A Daisyworld Ecological Parable Including the Revenge of Gaia and Greenhouse Effect

fractal and fractional

Article
A Daisyworld Ecological Parable Including the Revenge of Gaia
and Greenhouse Effect
Marcelo A. Savi * and Flavio M. Viola

 Center for Nonlinear Mechanics, COPPE—Department of Mechanical Engineering,
 Universidade Federal do Rio de Janeiro, P.O. Box 68503, Rio de Janeiro 21941-901, Brazil
 * Correspondence: savi@mecanica.coppe.ufrj.br

 Abstract: The Daisyworld model illustrates the concept of biological homeostasis in the global
 environment by establishing a connection between the biota and environment, resulting in a single
 intertwined system known as Gaia. In essence, the Daisyworld model represents life by daisy
 populations whereas temperature represents the environment, establishing a population dynamics
 model to represent life–environment ecological interactions. The recent occurrence of extreme weather
 events due to climate change and the critical crises brought on by the COVID-19 pandemic are
 strengthening the arguments for the revenge of Gaia, a term used to describe the protective response
 of the global biota-environment system. This paper presents a novel Daisyworld parable to describe
 ecological life–environment interactions including the revenge of Gaia and the greenhouse effect.
 The revenge of Gaia refers to a change in the interplay between life and environment, characterized
 by the Gaia state that establishes the life-environment state of balance and harmony. This results in
 reaction effects that impact the planet’s fertile regions. On the other hand, the greenhouse effect is
 incorporated through the description of the interactions of greenhouse gases with the planet, altering
 its albedo. Numerical simulations are performed using a nonlinear dynamics perspective, showing
 different ecological scenarios. An investigation of the system reversibility is carried out together with
 critical life–environment interactions. This parable provides a qualitative description that can be
 useful to evaluate ecological scenarios.

 Keywords: ecology; Daisyworld; revenge of Gaia; greenhouse effect; nonlinear dynamics;
Citation: Savi, M.A.; Viola, F.M. A life–environment system
Daisyworld Ecological Parable
Including the Revenge of Gaia and
Greenhouse Effect. Fractal Fract. 2023,
7, 190. https://doi.org/10.3390/
 1. Introduction
fractalfract7020190
 Daisyworld is a model originally proposed by James Lovelock [1,2] that represents
Academic Editors: Haci
 the biological homeostasis of the global environment, describing the self-regulation of
Mehmet Baskonus and Golan Bel
 the planetary system. Daisyworld established the biota–environment coupling, which
Received: 23 January 2023 means that life and environment is a single coupled system: Gaia. Watson and Lovelock [3]
Revised: 2 February 2023 developed a mathematical model presented as a parable considering an imaginary planet
Accepted: 13 February 2023 because the Earth case is too complex to be modeled.
Published: 14 February 2023 The basic idea is that the environment is characterized by temperature and life is
 represented by daisy populations. Biodiversity is symbolized by the colors of daisies,
 each of which has its own albedo, the ability to reflect light and heat from the sun into
 the atmosphere. The simplest version of the Daisyworld model employs black and white
Copyright: © 2023 by the authors. daisies only. The Daisyworld defines the symbiotic interactions of daisy populations with
Licensee MDPI, Basel, Switzerland.
 the ambient in order to preserve temperature in the optimal conditions for life.
This article is an open access article
 Since its inception, the Daisyworld model has garnered significant attention from vari-
distributed under the terms and
 ous perspectives and has become an iconic representation of the Gaia theory. It symbolizes
conditions of the Creative Commons
 the fundamental principles of ecology theory. One of the most important characteristics of
Attribution (CC BY) license (https://
 the Daisyworld is the capability to describe either global or local phenomena [4]. Wood
creativecommons.org/licenses/by/
4.0/).

Fractal Fract. 2023, 7, 190. https://doi.org/10.3390/fractalfract7020190 https://www.mdpi.com/journal/fractalfract

Fractal Fract. 2023, 7, 190 2 of 14

 et al. [5] presented a general overview regarding the literature associated with the Daisy-
 world, emphasizing the model characteristics, different approaches for the analysis, spatial
 aspects and variant models.
 A debate that has arisen over the years concerns the adaptability of the system and how
 it shapes the evolution of life through natural selection. Stocker [6] discussed mutations
 in the Daisyworld considering grey daisies that appear between white and black daises.
 Saunders [7] discussed evolution without natural selection in the Daisyworld. Afterward,
 Darwinian aspects were investigated by Robertson and Robinson [8], allowing for the
 description of adaptive evolution where the daisy populations can change their optimal
 temperature. Lenton and Lovelock [9] included constraints on adaptation showing that
 Daisyworld operates in a Darwinian manner, where different types of life compete with
 one another, and exhibit heritable variations in traits, which is a form of natural selection.
 Sugimoto [10] presented mathematical analytical arguments showing that Gaia hy-
 pothesis and Darwinian evolution can coexist. Staley [11] proposed an alternative approach
 to the Daisyworld model that is based on conventional Darwinian principles, in which en-
 vironmental regulation is a result of population dynamics rather than Darwinian selection,
 which favors the organisms with the highest survival capabilities. The conclusions point
 out that, as the environment is influenced by life, there is a co-evolution until the optimal
 conditions for life coincide with the equilibrium conditions. Doolittle [12] presented a
 Darwinized Daisyworld model which discussed the apparent contradictions and proposed
 a Gaia theory in a neo-Darwinian framework.
 Pujol [13] discussed the thermodynamical aspects of the Daisyworld model employing
 the maximum entropy principle. Results enlarge the range of climate stability. Ackland [14]
 employs the principle of maximum energy production showing that Daisyworld follows
 evolutionary dynamics. Nevison et al. [15] incorporated the energy equation to assess
 the temperature of the planet, leading to the discovery of self-sustaining temperature
 oscillations in Daisyworld.
 Cohen and Rich [16] explored the interactions among the daisy populations and how
 they contribute to maintaining an appropriate temperature range. Boyle et al. [17] discussed
 the symbiotic physiology of the Daisyworld establishing qualitative comparisons with the
 system without symbiosis. Weaver and Dyke [18] discussed timescale perspective using an
 analytical approach.
 Climate changes can also be described by considering greenhouse gases. Viola
 et al. [19] incorporated greenhouse gases in Daisyworld, considering prescribed time
 series that alter system albedo. Climate variability was also of concern considering har-
 monic variation of the solar luminosity. Complex dynamical responses were observed
 presenting chaotic behavior. Paiva et al. [20] discussed climate changes from the perspective
 of thermal balance.
 A crucial aspect to consider in evaluating the ecological behavior of Daisyworld is the
 impact of human activities, which are often characterized by destruction, pollution, and other
 damaging actions. It is important to determine whether these actions are so severe as to
 be considered symbiotic. In this regard, Lovelock introduced the revenge of Gaia theory
 which explained that Earth is fighting back against human activities [21,22]. Climate change is
 probably the most emblematic effect related to this reaction, but human society is still acting
 with an attitude of denialism. Recently, the COVID-19 pandemic brought another point to be
 imagined as a possible consequence of the revenge of Gaia. Cazzolla-Gatti [23] clearly pointed
 out that “coronavirus outbreak is a symptom of Gaia’s sickness”.
 The literature points to various issues related to the revenge of Gaia. Jones [24] em-
 phasized a scenario of rapid global warming leading to human extinction within 500 years,
 which he referred to as “Gaia bites back.” Aside from the ecological issues, it seems that
 human behavior has also evolved in the last few centuries. Torres [25] presented a survey
 of arguments related to that which attempted to answer the question: “Who would de-
 stroy the world?”. Morgan [26] evaluated two scenarios of the world on fire: nuclear and
 climate change.

Fractal Fract. 2023, 7, 190 3 of 14

 Wilkinson [27] pointed that Daisyworld can be employed to describe catastrophic
 scenarios. Ackland et al. [28] discussed catastrophic desert formation, considering spa-
 tiotemporal aspects of the model. Chaotic behavior associated with the Daisyworld is
 another source of unpredictability. Although this is a controversial point, Viola et al. [19]
 showed chaos in the Daisyworld induced by climate variability.
 The literature on ecological modeling is vast and includes different perspectives. Phe-
 nomenological models are useful due to their simplicity. In this regard, besides Daisyworld
 models, there are distinct initiatives to describe biosphere processes where carbon-cycles are
 of special interest [29–31]. Equilibrium and stability are some aspects that can be analyzed
 employing, for example, the Le Chatelier principle [32,33].
 Time series analysis is another important subject with considerable impact because it
 seems to be more convincing for a non-technical audience. In this regard, different perspec-
 tives are employed analyzing temperature, sea levels, greenhouse gases, deforestation and
 pollution, among others. Recently, sophisticated nonlinear tools and artificial intelligence
 have been employed to analyze distinct aspects of climate change [34–38].
 This paper deals with a novel Daisyworld parable to incorporate the revenge of Gaia
 and greenhouse effects. The novel mathematical model is proposed considering the classical
 variables—black daisies, white daisies and temperature—that incorporate extra variables to
 represent the planet fertile area, the Gaia-state condition associated with life–environment
 harmony that defines the Gaia reaction level, and greenhouse gases and their interactions
 with the planet. On this basis, Gaia state introduces constraints for the daisy populations
 establishing new scenarios for the planet’s self-regulation. Numerical simulations are
 carried out that show the model capabilities using a nonlinear dynamics perspective which
 provides an understanding of system behavior. The reduction of life has an impact on the
 biota–environment balance, which shows the effect of the revenge of Gaia.

 2. Daisyworld Model: Gaia Revenge and Greenhouse Effect
 Gaia is represented by the Daisyworld model, a self-regulated planet that is composed
 of the interactions between the environment and life. The classical Daisyworld model
 considers that life is represented by colored daisy populations and the variety of colors
 represents the biodiversity. For the sake of simplicity, two colors only are enough for a
 first approach to description: white, W; and black, B. The environment is represented by
 the temperature, T, and the thermal balance of the Daisyworld is defined by the energy
 equation that includes radiation energy and greenhouse effect [15,39]. Other effects can
 be incorporated in the Daisyworld parable by considering that Gaia reacts due to the
 unbalance of the life–environment interactions. The revenge of Gaia can be associated
 with either thermal balance or life conditions. In this regard, it is defined a variable G
 that represents the Gaia state with respect to life–environment harmony. Besides, the area
 that is proper for life—the planet fertile area—is represented by variable P. The classical
 Daisyworld assumes that this is a constant parameter, and the novel model assumes that it
 varies due to the Gaia state. The effect of greenhouse effect is incorporated by considering
 variable H that represents any greenhouse gas. In general, different gases can be treated,
 but a single generic gas is assumed. The greenhouse effect evolves based on the emission
 of greenhouse gases, defined by a time history h(t), being controlled by the carbon cycles
 from forests and oceans represented by the term ϕ H H. In essence, greenhouse gases alter
 the albedo, reducing the planet’s self-regulation capacity.

Fractal Fract. 2023, 7, 190 4 of 14

 On this basis, Daisyworld is described by a population dynamics model that allows a
 qualitative description of planet self-regulation governed by the following equations:
 .  
 W = W α g β( TW ) − γ
 .  
 B = B α g β( TB ) − γ
 .
 P = γ P − γG G
 . .
 (1)
 G = γG G Ξ + ε P (1 − P) sign P
 .

 . H = h(t) − ϕ H H
 1
 SL(1 − A) − σT 4
 
 T= c

 where γ is the daisy mortality rate; γP is a devastation/recovery rate that defines the
 planet fertile area reduction/increase, depending on the parameter sign; γG establishes
 the influence of Gaia state on planet fertile area, which means that the increase in variable
 G reduces thefertile area; and ε P connects the planet devastation with Gaia state; in
 . . .
 addition, sign P = P/ P . It should be pointed out that Gaia state, G, affects the daisy
 populations (W and B) reducing planet fertile area (P). The governing equations are coupled
 by temperature that has an influence on all populations; the main aspects of these couplings
 can be understood with the definitions presented in the sequel.
 The variable αg is the fertile fractional area coverage of the planet, being represented by

 αg = P − W − B (2)

 where 0 ≤ P ≤ 1. Note that (1 − P) represents a desert area that is not proper for life.
 Therefore, the total covered planet area is given by:

 αtg = 1 − W − B (3)

 The mean planetary albedo, A, can be estimated from the individual albedo of each
 daisy population (aW and a B ), the ground albedo (a g ), and the greenhouse gas albedo (a H ),
 which is weighted by the term associated with the amount of greenhouse gases, κH:

 A = WaW + Ba B + αtg a g − κHa H (4)

 The function β defines the optimum environment, being usually assumed to be a
 symmetric single-peak function. Here, a Gaussian function is assumed as follows:
 1 2
 β( TW ) = β 0 e− k (TW −Topt ) (5)
 1 2
 β( TB ) = β 0 e− k (TB −Topt ) (6)
 where β 0 is a parameter related to environmental characteristics, Topt is the optimal tem-
 perature for life, usually considered as Topt = 295 K; k is the variance of the curve and
 usually it is assumed to be k = 35 K which means that temperature life conditions can vary
 from 277.5 K to 312.5 K. Figure 1 presents the optimum environment function β = β( T )
 considering two environmental characteristics represented by β 0 = 1 and 2.
 The local temperature of each population is given by:
 4 = q( A − a ) + T 4
 TW W
 TB4 = q( A − a B
) + T 4 (7)
 Tg4 = q A − a g + T 4

 where q is a constant used to calculate local temperature as a function of albedo. The
 temperature calculation considers L as the solar luminosity and S as the solar constant that

Fractal Fract. 2023, 7, x FOR PEER REVIEW 5 of 14

Fractal Fract. 2023, 7, 190 5 of 14

Fractal Fract. 2023, 7, x FOR PEER REVIEW 5 of 14
 establishes the average solar energy represented by SL; σ is the Stefan–Boltzmann constant;
 c is a measure of the average heat capacity or thermal inertia of the planet.

 Figure 1. Optimum environnent function = ( ).

 The local temperature of each population is given by:
 = ( − ) + 

 = ( − ) + (7)

 = − + 
 whereenvironnent
 q is a constant used βto=calculate
 ( T ). . local temperature as a function of albedo. The
 Figure 1.
 Figure Optimum
 1. Optimum function
 environnent function = β ( )
 temperature calculation considers L as the solar luminosity and S as the solar constant that
 Finally, establishesΞ the average
 = Ξof( Geach solartoenergy represented
 Gaiabystate
 SL; changes.
 σ is the Stefan–Boltzmann
 The localfunction
 temperature
 constant;
 ) is used
 c is a measure
 represent
 ofpopulation
 the averageis
 theby:
 given
 heat capacity
 The definition
 or thermal inertia of the planet.
 of this function is inspired by phase transformations that presents hysteretic behavior [40].
 Finally, function Ξ = =Ξ( ) is used to 
 of)harmonyrepresent the Gaia state changes. The definition
 On this basis, Gaia state varies from ( −
 a state + Ξ = 0, to a critical reaction state,
 of this function is inspired by phase transformations that presents hysteretic behavior [40].
 represented by Ξ = 1. Therefore, the following function
 On this basis, Gaia state varies from a state of harmony is proposed,
 Ξ = 0, to a critical reaction state,
 = ( − ) + (7)
 represented by Ξ = 1. Therefore, the following function is proposed,
 Ξ = 1 − exp[−µG + Gs ] + Ξ0 (8)
 = Ξ−= 1 −+exp[− + ]+Ξ (8)
 Note that, Ξ = 0 until = GS /µ, where µ is the parameter that defines the transforma-
 where q is Note that, Ξ
 a constant to=condition
 0 until =local / temperature
 , where μ isasthe parameterofthat defines the
 tion that starts at GS ; Ξused
 0 is the
 transformation that
 calculate
 starts at 
 at ;the
 Ξ
 beginning
 is the
 of theaat
 condition
 function
 state
 the beginning
 albedo.
 transformation.of the
 The
 The
 state
 temperature calculation considers L as the solar luminosity and S as the solar 2 ln(constant
 10) that
 transformation is completed
 transformation. when
 The Ξ = 1 or, considering
 transformation is completed Ξ when Ξ =G
 = 0.99, 1 for,
 = considering+ GµsΞ. = 0.99,
 establishes the average ( ) solar energy represented by SL; σ is the Stefan–Boltzmann
 µ

 constant; c is a measure
 The reverse=transformation
 + . on the Gaia state is represented by the following equation
 of the average heat capacity or thermal inertia of the planet.
 The reverse
 Finally, function transformation
 Ξ = Ξ( ) isΞused on the Gaia state is represented by the following equation
 = Ξto represent
 0 exp[− µG − Gs ]
 the Gaia state changes. The definition (9)
 of this function is inspired by phase transformations Ξ = Ξ expthat [− − ] hysteretic behavior [40].(9)
 presents
 On this
 Now, Ξ =
 basis, Gaia state G
 1Now,
 until varies
 = Gfrom /µ, aandstatethe harmony Ξ = 0,istocompleted
 of transformation when Ξstate,
 a critical reaction = 0
 Ξ = 1 untilS = / , and the transformation is completed when Ξ = 0 or,
 represented by Ξ = 1. Therefore,
 or, considering Ξ = 0.01, G = the
 2 ln(following
 10) ( G)s function is proposed,
 − . Figure 2 shows the phase transformation
 R , µ=
 considering Ξ = f0.01 µ − . Figure 2 shows the phase transformation
 function where it is observed
 function Ξ = 1 −aexp[− 
 where it isaobserved
 hysteretic ]+Ξ
 +characteristic.
 characteristic.
 hysteretic (8)

 Note that, Ξ = 0 until = / , where μ is the parameter that defines the
 transformation that starts at ; Ξ is the condition at the beginning of the state
 transformation. The transformation is completed when Ξ = 1 or, considering Ξ = 0.99,
 ( )
 = + .
 The reverse transformation on the Gaia state is represented by the following equation
 Ξ = Ξ exp [− − ] (9)

 Now, Ξ = 1 until = / , and the transformation is completed when Ξ = 0 or,
 ( )
 considering Ξ = 0.01 , = − . Figure 2 shows the phase transformation
 Figure 2. Gaia state transformation represented by function Ξ = Ξ( G ).
 function where it is observed a hysteretic characteristic.
 3. Numerical Simulations
 Numerical simulations are carried out considering the fourth order Runge–Kutta
 method. Initially, the classical Daisyworld is treated and afterward, the effect of the revenge
 of Gaia and greenhouse effect are of concern. The classical Daisyworld parameters are
 presented in Table 1. The other parameters related to the Gaia revenge and greenhouse
 effect are described in Table 2. Parameters can vary to characterize different scenarios, and
 such changes are noted when necessary.

Fractal Fract. 2023, 7, 190 6 of 14

 Table 1. Classical Daisyworld parameters.

 Description Parameter Value
 Optimum temperature Topt (K) 295
 Temperature range k (K) 308
 White daisy albedo aW 0.75
 Black daisy albedo aB 0.25
 Planetary albedo ag 0.50
 Daisy mortality rate γ  0.3
 Definition of the local temperature from albedo q K−4 2.06 × 109
  
 Stefan–Boltzmann constant σ K−4 5.67 × 10−8
 Solar constant S  915
 Thermal inertia of the planet c K−4 950

 Table 2. Gaia revenge parameters.

 Description Parameter Value
 Devastation rate γP −3 × 10−4
 Gaia scenario effect on the fertile area γG 2 × 10−4
 Devastation on Gaia scenario εP 5 × 10−3
 Phase transformation condition µ 1
 Start of the phase transformation GS 0.4
 Greenhouse gas absorption ϕH 0.5
 Influence of greenhouse gases on albedo κ 0.1
 Greenhouse gas albedo aH 0.25

 Classical Daisyworld simulations are of concern in order to show planet self-regulation.
 In this regard, it is assumed that parameters of Table 2 vanish (γP = γG = ε P = µ =
 ϕ H = h(t) = 0) and thermal inertia is neglected, which means that c = 0. The solar
 luminosity is assumed to have a linear increase, varying from 0.8 to 1.8 and ambient
 condition is represented by β 0 = 1. Figure 3 presents the evolution of daisy populations
 and temperature showing that daisy population interactions define a temperature that
 tends to be constant close to the optimum temperature, the best condition for life. It is also
 Fractal Fract. 2023,noticeable
 7, x FOR PEER that the increase in the luminosity reaches a critical value that causes planet
 REVIEW 7 of 14
 death, a situation where there is not life anymore.

 (a) (b)
 Figure 3. Life–environment
 Figure 3. Life–environment interactions
 interactions of the classical
 of the classical Daisyworld
 Daisyworld neglecting thermal
 neglecting thermal inertia: (a)
 inertia:
 daisy populations; (b) temperature.
 (a) daisy populations; (b) temperature.
 The effect of thermal inertia on the classical Daisyworld is presented in Figure 4. It
 The effect of thermal
 shows inertia
 the evolution on the
 of daisy classical
 populations and Daisyworld
 temperature andis presented in Figuresub-
 the daisy population 4.
 It shows the space.
 evolution of daisy populations and temperature and the daisy
 Thermal inertia causes oscillations of the daisy populations and temperaturepopulation
 subspace. Thermal inertia
 around the steady causes oscillations
 state solution of the daisy
 of the previous populations
 case. Figure 5 showsand temperature
 the same case show-
 ing a different ambient condition represented by = 2.
 around the steady state solution of the previous case. Figure 5 shows the same Similar qualitative results are
 case
 predicted, but it should be pointed out that, under these conditions, planet
 showing a different ambient condition represented by β 0 = 2. Similar qualitative results death is post-
 poned. From now on, all results are simulated considering an ambient condition repre-
 sented by = 2.

(a) (b)
 Figure 3. Life–environment interactions of the classical Daisyworld neglecting thermal inertia: (a)
 daisy populations; (b) temperature.

 The effect of thermal inertia on the classical Daisyworld is presented in Figure 4. It
Fractal Fract. 2023, 7, 190 7 of 14
 shows the evolution of daisy populations and temperature and the daisy population sub-
 space. Thermal inertia causes oscillations of the daisy populations and temperature
 around the steady state solution of the previous case. Figure 5 shows the same case show-
 are predicted,ingbut
 a different
 it shouldambient condition
 be pointed outrepresented
 that, under = 2.conditions,
 bythese Similar qualitative
 planet results
 death are
 is
 predicted, but it should be pointed out that, under these conditions, planet death is post-
 postponed. From now on, all results are simulated considering an ambient condition
 poned. From now on, all results are simulated considering an ambient condition repre-
 represented by β 0 =by2. = 2.
 sented

 (a) (b)

 (c)
 Figure 4. Life–environment interactions of the classical Daisyworld considering thermal inertia: (a)
 Fractal Fract. 2023,Figure
 7, x FOR4. Life–environment
 PEER REVIEW interactions of the classical Daisyworld considering thermal inertia: 8 of 14
 daisy populations; (b) temperature; (c) daisy population subspace.
 (a) daisy populations; (b) temperature; (c) daisy population subspace.

 (a) (b)

 (c)
 Figure 5. Life–environment interactions of the classical Daisyworld considering thermal inertia and
 Figure 5. Life–environment interactions of the classical Daisyworld considering thermal inertia and
 = 2: (a) daisy populations; (b) temperature; (c) daisy population subspace.
 β 0 = 2: (a) daisy populations; (b) temperature; (c) daisy population subspace.
 The effect of greenhouse gases is now of concern assuming a linear gas emission,
 The effect of greenhouse
 varying gases 6ispresents
 from 0 to 3. Figure now oftheconcern
 evolutionassuming a linear gas
 of daisy populations emission,
 and temperature
 varying from and
 0 to shows
 3. Figure 6 presents the evolution of daisy populations and temperature
 the daisy subspace also. Note that greenhouse effect anticipates planet death
 and shows the daisy the
 because subspace also.
 greenhouse Notereduces
 albedo that greenhouse effect anticipates
 the daisy populations. Therefore,planet death
 the greenhouse
 effect is similar
 because the greenhouse to thereduces
 albedo increasethe
 in black
 daisydaisy population,Therefore,
 populations. but they arethe
 notgreenhouse
 interacting to
 induce planet self-regulation. In other words, greenhouse effect influences planet har-
 mony, being associated with global warming when the gas emission is greater than the
 capacity of the planet to neutralize it.

(c)
 Figure 5. Life–environment interactions of the classical Daisyworld considering thermal inertia
 = 2: (a) daisy populations; (b) temperature; (c) daisy population subspace.

Fractal Fract. 2023, 7, 190 The effect of greenhouse gases is now of concern assuming a linear gas emiss
 8 of 14
 varying from 0 to 3. Figure 6 presents the evolution of daisy populations and temperat
 and shows the daisy subspace also. Note that greenhouse effect anticipates planet de
 because the greenhouse albedo reduces the daisy populations. Therefore, the greenho
 effect is similar to the increase in black daisy population, but they are not interacting to
 effect is similar to the increase in black daisy population, but they are not interactin
 induce planet self-regulation. In other words, greenhouse effect influences planet harmony,
 induce planet self-regulation. In other words, greenhouse effect influences planet h
 being associatedmony,
 with global
 being warming
 associatedwhen the gaswarming
 with global emissionwhen
 is greater than
 the gas the capacity
 emission is greater than
 of the planet to neutralize it.
 capacity of the planet to neutralize it.

 Fractal Fract. 2023, 7, x FOR PEER REVIEW 9 o

 (a) (b)

 (c)
 Figure 6. Life–environment
 Figure 6. Life–environment interactions
 interactions of the of the classical
 classical Daisyworld Daisyworld
 considering considering
 greenhouse greenhouse g
 gases
 and = 2: (a) daisy populations; (b) temperature; (c) daisy population subspace.
 and β 0 = 2: (a) daisy populations; (b) temperature; (c) daisy population subspace.

 The effect of theThe effectof
 revenge of Gaia
 the revenge
 is now of
 in Gaia
 focusisby
 now in focus byluminosity
 considering consideringvarying
 luminosity vary
 from 0.8 to 1.4, = 2.
 from 0.8 to 1.4, β 0 = 2. The greenhouse effect is neglected to mainly focus on the focus
 The greenhouse effect is neglected to mainly Gaia on the G
 revenge phenomenon. Figure 7 presents the Daisyworld behavior showing the statethe state v
 revenge phenomenon. Figure 7 presents the Daisyworld behavior showing
 ables represented
 variables represented by the of
 by the evolution evolution of daisy populations,
 daisy populations, temperature,
 temperature, planet fertile a
 planet fertile
 and Gaia state. Gaia transformation variable Ξ and daisy population subspace are a
 area and Gaia state. Gaia transformation variable Ξ and daisy population subspace are
 presented. The Gaia reaction promotes a reduction in the planet fertile area, which redu
 also presented. The Gaia reaction promotes a reduction in the planet fertile area, which
 the daisy populations and, due to that, temperature dramatically increases. It is also
 reduces the daisy populations and, due to that, temperature dramatically increases. It is
 ticeable that the planet becomes a dead planet at the end of this scenario because =
 also noticeable that the planet becomes a dead planet at the end of this scenario because
 Similar behavior is observed when greenhouse gases are considered, anticipating pla
 P = 0. Similar behavior is observed when greenhouse gases are considered, anticipating
 death even before the critical point where = 0.
 planet death even before the critical point where P = 0.
 The reversibility of the revenge of Gaia is evaluated with a devastation reversion
 represented by positive values of the parameter γP . This is expressed by considering
 that γP = −3 × 10−4 for t < 900, becoming γP = +2 × 10−4 for t > 1000, with a linear
 transition between both values. Figure 8 presents the Daisyworld behavior showing the
 state variables. Note that there is a time instant that planet fertile area starts to increase
 again, opening space for the growth of daisy populations and therefore, the temperature
 stabilization around optimum values. This behavior is associated with the reduction of
 the Gaia reaction, which means a recovery. Nevertheless, there is a critical point where,
 after that, the reversibility is not possible anymore. Figure 9 shows a scenario where the
 devastation reversion starts for t > 1000 (instead of 900 of the previous simulation) showing
 that the reversion is not
 (a)possible, and planet death occurs although(b) the fertile area does not
 vanish yet. This result shows the nonlinear sensitivity of this complex system.

ables represented by the evolution of daisy populations, temperature, planet fertile ar
 and Gaia state. Gaia transformation variable Ξ and daisy population subspace are al
 presented. The Gaia reaction promotes a reduction in the planet fertile area, which reduc
 the daisy populations and, due to that, temperature dramatically increases. It is also n
 ticeable that the planet becomes a dead planet at the end of this scenario because =
Fractal Fract. 2023, 7, 190 9 of 14
 Similar behavior is observed when greenhouse gases are considered, anticipating plan
 death even before the critical point where = 0.

 (a) (b)

 Fractal Fract. 2023, 7, x FOR PEER REVIEW 10 of 1

 (c) (d)

 (e) (f)
 Figure 7. Life–environment
 Figure 7. Life–environment interactions
 interactions in in the Daisyworld
 the Daisyworld associated associated with the of
 with the revenge revenge
 Gaia: of Gaia: (
 daisy populations; (b) temperature; (c) planet fertile area; (d) Gaia state; (e) Gaia phase transfo
 (a) daisy populations; (b) temperature; (c) planet fertile area; (d) Gaia state; (e) Gaia phase transfor-
 mation; (f) daisy population subspace.
 mation; (f) daisy population subspace.
 The reversibility of the revenge of Gaia is evaluated with a devastation reversion rep
 Although the previous simulations present a general idea about the reversibility of the
 resented by positive values of the parameter . This is expressed by considering tha
 revenge of Gaia, the hysteretic characteristic of the Gaia state transformation introduces a
 = −3 × 10 for < 900, becoming = +2 × 10 for > 1000, with a linear tran
 more complex behavior associated with this issue. In this regard, a scenario of destruction
 sition between both values. Figure 8 presents the Daisyworld behavior showing the stat
 is evaluated by examining a period of devastation followed by a recovery phase, and then
 variables. Note that there is a time instant that planet fertile area starts to increase again
 another episode of destruction (Figure 10). In order to focus exclusively on the reversibility
 opening space for the growth of daisy populations and therefore, the temperature stabil
 effect, luminosity is assumed to be constant L = 1 and greenhouse effect is neglected.
 zation around optimum values. This behavior is associated with the reduction of the Gai
 Figure 12 presents state variable evolution showing that planet fertile area does not return
 reaction, which means a recovery. Nevertheless, there is a critical point where, after tha
 to the originalthe
 state after the revenge
 reversibility of Gaia,
 is not possible even though
 anymore. Figure 9the recovery
 shows is similar
 a scenario wheretothe
 the
 devastatio
 devastation. Therefore, hysteretic characteristics introduce difficulties for the reversion
 reversion starts for > 1000 (instead of 900 of the previous simulation) showing that th
 of the revengereversion
 of Gaia iseffects. At theand
 not possible, end, the death
 planet planetoccurs
 deathalthough
 is reached due to
 the fertile thedoes
 area lastnot vanis
 destruction phase.
 yet. This result shows the nonlinear sensitivity of this complex system.

opening space for the growth of daisy populations and therefore, the temperature stab
 zation around optimum values. This behavior is associated with the reduction of the Ga
 reaction, which means a recovery. Nevertheless, there is a critical point where, after th
 the reversibility is not possible anymore. Figure 9 shows a scenario where the devastati
 reversion starts for > 1000 (instead of 900 of the previous simulation) showing that t
Fractal Fract. 2023, 7, 190 10 of 14
 reversion is not possible, and planet death occurs although the fertile area does not vani
 yet. This result shows the nonlinear sensitivity of this complex system.

 (a) (b)

 Fractal Fract. 2023, 7, x FOR PEER REVIEW 11 of
 Fractal Fract. 2023, 7, x FOR PEER REVIEW 11 of

 (c) (d)

 (e) (f)
 (e) (f)
 Figure 8. Life–environment
 Figure 8. Life–environment interactions
 interactions in the Daisyworldin the Daisyworld
 showing showing the of
 the reversibility reversibility
 the revenge of the reven
 Figure
 of 8. (a)
 Gaia: Life–environment
 daisy interactions
 populations; (b) in the Daisyworld
 temperature; (c) planet showing
 fertile the(d)
 area; reversibility
 Gaia state;of
 (e)the reveng
 Gaia pha
 of Gaia: (a) daisy
 ofpopulations;
 Gaia: (a) (b)
 daisy temperature;
 populations; (b) (c) planet fertile
 temperature; (c) area; (d)
 planet Gaiaarea;
 fertile state;
 (d)(e) Gaia
 Gaia phase
 state; (e) Gaia phas
 transformation; (f) daisy population subspace.
 transformation; transformation;
 (f) daisy population subspace.
 (f) daisy population subspace.

 (a) (b)
 (a) (b)
 Figure 9. Cont.

 (c) (d)
 (c) (d)

Fractal Fract. 2023, 7, 190 11 of 14

 (a) (b)

 (c) (d)

Fractal Fract. 2023, 7, x FOR PEER REVIEW 12 of 14

 Fractal Fract. 2023, 7, x FOR PEER REVIEW 12 of

 Although the previous simulations present a general idea about the reversibility of
 the revenge of Gaia, the hysteretic
 Although characteristic
 the previous simulationsof the Gaia astate
 present transformation
 general idea about the intro-
 reversibility
 duces a more complex
 the revenge behavior
 of Gaia, associated withcharacteristic
 the hysteretic this issue. Inofthis theregard,
 Gaia state a scenario of
 transformation intr
 destruction is evaluated
 duces a more by complex
 examining a period
 behavior of devastation
 associated with thisfollowed
 issue. In by thisa regard,
 recovery a scenario
 phase, and then destruction is evaluated
 another episode by examining
 of destruction (Figure a period
 10). Inoforder
 devastation
 to focusfollowed
 exclusively by a recover
 phase,
 on the reversibility (e) and then
 effect, another episode
 luminosity is assumed of destruction
 to be constant (f) =
 (Figure 10).1 In
 andorder to focus exclusive
 greenhouse
 effect on the
 is neglected. reversibility
 Figure 11 presents effect, luminosity
 state variable isevolution
 assumedshowing
 to be constant
 that = 1 fertile
 planet and greenhou
 Figure
 Figure 9. Life–environment 9. Life–environment
 interactions in the interactions
 Daisyworld in the Daisyworld
 showing showing
 a situation wherea situation where the revers
 the reversibility
 area effect
 does not return
 bility is neglected.
 oftothe
 the original
 revenge Figure
 state
 of Gaia 11 presents
 is after
 not the state
 revenge
 possible variable
 even with evolution
 ofreversion:
 aGaia, even
 devastation showing
 though the that
 daisyplanet
 (a)recov- ferti
 of the revenge of Gaia is not possible even with a devastation (a) reversion:
 daisy populations; population
 ery area
 is similar to(b)
 the does not
 devastation.
 temperature; return to the
 Therefore,
 (c) planet original state
 fertilehysteretic
 area; (d) Gaiaafter the revenge
 characteristics of Gaia,
 introduce
 situation; (e) Gaia even though
 difficulties
 phase transformation; the reco
 (f) dais
 (b) temperature;ery(c) is
 planet
 similarfertile area;
 to the (d) Gaia situation;
 devastation. Therefore, (e) hysteretic
 Gaia phasecharacteristics
 transformation; (f) daisydifficulti
 introduce
 for the reversionpopulation
 of the subspace.
 revenge of Gaia effects. At the end, the planet death is reached due
 population subspace.
 for the reversion
 to the last destruction phase. of the revenge of Gaia effects. At the end, the planet death is reached du
 to the last destruction phase.

 Figure 10.
 Figure Devastation
 10. Devastation scenario
 Figure scenario
 10. represented
 Devastation byrepresented
 scenario
 represented by parameter γ P .parameter .
 by

 (a) (b)
 (a) (b)
 Figure 11. Cont.

Fractal Fract. 2023, 7, 190 12 of 14

 (a) (b)

 Fractal Fract. 2023, 7, x FOR PEER REVIEW 13 of

 (c) (d)

 (e) (f)
 Figure 11. Irreversibility
 Figure 12. Irreversibility of life-environment
 of life-environment interactionsinteractions in the Daisyworld
 in the Daisyworld due thedue the revenge
 revenge of of Gai
 (a) daisy populations; (b) temperature; (c) planet fertile area; (d) Gaia situation; (e) Gaia phase tran
 Gaia: (a) daisy populations; (b) temperature; (c) planet fertile area; (d) Gaia situation; (e) Gaia phase
 formation; (f) daisy population subspace.
 transformation; (f) daisy population subspace.

 4. Conclusions4. Conclusions
 This paper presents a novel Daisyworld model to describe the revenge of Gaia an
 This paper presents a novel Daisyworld model to describe the revenge of Gaia and the
 the greenhouse effect. Besides black daisies, white daisies and temperature, the mod
 greenhouse effect. Besides black daisies, white daisies and temperature, the model incor-
 incorporates the planet's fertile area, the Gaia state condition associated with life–environ
 porates the planet’s fertile area, the Gaia state condition associated with life–environment
 ment harmony and the greenhouse effect. Numerical simulations are carried out employ
 harmony and the greenhouse effect. Numerical simulations are carried out employing the
 ing the fourth order Runge–Kutta method and a nonlinear dynamics perspective. Resul
 fourth order Runge–Kutta method and a nonlinear dynamics perspective. Results show
 show scenarios where the revenge of Gaia reduces life through the reduction in th
 scenarios where the revenge of Gaia reduces life through the reduction in the planet’s fertile
 planet's fertile area. The greenhouse effect also reduces the self-regulation capacit
 area. The greenhouse effect also reduces the self-regulation capacity, changing the planetary
 changing the planetary albedo. In other words, greenhouse effect influences planet ha
 albedo. In other words,
 mony, being greenhouse
 associated effect
 withinfluences planet harmony,
 global warming when the being associated
 gas emission with than th
 is greater
 global warming when the gas emission is greater than the planet’s capacity to neutralize
 planet's capacity to neutralize it. The reversibility of the Gaia revenge is possible but ther
 it. The reversibility of thepoint
 is a turning Gaiawhere,
 revenge is that,
 after possible but thereisisnot
 the reversion a turning
 possible,point where,
 showing the nonlinea
 after that, the reversion
 sensitivityisofnot
 thispossible,
 complex showing the nonlinear
 system. Planet death cansensitivity
 occur due to ofGaia
 this reaction
 complexeven whe
 system. Planetthedeath canarea
 fertile occur
 stilldue to Gaia
 exists. reaction
 Besides, even when
 hysteretic the fertile
 characteristics area stilldifficulties
 introduce exists. for th
 Besides, hysteretic characteristics introduce difficulties for the reversion of the
 reversion of the effects of the revenge of Gaia, which make its consequences effects of theeven mor
 revenge of Gaia, which make its consequences even more dramatic. The authors believe
 dramatic. The authors believe that this new parable matches exactly Lovelock’s ideas o
 that this new parable
 the revengematches exactly
 of Gaia, Lovelock’s
 and the ideas of theof
 critical consequences revenge of Gaia, and the
 the anthropomorphic action on th
 critical consequences of the anthropomorphic action on the planet. Furthermore,
 planet. Furthermore, it shows the greenhouse effect in the planet’s thermal it shows balance, als
 the greenhousecollaborating
 effect in the toplanet’s thermal balance,
 the anticipation also collaborating
 of the planet's death. to the anticipation
 of the planet’s death.
 Author Contributions: Conceptualization, M.A.S.; methodology, M.A.S.; software, M.A.S. and F.M.V
 validation,
 Author Contributions: M.A.S. and F.M.V.;
 Conceptualization, formalmethodology,
 M.A.S.; analysis, M.A.S.; investigation,
 M.A.S.; software,M.A.S.;
 M.A.S. resources,
 and F.M.V.;M.A.S.; wr
 ing—original
 validation, M.A.S. draft
 and F.M.V.; preparation,
 formal M.A.S.
 analysis, and F.M.V.;
 M.A.S.; writing—review
 investigation, M.A.S.;and editing, M.A.S.;
 resources, M.A.S.;visualizatio
 writing—original M.A.S.
 draftand F.M.V.; supervision,
 preparation, M.A.S. and M.A.S.; project
 F.M.V.; administration,
 writing—review andM.A.S.; funding
 editing, acquisition,
 M.A.S.; visual- M.A.S.. A
 authors
 ization, M.A.S. and havesupervision,
 F.M.V.; read and agreed to theproject
 M.A.S.; published version of theM.A.S.;
 administration, manuscript.
 funding acquisition,
 M.A.S. All authors have read
 Funding: The and agreed
 authors to the
 would likepublished versionthe
 to acknowledge of support
 the manuscript.
 of the Brazilian Research Agenci
 CNPq, CAPES and FAPERJ.
 Funding: The authors would like to acknowledge the support of the Brazilian Research Agencies
 CNPq, CAPES andData Availability Statement: The datasets are available from the corresponding author on reaso
 FAPERJ.
 able request.
 Conflicts of Interest: The authors declare no conflict of interest.

 References
 1. Lovelock, J.E. Gaia as seen through the atmosphere. In Biomineralization and Biological Metal Accumulation; Westbroek, P., d Jon
 E.E., Eds.; D. Reidei Publishing Company: Dordrecht, The Netherlands, 1983; pp. 15–25.

Fractal Fract. 2023, 7, 190 13 of 14

 Data Availability Statement: The datasets are available from the corresponding author on
 reasonable request.
 Conflicts of Interest: The authors declare no conflict of interest.

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