accepted Atness of each population in this model; Hold: the longest time of a wolf’s being hungry that is accepted in the model mi : the mutation rate of population i; Ri : the minimum reproduction age of a wolf or a sheep; Then we give the rules of the model for the wolves and sheep, respectively: We search to handle the wolves preferentially: For the wolf at the location j: First A1j is increased by 1. Then calculate the number of diseases su1ered until the present age of a wolf as T1j =
A1j
B1j (k) :
k=1
If T1j ¿ T1 or T1j ¿ 32 (the maximum age of an individual) or Hjt ¿ Hold, the wolf dies. Otherwise, Hjt is increased by 1. Then when there is one or more sheep in its four Von Neumann neighbours, it will choose the one of the maximum T2 to eat and is
M. He et al. / Physica A 312 (2002) 243 – 250
245
located at the chosen site on the next iteration lattice. Hjt becomes 0. (If there are more than one sheep having the maximum T2 , the choice will be random.) When its Von Neumann neighbours are all wolves, it will remain they on the next iteration lattice. But when they are not all wolves and not any sheep, it will choose one empty sites in the four neighbours to move onto the next iteration. At last we consider the reproduction: compare A1j with R1 , if A1j ¿ R1 , the wolf generates b1 o1spring. And since then, it will generates b1 o1spring every r1 years. Each o1spring inherits a bit-string which is identical to its mother’s bit-string except for m randomly chosen additional deleterious mutations. The age for a new born baby is set to 1. The o1spring will be located randomly at one of the empty ones of the eight more neighbours of its mother on the next iteration lattice. So if there are no empty ones, it will die. If there has been a new-born wolf at the chosen site then an extra random toss decides which baby is alive. Then we start to handle the sheep whose rules are similar to those of wolves but a little more simple. For the sheep at the location j: First A2j is increased by 1. Then calculate the number of su1ered diseases until the present age of a sheep as T2j =
A2j
B2j (k) :
k=1
If T2j ¿ T2 or T2j ¿ 32 (the maximum age of an individual), the sheep dies. When its Von Neumann neighbours are all wolves and sheep, it will remain there on the next iteration lattice. But when there are not all wolves or sheep, it will choose one empty site in the four neighbours to move onto the next iteration. Then we consider the reproduction: compare A2j with R2 , if A2j ¿ R2 , the wolf generates b2 o1spring. And since then, it will generates b2 o1spring every r2 years. Each o1spring inherits a bit-string which is identical to its mother’s bit-string except for m randomly chosen additional deleterious mutations. The age for a new-born baby is set to 1. The o1spring will be located randomly at one of the empty ones of the four Von Neumann neighbours of its mother on the next iteration lattice. And so if there are no empty ones, it will die. If there has been a new born wolf at the chosen site, it will die. But if there has been a new born sheep at the chosen site then an extra random toss decides which baby is alive. With respect to the rating of the wolves being higher than that of the sheep in the food chain, we have assumed some priorities for the wolves such as the new-born wolf has some advantages in the competition with the new-born sheep and the birthing range of the wolves is larger than that of the sheep. Meanwhile, choosing the sheep with that of the maximum T2 to eat is based on such hypothesis: the more diseases one has experienced the easier it is to be caught. 4. Results In our simulation, L = 50, N0 (w) = 100, N0 (s) = 1000. r2 is Axed as 1 and let r1 change. As in the Penna model, a 1 in an individual’s bit string which
246
M. He et al. / Physica A 312 (2002) 243 – 250
Fig. 1. (a) Hold = 3; R1 = 4; r1 = 4; R2 = 3; r2 = 1; T1 = 7; T2 = 3; m1 = m2 = 1; b1 = 2; b2 = 1; The fuscous spots represent the wolves while the light-coloured ones represent the sheep. The abscissa represents the time (iteration) while the ordinate represents the quantity. The wolves’ quantity and the sheep’s quantity raise and reduce alternately. The average quantity of the sheep is about 639. (b) increase the T2 by 1 and keep the other parameters unchangeable. It means that the sheep will live two more years on average. This time the 5uctuation also shows distinct periodicity, but the quantity of the sheep raise obviously (above 700).
corresponds to a genetic disease that a1ects this individual’s life at the age corresponding to the position of this 1 in the bit-string. Initially, the individual distribution of bit-strings is random. The evolution is a synchronized iteration. It means that, according to the evolution in cellular automata, the results of application of a local rule do not in5uence the current population state, but are collected in a copy of a lattice to be used in the subsequent iteration. And also we use the periodic boundary. First, we make the Hold = 3; R1 = 4; r1 = 4; R2 = 3; r2 = 1; T1 = 7; T2 = 3; m1 = m2 = 1; b1 = 2; b2 = 1. Then we get Fig. 1(a) of the quantity’s 5uctuation of the
M. He et al. / Physica A 312 (2002) 243 – 250
247
Fig. 2. (a) and (b) correspond to Fig. 1(a) and (b), respectively. All the parameters are the same except for the expansion of the range of the wolves’ predation from the four Von Neumann neighbours to the eight more neighbours.This time the quantity’s 5uctuation is obviously larger and more similar to the results of the Lotka–Volterra model.
wolves and sheep. Second, we increase the T2 by 1 and keep the other parameters unchanged. It means that the sheep will live two more years on average. Then we get Fig. 1(b). This time the 5uctuation also shows distinct periodicity, but the quantity of the sheep raise obviously. This shows that the T2 is sensitive in our model. And so is T1 . Further we expand the preying range of the wolves. Let it change from the four Von Neumann neighbours to the eight more neighbours. Using the same parameters as before, we get Fig. 2(a) and (b) corresponding to Fig. 1(a) and (b) in which the quantity’s 5uctuation is obviously larger. It is more similar to the result of Lotka– Volterra model [4]. In fact, according to the idiographic situation, the preying range can be adjusted then.
248
M. He et al. / Physica A 312 (2002) 243 – 250
Finally, we illustrate the two populations’ 5uctuation and the process of their evolution through several snapshots. Using the same parameters as in Fig. 2(b), we get seven pictures in Fig. 3. Although the beginning is random, after some steps they come to be in order.
Fig. 3. (a) – (g), Hold = 3; R1 = 4; r1 = 4; R2 = 3; r2 = 1; T1 = 7; T2 = 4; m1 = m2 = 1; b1 = 2; b2 = 1, and the wolves’ preying range is the 8 more neighbours. The empty lattice represents the empty land. (a) is the initial state in which there are 100 wolves and 1000 sheep locating on the lattice randomly. After 20 steps (b), there are few wolves and only a few sheeps on the lattice. But after 10 steps (c), the number of sheep increased and the number of wolves become a little more ready for the increase in the following steps. So in step 60 (d), the wolves have expanded their dimensions. The ones on the edges are chasing sheep while the ones at the centre will die in the subsequent steps because of lacking in sheep. When it comes to step 70 (e), the dimensions of wolves has become quite large. And in step 90 (f) it even forms an isolated belt in which only few sheep and wolves exist. But just the few sheep occupy the belt after 20 steps (g) newly. And at this time the wolves start to besiege the sheep again in the top part of the lattice. So they keep multiplying just like this.
M. He et al. / Physica A 312 (2002) 243 – 250
249
Fig. 3. Continued.
5. Expectation Recently, more and more methods which use the computers to simulate the evolution of the biological populations have been introduced. Professor J. McGlade in Warwick College once introduced one of his research: simulating a small ecological system which consists of foxes, rabbits, grass and stones. The results are exciting. In our opinion, the most advantage of using the computers to simulate is to save large amounts of the money for constructing the real environment. And it can also prove some hypotheses and get some new ideas. We only simulate two species, but as long as the memory of the computer permits, more species can be simulated.
250
M. He et al. / Physica A 312 (2002) 243 – 250
References [1] [2] [3] [4]
T.J.P. Penna, J. Stat. Phys. 78 (1995) 1629. S. Moss de Oliveira, T.J.P. Penna, D. Stau1er, Physica 215 (1995) 298–304. D. Makowiec, Physica A 289 (2001) 208–222. Zhenji Li, Xiaolin Chen, Hailei Zheng, Yuwu Lian, Ecology, Science Press, Beijing, 2000 (in Chinese).