End Tidal Co2 Monitor

Luis Augusto Trevisan

Departamento de Matemática e Estatística, Universidade Estadual de Ponta Grossa, Avenida Carlos Cavalcante

Fabiano Meira De Moura Luz

Departamento de Física, Universidade Estadual de Ponta Grossa, Avenida Carlos Cavalcante 4748 Ponta Grossa, Paraná, ZIP 84030-000, Brazil. In this work we propose a mathematical model, based in a modified version of the Lotka-Volterra prey-predator equations, to predict the increasing in CO2 atmospheric concentration. We consider how the photosynthesis rate has changed with the increase of CO2 and how this affects plant reproduction and CO2 absorptions rates. Total CO2 emissions (natural and manmade) and biomass numerical parameter changes are considered. It is shown that the atmospheric system can be in equilibrium under some specific conditions, and also some comparisons with historical are done.

Keywords: Prey Predator model, Photosynthesis rate, CO2 concentration.

1. Introduction

Several billion of years ago, special terrestrial conditions made the formation of life possible. Since its formation, the biosphere has played an active role in controlling environmental conditions. An interaction has developed between the evolution of living species and the environment. Changes in environmental conditions modified the biosphere and vice-versa. When life began, the Earth’s atmosphere was not similar to the present air. The main peculiarity of our atmosphere - the presence of oxygen – is the result of the biospheres evolution (Meszaros). Natural changes in atmospheric composition and climate are slow processes when compared with typical human time scales. During the last 8.000 to 10.000 years, the climate has been stable. Such stability has been favorable for humans and made social and economic development possible. In the present industrial era, this development has reached such a level that human activities have become able to modify environmental conditions on a time scale ( ≈ 100 years ) that is quite shorter than periods of natural changes. It is well known that the CO2 emission due to human activities has contributed to cause the “greenhouse effect”, a warning in the Earth mean temperature. Another important effect of the increase of the CO2 concentration is acceleration in the photosynthesis rate, a subject widely studied (Kirsshbaum)(Laisk and Edwards).

With a higher photosynthesis rate, plants can absorb more CO2, and have a faster growing and reproduction rate, again consuming more CO2. In this work we present a model to predict the increase of CO2 atmospheric concentration, considering the plant-atmospheric carbon interaction. Basically, we consider the increase of photosynthesis rates, that is a consequence of the increase of Ca (CO2 atmospheric concentration), and apply a modified version of Lotka-Volterra predator-prey model to

=

describe a possible time evolution of Ca and a numerical biomass parameter. This parameter includes all life beings that use photosynthesis, even if they are not vegetables. 2. Predator-prey model adapted to plant-atmospheric carbon interaction The original model by Lotka-Volterra (Lotka,Volterra) was proposed in the twenty's, and have been widely used in ecological studies and also in another fields of knowledge. In our model, adapted to the problem of plants CO2 interaction, we will consider the CO2 as the inorganic prey (not reproducing) and plants are predator. The CO2 molecule does not reproduce by itself, therefore its concentration evolution, denoted by Ca (in ppm), depends only on emissions (i.e. combustion reactions) denoted by Q(t), and absorptions (i.e photosynthesis), assumed to be proportional to vegetal biomass surface, denoted by P(t) and also to the relative photosynthesis rate A(Ca , T), where T is the temperature. In this way, we have

(1)

The relative (in comparison with the maximum possible) photosynthesis rate is given, at temperature of

(2)

To estimate k, we consider the work by P.Tans, F.Y.Inez and T.Takahashi , that studied emissions and concentration during the 80’s, the value is obtained considering the annual average increase on concentration of 1.4ppm and difference between emissions and absorption, which the mean value is 3GtC/year. The ratio reads k= 0.47. We also consider that the oceans are a balanced system, that absorbs all the CO2 they emit and is also able to absorb around 30% of human emission (Sabine et.al).

Equation 1 says that the growing of the numerical value for P is proportional to itself and also to photosynthesis rate. The last equation is rewritten

B

. The total CO2, Q(t) emission can be separate in natural emissions Qn and human emission Qh(t).

(4)

We have important remarks about the balance in the system. First, we note that if the system is

=

, the photosynthesis rate is equal to vegetable death rate (b factor). Then, the

(5)

This fact is explored in this work, to show how we can use the parameter b to study possible stabilization scenarios in the atmospheric CO2 concentration. When:

Becomes Smaller,

approaching zero and a final balance condition. On the other hand, if

(7)

a necessary condition to reach balance conditions is:

=

that is, the absorptions must increase faster than emissions, until a balanced system (

Dca /

) is reached. The b factor is time dependent (more exactly, depends on the historical moment) and is also related to the human emissions, since some fraction of the vegetable death and human emission are due to the same fact: burning. Deforesting and urbanization also contribute to increase the b factor. In this way

(9)

Besides, the f value can be used to help to fit desirable value of the derivative. In this way, the model has only two free parameters, b(t) and f that could be estimate using environmental data.

To better understand the role of the b factor, we should observe figures 1 and 2 below: In figure 1, we use the same human emissions (Qh = 7.5GtC/Y, which taking into account ocean absorption, should be read 0.7x7.5=5,25GtC/Y). In cases 1, 3 and 5 we have used the same f value (f =0.0185), the value of the b factor is supposed constant and fitted to given three different final CO2 concentrations; 500ppm; 600ppm and 700ppm, corresponding to b=0.70; b=0.74 and b= 0.77 respectively. The initial concentration is Ca = 345ppm. Therefore there is a relantionship between the vegetal death rate (b) and

6

Besides this, in curves 2 and 4 , we have used the same b value of curve 3, b=0.74, with different values for f ( f= 0.03 in curve 2 and f=0.01 in curve 4, while f=0.0185 in curve 3). We observe that, using the values f=0.0185, and b=0.74 (which corresponds to final concentration of 600ppm) we may fit the data between 1985 (345ppm, jan) and 1995 (360ppm, jan ) .

5

Figure 1. CO2 concentration versus time for different b and f constant values. Curve 1: b=0.77, f=0.0185; curve 2: b=0.74, f=0.03, curve 3:b=0.74, f=0.0185, curve 4: b=0.74, f=0.01; curve 5 :b=0.70, f=0.0185.

7

In figure 2 below, we used the same constant b from figure 1 and show the time evolution of P/P0, being P0 the initial value. We also observe stabilization, after some decreasing period.

P/P0

Figure 2. Normalized biomass parameter evolution, for different constant values of the death rate b. Circles mean b=0.70, crosses mean b=0.74 and squares mean b=0.77.

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