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Ecological competition and the role of an apex predator
Physica A 389 (2010) 4075–4080
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Ecological competition and the role of an apex predator G. Camargo-Gamboa a , R. Huerta-Quintanilla a , M. Rodríguez-Achach b,∗ a
Departamento de Física Aplicada, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Unidad Mérida, Mérida, Yucatán, Mexico
b
Facultad de Física e Inteligencia Artificial, Universidad Veracruzana, Lomas del Estadio S/N, Xalapa, Veracruz, Mexico
article
info
Article history: Received 4 May 2009 Received in revised form 9 March 2010 Available online 4 June 2010 Keywords: Ecological competition Prey Predator Ecosystem
abstract In this work we propose an agent-based model to study the process of ecological competition between two species coexisting with a common predator, including inter and intra-specific competition. Without the predator, the ecosystem always reaches a configuration in which one of the prey species faces extinction and the other survives, as is well known. However, we find that there is an optimum value for the amount of food consumed by the prey species for which the mean extinction time (MET) is maximum. In the presence of predators, the average extinction time may increase or decrease with respect to the non-predator case, depending on the food availability. It is also observed that the average extinction time has an optimum value depending on the efficiency of prey of the predator. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Almost all known ecosystems in nature are comprised of several different species which have predator–prey relations between them and coexist in equilibrium [1–3]. Theoretical studies of the dynamics of these systems show a very rich behavior, like the existence of phase transitions from mixed to segregated states [4–6]; the spontaneous emergence of spatial patterns [3,7,8]; complexity [9,10]. Noise-induced effects are also frequently studied. Noise can indeed guide some evolutionary processes depending on its intensity [11], and produce interesting effects such as pattern formation, resonances, temporal oscillations and Hopf bifurcations [12–15]. The predator–prey relations generate structures known as trophic pyramids [16], with the predators sitting on the upper levels. The predator on top of the pyramid is known as the apex predator and is known to have a crucial role in maintaining the health and stability of the ecosystem [17,18]. Without the predator’s influence, the ecosystem would be destabilized and the prey will become involved in an ‘‘arms race’’ which eventually drives one of the competing species into extinction [19–21]. The conditions that lead to equilibrium or stability of ecosystems are not fully understood; biodiversity for example, seems to be a key ingredient for stability. The underlying structure of the food web also plays an important role, among many other things. There is still a lot of discussion about the leading dynamics on ecosystems [22], specifically on how some species can form stable ecosystems but, at the same time, other species face instability and extinction. Defining stability for an ecosystem can be done in several ways, and there are various types of stability [23]. One can define it by the response of the system under small perturbations, by the number of stable states of the system, by the amplitude of fluctuations or by the response of the system to the extinction of species, among other criteria [24–26]. For a toy ecosystem like the one studied here, where three species compose the total and most of the subtleties of real ecosystems are excluded, the longer the time it takes for an extinction to occur, the more stable the system is.
∗
Corresponding author. Tel.: +52 228421747; fax: +52 2288178209. E-mail address: [email protected] (M. Rodríguez-Achach).
0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.05.039
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G. Camargo-Gamboa et al. / Physica A 389 (2010) 4075–4080
Predator and prey systems, since the pioneering work of Lotka and Volterra [27,28], have traditionally been studied using mean field type models, i.e., differential and difference equations. These ignore space correlations and replace local interactions by uniform long-range ones. However, they are very useful as a first approach to understand the general behavior of a complex system. Agent-based models take explicitly into account spatial effects and local interactions, and therefore we believe they are better suited for describing an ecosystem than mean field models, especially when we are interested in local interactions like competition for space and food, aggression between neighbors and the like. In this work, we studied the processes of competition between two herbivore species coexisting in the same habitat, and competing for the same resources, both in absence and in presence of a third species (the apex predator) which is allowed to prey on them. 2. The model We use an agent-based model consisting of three species: two competing herbivore species (A and B) which feed on grass, and a predator species (C ) which prey on both herbivores. Models with more than three species have also been studied by other authors [11,29–31] in a different context. The prey are identical to the predator so it has no preference over any of them. All three species can live indefinitely, and death arises only as the result of aggression or starvation (when the predators cannot find prey or the herbivores do not have enough grass to eat). The habitat is a square lattice of size N × N with periodic boundary conditions, and at each site on the lattice there is grass constantly growing at a fixed rate, so it is an abiotic resource. The prey consumes grass at a certain rate a, called the environmental limitation factor. Higher values of a thus mean that it is harder for the herbivores to find enough food to survive. The competition process between herbivores consists of attacks or aggression from one prey species to the other (interspecific competition), that is, members from species A (B) are allowed to attack members belonging to species B (A) with an aggression intensity (probability of attack) σ12 (σ21 ). Note that since both herbivores compete for a common limiting resource – in the form of grass – we also have exploitation competition in the model. We also included intraspecific competition, where members of the same prey species A (B) fight with each other with an aggression intensity σ11 (σ22 ). The only interaction between prey and predators is the trophic relation: predators can attack prey with an aggression intensity ρ , but prey are not allowed to attack predators in response. At the beginning of each simulation, elements from all species are randomly distributed on the lattice. At each time step an individual is picked at random, and one of its neighbors from the von Neumann neighborhood is also randomly chosen. Allowing the individuals to interact only in the immediate neighborhood restricts the model to species that live in closed groups, as is frequently the case for many herbivores. Then the interactions proceed as follows: 1. If the individual is a prey and its neighbor is an empty site then, if there is enough grass to eat, it survives, and it can also reproduce asexually with reproductive rate b. In this case the new element is placed on the empty site. If the prey does not reproduce, it moves to the empty site. In all the following cases when the individual chosen is one of the prey, it has to be able to eat enough grass and reproduce in order to survive, or die, as in this case, so we will not mention it again. 2. If the individual is a prey A (B) and its neighbor is another prey B (A), then the neighbor is killed with probability σ12 (σ21 ). 3. If the individual is a prey A (B) and its neighbor is of the same species, then the neighbor is killed with probability σ11 (σ22 ). 4. If the individual is a predator and its neighbor is a prey, then the predator attacks the prey who dies with probability ρ . If the prey is killed (eaten), the predator can asexually reproduce, otherwise it dies with probability d. 5. If the individual is a predator and its neighbor is an empty site then it dies with probability d. Otherwise, with probability (1 − d), the predator moves to the empty site. 6. If the individual is a predator and its neighbor is another predator then it dies with probability d. The above steps are repeated n times, where n is the total number of individuals, which constitutes a Monte Carlo step (MCS). After each MCS, the grass grows by a fixed amount at every lattice site. The prey feeds on the grass at a rate a, which constitutes the environmental limitation factor for prey species. In the absence of the predator, the habitat only holds prey species A and B, and we only consider the competition processes between prey. In that case, steps 4, 5 and 6 are no longer taken into account. 3. Results A lattice size of 60 × 60 is considered in all simulations, and averages are taken over at least 5000 realizations. The initial population densities are xA = 0.20, xB = 0.20 and xC = 0.15, where xi is the population density of species i. The results of the whole simulations are robust against changes in initial densities, unless the differences are too large. The parameters b and d are fixed at 0.25 and 0.03 respectively. In Fig. 1(a) we show the results for the interaction of the two prey species in absence of the predator (xC = 0). Our results confirm the well known fact that, when two species compete for the same resource, other things being equal, one of them becomes extinct (in community ecology, this is known as Gause’s law of competitive exclusion [32]). Therefore in Fig. 1(a), we record the time elapsed until the first species becomes extinct. Note that when no aggression exists between them
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Fig. 1. (a) Mean extinction time for prey as function of the environmental limitation factor a. No predator is included and we set σ12 = σ21 = σ . From top to bottom, the values of prey-to-prey aggression intensity σ are 0.0, 0.02, 0.05, 0.10 and 0.99. (b) Results with intraspecific competition included. σ is fixed at 0.1. Values for σ11 = σ22 are: 0.0 (crosses), 0.05 (diamonds), 0.07 (squares) and 0.10 (circles).
(σ = 0), the MET grows monotonically when the environmental limitation factor a decreases, i.e., there is more available food for the prey. However, as soon as there is competition between them (σ > 0), the MET shows a local maximum. This behavior arises as the result of two processes: suppose we are located at the local maximum of the curve for which σ = 0.10 (approximately at a = 0.11) in Fig. 1(a). If a increases, there is less food available and the MET decreases because of starvation of the prey. On the other hand, if a decreases, there is more food, but the population density increases (for the case of σ = 0.10, the mean population density increases from 18% at a = 0.14 to 83% at a = 0.06), therefore there are more attacks between the prey, and this ‘‘territorial fight’’ also decreases the MET. If a decreases even more, there is so much food available that the effect of prey fights is not significant and the MET increases again until it saturates. When intraspecific competition is included, the behavior becomes more complicated, as shown in Fig. 1(b). There are two regimes separated at a value of the environmental limitation factor a of approximately 0.07. When a > 0.07, we observe that as the intraspecific competition increases, so does the MET. This is because the increased competition decreases the population density, which gives the prey more available food. On the other hand, in the region a < 0.07, when intraspecific competition increases, the population density also decreases, but now the increment in available food is not important since in this region we already have plenty of food, therefore the MET initially decreases since attacks are more frequent. However, when the intraspecific competition increases even more, the MET increases greatly (curve with circles). This effect comes from the fact that at higher values of competition the density becomes very low, therefore encounters between prey become rare, with the result that attacks are less frequent, hence the MET shows an increase. Despite the fact that a lattice of size 60 × 60 can be considered small enough to introduce some simulation artifacts, we have tested that it is not the case. Fig. 2 shows that the results from Fig. 1(a) do not qualitatively change for larger lattice sizes. Similarly, other system behaviors are maintained as the lattice size is increased. The behavior described above changes when predators are included in the habitat. In Fig. 3, we show the MET, defined as the time when the first herbivore species becomes extinct. After that, we get the well known population oscillations of a single predator–single prey model. We use a fixed value of σ12 = σ21 = 0.10. When no predators are present (ρ = 0, circles), the behavior is the same as in the curve with crosses in Fig. 1(a). The introduction of predators makes the MET increase monotonically, as more food is available for the prey. Thus the process by which the MET decreases because of prey-to-prey aggression no longer produces the maxima and minima in the curves. The inset shows how the curve for ρ = 0.04 is modified when intraspecific competition is present between the prey: we get higher METs, as the intraspecific competition increases, which is a result of the lower population density induced by competition. Returning to the main figure, note how depending on the value of the environmental limitation factor, the presence of predators can make the MET increase or decrease. In order to investigate this behavior in more detail, we studied the MET
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Fig. 2. Same as the fourth curve (from top to bottom) in Fig. 1 for different lattice sizes: 60 × 60 (circles), 100 × 100 (squares) and 150 × 150 (asterisks). The qualitative behavior of the system does not change.
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Fig. 3. Mean extinction time for prey as function of the environmental limitation factor a, with predators included. The curve with circles is for ρ = 0.0, the diamonds curve is for ρ = 0.03 and the squares curve is for ρ = 0.04. The value of the prey-to-prey aggression intensity is fixed at σ12 = σ21 = 0.1 and σ12 = σ21 = 0.1.
as function of the predator-to-prey aggression intensity ρ . The results are shown in Fig. 4. Note that when the prey-to-prey aggression intensity σ is small (upper curves), the MET decreases monotonically when the predator-to-prey aggression intensity ρ increases. This is natural since prey die sooner because the chance of being killed is higher as ρ increases. However, for higher values of σ (lower curves), the MET has a maximum at intermediate values of the predator-to-prey aggression intensity ρ . This counterintuitive behavior arises again due to the influence of prey-to-prey aggression effects. When σ is small, lower values of predator aggression lead to higher extinction times as expected, but when σ is higher, lower values of predator aggression also yield higher densities of prey which in turn gives rise to more prey-to-prey aggressions, and the result is a lower MET. A higher extinction time can be related to a higher prey population in a real ecosystem. Therefore we see that the rate of consumption of predators should not be too large, which is obvious, but it cannot be too low either. This is an example of the many subtle balances that we encounter in nature. A similar result has been obtained by Grasman et al. [33], using a mean field model. They find a monotonic increase in the extinction times of prey as the hunting effort of the predator over one of two prey species grows. The key difference is that the density effects are absent in a model based on differential equations, and we have seen that it has a crucial role in the dynamics. As an illustration of the importance of spatial effects, Fig. 5 shows how the MET decays rapidly when the individuals, instead of looking only to their first neighbors, are given a larger neighborhood. This situation results in mixing of the species, thereby increasing the number of deaths due to interspecific competition. As the neighborhood size gets larger, we can expect a behavior equivalent to a mean field approximation, where any two individuals in the lattice can interact. A similar situation will arise if we include shortcuts for the species interactions (as in a small-world lattice). This will induce mixing and, in general, will increase the effects of interspecific competition.
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Fig. 4. Mean extinction time as function of the predator-to-prey aggression intensity ρ . From top to bottom, the values of prey-to-prey aggression intensity σ12 = σ21 = σ are 0.05, 0.08, 0.09, 0.10, 0.11, 0.15 and 0.20. The value for a is fixed at 0.10. 25000
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Fig. 5. Mean extinction time as function of the neighborhood size, expressed as a fraction of the lattice full size. In the insets it is clearly seen how the two species get mixed as the neighborhood increases. No predators are included and σ12 = σ21 = 0.1, σ11 = σ22 = 0.
4. Conclusions We constructed an agent-based model to study ecological competition and predator–prey relations. The simplest case of two prey that can fight for food and space through an aggression parameter shows nontrivial dynamics. We found that the mean extinction time depends on the quantity of available food and has a local maximum at intermediate values of food availability. This arises as a compromise between death by starvation or by competition, creating a balance at an optimum value of food availability. When a common predator, the apex predator, is introduced in the system, the above compromise is not observed, but the mean extinction time can increase or decrease depending on both the predator aggression intensity and the quantity of available food. Again we find that there is an optimum value, this time in the predator aggression intensity, for which the mean extinction time is maximum. This kind of complex interaction and unexpected behavior of communities of interacting species can be found in many real systems, for example, it has been observed that the abundance of a species of grasshopper (in direct competition with another grasshopper species) decreases as the food availability (grass) increases [34]. Also, in experiments of predator exclusion, it was found that an amphibod species, after exclosure from their predators (fish and shorebirds), had a decrease in their population [35]. It is surprising to find such a nontrivial and even counterintuitive behavior displayed by a model as simple as the one studied here. This highlights the intricate dynamics and delicate balances that exist in food chains, and the need to do research in order to understand the factors that govern the dynamics of trophic relations, which in turn will allow us to better take care of the world’s ecosystems. Acknowledgements We thank CONACyT (México) for financial support through grant No. 45782. References [1] R.M. May, Nature (London) 261 (1976) 459–467. [2] J.R. Beddington, C.A. Free, J.H. Lawton, The Journal of Animal Ecology 45 (3) (1976) 791–816.
4080 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
G. Camargo-Gamboa et al. / Physica A 389 (2010) 4075–4080 E.M. Rauch, Y. Bar-Yam, Physical Review E 73 (2006) 020903(R). G. Szabó, A. Szolnoki, Physical Review E 77 (2008) 011906. T. Antal, M. Droz, Physical Review E 63 (2001) 056119. M. Opper, S. Diederich, Physical Review Letters 69 (1992) 1616–1619. M.V. Carneiro, I.C. Charret, Physical Review E 76 (2007) 061902. G. Szabó, A. Szolnoki, I. Borsos, Physical Review E 77 (2008) 041919. W. Wang, Q. Liu, Z. Jin, Physical Review E 75 (2007) 051913. N. Guttenberg, N. Goldenfeld, Physical Review Letters 100 (2008) 058102. M. Perc, A. Szolnoki, New Journal of Physics 9 (2007) 267. B. Spagnolo, M. Cirone, A. La Barbera, F. de Pasquale, Journal of Physics: Condensed Matter 14 (2002) 2247–2255. R. Mankin, T. Laas, A. Ainsaar, Physical Review E 74 (2006) 021101. T. Butler, N. Goldenfeld, Physical Review E 80 (2009) 030902. M. Perc, M. Marhl, New Journal of Physics 8 (2006) 142. J. Hofbauer, K. Sigmund, The Theory of Evolution and Dynamics Systems, Cambridge University Press, Cambridge, 1988. J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982. M.L. Pace, J.J. Cole, et al., Trends in Ecology and Evolution 14 (12) (1999) 483–488. T. Cheon, S. Ohta, Physical Review E 70 (2004) 021913. R.M. May, Theoretical Ecology: Principles and Applications, 2nd ed., Sinauer, Sunderland, Massachusetts, 1981, pp. 5–29. T. Yoshida, L.E. Jones, S.P. Ellner, G.F. Fussman, N.G. Hairston Jr., Nature 424 (2003) 303–306. R. Solé, B. Goodwin, Signs of Life, Basic Books, New York, 2000. A.R. Ives, S.R. Carpenter, Science 317 (5834) (2007) 58–62. M. Loreau, et al., in: M. Loreau, S. Naeem, P. Inchausti (Eds.), Biodiversity and Ecosystem Functioning: Synthesis and Perspectives, Oxford Univ. Press, Oxford, 2002, pp. 79–91. V. Grimm, C. Wissel, Oecologia 109 (1997) 323. S.L. Pimm, Nature 307 (1984) 321. A.J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925. Reprinted under the new title: Elements of Mathematical Biology, Dover, New York, 1956. V. Volterra, Rendiconti dell’Accademia dei Lincei 6 (2) (1926) 31–113; An abridged english version can be found in: Nature 118 (1926) 558–560. G. Szabó, G.A. Sznaider, Physical Review E 69 (2004) 031911. F. Chen, Applied Mathematics and Computation 171 (2005) 492. M. Perc, A. Szolnoki, G. Szabó, Physical Review E 75 (2007) 052102. G. Hardin, Science 131 (1960) 1292–1297. J. Grasman, F. van den Bosch, O.A. van Herwaarden, Bulletin of Mathematical Biology 63 (2001) 259–269. A.P. Beckerman, Ecology 81 (2000) 948–957. W.H. Wilson Jr., Journal of Experimental Marine Biology and Ecology 132 (1989) 221–245.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Ecological competition and the role of an apex predator G. Camargo-Gamboa a , R. Huerta-Quintanilla a , M. Rodríguez-Achach b,∗ a
Departamento de Física Aplicada, Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Unidad Mérida, Mérida, Yucatán, Mexico
b
Facultad de Física e Inteligencia Artificial, Universidad Veracruzana, Lomas del Estadio S/N, Xalapa, Veracruz, Mexico
article
info
Article history: Received 4 May 2009 Received in revised form 9 March 2010 Available online 4 June 2010 Keywords: Ecological competition Prey Predator Ecosystem
abstract In this work we propose an agent-based model to study the process of ecological competition between two species coexisting with a common predator, including inter and intra-specific competition. Without the predator, the ecosystem always reaches a configuration in which one of the prey species faces extinction and the other survives, as is well known. However, we find that there is an optimum value for the amount of food consumed by the prey species for which the mean extinction time (MET) is maximum. In the presence of predators, the average extinction time may increase or decrease with respect to the non-predator case, depending on the food availability. It is also observed that the average extinction time has an optimum value depending on the efficiency of prey of the predator. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Almost all known ecosystems in nature are comprised of several different species which have predator–prey relations between them and coexist in equilibrium [1–3]. Theoretical studies of the dynamics of these systems show a very rich behavior, like the existence of phase transitions from mixed to segregated states [4–6]; the spontaneous emergence of spatial patterns [3,7,8]; complexity [9,10]. Noise-induced effects are also frequently studied. Noise can indeed guide some evolutionary processes depending on its intensity [11], and produce interesting effects such as pattern formation, resonances, temporal oscillations and Hopf bifurcations [12–15]. The predator–prey relations generate structures known as trophic pyramids [16], with the predators sitting on the upper levels. The predator on top of the pyramid is known as the apex predator and is known to have a crucial role in maintaining the health and stability of the ecosystem [17,18]. Without the predator’s influence, the ecosystem would be destabilized and the prey will become involved in an ‘‘arms race’’ which eventually drives one of the competing species into extinction [19–21]. The conditions that lead to equilibrium or stability of ecosystems are not fully understood; biodiversity for example, seems to be a key ingredient for stability. The underlying structure of the food web also plays an important role, among many other things. There is still a lot of discussion about the leading dynamics on ecosystems [22], specifically on how some species can form stable ecosystems but, at the same time, other species face instability and extinction. Defining stability for an ecosystem can be done in several ways, and there are various types of stability [23]. One can define it by the response of the system under small perturbations, by the number of stable states of the system, by the amplitude of fluctuations or by the response of the system to the extinction of species, among other criteria [24–26]. For a toy ecosystem like the one studied here, where three species compose the total and most of the subtleties of real ecosystems are excluded, the longer the time it takes for an extinction to occur, the more stable the system is.
∗
Corresponding author. Tel.: +52 228421747; fax: +52 2288178209. E-mail address: [email protected] (M. Rodríguez-Achach).
0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.05.039
4076
G. Camargo-Gamboa et al. / Physica A 389 (2010) 4075–4080
Predator and prey systems, since the pioneering work of Lotka and Volterra [27,28], have traditionally been studied using mean field type models, i.e., differential and difference equations. These ignore space correlations and replace local interactions by uniform long-range ones. However, they are very useful as a first approach to understand the general behavior of a complex system. Agent-based models take explicitly into account spatial effects and local interactions, and therefore we believe they are better suited for describing an ecosystem than mean field models, especially when we are interested in local interactions like competition for space and food, aggression between neighbors and the like. In this work, we studied the processes of competition between two herbivore species coexisting in the same habitat, and competing for the same resources, both in absence and in presence of a third species (the apex predator) which is allowed to prey on them. 2. The model We use an agent-based model consisting of three species: two competing herbivore species (A and B) which feed on grass, and a predator species (C ) which prey on both herbivores. Models with more than three species have also been studied by other authors [11,29–31] in a different context. The prey are identical to the predator so it has no preference over any of them. All three species can live indefinitely, and death arises only as the result of aggression or starvation (when the predators cannot find prey or the herbivores do not have enough grass to eat). The habitat is a square lattice of size N × N with periodic boundary conditions, and at each site on the lattice there is grass constantly growing at a fixed rate, so it is an abiotic resource. The prey consumes grass at a certain rate a, called the environmental limitation factor. Higher values of a thus mean that it is harder for the herbivores to find enough food to survive. The competition process between herbivores consists of attacks or aggression from one prey species to the other (interspecific competition), that is, members from species A (B) are allowed to attack members belonging to species B (A) with an aggression intensity (probability of attack) σ12 (σ21 ). Note that since both herbivores compete for a common limiting resource – in the form of grass – we also have exploitation competition in the model. We also included intraspecific competition, where members of the same prey species A (B) fight with each other with an aggression intensity σ11 (σ22 ). The only interaction between prey and predators is the trophic relation: predators can attack prey with an aggression intensity ρ , but prey are not allowed to attack predators in response. At the beginning of each simulation, elements from all species are randomly distributed on the lattice. At each time step an individual is picked at random, and one of its neighbors from the von Neumann neighborhood is also randomly chosen. Allowing the individuals to interact only in the immediate neighborhood restricts the model to species that live in closed groups, as is frequently the case for many herbivores. Then the interactions proceed as follows: 1. If the individual is a prey and its neighbor is an empty site then, if there is enough grass to eat, it survives, and it can also reproduce asexually with reproductive rate b. In this case the new element is placed on the empty site. If the prey does not reproduce, it moves to the empty site. In all the following cases when the individual chosen is one of the prey, it has to be able to eat enough grass and reproduce in order to survive, or die, as in this case, so we will not mention it again. 2. If the individual is a prey A (B) and its neighbor is another prey B (A), then the neighbor is killed with probability σ12 (σ21 ). 3. If the individual is a prey A (B) and its neighbor is of the same species, then the neighbor is killed with probability σ11 (σ22 ). 4. If the individual is a predator and its neighbor is a prey, then the predator attacks the prey who dies with probability ρ . If the prey is killed (eaten), the predator can asexually reproduce, otherwise it dies with probability d. 5. If the individual is a predator and its neighbor is an empty site then it dies with probability d. Otherwise, with probability (1 − d), the predator moves to the empty site. 6. If the individual is a predator and its neighbor is another predator then it dies with probability d. The above steps are repeated n times, where n is the total number of individuals, which constitutes a Monte Carlo step (MCS). After each MCS, the grass grows by a fixed amount at every lattice site. The prey feeds on the grass at a rate a, which constitutes the environmental limitation factor for prey species. In the absence of the predator, the habitat only holds prey species A and B, and we only consider the competition processes between prey. In that case, steps 4, 5 and 6 are no longer taken into account. 3. Results A lattice size of 60 × 60 is considered in all simulations, and averages are taken over at least 5000 realizations. The initial population densities are xA = 0.20, xB = 0.20 and xC = 0.15, where xi is the population density of species i. The results of the whole simulations are robust against changes in initial densities, unless the differences are too large. The parameters b and d are fixed at 0.25 and 0.03 respectively. In Fig. 1(a) we show the results for the interaction of the two prey species in absence of the predator (xC = 0). Our results confirm the well known fact that, when two species compete for the same resource, other things being equal, one of them becomes extinct (in community ecology, this is known as Gause’s law of competitive exclusion [32]). Therefore in Fig. 1(a), we record the time elapsed until the first species becomes extinct. Note that when no aggression exists between them
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Fig. 1. (a) Mean extinction time for prey as function of the environmental limitation factor a. No predator is included and we set σ12 = σ21 = σ . From top to bottom, the values of prey-to-prey aggression intensity σ are 0.0, 0.02, 0.05, 0.10 and 0.99. (b) Results with intraspecific competition included. σ is fixed at 0.1. Values for σ11 = σ22 are: 0.0 (crosses), 0.05 (diamonds), 0.07 (squares) and 0.10 (circles).
(σ = 0), the MET grows monotonically when the environmental limitation factor a decreases, i.e., there is more available food for the prey. However, as soon as there is competition between them (σ > 0), the MET shows a local maximum. This behavior arises as the result of two processes: suppose we are located at the local maximum of the curve for which σ = 0.10 (approximately at a = 0.11) in Fig. 1(a). If a increases, there is less food available and the MET decreases because of starvation of the prey. On the other hand, if a decreases, there is more food, but the population density increases (for the case of σ = 0.10, the mean population density increases from 18% at a = 0.14 to 83% at a = 0.06), therefore there are more attacks between the prey, and this ‘‘territorial fight’’ also decreases the MET. If a decreases even more, there is so much food available that the effect of prey fights is not significant and the MET increases again until it saturates. When intraspecific competition is included, the behavior becomes more complicated, as shown in Fig. 1(b). There are two regimes separated at a value of the environmental limitation factor a of approximately 0.07. When a > 0.07, we observe that as the intraspecific competition increases, so does the MET. This is because the increased competition decreases the population density, which gives the prey more available food. On the other hand, in the region a < 0.07, when intraspecific competition increases, the population density also decreases, but now the increment in available food is not important since in this region we already have plenty of food, therefore the MET initially decreases since attacks are more frequent. However, when the intraspecific competition increases even more, the MET increases greatly (curve with circles). This effect comes from the fact that at higher values of competition the density becomes very low, therefore encounters between prey become rare, with the result that attacks are less frequent, hence the MET shows an increase. Despite the fact that a lattice of size 60 × 60 can be considered small enough to introduce some simulation artifacts, we have tested that it is not the case. Fig. 2 shows that the results from Fig. 1(a) do not qualitatively change for larger lattice sizes. Similarly, other system behaviors are maintained as the lattice size is increased. The behavior described above changes when predators are included in the habitat. In Fig. 3, we show the MET, defined as the time when the first herbivore species becomes extinct. After that, we get the well known population oscillations of a single predator–single prey model. We use a fixed value of σ12 = σ21 = 0.10. When no predators are present (ρ = 0, circles), the behavior is the same as in the curve with crosses in Fig. 1(a). The introduction of predators makes the MET increase monotonically, as more food is available for the prey. Thus the process by which the MET decreases because of prey-to-prey aggression no longer produces the maxima and minima in the curves. The inset shows how the curve for ρ = 0.04 is modified when intraspecific competition is present between the prey: we get higher METs, as the intraspecific competition increases, which is a result of the lower population density induced by competition. Returning to the main figure, note how depending on the value of the environmental limitation factor, the presence of predators can make the MET increase or decrease. In order to investigate this behavior in more detail, we studied the MET
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Fig. 2. Same as the fourth curve (from top to bottom) in Fig. 1 for different lattice sizes: 60 × 60 (circles), 100 × 100 (squares) and 150 × 150 (asterisks). The qualitative behavior of the system does not change.
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Fig. 3. Mean extinction time for prey as function of the environmental limitation factor a, with predators included. The curve with circles is for ρ = 0.0, the diamonds curve is for ρ = 0.03 and the squares curve is for ρ = 0.04. The value of the prey-to-prey aggression intensity is fixed at σ12 = σ21 = 0.1 and σ12 = σ21 = 0.1.
as function of the predator-to-prey aggression intensity ρ . The results are shown in Fig. 4. Note that when the prey-to-prey aggression intensity σ is small (upper curves), the MET decreases monotonically when the predator-to-prey aggression intensity ρ increases. This is natural since prey die sooner because the chance of being killed is higher as ρ increases. However, for higher values of σ (lower curves), the MET has a maximum at intermediate values of the predator-to-prey aggression intensity ρ . This counterintuitive behavior arises again due to the influence of prey-to-prey aggression effects. When σ is small, lower values of predator aggression lead to higher extinction times as expected, but when σ is higher, lower values of predator aggression also yield higher densities of prey which in turn gives rise to more prey-to-prey aggressions, and the result is a lower MET. A higher extinction time can be related to a higher prey population in a real ecosystem. Therefore we see that the rate of consumption of predators should not be too large, which is obvious, but it cannot be too low either. This is an example of the many subtle balances that we encounter in nature. A similar result has been obtained by Grasman et al. [33], using a mean field model. They find a monotonic increase in the extinction times of prey as the hunting effort of the predator over one of two prey species grows. The key difference is that the density effects are absent in a model based on differential equations, and we have seen that it has a crucial role in the dynamics. As an illustration of the importance of spatial effects, Fig. 5 shows how the MET decays rapidly when the individuals, instead of looking only to their first neighbors, are given a larger neighborhood. This situation results in mixing of the species, thereby increasing the number of deaths due to interspecific competition. As the neighborhood size gets larger, we can expect a behavior equivalent to a mean field approximation, where any two individuals in the lattice can interact. A similar situation will arise if we include shortcuts for the species interactions (as in a small-world lattice). This will induce mixing and, in general, will increase the effects of interspecific competition.
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Fig. 4. Mean extinction time as function of the predator-to-prey aggression intensity ρ . From top to bottom, the values of prey-to-prey aggression intensity σ12 = σ21 = σ are 0.05, 0.08, 0.09, 0.10, 0.11, 0.15 and 0.20. The value for a is fixed at 0.10. 25000
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4. Conclusions We constructed an agent-based model to study ecological competition and predator–prey relations. The simplest case of two prey that can fight for food and space through an aggression parameter shows nontrivial dynamics. We found that the mean extinction time depends on the quantity of available food and has a local maximum at intermediate values of food availability. This arises as a compromise between death by starvation or by competition, creating a balance at an optimum value of food availability. When a common predator, the apex predator, is introduced in the system, the above compromise is not observed, but the mean extinction time can increase or decrease depending on both the predator aggression intensity and the quantity of available food. Again we find that there is an optimum value, this time in the predator aggression intensity, for which the mean extinction time is maximum. This kind of complex interaction and unexpected behavior of communities of interacting species can be found in many real systems, for example, it has been observed that the abundance of a species of grasshopper (in direct competition with another grasshopper species) decreases as the food availability (grass) increases [34]. Also, in experiments of predator exclusion, it was found that an amphibod species, after exclosure from their predators (fish and shorebirds), had a decrease in their population [35]. It is surprising to find such a nontrivial and even counterintuitive behavior displayed by a model as simple as the one studied here. This highlights the intricate dynamics and delicate balances that exist in food chains, and the need to do research in order to understand the factors that govern the dynamics of trophic relations, which in turn will allow us to better take care of the world’s ecosystems. Acknowledgements We thank CONACyT (México) for financial support through grant No. 45782. References [1] R.M. May, Nature (London) 261 (1976) 459–467. [2] J.R. Beddington, C.A. Free, J.H. Lawton, The Journal of Animal Ecology 45 (3) (1976) 791–816.
4080 [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
G. Camargo-Gamboa et al. / Physica A 389 (2010) 4075–4080 E.M. Rauch, Y. Bar-Yam, Physical Review E 73 (2006) 020903(R). G. Szabó, A. Szolnoki, Physical Review E 77 (2008) 011906. T. Antal, M. Droz, Physical Review E 63 (2001) 056119. M. Opper, S. Diederich, Physical Review Letters 69 (1992) 1616–1619. M.V. Carneiro, I.C. Charret, Physical Review E 76 (2007) 061902. G. Szabó, A. Szolnoki, I. Borsos, Physical Review E 77 (2008) 041919. W. Wang, Q. Liu, Z. Jin, Physical Review E 75 (2007) 051913. N. Guttenberg, N. Goldenfeld, Physical Review Letters 100 (2008) 058102. M. Perc, A. Szolnoki, New Journal of Physics 9 (2007) 267. B. Spagnolo, M. Cirone, A. La Barbera, F. de Pasquale, Journal of Physics: Condensed Matter 14 (2002) 2247–2255. R. Mankin, T. Laas, A. Ainsaar, Physical Review E 74 (2006) 021101. T. Butler, N. Goldenfeld, Physical Review E 80 (2009) 030902. M. Perc, M. Marhl, New Journal of Physics 8 (2006) 142. J. Hofbauer, K. Sigmund, The Theory of Evolution and Dynamics Systems, Cambridge University Press, Cambridge, 1988. J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982. M.L. Pace, J.J. Cole, et al., Trends in Ecology and Evolution 14 (12) (1999) 483–488. T. Cheon, S. Ohta, Physical Review E 70 (2004) 021913. R.M. May, Theoretical Ecology: Principles and Applications, 2nd ed., Sinauer, Sunderland, Massachusetts, 1981, pp. 5–29. T. Yoshida, L.E. Jones, S.P. Ellner, G.F. Fussman, N.G. Hairston Jr., Nature 424 (2003) 303–306. R. Solé, B. Goodwin, Signs of Life, Basic Books, New York, 2000. A.R. Ives, S.R. Carpenter, Science 317 (5834) (2007) 58–62. M. Loreau, et al., in: M. Loreau, S. Naeem, P. Inchausti (Eds.), Biodiversity and Ecosystem Functioning: Synthesis and Perspectives, Oxford Univ. Press, Oxford, 2002, pp. 79–91. V. Grimm, C. Wissel, Oecologia 109 (1997) 323. S.L. Pimm, Nature 307 (1984) 321. A.J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925. Reprinted under the new title: Elements of Mathematical Biology, Dover, New York, 1956. V. Volterra, Rendiconti dell’Accademia dei Lincei 6 (2) (1926) 31–113; An abridged english version can be found in: Nature 118 (1926) 558–560. G. Szabó, G.A. Sznaider, Physical Review E 69 (2004) 031911. F. Chen, Applied Mathematics and Computation 171 (2005) 492. M. Perc, A. Szolnoki, G. Szabó, Physical Review E 75 (2007) 052102. G. Hardin, Science 131 (1960) 1292–1297. J. Grasman, F. van den Bosch, O.A. van Herwaarden, Bulletin of Mathematical Biology 63 (2001) 259–269. A.P. Beckerman, Ecology 81 (2000) 948–957. W.H. Wilson Jr., Journal of Experimental Marine Biology and Ecology 132 (1989) 221–245.