Deforestation and world population sustainability: a quantitative analysis
aa r X i v : . [ q - b i o . P E ] J un Deforestation and world population sustainability: aquantitative analysis
Mauro Bologna and Gerardo Aquino Departamento de Ingenier´ıa El ´ectrica-Electr ´onica, Universidad de Tarapac ´a The Alan Turing Institute, London, UK University of Surrey, Guildford, UK. Goldsmiths, University of London, UK * [email protected] + these authors contributed equally to this work ABSTRACT
In this paper we afford a quantitative analysis of the sustainability of current world population growth in relation to the paralleldeforestation process adopting a statistical point of view. We consider a simplified model based on a stochastic growthprocess driven by a continuous time random walk, which depicts the technological evolution of human kind, in conjunctionwith a deterministic generalised logistic model for humans-forest interaction and we evaluate the probability of avoiding theself-destruction of our civilisation. Based on the current resource consumption rates and best estimate of technological rategrowth our study shows that we have very low probability, less than in most optimistic estimate, to survive without facinga catastrophic collapse.
Introduction
In the last few decades, the debate on climate change has assumed global importance with consequences on national andglobal policies. Many factors due to human activity are considered as possible responsible of the observed changes: amongthese water and air contamination (mostly greenhouse effect) and deforestation are the mostly cited. While the extent ofhuman contribution to the greenhouse effect and temperature changes is still a matter of discussion, the deforestation is anundeniable fact. Indeed before the development of human civilisations, our planet was covered by 60 million square kilometresof forest . As a result of deforestation, less than 40 million square kilometres currently remain . In this paper, we focus onthe consequence of indiscriminate deforestation.Trees’ services to our planet range from carbon storage, oxygen production to soil conservation and water cycle regula-tion. They support natural and human food systems and provide homes for countless species, including us, through buildingmaterials. Trees and forests are our best atmosphere cleaners and, due to the key role they play in the terrestrial ecosystem, itis highly unlikely to imagine the survival of many species, including ours, on Earth without them. In this sense, the debate onclimate change will be almost obsolete in case of a global deforestation of the planet.Starting from this almost obvious observation, we investigate the problem of the survival of humanity from a statisticalpoint of view. We model the interaction between forests and humans based on a deterministic logistic-like dynamics, while weassume a stochastic model for the technological development of the human civilisation. The former model has already beenapplied in similar contexts while the latter is based on data and model of global energy consumption used as a proxy forthe technological development of a society. This gives solidity to our discussion and we show that, keeping the current rateof deforestation, statistically the probability to survive without facing a catastrophic collapse, is very low. We connect suchprobability to survive to the capability of humankind to spread and exploit the resources of the full solar system. Accordingto Kardashev scale , which measures a civilisation’s level of technological advancement based on the amount of energy theyare able to use, in order to spread through the solar system we need to be able to harness the energy radiated by the Sun at arate of ≈ × Watt. Our current energy consumption rate is estimated in ≈ Watt . As showed in the subsections"Statistical Model of technological development" and "Numerical results" of the following section, a successful outcome hasa well defined threshold and we conclude that the probability of avoiding a catastrophic collapse is very low, less than 10% inthe most optimistic estimate. odel and Results Deforestation
The deforestation of the planet is a fact . Between 2000 and 2012, 2.3 million Km of forests around the world were cutdown which amounts to 2 × Km per year. At this rate all the forests would disappear approximatively in 100 − . We therefore use the model introduced in that reference to roughly describe the humans-forest interaction.Admittedly, we are not aiming here for an exact exhaustive model. It is probably impossible to build such a model. Whatwe propose and illustrate in the following sections, is a simplified model which nonetheless allows us to extrapolate the timescales of the processes involved: i.e. the deterministic process describing human population and resource (forest) consumptionand the stochastic process defining the economic and technological growth of societies. Adopting the model in (see also )we have for the humans-forest dynamics ddt N ( t ) = rN ( t ) (cid:20) − N ( t ) β R ( t ) (cid:21) , (1) ddt R ( t ) = r ′ R ( t ) (cid:20) − R ( t ) R c (cid:21) − a N ( t ) R ( t ) . (2)where N represent the world population and R the Earth surface covered by forest. β is a positive constant related to thecarrying capacity of the planet for human population, r is the growth rate for humans (estimated as r ∼ . − ) , a may be identified as the technological parameter measuring the rate at which humans can extract the resources from theenvironment, as a consequence of their reached technological level. r ′ is the renewability parameter representing the capabilityof the resources to regenerate, (estimated as r ′ ∼ .
001 years − ) , R c the resources carrying capacity that in our case may beidentified with the initial 60 million square kilometres of forest.A closer look at this simplified model and at the analogy with Easter Island on which is based, shows nonetheless, strongsimilarities with our current situation. Like the old inhabitants of Easter Island we too, at least for few more decades, cannotleave the planet. The consumption of the natural resources, in particular the forests, is in competition with our technologicallevel. Higher technological level leads to growing population and higher forest consumption (larger a ) but also to a moreeffective use of resources. With higher technological level we can in principle develop technical solutions to avoid/preventthe ecological collapse of our planet or, as last chance, to rebuild a civilisation in the extraterrestrial space ( see section onthe Fermi paradox). The dynamics of our model for humans-forest interaction in Eqs. (1 , 2), is typically characterised bya growing human population until a maximum is reached after which a rapid disastrous collapse in population occurs beforeeventually reaching a low population steady state or total extinction. We will use this maximum as a reference for reaching adisastrous condition. We call this point in time the "no-return point" because if the deforestation rate is not changed beforethis time the human population will not be able to sustain itself and a disastrous collapse or even extinction will occur. As afirst approximation , since the capability of the resources to regenerate, r ′ , is an order of magnitude smaller than the growingrate for humans, r , we may neglect the first term in the right hand-side of Eq. (2). Therefore, working in a regime of theexploitation of the resources governed essentially by the deforestation, from Eq. (2) we can derive the rate of tree extinctionas 1 R dRdt ≈ − a N . (3)The actual population of the Earth is N ∼ . × inhabitants with a maximum carrying capacity estimated of N c ∼ inhabitants. The forest carrying capacity may be taken as R c ∼ × Km while the actual surface of forest is R . × Km . Assuming that β is constant, we may estimate this parameter evaluating the equality N c ( t ) = β R ( t ) at the time when theforests were intact. Here N c ( t ) is the instantaneous human carrying capacity given by Eq. (1). We obtain β ∼ N c / R c ∼ β using actual data of the population growth and inserting it in Eq. (1). In this case weobtain a range 700 . β .
900 that gives a slightly favourable scenario for the human kind (see below and Fig. 4). We stressanyway that this second scenario depends on many factors not least the fact that the period examined in is relatively short.On the contrary β ∼
170 is based on the accepted value for the maximum human carrying capacity. With respect to the value f parameter a , adopting the data relative to years 2000-2012 of Ref. , we have1 R ∆ R ∆ t ≈ × . × ≈ − a N ⇒ a ∼ − years − (4)The time evolution of system (1)-(2) is plotted in Figs. 1 and 2. We note that in Fig. 1 the numerical value of the maximum ofthe function N ( t ) is N M ∼ estimated as the carrying capacity for the Earth population . Again we have to stress that it isunrealistic to think that the decline of the population in a situation of strong environmental degradation would be a non-chaoticand well-ordered decline, that is also way we take the maximum in population and the time at which occurs as the point ofreference for the occurrence of an irreversible catastrophic collapse, namely a ’no-return’ point. t (cid:215) ✾ (cid:215) ✾ (cid:215) ✾ (cid:215) ✾ (cid:215) N ( t ) t × × (cid:0) × (cid:1) × (cid:2) R ( t ) Figure 1.
On the left: plot of the solution of Eq. (1) with the initial condition N = × at initial time t = R = × . Here β =
700 and a = − t (cid:215) ✾ (cid:215) ✾ (cid:215) ✾ (cid:215) ✾ (cid:215) ✾ (cid:215) ✾ N ( t ) t × × × × R ( t ) Figure 2.
On the left: plot of the solution of Eq. (1) with the initial condition N = × at initial time t = R = × . Here β =
170 and a = − Statistical Model of technological development
According to Kardashev scale , in order to be able to spread through the solar system, a civilisation must be capable to builda Dyson sphere , i.e. a maximal technological exploitation of most the energy from its local star, which in the case of theEarth with the Sun would correspond to an energy consumption of E D ≈ × Watts, we call this value Dyson limit. Ouractual energy consumption is estimated in E c ≈ Watts (Statistical Review of World Energy source) . To describe ourtechnological evolution, we may roughly schematise the development as a dichotomous random process ddt T = α T ξ ( t ) . (5)where T is the level of technological development of human civilisation that we can also identify with the energy consumption. α is a constant parameter describing the technological growth rate (i.e. of T ) and ξ ( t ) a random variable with values 0 ,
1. We onsider therefore, based on data of global energy consumption an exponential growth with fluctuations mainly reflectingchanges in global economy. We therefore consider a modulated exponential growth process where the fluctuations in thegrowth rate are captured by the variable ξ ( t ) . This variable switches between values 0 , ψ ( t ) . When ξ ( t ) = ξ switches to ξ ( t ) =
1. If we consider T more strictly as describing the technological development, ξ ( t ) reflects the fact that investments in research can haveinterruptions as a consequence of alternation of periods of economic growth and crisis. With the following transformation, W = log (cid:18) TT (cid:19) / α − h ξ i t , (6)differentiating both sides respect to t and using Eq. (5), we obtain for the transformed variable Wddt W = ¯ ξ ( t ) (7)where ¯ ξ ( t ) = [ ξ ( t ) − h ξ i ] and h ξ i is the average of ξ ( t ) so that ¯ ξ ( t ) takes the values ± P ( W , t ) generatedby a generic waiting time distribution can be found in literature . Knowing the distribution we may evaluate the first passagetime distribution in reaching the necessary level of technology to e.g. live in the extraterrestrial space or develop any otherway to sustain population of the planet. This characteristic time has to be compared with the time that it will take to reachthe no-return point. Knowing the first passage time distribution we will be able to evaluate the probability to survive for ourcivilisation.If the dichotomous process is a Poissonian process with rate γ then the correlation function is an exponential, i.e. h ¯ ξ ( t ) ¯ ξ ( t ′ ) i = exp [ − γ | t − t ′ | ] (8)and Eq. (7) generates for the probability density the well known telegrapher’s equation ∂ ∂ t P ( W , t ) + γ ∂∂ t P ( W , t ) = ∂ ∂ x P ( W , t ) (9)We note that the approach that we are following is based on the assumption that at random times, exponentially distributedwith rate γ , the dichotomous variable ¯ ξ changes its value.With this assumption the solution to Eq. (9) is P ( W , t ) =
12 exp h − γ t i δ ( t − | W | ) + γ I h γ p t − W i + tI h γ √ t − W i √ t − W θ ( t − | W | ) , (10)where I n ( z ) are the modified Bessel function of the first kind. Transforming back to the variable T we have P ( T , t ) = Je − γ t δ ( t − x ) + δ ( x ) + γ I h γ p ( t − x ) x i + tI h γ p ( t − x ) x (cid:17)p x ( t − x ) θ (cid:16) t − x (cid:17) θ ( x ) (11)where for sake of compactness we set x = log ( T / T ) / α , J = dWdT = α T (12)In Laplace transform we haveˆ P ( T , s ) = J (cid:0) γ + s (cid:1) " δ ( x ) + ( γ + s ) ( γ + s ) exp " − sx ( γ + s ) (cid:0) γ + s (cid:1) . (13)
20 40 60 t f T t f T Figure 3. (Left) Comparison between theoretical prediction of Eq. (15) (black curve) and numerical simulation of Eq. (3)(cyan curve) for γ = γ = / ˆ f T ( s ) = ˆ P ( x , s ) ˆ P ( x , s ) = exp " − s ( γ + s )( x − x ) (cid:0) γ + s (cid:1) , x > x . (14)Inverting the Laplace transform we obtain f T ( t ) = exp h − γ t i γ √ x − x I √ x − x q t − x − x γ √ ! √ q t − x − x θ (cid:20) t − x − x (cid:21) + exp h − γ t i δ (cid:18) t − x − x (cid:19) , (15)which is confirmed (see Fig. 3) by numerical simulations. The time average to get the point x for the first time is given by h t i = ∞ Z t f T ( t ) dt = x − x = log ( T / T ) / α − log ( T / T ) / α = α log (cid:18) TT (cid:19) , (16)which interestingly is double the time it would take if a pure exponential growth occurred, depends on the ratio between finaland initial value of T and is independent of γ . We also stress that this result depends on parameters directly related to the stageof development of the considered civilisation, namely the starting value T , that we assume to be the energy consumption E c of the fully industrialised stage of the civilisation evolution and the final value T , that we assume to be the Dyson limit E D , andthe technological growth rate α . For the latter we may, rather optimistically, choose the value α = . (see next section). Using the data above, relative to our planet’s scenario, we obtain the estimate of h t i ≈
180 years.From Figs 1 and 2 we see that the estimate for the no-return time are 130 and 22 years for β =
700 and β =
170 respectively,with the latter being the most realistic value. In either case, these estimates based on average values, being less than 180 years,already portend not a favourable outcome for avoiding a catastrophic collapse. Nonetheless, in order to estimate the actualprobability for avoiding collapse we cannot rely on average values, but we need to evaluate the single trajectories, and countthe ones that manage to reach the Dyson limit before the ’no-return point’. We implement this numerically as explained in thefollowing.
Numerical Results
We run simulations of Eqs. (1), (2) and (5) simultaneously for different values of of parameters a and α for fixed β and wecount the number of trajectories that reach Dyson limit before the population level reaches the "no-return point" after which apid collapse occurs. More precisely, the evolution of T is stochastic due to the dichotomous random process ξ ( t ) , so wegenerate the T ( t ) trajectories and at the same time we follow the evolution of the population and forest density dictated by thedynamics of Eqs. (1), (2) until the latter dynamics reaches the no-return point (maximum in population followed by collapse).When this happens, if the trajectory in T ( t ) has reached the Dyson limit we count it as a success, otherwise as failure. Thisway we determine the probabilities and relative mean times in Figs 5,6 and 7. Adopting a weak sustainability point of viewour model does not specify the technological mechanism by which the successful trajectories are able to find an alternative toforests and avoid collapse, we leave this undefined and link it exclusively and probabilistically to the attainment of the Dysonlimit. It is important to notice that we link the technological growth process described by Eq.(5) to the economic growthand therefore we consider, for both economic and technological growth, a random sequence of growth and stagnation cycles,with mean periods of about 1 and 4 years in accordance with estimates for the driving world economy, i.e. the United Statesaccording to the National Bureau of Economic Research. .In Eqs. (1, 2) we redefine the variables as N ′ = N / R W and R ′ = R / R W with R W ≃ × Km the total continentalarea, and replace parameter a accordingly with a = a × R W = . × − Km ys − . We run simulations accordingly startingfrom values R ′ and N ′ , based respectively on the current forest surface and human population. We take values of a from 10 − to 3 × − Km ys − and for α from 0 .
01 ys − to 4 . − . Results are shown in Figs. 4 and 6. Figure 4 shows a thresholdvalue for the parameter α , the technological growth rate, above which there is a non-zero probability of success. This thresholdvalue increases with the value of the other parameter a . As shown in Figs. 7 this values depends as well on the value of β and higher values of β correspond to a more favourable scenario where the transition to a non-zero probability of successoccurs for smaller α , i.e. for smaller, more accessible values, of technological growth rate. More specifically, left panel ofFig. 4 shows that, for the more realistic value β = α larger than 0.5. Even assuming that the technological growth rate be comparable to thevalue α = log ( ) / = .
345 ys − , given by the Moore Law (corresponding to a doubling in size every two years), therefore,it is unlikely in this regime to avoid reaching the the catastrophic ’no-return point’. When the realistic value of a = . × Km ys − estimated from Eq.(4), is adopted, in fact, a probability less than 10% is obtained for avoiding collapse with a Mooregrowth rate, even when adopting the more optimistic scenario corresponding to β =
700 (black curve in right panel of Fig4). While an α larger than 1 . β =
170 (red curve, samepanel).As far as time scales are concerned, right panel of Fig 5 shows for β =
170 that even in the range α > .
5, corresponding toa non-zero probability of avoiding collapse, collapse is still possible, and when this occurs, the average time to the ’no-returnpoint’ ranges from 20 to 40 years. Left panel in same figure, shows for the same parameters, that in order to avoid catastrophe,our society has to reach the Dyson’s limit in the same average amount of time of 20-40 years.
0 1 2 3 4 0 1 2 3 0 0.5 1 a ( ys -1 ) a ( · -4 ) p suc a (ys) -1 p suc Figure 4. (Left panel) Probability p suc of reaching Dyson value before reaching "no-return" point as function of α and a for β = a is expressed in Km ys − . (Right panel) 2D plot of p suc for a = . × − Km ys − as a functionof α . Red line is p suc for β = p suc for β =
300 and 700 repsectively (seealso Fig 6). Green dashed line indicates the value of α corresponding to Moore’s law.In Fig. 7 we show the dependence of the model on the parameter β for a = . × − .
0 1 2 3 4 0 1 2 3 0 25 50 a ( ys -1 ) a ( · -4 ) t (ys)
0 1 2 3 4 0 1 2 3 0 20 40 a ( ys -1 ) a ( · -4 ) t (ys) Figure 5.
Average time τ (in years) to reach Dyson value before hitting "no-return" point (success, left) and withoutmeeting Dyson value (failure, right) as function of α and a for β = τ ≥
50 correspondsto diverging τ , i.e. Dyson value not being reached before hitting "no-return" point and therefore failure. Plauteau region at τ = a is expressed in Km ys − .
0 1 2 3 4 0 1 2 3 0 0.5 1 a ( ys -1 ) a ( · -4 ) p suc
0 1 2 3 4 0 1 2 3 0 0.5 1 a ( ys -1 ) a ( · -4 ) p suc Figure 6.
Probability p suc of reaching Dyson value before hitting "no-return" point as function of α and a for β =
300 (left)and 700 (right). Parameter a is expressed in Km ys − .
0 1 2 3 4 0 1 2 3 0 0.5 1 a ( ys -1 ) b ( · p suc Figure 7.
Probability of reaching Dyson value p suc before reaching "no-return" point as function of β and α for a = . × − Km ys − . Methods
We run simulations of Eqs. (1), (2) and (5) simultaneously for different values of of parameters a and α depending on β asexplained in Methods and Results to generate Figs. 5,6 and 7. Eqs. (1), (2) are integrated via standard Euler method. Eq. (5)is integrated as well via standard Euler method between the random changes of the variable ξ . The stochastic dichotomousprocess ξ is generated numerically in the following way: using the random number generator from gsl library we generate thetimes intervals between the changes of the dichotomous variable ξ = ,
1, with an exponential distribution(with mean values f 1 and 4 years respectively), we therefore obtain a time series of 0 and 1 for each trajectory. We then integrate Eq. (5) intime using this time series and we average over N = Fermi Paradox
In this section we briefly discuss a few considerations about the so called Fermi paradox that can be drawn from our model. Wemay in fact relate the Fermi paradox to the problem of resource consumption and self destruction of a civilisation. The originof Fermi paradox dates back to a casual conversation about extraterrestrial life that Enrico Fermi had with E. Konopinski, E.Teller and H. York in 1950, during which Fermi asked the famous question: "where is everybody?", since then become epony-mous for the paradox. Starting from the closely related Drake equation , used to estimate the number of extraterrestrialcivilisations in the Milky Way, the debate around this topic has been particularly intense in the past (for a more comprehensivecovering we refer to Hart , Freitas and reference therein). Hart’s conclusion is that there are no other advanced or ’techno-logical’ civilisations in our galaxy as also supported recently by based on a careful reexamination of Drake’s equation. Inother words the terrestrial civilisation should be the only one living in the Milk Way. Such conclusions are still debated, butmany of Hart’s arguments are undoubtedly still valid while some of them need to be rediscussed or updated. For example,there is also the possibility that avoiding communication might actually be an ’intelligent’ choice and a possible explanation ofthe paradox. On several public occasions, in fact, Professor Stephen Hawking suggested human kind should be very cautiousabout making contact with extraterrestrial life. More precisely when questioned about planet Gliese 832c’s potential for alienlife he once said: "One day, we might receive a signal from a planet like this, but we should be wary of answering back". Hu-man history has in fact been punctuated by clashes between different civilisations and cultures which should serve as caveat.From the relatively soft replacement between Neanderthals and Homo Sapiens (Kolodny ) up to the violent confrontationbetween native Americans and Europeans, the historical examples of clashes and extinctions of cultures and civilisations havebeen quite numerous. Looking at human history Hawking’s suggestion appears as a wise warning and we cannot role out thepossibility that extraterrestrial societies are following similar advice coming from their best minds.With the help of new technologies capable of observing extrasolar planetary systems, searching and contacting alien life isbecoming a concrete possibility (see for example Grimaldi for a study on the chance of detecting extraterrestrial intelligence),therefore a discussion on the probability of this occurring is an important opportunity to assess also our current situation asa civilisation. Among Hart’s arguments, the self-destruction hypothesis especially needs to be rediscussed at a deeper level.Self-destruction following environmental degradation is becoming more and more an alarming possibility. While violentevents, such as global war or natural catastrophic events, are of immediate concern to everyone, a relatively slow consumptionof the planetary resources may be not perceived as strongly as a mortal danger for the human civilisation. Modern societiesare in fact driven by Economy, and, without giving here a well detailed definition of "economical society", we may agreethat such a kind of society privileges the interest of its components with less or no concern for the whole ecosystem thathosts them (for more details see for a review on Ecological Economics and its criticisms to mainstream Economics) . Clearexamples of the consequences of this type of societies are the international agreements about Climate Change. The Parisclimate agreement is in fact, just the last example of a weak agreement due to its strong subordination to the economicinterests of the single individual countries. In contraposition to this type of society we may have to redefine a different modelof society, a "cultural society", that in some way privileges the interest of the ecosystem above the individual interest ofits components, but eventually in accordance with the overall communal interest. This consideration suggests a statisticalexplanation of Fermi paradox: even if intelligent life forms were very common (in agreement with the mediocrity principle inone of its version : "there is nothing special about the solar system and the planet Earth") only very few civilisations wouldbe able to reach a sufficient technological level so as to spread in their own solar system before collapsing due to resourceconsumption.We are aware that several objections can be raised against this argument and we discuss below the one that we believe tobe the most important. The main objection is that we do not know anything about extraterrestrial life. Consequently, we donot know the role that a hypothetical intelligence plays in the ecosystem of the planet. For example not necessarily the planetneeds trees (or the equivalent of trees) for its ecosystem. Furthermore the intelligent form of life could be itself the analogousof our trees, so avoiding the problem of the "deforestation" (or its analogous). But if we assume that we are not an exception(mediocrity principle) then independently of the structure of the alien ecosystem, the intelligent life form would exploit everykind of resources, from rocks to organic resources (animal/vegetal/etc), evolving towards a critical situation. Even if we are atthe beginning of the extrasolar planetology, we have strong indications that Earth-like planets have the volume magnitude ofthe order of our planet. In other words, the resources that alien civilisations have at their disposal are, as order of magnitude,the same for all of them, including ourselves. Furthermore the mean time to reach the Dyson limit as derived in Eq. 6 dependsonly on the ratio between final and initial value of T and therefore would be independent of the size of the planet, if we assumeas a proxy for T energy consumption (which scales with the size of the planet), producing a rather general result which can be xtended to other civilisations. Along this line of thinking, if we are an exception in the Universe we have a high probability tocollapse or become extinct, while if we assume the mediocrity principle we are led to conclude that very few civilisations areable to reach a sufficient technological level so as to spread in their own solar system before the consumption of their planet’sresources triggers a catastrophic population collapse. The mediocrity principle has been questioned (see for example Kukla for a critical discussion about it) but on the other hand the idea that the humankind is in some way "special" in the universe hashistorically been challenged several times. Starting with the idea of the Earth at the centre of the universe (geocentrism), thenof the solar system as centre of the universe (Heliocentrism) and finally our galaxy as centre of the universe. All these beliefshave been denied by the facts. Our discussion, being focused on the resource consumption, shows that whether we assumethe mediocrity principle or our "uniqueness" as an intelligent species in the universe, the conclusion does not change. Givinga very broad meaning to the concept of cultural civilisation as a civilisation not strongly ruled by economy, we suggest foravoiding collapse that only civilisations capable of such a switch from an economical society to a sort of "cultural" societyin a timely manner, may survive. This discussion leads us to the conclusion that, even assuming the mediocrity principle, theanswer to "Where is everybody?" could be a lugubrious " (almost) everyone is dead". Conclusions
In conclusion our model shows that a catastrophic collapse in human population, due to resource consumption, is the mostlikely scenario of the dynamical evolution based on current parameters. Adopting a combined deterministic and stochasticmodel we conclude from a statistical point of view that the probability that our civilisation survives itself is less than 10% in themost optimistic scenario. Calculations show that, maintaining the actual rate of population growth and resource consumption,in particular forest consumption, we have a few decades left before an irreversible collapse of our civilisation (see Fig 5).Making the situation even worse, we stress once again that it is unrealistic to think that the decline of the population in asituation of strong environmental degradation would be a non-chaotic and well-ordered decline. This consideration leads to aneven shorter remaining time. Admittedly, in our analysis, we assume parameters such as population growth and deforestationrate in our model as constant. This is a rough approximation which allows us to predict future scenarios based on currentconditions. Nonetheless the resulting mean-times for a catastrophic outcome to occur, which are of the order of 2-4 decades(see Fig. 5) , make this approximation acceptable, as it is hard to imagine, in absence of very strong collective efforts, bigchanges of these parameters to occur in such time scale. This interval of time seems to be out of our reach and incompatiblewith the actual rate of the resource consumption on Earth, although some fluctuations around this trend are possible notonly due to unforeseen effects of climate change but also to desirable human-driven reforestation. This scenario offers aswell a plausible additional explanation to the fact that no signals from other civilisations are detected. In fact according to Eq.(16) the mean time to reach Dyson sphere depends on the ratio of the technological level T and therefore, assuming energyconsumption (which scales with the size of the planet) as a proxy for T , such ratio is approximately independent of the sizeof the planet. Based on this observation and on the mediocrity principle, one could extend the results shown in this paper,and conclude that a generic civilisation has approximatively two centuries starting from its fully developed industrial age toreach the capability to spread through its own solar system. In fact, giving a very broad meaning to the concept of culturalcivilisation as a civilisation not strongly ruled by economy, we suggest that only civilisations capable of a switch from aneconomical society to a sort of "cultural" society in a timely manner, may survive. Acknowledgements (not compulsory)
M.B. and G.A. acknowledge Phy.C.A for logistical support.
Author contributions statement
M.B. and G.A. equally contributed and reviewed the manuscript.
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Competing interests
The authors declare no competing interests.
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