Analysis of a fractional order climate model due to excessive emission and accumulation of carbon dioxide in the atmosphere

The devastating consequences of climate change on our planet cannot be taken lightly. Greenhouse gas emissions due to the various activities of the increasing human population are solely responsible for this change. The chief and most significant of these gases is carbon dioxide. A fractional-order model of five compartments is considered. The uniqueness and existence, the positivity, and the boundedness of the model solution are established. The equilibrium points of the model are given. By formulating di ff erent Lyapunov functions, the global stability of the four equilibrium points was determined. The numerical simulation of the model was done using the Predict-Evaluate-Correct-Evaluate method of Adam-Bashforth-Moulton by considering four di ff erent orders of 0.7, 0.8, 0.9, and 1.0. According to the results, excessive concentrations of carbon dioxide in the atmosphere can be reduced by the joint employment of mitigation measures. 10.46481 / jnsps.2023.1749


Introduction
The issue of climate change has become a global problem beside other global challenges such as terrorism, environmental degradation, hunger, poverty, and many others which are highlighted and enshrined in the Sustainable Development Goals (SDGs) projected to be achieved in 2030.The current global increasing trends in carbon dioxide emission and accumulation can be attributed to factors such as per capita GDP, urbanization level, technology level, trade openness, energy consumption intensity, population growth, energy consumption structure and industrial structure [1,2].About 50% of the global total carbon emissions by volume in countries in the "Belt and Road Initiative (BRI)" are majorly from transportation industry, which is about 23.96% of the total emissions from all industries in the BRI countries [2].Global climate change is one of the burning issues posing threats to human existence and the ecosystem primarily caused by the emission of carbon dioxide into the atmosphere from energy generating and transport industries [3].There is a strong correlation between increase in global temperature and the accumulation of carbon dioxide in the atmosphere, which is contributing greatly to climate change [4].
The amount of carbon dioxide in the atmosphere is increased by the combustion of fossil fuels [4].
Carbon dioxide is a colourless and odourless gas that traps heat from the sun and warms the earth through a natural process known as the Greenhouse Effect [4].The power stations that use coal to generate electricity contribute to about 83% of the current emission of carbon dioxide [4].Carbon dioxide can exist in three different forms in sea water (solute carbon dioxide, hydrogen carbonate and carbonate) making the seawater capacity to store carbon dioxide superior to that of the atmosphere [4].Gas pollutants like nitrogen dioxide have negative consequences like depletion of ozone layer, damaging green plants and trees, acid rain, hazardous fog and clouds as well as contamination of coastal water bodies [5].Climate change due to large amounts of carbon monoxide lead to change in land and sea temperature which can further lead to extreme weather conditions [5].It is very important to capture and remove about 20 Gt of carbon dioxide annually in compliance with the Paris agreement of keeping the global temperature increase below a value of 2 • C [6].
Research have revealed that excessive emission and accumulation of greenhouse gases into our environment is responsible for the current world climate change.Due to the visible consequences of climate change that is affecting global economy, different parties with interest in seeking for solution to this ravaging and burning issue of concern are looking for solutions in forms of policies, programmes and clean technologies that would help in reducing excessive emission of the greenhouse gases [7].Mathematical modelling is very useful in providing solutions to human problems such as dynamics of global infections [8], toxic effects of cytotoxic and hemotoxic snake venom [9], honeybee farming improvement [10] and many other fields.As part of the global solution-seeking process, there is the need to develop some mathematical models of climate change due to carbon dioxide emission and accumulation by incorporating some mitigation measures into the model.A number of mathematical models on climate change attributable to excessive emission of carbon dioxide have been developed [11][12][13][14][15][16][17].
Most of them considered one or two mitigation measure (s) in their model dynamics for the reduction of carbon dioxide.Leveraging on the idea that a combination of more simple, viable and easy to implement mitigation measures could bring a much better considerable reduction of carbon dioxide emission and accumulation in the atmosphere, a mathematical model involving three mitigation measures was developed [18].In this current work, we extend this model in Ref. [18] using the concept of fractional calculus.Fractional calculus has become very crucial and an integral tool in the last three decades in fields such as biomathematics, economics, physics, finance, engineering and many others as it provides different scenarios and more accurate results compared to other techniques [8].Almeida [19] in his work presented two reasons for the preference of fractional calculus over the classical (traditional or ordinary) calculus.
Fractional derivatives are nonlocal operators and may be more suitable for long-term behavioural studies.Thus, they contain memory (solution depends on previous instant) unlike integer-order derivatives.Secondly, the consideration of the order of the derivative as an arbitrary real number which is not necessarily an integral value, enables modelling more efficiently real data compared to the theoretical model.Yuan et al. [20] in studying long term memory (LTM) in climate variability using fractional integral technique, considered Fractional Integral Statistical Model (FISM) that could estimate the long-lasting effects of historical climate states on the contemporary time quantitatively and also that of climate memory signals.They concluded that one could ascertain the trends in time series change using extracted climate memory signals, which gave a new dimension to the narrative of prediction of climate.
Kumar et al. [21] proposed and analyzed a fractional order nonlinear mathematical model using the generalized form of the Caputo fractional derivative to describe the dynamics of the problem.They presented their novel results graphically and recommended the usefulness of the fractional approach in solving real-world phenomena.Eze & Oyesanya [22] presented fractional model on the impact of climate change with dominant earth's fluctuations.Their model was a second order ODE that was solved using modified Laplace Adomian decomposition method.They compared the results of the fractional solutions with the integer solutions.Their findings revealed that the fractional model gives a better situation compared to the integer solution.
Xie et al. [23] developed a novel continuous fractional non-linear grey Bernoulli model with Grey Wolf Optimizer for forecasting fuel combustion-related CO 2 emissions in China.Their findings revealed that their developed model produced better results than other available similar models and that about 1 × 10 10 tonnes of fuel combustion-related CO 2 will be emitted by 2023.Based on their forecast results, they recommended developing low-carbon technologies, acceleration of the promotion of the national carbon market and strengthening citizens environmental awareness.Ilhan et al. [24] found a series solution for a system of fractional ODE describing the atmospheric dynamics of carbon dioxide using the q-homotopy analysis transform method (q-HAM).Their results revealed that the fractional scheme is highly methodical and very effective in analyzing the nature of the system of arbitrary order differential equations in daily life.
Ozarslan & Sekerci [25] developed a fractional order differential model on oxygen-plankton-zooplanktoon system under climate change using the Caputo fractional operator and considering a temperature function to represent the production rate of photosynthesis.Their findings revealed that the effect of global warming on the rate of oxygen production was severe, leading to a decrease in oxygen level which could further cause plankton extinction.Qureshi & Yusuf [26] worked on mathematical modelling for the impacts of deforestation on wildlife species using Caputo differential operator.Their results were displayed graphically and they concluded that the fractional model was better in performance compared to the usual classical model, as it captured all the historical information of the considered system unlike the corresponding classical system.
Sekerci & R. Ozarslan [27] studied a fractional oxygenplankton-zooplankton mathematical model with climate change effect by employing nonsingular fractional operators, Caputo-Fabrizio (CF) and Atangana-Baleanu (ABC).In conclusion, they averred that increase in global warming adversely affect the rate of oxygen production, leading to depletion in the level of oxygen and the extinction of plankton.Din et al. [28] worked on the mathematical study of climate change model under nonlocal fractional derivatives by considering a threecompartment of oxygen concentration, phytoplankton and zooplankton.Their findings revealed that firstly, there was a reduction in the level of oxygen concentration whose value tended to the free equilibrium point.Secondly, there was a decrease in the number of phytoplankton also due to decrease in the concentration of oxygen.Lastly, the number of zooplankton reached its maximum value due to the consumption of the oxygen and then became stable thereafter.In conclusion, they opined that the fractional model gives a better situation for climate control compared to the classical case.
Machado & Lopes [29] studied global warming patterns using the concepts of dynamical systems and fractional calculus.From their findings, an assertive representation of the global warming dynamics and a simpler analysis of its characteristics was revealed by the application of Fourier transforms and power law trend lines.Furthermore, they opined that fractional calculus took into account long range effects usually ignored by classical models.
Xu et al. [30] proposed and analyzed the correlation that existed between energy consumption and carbon dioxide emissions using Non-equigap GM(1,1) model with conformable fractional accumulation [CFNGM (1,1)].From their findings, the carbon dioxide emissions of 30 out of the 53 countries studied had risen to different levels, the top three being China, USA and India.They recommended that particular attention should be paid mostly to emission trends of China.They also showed that conformal fractional order had a better prediction effect when the order was smaller.Considering the current emergency caused by climate change and the need to seek for solutions from all possible sources, there is a motivation to formulate a mathematical model that combines a number of mitigations that are feasible and realistic in mitigating against excessive emission of carbon dioxide.Furthermore,the use of fractional calculus approach gives more realistic scenarios compared to use of integer order ordinary equations.
The rest of the paper is organized as follows: In Section 2, related mathematical preliminaries are presented.In Section 3, the Caputo fractional model equations are given and their corresponding mathematical properties determined.In Section 4, the stability analysis of the model is done.In Section 5, the numerical simulation results of the model are presented.In Section 6, the discussion of the results is done.The conclusion is presented in Section 7.

Mathematical Preliminaries
Definition 1 (Lipschitz Condition).[31].A function g (t, y(t)) is said to be Lipschitz if for some constants L > 0 which does not depend on t, y and w.
Lemma 2. [13,32].Let p(t) ∈ C(ℜ + ) and its fractional derivatives of order r exist for any 0 < r ≤ 1.Then, for any t > 0: and Q be the greatest invariant set of F. Then every solution p(t) starting in Φ approaches Q as t → ∞.

The Fractional Order Model and its Properties
Using the Caputo derivative operator, the fractional model of order r (the order not visibly written on the RHS for the sake of ease of writing, reading and simplicity), where 0 < r ≤ 1 of the model equations in Ref. [18] become: The initial conditions of these equations are: In subsequent writings, the order, r, on the right hand side of equations ( 1)-( 5) would be omitted for ease of writing and simplification.Hence, it would be implied.

Positivity of the Model Solution
The biological meaningfulness of the model system given by equations (1 ) to ( 5) is dependent importantly on establishing that all its state variables are non-negative for all time, (t).In other words, it suffices to proving that for positive initial data, the solutions of the model system will remain positive for all time t > 0.
Since the system represents a biological system and there is continuous emission and accumulation of carbon dioxide into the atmosphere, we assume that all derivatives are positive.Thus, for t 1 > 0: Taking the Laplace transform of both sides of the inequality in (18): Taking the inverse Laplace transform of the inequality in (19): Taking the second equation given by equation ( 2): Taking the Laplace transform of both sides of the inequality in (21): Taking the inverse Laplace transform of both sides of the inequality in (22): Since P 0 > 0 and the Mittag-Leffler function E r , 1 (−(µ 1 + µ 2 )t r ) > 0, then P(t) > 0, ∀t > 0.
Taking the third equation given by equation ( 3): Taking the Laplace transform of both sides of the inequality in (24): Taking the inverse Laplace transform of both sides of the inequality in (25): Since R 0 > 0 and the Mittag-Leffler function E r , 1 (−a 0 t r ) > 0, then R(t) > 0, ∀t > 0.
Taking the fourth equation given by equation ( 4): Taking the Laplace transform of both sides of ( 27): Taking the inverse Laplace transform of both sides of the inequality (28): Since E 0 > 0 and the Mittag-Leffler function E r , 1 (−b 0 t r ) > 0, then E(t) > 0, ∀t > 0.
Taking the fifth equation given by equation ( 5): Taking the Laplace transform of both sides of the inequality (30): Taking the inverse Laplace transform of both sides of the inequality (31): Since T 0 > 0 and the Mittag-Leffler function E r , 1 (−m 0 t r ) > 0, then T (t) > 0, ∀t > 0.
+ is positively-invariant for the model system given by equations ( 1) to (5) with initial conditions in ℜ 5 + , that is Proof 2. Using the approach of getting boundedness of a model as presented in the works of Refs.[15,17,43,44], the regions or domains of positive attraction (invariant regions) are obtained as follows: Since the model must be positive for the biological meaningfulness of the model which has been established in the proof of lemma 7, then we have: Taking equation ( 1): From the definition or statement of lemma 8, C 0 > 0 implies that Taking equation (4): From equation ( 33): Taking equation ( 5): since from equation (33 Considering equation ( 3): From the inequality results in ( 33) and ( 35): Lastly, taking equation ( 2): Substituting the inequality results in (33) and (38) into this, we have: From the results got in equations ( 34),( 40), ( 39), ( 36) and ( 37), the region or domain of attraction of the model solution with given positive initial data is Φ={(C, P, R, E, T + and it is positivelyinvariant, where β > Θ.

Equilibrium Points
The equilibrium points are obtained by putting each of the expressions on the RHS of equations ( 1) to ( 5) to zero and solving.That is, D r C(t) = 0, D r P(t) = 0, D r R(t) = 0, D r E(t) = 0 and D r T (t) = 0 are solved to obtain the equilibrium points.The equilibrium points of the model (as also obtained in Ref. [18]) are: ], 0, 0, 0 .Where,
Next, we establish the last axiom as given thus: Using lemma 3, equation ( 41) becomes: Substituting equations ( 1)-( 5) into the inequality (42) and simplifying; Hence, V 1 is a Lyapunov function in Φ and the largest compact invariant set in {C, P, R, E, T : D r V 1 ≤ 0} is the singleton {ε 0 }.Employing the Generalized LaSalle Invariance Principle as given by lemma 5, every solution of the model system given by equations ( 1)-( 5) with initial conditions in Φ, tends to ε 0 as t and (µ 1 + µ 2 ) ≥ ω.Hence, the equilibrium point, ε 0 , is globally asymptotically stable.
Theorem 2. The equilibrium point ε 1 = (C * , P * , R * , E * , T * ) is globally asymptotically stable (GAS) if T * ln T T * be a hybrid Lyapunov function.Then we establish that it satisfies all the axioms that guarantees the global stability of ε 1 = (C * , P * , R * , E * , T * ) using it.Obviously, V 2 > 0 for C C * , P P * , R R * , E E * , T T * .Thus, the first axiom is satisfied.Also, Hence, the second axiom for V 2 being a Lyapunov function is satisfied.
Theorem 3. The equilibrium point T 2 be a hybrid Lyapunov function.Then we establish that it satisfies all the axioms that guarantee the global stability of ε 2 = (C * * , P * * , R * * , E * * , T * * ) using it.Obviously, V 3 > 0 for C C * * , P P * * , R R * * , E E * * , T T * * .Thus, the first axiom is satisfied.Also, Hence, the second axiom for V 3 being a Lyapunov function is satisfied.Next, we establish the last axiom.
Using lemmas 2 and 3, equation ( 43) becomes: Substituting equations ( 1)-( 5) into the inequality (44) and simplifying using the equilibrium point: . Thus, V 3 is a Lyapunov function in Φ and the largest compact invariant set in {C, P, R, E, T : D r V 3 ≤ 0} is the singleton {ε 2 }.Employing the Generalized LaSalle Invariance Principle as given by lemma 5, every solution of the model system given by equations ( 1)-( 5) with initial conditions in Φ, approaches ε 1 as t → ∞ provided d 1 ≥ ϕ and (ϕP . Hence, the equilibrium point, ε 2 , is globally asymptotically stable.Using the result given by lemma 2: Substituting equations ( 1),( 2), ( 3), ( 4) and ( 5) into this and simplifying: is the singleton {ε 3 }, where ε 3 is the fourth equilibrium point of the model system given by equations ( 1) to (5).Hence, using the Lasalle's Invariance Principle given in lemma 5, we conclude that ε 3 is globally asymptotically stable in Φ if G < H.

Parameters and their Values
The following parameter values (as also presented in Ref. [18] for the classical case) were used for the numerical simulation of the model in MATLAB using the Predict-Evaluate-Correct-Evaluate method of Adams-Bashforth-Moulton [53].The initial conditions of the model were set as: C 0 = 1, P 0 = 4, R 0 = 2, E 0 = 5 and T 0 = 7.

Discussion
Three cases between the accumulation rate of carbon dioxide, β, and the intrinsic growth rate of the photosynthetic biomass, ω, for four different orders of the model are depicted in Figure 1.By varying both the intrinsic accumulation rate of carbon dioxide, β, and the intrinsic growth rate of the photosynthetic biomass, ω, for different orders of the model of r = 1.0, r = 0.9, r = 0.8 and r = 0.7, the maximum excessive concentration of carbon dioxide corresponding to these orders obtained were (2.2759, 1.9971, 1.5504), (1.6862, 1.554, 1.3069), (1.3311, 1.2759, 1.1475) and (1.1281, 1.1098, 1.0507) respectively.From these results, lower values of the excessive concentration were obtained as the order reduces.The excessive concentration of carbon dioxide was less for β < ω compared to when it was β > ω. β < ω is supposed to be the idea situation, as the expectation would be that emitted carbon dioxide should be removed by the natural mechanism in place (photosynthetic biomass).However, reality is represented by β > ω, as the natural mechanism is overwhelmed and thereby leading to excessive accumulation of this greenhouse gas.Hence, other mitigation measures are needed.
In Figure 2, the effect of both accumulation rate of carbon dioxide and the photosynthetic rate for different orders of r = 1.0, 0.9, 0.8, 0.7 are simulated.The natural mechanism (sink) in place to regulate the concentration of carbon dioxide are the photosynthetic biomass.For four different orders of r = 1.0, r = 0.9, r = 0.8 and r = 0.7, three cases of relationships between the intrinsic accumulation rate of carbon dioxide, β and the rate of reduction of carbon dioxide by the photosynthetic biomass, d 1 , were shown.Corresponding to these model orders of r = 1.0, r = 0.9, r = 0.8 and r = 0.7, the values of the excessive concentration obtained were (6.347, 1.9971, 1.5504), (1.554, 1, 1), (1.2759, 1, 1) and (1.1098, 1, 1) respectively.As seen in Figure 2, β > d 1 means that the rate of accumulation of carbon dioxide surpasses the rate at which the same gas is removed from the atmosphere.Hence, something needs to be done to reduce this accumulation into the atmosphere.β < d 1 represents a scenario where the excessive concentration of carbon dioxide is effectively checked by the photosynthetic biomass.Unfortunately, this scenario does not represent the current reality.
The variation of the accumulation rate of carbon dioxide for different orders of the model are shown in Figure 3. Three different values of β = 6, β = 3 and β = 1.5 for the model orders r = 1.0, r = 0.9, r = 0.8 and r = 0.7 correspondingly produced values of the maximum excessive concentration of carbon dioxide as (19.651, 11.625, 3.8053), (15.055, 7.3906, 2.5096), (9.948, 4.2791, 1.747) and (5.5851, 2.4337, 1.3241) respectively.These represent reduction in the maximum excessive concentration of carbon dioxide in the atmosphere to about (40.84%, 80.03%), (50.91%, 83.33%), (56.99%, 82.44%) and (56.43%, 76.29%) respectively.The lesser the value of the accumulation rate, β, the lower the concentration of carbon dioxide in the atmosphere.This can be achieved by putting effective and efficient mitigation measures in place to reduce the excessive concentration.Such mitigations could include: good conservations policies, enlightenment programmes and the use of technology that can capture and store the carbon dioxide.
By keeping the natural depletion rate of the photosynthetic biomass constant in Figure 6, the effect of variation of only the artificial depletion rate of the photosyn-             Three different values of the natural depletion rate µ 1 = 0.02, µ 1 = 0.002 and µ 1 = 0.0001 were simulated for the maximum excessive concentration of carbon dioxide while keeping the artificial depletion rate at a fixed value of µ 2 = 0.04.The corresponding values of the maximum excessive concentration obtained for the different orders r = 1.0, r = 0.9, r = 0.8 and r = 0.7 were (19.432, 10.197 7, led to a proportional reduction in excessive concentration of carbon dioxide in the atmosphere.Hence, the higher the rate of natural depletion of the photosynthetic biomass, the more the concentration of carbon dioxide that would be present in the atmosphere and vice versa. In Figure 8, the effect of photosynthetic biomass density on the dynamics of reducing the excessive concentration of carbon dioxide in the atmosphere is depicted.By simulating for the different orders r = 1.0, r = 0.9, r = 0.8 and r = 0.The effect of the combination of the photosynthetic biomass and good conservation policies rates are presented in Figure 9.A variation of three different values of (d 1 , d 2 ) = {(0.05,0.013), (0.15, 0.089), (0.33, 0.19)} for the different orders r = 1.0, r = 0.9, r = 0.8 and r = 0.7 were simulated.
The effect of combining the photosynthetic biomass, good conservation policies and enlightenment programmes are presented in Figure 10 for four different orders of the model.
In Figure 12, we look at different scenarios involving the excessive accumulation of carbon dioxide into the atmosphere and the collective combination of the mitigation measures.

Conclusion
A fractional order deterministic model on climate change of five compartments is presented.The compartments comprises: Excessive Concentration of Carbon Dioxide, Photosynthetic Biomass, Good Conservation Policies, Enlightenment Programmes and Direct Air Capture Technology.The proofs of the uniqueness and existence of the model solution, the positivity of the model solution as well as the boundedness of the model solution were shown.By constructing different Lyapunov functions, the global stability analysis of the four equilibrium points was done.The simulation of the model for four different orders of 1.0, 0.9, 0.8 and 0.7 was done in Matlab using the Predict-Evaluate-Correct-Evaluate (PECE) method of Adam-Bashforth-Moulton.The results obtained were presented graphically and discussed.Findings revealed that the excessive concentration of carbon dioxide can be reduced by effective implementation of mitigation measures.

Figure 1 :
Figure 1: Effect of comparing accumulation rate of carbon dioxide with intrinsic growth rate of photosynthetic biomass for different orders of the model.

Figure 2 :
Figure 2: Effect of comparing the accumulation rate of carbon dioxide with the photosynthetic rate of the photosynthetic biomass for a variation of the order of the model

Figure 3 :
Figure 3: Effect of the accumulation rate of carbon dioxide on the excessive concentration of carbon dioxide for a variation of the order of the model

Figure 4 :
Figure 4: Effect of variation of maximum tolerated concentration of carbon dioxide on the excessive concentration of carbon dioxide for different orders of the model

Figure 5 :
Figure 5: Effect of variation of both the natural and artificial depletion of the photosynthetic biomass on the excessive concentration of carbon dioxide for variation of the order of the model

Figure 6 :
Figure 6: Effect of variation of only the artificial depletion of the photosynthetic biomass on excessive concentration of carbon dioxide for a variation of the order of the model

Figure 7 :
Figure 7: Effect of varying only the natural depletion rate of the photosynthetic biomass on excessive concentration of carbon dioxide for a variation of the model order

Figure 8 :
Figure 8: Effect of the photosynthetic biomass on excessive concentration of carbon dioxide for a variation of the model order

Figure 9 :
Figure 9: Effect of photosynthetic biomass and good conservation policies on excessive concentration of carbon dioxide for a variation of the order of the model

Figure 10 :
Figure 10: Effect of photosynthetic biomass, good conservation policies and enlightenment programmes on excessive concentration of carbon dioxide for a variation of the order of the model

Figure 11 :
Figure 11: Effect of photosynthetic biomass, good conservation policies, enlightenment programmes and direct air capture technology on excessive concentration of carbon dioxide for a variation of the order of the model

Figure 12 :
Figure 12: Effect of comparing the accumulation rate of CO 2 with combined proportions of success of photosynthetic biomass, good conservation policies, enlightenment programmes and direct air capture technology on excessive concentration of carbon dioxide for a variation of the order of the model By taking different parameter values forβ > d 1 + d 2 + d 3 + d 4 , β = d 1 + d 2 + d 3 + d 4 , β < d 1 + d 2 + d 3 + d 4and for the different orders of r = 1.0, r = 0.9, r = 0.8 and r = 0.7, the maximum excessive concentration of carbon dioxide obtained were (1.0019, 1, 1), (1, 1, 1), (1, 1, 1) and (1, 1, 1) respectively.As seen from the figure, more concentration of carbon dioxide was available in the atmosphere when β > d 1 + d 2 + d 3 + d 4 compared to when β < d 1 + d 2 + d 3 + d 4 which would represent a logical explanation of what should be expected.

Table 1 :
[18]riptions of Model Variables and Parameters[18]Rate of decrease in CO 2 concentration due to interaction between CO 2 and the photosynthetic biomassd 2Rate of decrease in CO 2 concentration due to implementation of recycling and good conservation policiesd 3Rate of decrease in CO 2 concentration due to implementation of enlightenment/awareness programmes d 4 Rate of decrease in CO 2 concentration due to implementation of direct air capture technology µ 0 Natural rate of depletion in concentration of CO 2 ω Intrinsic rate of growth of the photosynthetic biomass N Carrying capacity for the photosynthetic biomass ϕ Rate of increase in the photosynthetic biomass due to the interaction between this biomass and CO 2 τ Rate of increase in photosynthetic biomass due to interaction between good conservation policies and the photosynthetic biomass µ 1 Rate of decrease in photosynthetic biomass due to natural phenomena µ 2 Rate of decrease in photosynthetic biomass due to human activities a 1 Rate of success of recycling and good conservation policies a 0 Rate of negligence or evasion of recycling and good conservation policies b 1 Rate of success of enlightenment/ awareness programmes b 0 Rate of Ignorance, negligence and evasion of the enlightenment programmes m 1 Rate of success of direct air capture technology m 0 Rate of decline in the implementation of direct air capture technology