BioSystems 125 (2014) 1–15
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BioSystems journal homepage: www.elsevier.com/locate/biosystems
Global warming description using Daisyworld model with greenhouse gases Susana L.D. Paiva, Marcelo A. Savi *, Flavio M. Viola, Albino J.K. Leiroz Universidade Federal do Rio de Janeiro, COPPE – Department of Mechanical Engineering, P.O. Box 68.503, 21.941.972 Rio de Janeiro, Brazil
A R T I C L E I N F O
A B S T R A C T
Article history: Received 23 January 2014 Received in revised form 9 September 2014 Accepted 10 September 2014 Available online 16 September 2014
Daisyworld is an archetypal model of the earth that is able to describe the global regulation that can emerge from the interaction between life and environment. This article proposes a model based on the original Daisyworld considering greenhouse gases emission and absorption, allowing the description of the global warming phenomenon. Global and local analyses are discussed evaluating the influence of greenhouse gases in the planet dynamics. Numerical simulations are carried out showing the general qualitative behavior of the Daisyworld for different scenarios that includes solar luminosity variations and greenhouse gases effect. Nonlinear dynamics perspective is of concern discussing a way that helps the comprehension of the global warming phenomenon. ã 2014 Elsevier Ireland Ltd. All rights reserved.
Keywords: Daisyworld Global warming Carbon cycle Ecology Nonlinear dynamics
1. Introduction Climate change is a growing global concern, and a topic of considerable research. Global warming is within this general issue being related to the increase of earth’s temperature (Houghton, 2005). While the consequences of global temperature changes remain unknown, it is possible to use mathematical models to evaluate different scenarios. There are several general modeling approaches (Alexiadis, 2007). A first strategy makes use of timeseries analysis to build predictive models of future temperature change (Viola et al., 2010). A second approach makes use of mathematical models to describe system behavior using governing equations based on fundamental principles of involved phenomena (Jorgensen, 1999). Phenomenological models are a considerably simpler approach, useful for some descriptions (Kay et al., 2009). Daisyworld is an archetypal model that represents the earth describing the global regulation that can emerge from the interaction between life and environment. Daisyworld is a mathematical description of the Gaia theory of the earth representing life by daisy populations while the environment is represented by temperature. It was originally proposed by Watson and Lovelock (1983) in order to describe the self-regulation of the planetary system. Wood et al. (2008) presented a general overview
* *Corresponding author. E-mail addresses: [email protected] (S.L.D. Paiva), [email protected] ? (M.A. Savi), [email protected] (F.M. Viola), [email protected] (A.J.K. Leiroz). http://dx.doi.org/10.1016/j.biosystems.2014.09.008 0303-2647/ ã 2014 Elsevier Ireland Ltd. All rights reserved.
about the literature associated with the Daisyworld, emphasizing the model main characteristics and different analysis approaches. Spatial aspects and variants of the model are discussed. Adaptability is an important issue that defines system evolution by natural selection. In brief, it is possible to say that Daisyworld allows one to perform a qualitative description of the climate system, permitting the investigation of different aspects of the system behavior. Because of that, several researches are dedicated for variants and extensions of the Daisyworld. An important discussion about Daisyworld characteristics is related to the antagonism between altruism and competition of the daisy populations. In this regard, several researches were developed trying to investigate adaptive behavior of the Daisyworld, incorporating competition and mutation aspects of life. Lenton and Lovelock (2000) discussed some adaptive aspects of the Daisyworld based on the idea that it is a Darwinian model in the sense that there is a competition among different types of life with heritable variation in a trait, which represents a kind of natural selection (Robertson and Robinson, 1998). Daisy colors (or planetary albedo) and optimum growth temperature are the main parameters related to the Daisyworld adaptation. Restrictions on environmental conditions define some evolution changes. Cohen and Rich (2000) introduced an extra source of competition that describes the interaction among daisy species in order to retain temperature in an appropriate range in order to make the planet habitable. Boyle et al. (2011) pointed that homeosthasis of the Daisyworld can be promoted by symbiotic physiology. Qualitative comparisons are evaluated with and without the consideration of symbiotic physiology.
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S.L.D. Paiva et al. / BioSystems 125 (2014) 1–15 Table 1 Parameters of the Daisyworld. aw ab ag
0.75 0.25 0.5 0.3 2.06 109 K4 295.5 K
g
q Topt
Table 2 Parameters of the Daisyworld considering different timescales. Different timescales W m2
s S c
Fig. 1. Gaussian representing the optimum condition and limits of life.
Weaver and Dyke (2012) discussed Daisyworld model from timescale perspective. Basically, it is considered temperature, changes in insolation, and self-organisation of the biota using an analytical approach.
J day1 m2 8
5.67 10 915 950 (K1)
4
(K
)
4.90 103 (K4) 7.91 107 8.21 107(K1)
One of the most important characteristics of the Daisyworld is the capability to describe either global or local phenomena. Global analysis assumes that the translational movement of the planet is the main component of temperature variation, being the planet subjected to temperature changes according to seasons of the year or aspects of the planet. Under this assumption, Daisyworld is related to the entire planet. On the other hand, local analysis defines a
Fig. 2. Classical Daisyworld response (Nevison et al., 1999). (a) Daisy dynamics; (b) temperature dynamics; (c) projection of the state space showing the daisy subspace.
S.L.D. Paiva et al. / BioSystems 125 (2014) 1–15
specific region of the planet. Therefore, either rotation or translation of the planet influence solar energy input to the planet since a specific region is subjected to temperature changes due to solar luminosity variations associated with days and nights, and also with seasons. Staley (2002) and Charlson et al. (1987) discussed the differences between the global and local effects on the planet dynamics. The Daisyworld model was also studied considering spatio-temporal behavior. Adams et al. (2003) proposed a one-dimensional model incorporating a distribution of incoming solar radiation and heat diffusion consistent with a spherical planet. Biton and Gildor (2012) added seasonal solar radiation forcing to the latitudinal-depended one-dimensional Daisyworld model. Wood et al. (2006) included a mutation of the optimal growth temperature and of the albedo in a two-dimensional Daisyworld model determining the temperature oscillations. Ackland (2004) used a two-dimensional Daisyworld model considering the temperatures and albedos on a grid indexed by longitude and latitude positions.
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Daisyworld can be employed for global warming description. In essence, greenhouse gases affect Daisyworld behavior in a similar way of the black daisies in terms of radiation absortion. Nevetheless, the interaction with environment has an important difference. Viola et al. (2013) incorporated greenhouse gases in the Daisyworld considering prescribed time series that alter system albedo. Besides, a sinusoidal variation of the solar luminosity represents climate variability. Hence, the Daisyworld is employed for global warming description and complex dynamical responses are observed. Chaotic behavior was of concern showing that this kind of response is possible under certain conditions. Zeng et al. (1990) and Wood et al. (2008) also investigated possibilities related to the chaotic behavior of the Daisyworld. This paper deals with the description of the global warming using the Daisyworld model considering that greenhouse gases are included in the model with the aid of an extra equation motivated by the carbon cycle dynamics and predator–prey models. The
Fig. 3. Daisyworld with greenhouse effect where a = 104 and b = 0. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing the daisy-greenhouse effect subspace; (d) subspace projections.
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Fig. 4. Daisyworld with greenhouse effect where a = b = 104. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing the daisy-greenhouse effect subspace; (d) subspace projections.
temperature dependence of life is represented by a Gaussian function avoiding life extinction observed on the classical peak function of the Daisyworld model. Emissions and absorptions of greenhouse gases are treated showing the capability of the model to describe global warming scenarios. Numerical simulations show this general qualitative capability presenting several simulations related to greenhouse gases effect and also changes in solar luminosity. Results are analyzed considering a nonlinear dynamics perspective that provides a comprehension about system behavior. 2. Daisyworld model and the carbon cycle The climate system is a complex system based on several phenomena. The carbon cycle is associated with chemical and biological interactions of great importance to define the origins and the destinations of this element. Carbon is stored in four main reservoirs of the earth: atmosphere, lithosphere, biosphere and hydrosphere. Each reservoir contains a variety of carbon compounds
of organic and inorganic nature. Furthermore, the transfer time and storage of carbon compounds in each reservoir can vary from years to millennia. For example, the lithosphere contains a large amount of carbon trapped in sedimentary rocks formations of mineral carbonates and organic compounds such as oil, natural gas and coal. The redistribution from the lithosphere reservoir to another is associated with millions of years, until all the geological process, such as chemical decomposition and deposition, occurs. Therefore, the lithosphere is considered a relatively inactive component of the carbon cycle when preserved in its natural state. The active reservoirs are divided between the atmosphere, the terrestrial biosphere and the hydrosphere. As the absolute sum of the amounts of active carbon reservoir is maintained close to the steady state value by slow geological processes, biogeochemical processes that lead to redistribution of carbon between active reservoirs occur more rapidly. The Daisyworld represents self-regulation of the planet that is basically composed of the environment, represented by the temperature, and life, represented by daisy populations.
S.L.D. Paiva et al. / BioSystems 125 (2014) 1–15
Daisyworld model represents life by daisy populations, which evolution is described by the following equations: (1) a_ w ¼ aw ag bðT w Þ g a_ b ¼ ab ag bðT b Þ g
A ¼ a w aw þ a b ab þ a g ag
(4)
The functional form for b is usually assumed to be a symmetric single-peak function. Here, a Gaussian function is assumed as follows:
bðT i Þ ¼ Be s ðT i T opt Þ 1
(2)
5
2
(5)
For the sake of simplicity, only two daisy populations are considered: white (aw) and black (ab) daisies. The dot represents time derivative and b = b(Ti) represents the growth rate that is temperature dependent and g is the population death rate. The variable ag is the fractional area coverage of the planet represented by
where B is a parameter related to environmental characteristics, s is the variance and Topt is the optimal temperature, usually considered as Topt = 295 K = 22.5 C. The Gaussian form of this function is shown in Fig. 1 and it is convenient to avoid a drastic population extinction observed on the classical peak function of the Daisyworld model. The local temperature of each population is defined as follows:
ag ¼ p aw ab
T 4w ¼ qðA aw Þ þ T 4
(6)
T 4b ¼ qðA ab Þ þ T 4
(7)
(3)
where p = 1 represents the portion of land suitable for the growth of daisies. The mean planetary albedo of the Daisyworld, A, can be estimated from the individual albedo of each daisy population (aw and ab), and from the ground albedo (ag):
Fig. 5. Daisyworld with greenhouse effect where a = 104 and b = 103. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing daisy-greenhouse effect subspace; (d) subspace projections.
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T 4g ¼ qðA ag Þ þ T 4
S.L.D. Paiva et al. / BioSystems 125 (2014) 1–15
(8)
where T is the globally-averaged temperature of the Daisyworld, and q is a constant used to calculate local temperature as a function of albedo. Finally, it is important to establish the thermal balance of the Daisyworld (Foong, 2006). Therefore, the absorbed energy is given by (Nevison et al., 1999): 1 T_ ¼ ½SLð1 AÞ s T 4 ð1 EÞ c
(9)
where L is the solar luminosity and S is the solar constant that establishes the average solar energy represented by SL; s is the Stefan–Boltzmann constant; c is a measure of the average heat capacity or thermal inertia of the planet. Finally, E represents the greenhouse effect that alters the thermal balance, described by the following differential equation. E_ ¼ a bEðaw þ ab Þ
(10)
This new equation establishes the interaction between gases and daisy populations being motivated by the general predator– prey equation and the carbon cycle where life is responsible for the absorption of the gases (Novak, 2006). Parameter a is related to gas
emission while b represents the absorption. Note that when b vanishes, no interaction occurs and the effect of greenhouse gases is only related to the emissions. From now on, the greenhouse variable E is referred as greenhouse effect. 3. Numerical simulations The Daisyworld model can represent global or local aspects of the planet dynamics. Global analysis considers that Daisyworld is related to the entire planet. On the other hand, local analysis defines a specific region of the planet. Numerical simulations are carried out considering both situations and different scenarios. Initially, numerical simulations are performed considering a global analysis. Three different models are treated: the original, without greenhouse gases; a model with greenhouse gases; and a model with emission and absorption of greenhouse gases. Moreover, luminosity variations are also investigated representing the profile of solar activity. In this regard, three different scenarios are considered: constant value; linear increase; and climate variability represented by a sinusoidal modulation over a linear growth. The Daisyworld model can be simulated using classical procedures for numerical integration. Here, a fourth order
Fig. 6. Daisyworld with greenhouse effect where a = b = 0. (a) Daisy dynamics; (b) temperature dynamics; (c) projection of the state space showing the daisy subspace.
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Runge–Kutta method associated with an iterative process is employed. Time steps smaller than 102 are assumed for the global analysis while local analysis assumes time steps smaller than 103. These time-step values are defined after convergence analysis. Tables 1 and 2 show the values of the parameters employed in all simulations. 3.1. Global analysis This section presents the global analysis of the Daisyworld. Three different luminosity scenarios are discussed: constant value; linear increase; and climate variability. Each of these analyses may represent a scenario related to different timescales, for instance. For each one of them, three different situations are treated: the original model, without greenhouse gases; a model with greenhouse gases; a model with emission and absorption of greenhouse gases. 3.1.1. Constant luminosity A situation where greenhouse gases are not involved is used to start the analysis. Therefore, emissions or absorptions of
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greenhouse gases are not considered in the model, which is represented by a = b = 0. Fig. 2 presents Daisyworld response showing consistent results compared with the classical Daisyworld (Nevison et al., 1999). Fig. 2a shows the dynamics of daisy populations while Fig. 2b shows the temperature dynamics. Fig. 2c presents a projection of the state space in daisy subspace. Note that the interaction between the two daisy species, white and black, limits temperature values within a proper range for life existence. The increase of white daisies tends to decrease of temperature, while the increase of black daisies causes the increase of temperature due to the respective albedo. A scenario with greenhouse gases emissions is now investigated, but the planet is not able to absorb these gases. This analysis is accomplished by considering a = 104 and b = 0. Fig. 3 presents results of this case showing that the increase of gases causes a general increase of planet temperature. Basically, Fig. 3a shows the time history of daisy populations while Fig. 3b shows the temperature and the greenhouse effect dynamics (represented by variable E). Fig. 3c shows a projection of the state space considering daisy populations and greenhouse effect. Fig. 3d presents some projections of the subspace of Fig. 3c. Results show
Fig. 7. Daisyworld with greenhouse effect where a = 104 and b = 0. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing the daisy-greenhouse effect subspace; (d) subspace projections.
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Fig. 8. Daisyworld with greenhouse effect where a = 104 and b = 101. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing the daisy-greenhouse effect subspace; (d) subspace projections.
that in this situation, the balance between life and environment is lost, and daisy populations cannot compensate the greenhouse effect. It is noticeable the persistence of white daisies for a longer time, acting to delay the life extinction caused by the excessive increase of temperature. It should be highlighted that the extinction is related to an escape imposed by the variable E, clearly shown in state space projection (Fig. 3d). The planet capacity to absorb greenhouse gases is now of concern by assuming a = b = 104. In this scenario, daisy populations contribute to gas absorption. Fig. 4 presents the response of the Daisyworld for this case. It is clear the increase of the self-regulation is caused by the absorption capacity. Nevertheless, this is not enough to avoid the extinction of life on the planet. The increase of the absorption capacity can be represented by considering a = 104 and b = 103. This increase avoids the extinction as shown in Fig. 5. Note that this new scenario is related to a stabilization of the system response for a specific variable E, which is clear in the state space projections (Fig. 5d).
Fig. 9. Luminosity representing climate variability.
S.L.D. Paiva et al. / BioSystems 125 (2014) 1–15
3.1.2. Linear increase of the luminosity In order to represent a more realistic solar activity, a linear increase of the luminosity is considered as: L ¼ 1:9 104 t þ 0:75
(11)
Once again, three scenarios are treated: the original model, without greenhouse gases; a model with greenhouse gases emissions; and a model with greenhouse gases emission and absorption. Initially, the dynamic behavior of the Daisyworld without greenhouse gases, represented by a = b = 0, is observed. Fig. 6 presents the general behavior of the Daisyworld where an increase of luminosity causes a temperature increase, which leads to unfavorable environmental conditions, causing the life extinction. The emission of greenhouse gases (a = 104 and b = 0) accelerates this tendency even more, as shown in Fig. 7. Once again, it is possible to associate the extinction with the greenhouse effect variable escape observed in the projections of the state space, Fig. 7d. The absorption capacity can change the scenario associated with emissions. By considering a = 104 and b = 101 (Fig. 8), the Daisyworld presents similar results of the one presented in Fig. 6,
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showing the system capacity to avoid extreme situations. Note however, that there is still an extinction of life that occurs when the luminosity reaches a critical value. 3.1.3. Luminosity representing climate variability Climate variability is now in focus assuming a sinusoidal variation of the luminosity over a linear increase as follows: ðSL Þglobal ¼ S þ ðNsinwtÞ
(12)
where (SL)global is the global solar energy and corresponds to SL, where S is the solar constant and L is the luminosity; N is the amplitude and w is the forcing frequency. Here, it is considered N = 2.89 1012 and w = 0.01 that is related to a behavior presented in Fig. 9. Initially, situations without greenhouse gases (a = b = 0) are considered. Under this condition, the dynamics of daisy populations and temperature levels are presented in Fig. 10. Note that Daisyworld presents a more complex behavior due to sinusoidal excitation. A situation with greenhouse gases emission but without absorption, represented by a = 104 and b = 0, is now considered.
Fig. 10. Daisyworld without greenhouse effect where a = b = 0. (a) Daisy dynamics; (b) temperature dynamics; (c) projection of the state space showing the daisy subspace.
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Fig. 11. Daisyworld with greenhouse effect where a = 104 and b = 0. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing the daisy-greenhouse effect subspace; (d) subspace projections.
Fig. 11 presents the Daisyworld behavior showing the same complex response but causing the extinction event earlier when compared with the previous case, showed in Fig. 10. The inclusion of absorption, assuming a = 104 and b = 101, changes the system dynamics as shown in Fig. 12. Note that life is able to persist for a longer time, causing a regulation of the temperature, similar to the situation without greenhouse gases. For later times, when luminosity reaches critical values, the life ability for system regulation saturates and life extinction is achieved. 3.2. Local analysis One of the characteristics of the Daisyworld model is the capability to represent either global or local analysis (Wood et al., 2008). This section treats the local analysis by assuming that the solar energy input occurs in an isolated point of the planet. A proper description of this luminosity needs to be proposed. For example, in a region near the Equator, the solar luminosity is zero
over the nights and varies from zero to its maximum value during the day. For the sake of simplicity, it is possible to consider that luminosity intensity vanishes for twelve hours and varies from zero to its maximum value through the day. Here, it is assumed the following function to represent day–night behavior together with seasons: ðSL Þlocal ¼ S½sinð2ptÞ½M1 sinðwtÞ þ M2
(13)
where (SL)local is the incident solar energy and corresponds to SL, where S is a solar constant equal to 7.9 107 J day1 m2 and L is the luminosity; M2 = 2.8 107 and M1 = 32 107 are the amplitudes; w = 0.0175, such that sin(wt) corresponds to one cycle of the year. Fig. 13 shows the general behavior of the luminosity. Once again, three scenarios are of concern: a model without gases; a model with gases emissions; and a model with emission and absorption of gases. Initially, consider a situation without greenhouse gases, a = b = 0. Daisyworld behavior is presented in Fig. 14. Fig. 14a shows the time history of daisy populations while Fig. 14b presents the projection of the state space showing the
S.L.D. Paiva et al. / BioSystems 125 (2014) 1–15
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Fig. 12. Daisyworld with greenhouse effect where a = 104 and b = 101. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing the daisy-greenhouse effect subspace; (d) subspace projections.
Fig. 13. Dynamics of local luminosity and its highlight in 3 days.
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Fig. 14. Daisyworld without greenhouse effect (a = b = 0). (a) Daisy dynamics; (b) temperature and luminosity dynamics; (c) projection of the state space showing the daisy subspace; (d) projection of the state space showing the black daisy-temperature subspace;(e) projection of the state space showing the white daisy-temperature subspace.
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Fig. 15. Daisyworld with greenhouse effect where a = 5 105 and b = 0. (a) Daisy dynamics; (b) projection of the state space showing the daisy subspace; (c) temperature and luminosity dynamics; (d) temperature and greenhouse effect dynamics.
daisy subspace. Fig. 14c shows temperature and luminosity dynamics. Fig. 14d–e presents different projections of the state space showing respectively, the black daisy-temperature and white daisy-temperature subspaces. Two important aspects should be observed: the stabilization of the system behavior and the change in representing the changes through the year. The introduction of greenhouse gases in the analysis (a = 5 105 and b = 0) changes the general behavior, causing an extinction event (Fig. 15). Now temperature increases dramatically after years of small increases in temperature. The state space shows a tendency to go to a stationary response at the origin, associated with the life extinction. The effect of gas absorption changes again this general behavior, bringing a proper balance through the system. By assuming a = 5 105 and b = 103, Daisyworld behavior is presented in Fig. 16. The temperature regulation of the environment is noticeable, but temperature also has a general
increase over time. The state space allows one to identify a creation of an equilibrium point with preponderancy of white daisies. 4. Conclusions The Daisyworld model aims to illustrate the biota influence in their environment promoting self-regulation to benefit life. This paper proposes a way to include greenhouse gases emission and absorption into the model, allowing the description of the global warming and other related effects. Two different analyses are of concern: global and local. For each one of these, three scenarios of the Daisyworld are compared: a model without greenhouse gases; a model with greenhouse gases emissions; a model with greenhouse gases emissions and absorptions. Solar luminosity is also changed in order to contemplate different scenarios. Basically, global analysis considers three situations related to solar
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Fig. 16. Daisyworld with greenhouse effect where a = 5 105 and b = 103. (a) Daisy dynamics; (b) projection of the state space showing the daisy subspace; (c) temperature and luminosity dynamics; (d) temperature and greenhouse effect dynamics.
luminosity: constant value; linear increase; sinusoidal variation over a linear increase. Local analysis, on the other hand, simulates day and night luminosity and season variations. The greenhouse effect acts to retain heat in the environment and reducing the lifetime of daisies. On the other hand, the absorption capacity of the planet tends to extend daisy lifetimes, increasing the self-regulation. Daisy populations tend to vary in order to regulate temperature in benefit of life. White daisies tend to increase when greenhouse effect increases while the opposite occurs with the black daisies. In general, the model is able to describe the influence of greenhouse effect in a qualitative point of view. In this regard, it allows one to investigate different qualitative scenarios related to global warming. Acknowledgements The authors would like to acknowledge the support of the Brazilian Research Agencies CNPq, CAPES and FAPERJ and through the INCT-EIE (National Institute of Science and Technology – Smart Structures in Engineering) the CNPq and FAPEMIG. The Air Force Office of Scientific Research (AFOSR) is also acknowledged.
References Ackland, G.J., 2004. Maximization principles and Daisyworld. J. Theor. Biol. 227, 121– 128. Adams, B., Carr, J., Lenton, T.M., White, A., 2003. One-dimensional Daisyworld: spatial interactions and pattern formation. J. Theor. Biol. 223, 505–513. Alexiadis, A., 2007. Global warming and human activity: a model for studying the potential instability of the carbon dioxide/temperature feedback mechanism. Ecol. Modell. 203, 243–256. Biton, E., Gildor, H., 2012. The seasonal effect in one-dimensional Daisyworld. J. Theor. Biol. 314, 145–156. Boyle, R.A., Lenton, T.M., Watson, A.J., 2011. Symbiotic physiology promotes homeostasis in Daisyworld. J. Theor. Biol. 274, 170–182. Charlson, R.J., Lovelock, J.E., Andreae, M.O., Warren, S.G., 1987. Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate. Nature 326, 655–661. Cohen, J.E., Rich, A.D., 2000. Interspecific competition affects temperature stability in Daisyworld. Tellus 52B, 980–984. Foong, S.K., 2006. An accurate analytical solution of a zero-dimensional greenhouse model for global warming. Eur. J. Phys. 27, 933–942. Houghton, J., 2005. Global warming. Rep. Prog. Phys. 68, 1343–1403. Jorgensen, S.E., 1999. State-of-the-art of ecological modelling with emphasis on development of structural dynamic models. Ecol. Modell. 120 (2–3), 75–96. Kay, A.L., Davies, H.N., Bell, V.A., Jones, R.G., 2009. Comparison of uncertainty sources for climate change impacts: flood frequency in England. Clim. Change 92, 41–63. Lenton, T.M., Lovelock, J.E., 2000. Daisyworld is Darwinian: constraints on adaptation are important for planetary self-regulation. J. Theor. Biol. 206, 109–114.
S.L.D. Paiva et al. / BioSystems 125 (2014) 1–15 Nevison, C., Gupta, V., Klinger, L., 1999. Self-sustained temperature oscillations on Daisyworld. Tellus 51B, 806–814. Novak, M.A., 2006. Evolutionary Dynamics: Exploring the Equations of Life. Belknap Press of Harvard University Press. Robertson, D., Robinson, J., 1998. Darwinian Daisyworld. J. Theor. Biol. 195, 129–134. Staley, M., 2002. Darwinian selection leads to Gaia. J. Theor. Biol. 218, 35–46. Viola, F.M., Savi, M.A., Paiva, S.L.D., Brasil Jr., F.M., 2013. Nonlinear dynamics and chaos of the Daisyworld employed for global warming description. Appl. Ecol. Environ. Res. 11 (3), 463–490. Viola, F.M., Paiva, S.L.D., Savi, M.A., 2010. Analysis of the global warming dynamics from temperature time series. Ecol. Modell. 221 (16), 1964–1978.
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Watson, A.J., Lovelock, J.E., 1983. Biological homeostasis of the global environment: the parable of Daisyworld. Tellus 35B, 284–289. Weaver, I.S., Dyke, J.G., 2012. The importance of timescales for the emergence of environmental self-regulation. J. Theor. Biol. 313, 172–180. Wood, A.J., Ackland, G.J., Lenton, T.M., 2006. Mutation of albedo and growth response produces oscillations in a spatial Daisyworld. J. Theor. Biol. 242, 188–198. Wood, A.J., Ackland, G.J., Dyke, J.G., Williams, H.T.P., Lenton, T.M., 2008. Daisyworld: a review. Rev. Geophys. 46, RG1001. doi:http://dx.doi.org/10.1029/ 2006RG000217. Zeng, X., Pielke, R.A., Eykholt, R., 1990. Chaos in Daisyworld. Tellus 42B, 309–318.
Contents lists available at ScienceDirect
BioSystems journal homepage: www.elsevier.com/locate/biosystems
Global warming description using Daisyworld model with greenhouse gases Susana L.D. Paiva, Marcelo A. Savi *, Flavio M. Viola, Albino J.K. Leiroz Universidade Federal do Rio de Janeiro, COPPE – Department of Mechanical Engineering, P.O. Box 68.503, 21.941.972 Rio de Janeiro, Brazil
A R T I C L E I N F O
A B S T R A C T
Article history: Received 23 January 2014 Received in revised form 9 September 2014 Accepted 10 September 2014 Available online 16 September 2014
Daisyworld is an archetypal model of the earth that is able to describe the global regulation that can emerge from the interaction between life and environment. This article proposes a model based on the original Daisyworld considering greenhouse gases emission and absorption, allowing the description of the global warming phenomenon. Global and local analyses are discussed evaluating the influence of greenhouse gases in the planet dynamics. Numerical simulations are carried out showing the general qualitative behavior of the Daisyworld for different scenarios that includes solar luminosity variations and greenhouse gases effect. Nonlinear dynamics perspective is of concern discussing a way that helps the comprehension of the global warming phenomenon. ã 2014 Elsevier Ireland Ltd. All rights reserved.
Keywords: Daisyworld Global warming Carbon cycle Ecology Nonlinear dynamics
1. Introduction Climate change is a growing global concern, and a topic of considerable research. Global warming is within this general issue being related to the increase of earth’s temperature (Houghton, 2005). While the consequences of global temperature changes remain unknown, it is possible to use mathematical models to evaluate different scenarios. There are several general modeling approaches (Alexiadis, 2007). A first strategy makes use of timeseries analysis to build predictive models of future temperature change (Viola et al., 2010). A second approach makes use of mathematical models to describe system behavior using governing equations based on fundamental principles of involved phenomena (Jorgensen, 1999). Phenomenological models are a considerably simpler approach, useful for some descriptions (Kay et al., 2009). Daisyworld is an archetypal model that represents the earth describing the global regulation that can emerge from the interaction between life and environment. Daisyworld is a mathematical description of the Gaia theory of the earth representing life by daisy populations while the environment is represented by temperature. It was originally proposed by Watson and Lovelock (1983) in order to describe the self-regulation of the planetary system. Wood et al. (2008) presented a general overview
* *Corresponding author. E-mail addresses: [email protected] (S.L.D. Paiva), [email protected] ? (M.A. Savi), [email protected] (F.M. Viola), [email protected] (A.J.K. Leiroz). http://dx.doi.org/10.1016/j.biosystems.2014.09.008 0303-2647/ ã 2014 Elsevier Ireland Ltd. All rights reserved.
about the literature associated with the Daisyworld, emphasizing the model main characteristics and different analysis approaches. Spatial aspects and variants of the model are discussed. Adaptability is an important issue that defines system evolution by natural selection. In brief, it is possible to say that Daisyworld allows one to perform a qualitative description of the climate system, permitting the investigation of different aspects of the system behavior. Because of that, several researches are dedicated for variants and extensions of the Daisyworld. An important discussion about Daisyworld characteristics is related to the antagonism between altruism and competition of the daisy populations. In this regard, several researches were developed trying to investigate adaptive behavior of the Daisyworld, incorporating competition and mutation aspects of life. Lenton and Lovelock (2000) discussed some adaptive aspects of the Daisyworld based on the idea that it is a Darwinian model in the sense that there is a competition among different types of life with heritable variation in a trait, which represents a kind of natural selection (Robertson and Robinson, 1998). Daisy colors (or planetary albedo) and optimum growth temperature are the main parameters related to the Daisyworld adaptation. Restrictions on environmental conditions define some evolution changes. Cohen and Rich (2000) introduced an extra source of competition that describes the interaction among daisy species in order to retain temperature in an appropriate range in order to make the planet habitable. Boyle et al. (2011) pointed that homeosthasis of the Daisyworld can be promoted by symbiotic physiology. Qualitative comparisons are evaluated with and without the consideration of symbiotic physiology.
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S.L.D. Paiva et al. / BioSystems 125 (2014) 1–15 Table 1 Parameters of the Daisyworld. aw ab ag
0.75 0.25 0.5 0.3 2.06 109 K4 295.5 K
g
q Topt
Table 2 Parameters of the Daisyworld considering different timescales. Different timescales W m2
s S c
Fig. 1. Gaussian representing the optimum condition and limits of life.
Weaver and Dyke (2012) discussed Daisyworld model from timescale perspective. Basically, it is considered temperature, changes in insolation, and self-organisation of the biota using an analytical approach.
J day1 m2 8
5.67 10 915 950 (K1)
4
(K
)
4.90 103 (K4) 7.91 107 8.21 107(K1)
One of the most important characteristics of the Daisyworld is the capability to describe either global or local phenomena. Global analysis assumes that the translational movement of the planet is the main component of temperature variation, being the planet subjected to temperature changes according to seasons of the year or aspects of the planet. Under this assumption, Daisyworld is related to the entire planet. On the other hand, local analysis defines a
Fig. 2. Classical Daisyworld response (Nevison et al., 1999). (a) Daisy dynamics; (b) temperature dynamics; (c) projection of the state space showing the daisy subspace.
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specific region of the planet. Therefore, either rotation or translation of the planet influence solar energy input to the planet since a specific region is subjected to temperature changes due to solar luminosity variations associated with days and nights, and also with seasons. Staley (2002) and Charlson et al. (1987) discussed the differences between the global and local effects on the planet dynamics. The Daisyworld model was also studied considering spatio-temporal behavior. Adams et al. (2003) proposed a one-dimensional model incorporating a distribution of incoming solar radiation and heat diffusion consistent with a spherical planet. Biton and Gildor (2012) added seasonal solar radiation forcing to the latitudinal-depended one-dimensional Daisyworld model. Wood et al. (2006) included a mutation of the optimal growth temperature and of the albedo in a two-dimensional Daisyworld model determining the temperature oscillations. Ackland (2004) used a two-dimensional Daisyworld model considering the temperatures and albedos on a grid indexed by longitude and latitude positions.
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Daisyworld can be employed for global warming description. In essence, greenhouse gases affect Daisyworld behavior in a similar way of the black daisies in terms of radiation absortion. Nevetheless, the interaction with environment has an important difference. Viola et al. (2013) incorporated greenhouse gases in the Daisyworld considering prescribed time series that alter system albedo. Besides, a sinusoidal variation of the solar luminosity represents climate variability. Hence, the Daisyworld is employed for global warming description and complex dynamical responses are observed. Chaotic behavior was of concern showing that this kind of response is possible under certain conditions. Zeng et al. (1990) and Wood et al. (2008) also investigated possibilities related to the chaotic behavior of the Daisyworld. This paper deals with the description of the global warming using the Daisyworld model considering that greenhouse gases are included in the model with the aid of an extra equation motivated by the carbon cycle dynamics and predator–prey models. The
Fig. 3. Daisyworld with greenhouse effect where a = 104 and b = 0. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing the daisy-greenhouse effect subspace; (d) subspace projections.
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Fig. 4. Daisyworld with greenhouse effect where a = b = 104. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing the daisy-greenhouse effect subspace; (d) subspace projections.
temperature dependence of life is represented by a Gaussian function avoiding life extinction observed on the classical peak function of the Daisyworld model. Emissions and absorptions of greenhouse gases are treated showing the capability of the model to describe global warming scenarios. Numerical simulations show this general qualitative capability presenting several simulations related to greenhouse gases effect and also changes in solar luminosity. Results are analyzed considering a nonlinear dynamics perspective that provides a comprehension about system behavior. 2. Daisyworld model and the carbon cycle The climate system is a complex system based on several phenomena. The carbon cycle is associated with chemical and biological interactions of great importance to define the origins and the destinations of this element. Carbon is stored in four main reservoirs of the earth: atmosphere, lithosphere, biosphere and hydrosphere. Each reservoir contains a variety of carbon compounds
of organic and inorganic nature. Furthermore, the transfer time and storage of carbon compounds in each reservoir can vary from years to millennia. For example, the lithosphere contains a large amount of carbon trapped in sedimentary rocks formations of mineral carbonates and organic compounds such as oil, natural gas and coal. The redistribution from the lithosphere reservoir to another is associated with millions of years, until all the geological process, such as chemical decomposition and deposition, occurs. Therefore, the lithosphere is considered a relatively inactive component of the carbon cycle when preserved in its natural state. The active reservoirs are divided between the atmosphere, the terrestrial biosphere and the hydrosphere. As the absolute sum of the amounts of active carbon reservoir is maintained close to the steady state value by slow geological processes, biogeochemical processes that lead to redistribution of carbon between active reservoirs occur more rapidly. The Daisyworld represents self-regulation of the planet that is basically composed of the environment, represented by the temperature, and life, represented by daisy populations.
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Daisyworld model represents life by daisy populations, which evolution is described by the following equations: (1) a_ w ¼ aw ag bðT w Þ g a_ b ¼ ab ag bðT b Þ g
A ¼ a w aw þ a b ab þ a g ag
(4)
The functional form for b is usually assumed to be a symmetric single-peak function. Here, a Gaussian function is assumed as follows:
bðT i Þ ¼ Be s ðT i T opt Þ 1
(2)
5
2
(5)
For the sake of simplicity, only two daisy populations are considered: white (aw) and black (ab) daisies. The dot represents time derivative and b = b(Ti) represents the growth rate that is temperature dependent and g is the population death rate. The variable ag is the fractional area coverage of the planet represented by
where B is a parameter related to environmental characteristics, s is the variance and Topt is the optimal temperature, usually considered as Topt = 295 K = 22.5 C. The Gaussian form of this function is shown in Fig. 1 and it is convenient to avoid a drastic population extinction observed on the classical peak function of the Daisyworld model. The local temperature of each population is defined as follows:
ag ¼ p aw ab
T 4w ¼ qðA aw Þ þ T 4
(6)
T 4b ¼ qðA ab Þ þ T 4
(7)
(3)
where p = 1 represents the portion of land suitable for the growth of daisies. The mean planetary albedo of the Daisyworld, A, can be estimated from the individual albedo of each daisy population (aw and ab), and from the ground albedo (ag):
Fig. 5. Daisyworld with greenhouse effect where a = 104 and b = 103. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing daisy-greenhouse effect subspace; (d) subspace projections.
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T 4g ¼ qðA ag Þ þ T 4
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(8)
where T is the globally-averaged temperature of the Daisyworld, and q is a constant used to calculate local temperature as a function of albedo. Finally, it is important to establish the thermal balance of the Daisyworld (Foong, 2006). Therefore, the absorbed energy is given by (Nevison et al., 1999): 1 T_ ¼ ½SLð1 AÞ s T 4 ð1 EÞ c
(9)
where L is the solar luminosity and S is the solar constant that establishes the average solar energy represented by SL; s is the Stefan–Boltzmann constant; c is a measure of the average heat capacity or thermal inertia of the planet. Finally, E represents the greenhouse effect that alters the thermal balance, described by the following differential equation. E_ ¼ a bEðaw þ ab Þ
(10)
This new equation establishes the interaction between gases and daisy populations being motivated by the general predator– prey equation and the carbon cycle where life is responsible for the absorption of the gases (Novak, 2006). Parameter a is related to gas
emission while b represents the absorption. Note that when b vanishes, no interaction occurs and the effect of greenhouse gases is only related to the emissions. From now on, the greenhouse variable E is referred as greenhouse effect. 3. Numerical simulations The Daisyworld model can represent global or local aspects of the planet dynamics. Global analysis considers that Daisyworld is related to the entire planet. On the other hand, local analysis defines a specific region of the planet. Numerical simulations are carried out considering both situations and different scenarios. Initially, numerical simulations are performed considering a global analysis. Three different models are treated: the original, without greenhouse gases; a model with greenhouse gases; and a model with emission and absorption of greenhouse gases. Moreover, luminosity variations are also investigated representing the profile of solar activity. In this regard, three different scenarios are considered: constant value; linear increase; and climate variability represented by a sinusoidal modulation over a linear growth. The Daisyworld model can be simulated using classical procedures for numerical integration. Here, a fourth order
Fig. 6. Daisyworld with greenhouse effect where a = b = 0. (a) Daisy dynamics; (b) temperature dynamics; (c) projection of the state space showing the daisy subspace.
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Runge–Kutta method associated with an iterative process is employed. Time steps smaller than 102 are assumed for the global analysis while local analysis assumes time steps smaller than 103. These time-step values are defined after convergence analysis. Tables 1 and 2 show the values of the parameters employed in all simulations. 3.1. Global analysis This section presents the global analysis of the Daisyworld. Three different luminosity scenarios are discussed: constant value; linear increase; and climate variability. Each of these analyses may represent a scenario related to different timescales, for instance. For each one of them, three different situations are treated: the original model, without greenhouse gases; a model with greenhouse gases; a model with emission and absorption of greenhouse gases. 3.1.1. Constant luminosity A situation where greenhouse gases are not involved is used to start the analysis. Therefore, emissions or absorptions of
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greenhouse gases are not considered in the model, which is represented by a = b = 0. Fig. 2 presents Daisyworld response showing consistent results compared with the classical Daisyworld (Nevison et al., 1999). Fig. 2a shows the dynamics of daisy populations while Fig. 2b shows the temperature dynamics. Fig. 2c presents a projection of the state space in daisy subspace. Note that the interaction between the two daisy species, white and black, limits temperature values within a proper range for life existence. The increase of white daisies tends to decrease of temperature, while the increase of black daisies causes the increase of temperature due to the respective albedo. A scenario with greenhouse gases emissions is now investigated, but the planet is not able to absorb these gases. This analysis is accomplished by considering a = 104 and b = 0. Fig. 3 presents results of this case showing that the increase of gases causes a general increase of planet temperature. Basically, Fig. 3a shows the time history of daisy populations while Fig. 3b shows the temperature and the greenhouse effect dynamics (represented by variable E). Fig. 3c shows a projection of the state space considering daisy populations and greenhouse effect. Fig. 3d presents some projections of the subspace of Fig. 3c. Results show
Fig. 7. Daisyworld with greenhouse effect where a = 104 and b = 0. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing the daisy-greenhouse effect subspace; (d) subspace projections.
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Fig. 8. Daisyworld with greenhouse effect where a = 104 and b = 101. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing the daisy-greenhouse effect subspace; (d) subspace projections.
that in this situation, the balance between life and environment is lost, and daisy populations cannot compensate the greenhouse effect. It is noticeable the persistence of white daisies for a longer time, acting to delay the life extinction caused by the excessive increase of temperature. It should be highlighted that the extinction is related to an escape imposed by the variable E, clearly shown in state space projection (Fig. 3d). The planet capacity to absorb greenhouse gases is now of concern by assuming a = b = 104. In this scenario, daisy populations contribute to gas absorption. Fig. 4 presents the response of the Daisyworld for this case. It is clear the increase of the self-regulation is caused by the absorption capacity. Nevertheless, this is not enough to avoid the extinction of life on the planet. The increase of the absorption capacity can be represented by considering a = 104 and b = 103. This increase avoids the extinction as shown in Fig. 5. Note that this new scenario is related to a stabilization of the system response for a specific variable E, which is clear in the state space projections (Fig. 5d).
Fig. 9. Luminosity representing climate variability.
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3.1.2. Linear increase of the luminosity In order to represent a more realistic solar activity, a linear increase of the luminosity is considered as: L ¼ 1:9 104 t þ 0:75
(11)
Once again, three scenarios are treated: the original model, without greenhouse gases; a model with greenhouse gases emissions; and a model with greenhouse gases emission and absorption. Initially, the dynamic behavior of the Daisyworld without greenhouse gases, represented by a = b = 0, is observed. Fig. 6 presents the general behavior of the Daisyworld where an increase of luminosity causes a temperature increase, which leads to unfavorable environmental conditions, causing the life extinction. The emission of greenhouse gases (a = 104 and b = 0) accelerates this tendency even more, as shown in Fig. 7. Once again, it is possible to associate the extinction with the greenhouse effect variable escape observed in the projections of the state space, Fig. 7d. The absorption capacity can change the scenario associated with emissions. By considering a = 104 and b = 101 (Fig. 8), the Daisyworld presents similar results of the one presented in Fig. 6,
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showing the system capacity to avoid extreme situations. Note however, that there is still an extinction of life that occurs when the luminosity reaches a critical value. 3.1.3. Luminosity representing climate variability Climate variability is now in focus assuming a sinusoidal variation of the luminosity over a linear increase as follows: ðSL Þglobal ¼ S þ ðNsinwtÞ
(12)
where (SL)global is the global solar energy and corresponds to SL, where S is the solar constant and L is the luminosity; N is the amplitude and w is the forcing frequency. Here, it is considered N = 2.89 1012 and w = 0.01 that is related to a behavior presented in Fig. 9. Initially, situations without greenhouse gases (a = b = 0) are considered. Under this condition, the dynamics of daisy populations and temperature levels are presented in Fig. 10. Note that Daisyworld presents a more complex behavior due to sinusoidal excitation. A situation with greenhouse gases emission but without absorption, represented by a = 104 and b = 0, is now considered.
Fig. 10. Daisyworld without greenhouse effect where a = b = 0. (a) Daisy dynamics; (b) temperature dynamics; (c) projection of the state space showing the daisy subspace.
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Fig. 11. Daisyworld with greenhouse effect where a = 104 and b = 0. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing the daisy-greenhouse effect subspace; (d) subspace projections.
Fig. 11 presents the Daisyworld behavior showing the same complex response but causing the extinction event earlier when compared with the previous case, showed in Fig. 10. The inclusion of absorption, assuming a = 104 and b = 101, changes the system dynamics as shown in Fig. 12. Note that life is able to persist for a longer time, causing a regulation of the temperature, similar to the situation without greenhouse gases. For later times, when luminosity reaches critical values, the life ability for system regulation saturates and life extinction is achieved. 3.2. Local analysis One of the characteristics of the Daisyworld model is the capability to represent either global or local analysis (Wood et al., 2008). This section treats the local analysis by assuming that the solar energy input occurs in an isolated point of the planet. A proper description of this luminosity needs to be proposed. For example, in a region near the Equator, the solar luminosity is zero
over the nights and varies from zero to its maximum value during the day. For the sake of simplicity, it is possible to consider that luminosity intensity vanishes for twelve hours and varies from zero to its maximum value through the day. Here, it is assumed the following function to represent day–night behavior together with seasons: ðSL Þlocal ¼ S½sinð2ptÞ½M1 sinðwtÞ þ M2
(13)
where (SL)local is the incident solar energy and corresponds to SL, where S is a solar constant equal to 7.9 107 J day1 m2 and L is the luminosity; M2 = 2.8 107 and M1 = 32 107 are the amplitudes; w = 0.0175, such that sin(wt) corresponds to one cycle of the year. Fig. 13 shows the general behavior of the luminosity. Once again, three scenarios are of concern: a model without gases; a model with gases emissions; and a model with emission and absorption of gases. Initially, consider a situation without greenhouse gases, a = b = 0. Daisyworld behavior is presented in Fig. 14. Fig. 14a shows the time history of daisy populations while Fig. 14b presents the projection of the state space showing the
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Fig. 12. Daisyworld with greenhouse effect where a = 104 and b = 101. (a) Daisy dynamics; (b) temperature and greenhouse effect dynamics; (c) projection of the state space showing the daisy-greenhouse effect subspace; (d) subspace projections.
Fig. 13. Dynamics of local luminosity and its highlight in 3 days.
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Fig. 14. Daisyworld without greenhouse effect (a = b = 0). (a) Daisy dynamics; (b) temperature and luminosity dynamics; (c) projection of the state space showing the daisy subspace; (d) projection of the state space showing the black daisy-temperature subspace;(e) projection of the state space showing the white daisy-temperature subspace.
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Fig. 15. Daisyworld with greenhouse effect where a = 5 105 and b = 0. (a) Daisy dynamics; (b) projection of the state space showing the daisy subspace; (c) temperature and luminosity dynamics; (d) temperature and greenhouse effect dynamics.
daisy subspace. Fig. 14c shows temperature and luminosity dynamics. Fig. 14d–e presents different projections of the state space showing respectively, the black daisy-temperature and white daisy-temperature subspaces. Two important aspects should be observed: the stabilization of the system behavior and the change in representing the changes through the year. The introduction of greenhouse gases in the analysis (a = 5 105 and b = 0) changes the general behavior, causing an extinction event (Fig. 15). Now temperature increases dramatically after years of small increases in temperature. The state space shows a tendency to go to a stationary response at the origin, associated with the life extinction. The effect of gas absorption changes again this general behavior, bringing a proper balance through the system. By assuming a = 5 105 and b = 103, Daisyworld behavior is presented in Fig. 16. The temperature regulation of the environment is noticeable, but temperature also has a general
increase over time. The state space allows one to identify a creation of an equilibrium point with preponderancy of white daisies. 4. Conclusions The Daisyworld model aims to illustrate the biota influence in their environment promoting self-regulation to benefit life. This paper proposes a way to include greenhouse gases emission and absorption into the model, allowing the description of the global warming and other related effects. Two different analyses are of concern: global and local. For each one of these, three scenarios of the Daisyworld are compared: a model without greenhouse gases; a model with greenhouse gases emissions; a model with greenhouse gases emissions and absorptions. Solar luminosity is also changed in order to contemplate different scenarios. Basically, global analysis considers three situations related to solar
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Fig. 16. Daisyworld with greenhouse effect where a = 5 105 and b = 103. (a) Daisy dynamics; (b) projection of the state space showing the daisy subspace; (c) temperature and luminosity dynamics; (d) temperature and greenhouse effect dynamics.
luminosity: constant value; linear increase; sinusoidal variation over a linear increase. Local analysis, on the other hand, simulates day and night luminosity and season variations. The greenhouse effect acts to retain heat in the environment and reducing the lifetime of daisies. On the other hand, the absorption capacity of the planet tends to extend daisy lifetimes, increasing the self-regulation. Daisy populations tend to vary in order to regulate temperature in benefit of life. White daisies tend to increase when greenhouse effect increases while the opposite occurs with the black daisies. In general, the model is able to describe the influence of greenhouse effect in a qualitative point of view. In this regard, it allows one to investigate different qualitative scenarios related to global warming. Acknowledgements The authors would like to acknowledge the support of the Brazilian Research Agencies CNPq, CAPES and FAPERJ and through the INCT-EIE (National Institute of Science and Technology – Smart Structures in Engineering) the CNPq and FAPEMIG. The Air Force Office of Scientific Research (AFOSR) is also acknowledged.
References Ackland, G.J., 2004. Maximization principles and Daisyworld. J. Theor. Biol. 227, 121– 128. Adams, B., Carr, J., Lenton, T.M., White, A., 2003. One-dimensional Daisyworld: spatial interactions and pattern formation. J. Theor. Biol. 223, 505–513. Alexiadis, A., 2007. Global warming and human activity: a model for studying the potential instability of the carbon dioxide/temperature feedback mechanism. Ecol. Modell. 203, 243–256. Biton, E., Gildor, H., 2012. The seasonal effect in one-dimensional Daisyworld. J. Theor. Biol. 314, 145–156. Boyle, R.A., Lenton, T.M., Watson, A.J., 2011. Symbiotic physiology promotes homeostasis in Daisyworld. J. Theor. Biol. 274, 170–182. Charlson, R.J., Lovelock, J.E., Andreae, M.O., Warren, S.G., 1987. Oceanic phytoplankton, atmospheric sulphur, cloud albedo and climate. Nature 326, 655–661. Cohen, J.E., Rich, A.D., 2000. Interspecific competition affects temperature stability in Daisyworld. Tellus 52B, 980–984. Foong, S.K., 2006. An accurate analytical solution of a zero-dimensional greenhouse model for global warming. Eur. J. Phys. 27, 933–942. Houghton, J., 2005. Global warming. Rep. Prog. Phys. 68, 1343–1403. Jorgensen, S.E., 1999. State-of-the-art of ecological modelling with emphasis on development of structural dynamic models. Ecol. Modell. 120 (2–3), 75–96. Kay, A.L., Davies, H.N., Bell, V.A., Jones, R.G., 2009. Comparison of uncertainty sources for climate change impacts: flood frequency in England. Clim. Change 92, 41–63. Lenton, T.M., Lovelock, J.E., 2000. Daisyworld is Darwinian: constraints on adaptation are important for planetary self-regulation. J. Theor. Biol. 206, 109–114.
S.L.D. Paiva et al. / BioSystems 125 (2014) 1–15 Nevison, C., Gupta, V., Klinger, L., 1999. Self-sustained temperature oscillations on Daisyworld. Tellus 51B, 806–814. Novak, M.A., 2006. Evolutionary Dynamics: Exploring the Equations of Life. Belknap Press of Harvard University Press. Robertson, D., Robinson, J., 1998. Darwinian Daisyworld. J. Theor. Biol. 195, 129–134. Staley, M., 2002. Darwinian selection leads to Gaia. J. Theor. Biol. 218, 35–46. Viola, F.M., Savi, M.A., Paiva, S.L.D., Brasil Jr., F.M., 2013. Nonlinear dynamics and chaos of the Daisyworld employed for global warming description. Appl. Ecol. Environ. Res. 11 (3), 463–490. Viola, F.M., Paiva, S.L.D., Savi, M.A., 2010. Analysis of the global warming dynamics from temperature time series. Ecol. Modell. 221 (16), 1964–1978.
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Watson, A.J., Lovelock, J.E., 1983. Biological homeostasis of the global environment: the parable of Daisyworld. Tellus 35B, 284–289. Weaver, I.S., Dyke, J.G., 2012. The importance of timescales for the emergence of environmental self-regulation. J. Theor. Biol. 313, 172–180. Wood, A.J., Ackland, G.J., Lenton, T.M., 2006. Mutation of albedo and growth response produces oscillations in a spatial Daisyworld. J. Theor. Biol. 242, 188–198. Wood, A.J., Ackland, G.J., Dyke, J.G., Williams, H.T.P., Lenton, T.M., 2008. Daisyworld: a review. Rev. Geophys. 46, RG1001. doi:http://dx.doi.org/10.1029/ 2006RG000217. Zeng, X., Pielke, R.A., Eykholt, R., 1990. Chaos in Daisyworld. Tellus 42B, 309–318.
