
Stability of the Biosphere Processes and the Le Chatellier Principle A.M. Tarko Presenting a paper from a journal to Internet On the investigating the global processes alongside with the traditionally used analysis of Lyapunov's stability it is useful to research the biosphere on the basis of the Le Chatellier principle. The Le Chatellier principle is formulated as follows: an external influence removing system from the steady state stimulates in it the processes aspiring to weaken the results of this influence. In different disciplines there are various treatments of this principle. In statistical physics the principle is considered for the thermodynamically closed system which is in the steady state in which the rule of a maximum of entropy takes place. The fulfillment of the Le Chatellier principle in this case is a consequence of the mentioned preconditions. The biosphere and its any part in sense of statistical physics is not the closed system. Therefore it we can not tell about necessity of fulfillment of the Le Chatellier principle in the mentioned above sense. However here its application can be useful not as a "rule" but as a property which fulfillment or nonfulfillment characterizes ability of biosphere to weaken influences. Let's consider fulfillment of the Le Chatellier principle in the biosphere. We are interested in processes with characteristic time are decades and hundred years, i.e. commensurable with duration of the industrial period. On such intervals of time it is possible to tell about a constant quantity of substance in biosphere if there are no anthropogenic releases. The most interesting is a global carbon cycle in the biosphere. The strongest global antropogenic impacts affecting the biosphere are fossil fuels burning, deforestation and soil erosion. Let's consider a case of change of substance quantity in a system in a general view. Let dynamics of the system is described by the system of ordinary differential equations under conditions of the theorem of existence and uniqueness with variables x_{i} (): .
(1)Let the system be in a steady state and in a result of one influence the value of variable l was instantly changed by . We are interesting what will be the change of stationary state of this variable . Let us the condition of fulfillment of the Le Chatellier principle in the system is the following: (2)It is possible to prove that for definition it is sufficient to solve a system of the equations for variables (): , , .It is obviously to generalize the Le Chatellier principle for whole the system as fulfillment of a condition (2) for all variables l = 1, :, n. It is easy to prove that the conditions (3)are equivalent to fulfillment of a condition (2) for all variables l = 1, :, n and hence are the expression of the Le Chatellier principle for whole the system. Such treatment of the Le Chatellier principle allows us to calculate its fulfillment in the systems with plenty of variables for which the analytical calculation of corresponding derivatives is practically impossible with the aid of the simple computer calculations. It is necessary to change instantly value of any one variable of the system and then to calculate whether positive or negative are changes of every variable in a new stationary state. According to (3) the fulfillment of the Le Chatellier principle for whole the system we can interpret as follows: if there is a change of quantity of substance in the system it should be accompanied by change of stationary values of all variables in the same direction. It is possible to prove that the condition (3) for whole the system is equivalent to the following conditions: (4)It means that projections of a line of equilibrum states on a plane (x_{i} ,x_{j}) (i, j = 1..., n; i > j) should be growing functions of one of the phase variables. Given condition in case of two variables n = 2 can be written as: (5)It means that the line of equilibrium states in a plane of variables (x_{1}, x_{2}) should be growing function of x_{1}. In the given treatment the Le Chatellier principle is stronger requirement than Lyapunov's satbility. Possibly system is stable according to Lyapunov's stability but the Le Chatellier principle is not fulfilled. Let us consider an example. The biosphere is represented by system atmosphere  grasslands  soil in which the quantity of carbon is constant in absence of CO_{2} releases. We have two variables: Z_{1}  quantity of carbon in the atmosphere and Z_{4}  quantity of carbon in soil humus. Weight of grassland vegetation can be neglected. We consider that the system in absence of influences is in a steady state. Let us consider a plane of variables (Z_{1} , Z_{4}) (fig. 1). The condition of a constancy of quantity of carbon in the system means that the trajectories of the system should be on lines of constancy Z_{1} + Z_{4} = M = const. If we increase instantly the quantity of atmospheric CO_{2} (fig. 1 à) the point representing a condition of system will be moved on a new state (from a point 1 to 2) corresponding a new line of constancy. Then it will move to a new steady state over line of constancy (point 3). So the contents CO_{2} in the atmosphere was instantly increased in result of external action and then decreased a little in result of internal ability. Result of action was weakened. Similarly system answers to change of quantity of humus. We must point out that the effect of influence is compensated not completely.
The Le Chatellier principle is fulfilled if the line of equilibrium state increases with the growth of Z_{1} (see condition (5) and fig. 1a). If the line decreases (fig. 1 b) the system answers to instant CO_{2} increase in the atmosphere so that the effect of influence either amplifies (when the quantity of carbon in the atmosphere increases) or becomes smaller that the initial state (when the quantity of humus increases). In this case the Le Chatellier principle is not fulfilled. Thus, at in result of CO_{2} emissions to the atmosphere in the system atmosphere  grasslands  soil the Le Chatellier principle can be fulfilled or not fulfilled. In the models describing the global biogeochemical carbon cycle under the nonlinear character of dependence of annual production on atmospheric CO_{2} concentration and on temperature at the small and moderate influences the principle is fulfilled and at the strong influences  is not fulfilled. In the latter case effect of influence amplifies. Calculations show that the present time the Le Chatellier principle concerning CO_{2} releases to the atmosphere will be fulfilled at least until the beginning of XXII century (fig. 1 a). But if we suppose that annual terrestrial plant production is decreased at CO_{2} concentration is more than nowadays by 1.5 times (such dependence was measures in some laboratories) Le Chatellier principle fulfillment would be ceased during 20502100 (fig 2.b). In this case land ecosystems becomes a source of CO_{2} . It should be noted that the last simulating example is presented to demonstrate idea of the Le Chatellier principle only and most possible can not be considered as a forecast of future of the biosphere.
Fig. 2 a
Fig 2 b
Copyright c A.M. Tarko, 2000
