The slash and burn agriculture Hugo Loza

1. The slash and burn agricultureHugo Loza Universidad Mayor, Real y Pontificia San Francisco Xavier de Chuquisaca Centro de Postgrado e Investigación Sucre, Bolivia 29 January 2004

2. Abstract • The slash and burn agriculture is a production system that is broadly diffused among the peasants who inhabit the low lands of Bolivia. • The study of this production system is important because it has an unfavorable impact on the farm economy as well as on the environment. • Nevertheless, it may represent only the first stage towards sustainable forms of production.

3. We use dynamical equations to represent and analyse the slash and burn agriculture. • We find that, under certain conditions, the farm can reach a stable equilibrium. • The equilibrium can be reached following three different kinds of paths, corresponding to three local forms of convergence for the economy.

4. We associate: • a slow evolution of the farm with an equilibrium, corresponding to two real eigen-values in the matrix of the linearized dynamic system; • a moderate motion with another equilibrium, corresponding to one eigen-value; • and an evolution with fluctuations to a third equilibrium, associated with two complexes and conjugated eigen-values.

5. 1. Introduction • Coming to the zone of colonization, the peasants receive, from the State, an endowment of woodland that they prepare and sow with the help of manual tools. • The agricultural works comprehend two kinds of tasks: • the deforestation of the plot; • the cultivation of rice properly speaking.

6. The culture of rice without rotation, on the same plot: • rapidly exhausts the natural fertility of soil, • which means: • the multiplication of brush competing with the principal culture for the soil nutrients.

7. and, as a consequence: • the yields fall; • the peasant prepares a new plot of woodland; • he abandon the previous parcel as fallow.

8. In this paper, • we present a model; • we examine the stability analysis; • we address the issue of sustainability in terms of the parameters of the model; • we illustrate the possible evolution of a farm in three different scenarios, corresponding to three local forms of convergence.

9. represents the constant rate of formation of fallow • The model • The first equation describes the evolution of the fertile land area available for production. • The change in this area F ensues from the balance between the new added parcels H and the exhausted plots put in fallow.

10. The second and third equations represent the production function of new parcels. • The available technology does not allow any replacement between factors. • It is described by means of a Leontief production function with constant coefficients. and capital (tools) .

11. The fourth equation shows the growth rate of labour devoted to land clearing, according to a linear decreasing function of the cultivable surface, where: • c1, represents the growth rate of the clearing labour in the absence of fertile soil available for production; • c2, the threshold parameter, could be interpreted as the expected farm size in terms of fertile soil area.

12. The fifth and sixth equations represent the production function for rice Y : • the farmer uses two substitutable production factors, labour L2and the cultivable area F, as well as a third complementary factor K2 ; • a Cobb-Douglas function drives the substitution between labour and cultivable area, where e represents the L2production elasticity.

13. The last equation establishes the budget balance of the farm: • equalling income to expenses (remuneration of production factors: labourevaluated at the wage rate w, and the two forms of capital).

14. Analysis of stability • We analyze the qualitative behaviour of the solutions considering the variables trajectories and their speeds in the space of phases. • We simplify this analysis by reducing the size of the system trough the elimination of an equal number of variables and equations. • We discard the variables others that F and L1.

15. Thus, we obtain the following system of differential equations:

16. This system has two equilibrium points:

17. The two equilibrium states of the system are at the intersection of the straight line of equation: with the axis of abscissas: and with the vertical straight line of equation:

18. Figure 3.1. Phase diagram F

19. We now demonstrate that the origin corresponds to: • a saddle point and, as a consequence, is an unstable equilibrium; • whereas the second point is: • an asymptotically stable equilibrium.

20. Indeed, the matrix of the linear form associated with the system could be written as follows:

21. The value of the determinant of this matrix at the point of coordinates (0, 0) is: and the origin is an unstable equilibrium.

22. On the other hand, if we calculate the determinant and the trace of the same matrix at the alternative equilibrium point, we obtain the following expressions: allowing us to affirm that this equilibrium point is asymptotically stable.

23. 4. The parameters of the model

26. The variables at the stable equilibrium • At the stable equilibrium, the fertile soil area equals the scale parameter of the farm:

27. Labour for deforestation purposes increases with: • the farm size; • the rate of formation of fallow; • with the loss of efficiency of the labourers.

28. The deforested area is proportional to: • the farm size; • the rate of loss of the natural soil fertility.

29. Capital expenses are represented as follows: • We observe an additional parameter representing the payment amortization for capital tools services. • It depends on factors such as the exchange rate or tariffs, as well as on the evolution of the industrial technology.

30. 63 54 45 36 27 18 9 0 0 16 32 48 64 80 96 Labour The feasibility condition for the slash and burn agriculture:

31. 5. The evolution of the farm In this section we first study the value determination of the last parameter, corresponding to the motion of the farm:

32. We refer to the eigen value calculation formula:

33. If we are interested in the values of c1 that cancel out the value of the discriminant, we have: • When the parameter takes this value, the characteristic polynomial has only one root and the farm moves at a moderate velocity.

34. When the value is at the left of the threshold value, there are two real eigen values and the farm moves at a slower velocity • When the value is at the right of the threshold value, there are two complexes conjugated eigen values, implying faster trajectories that are wrapped in spiral around the stable equilibrium.

35. Simulation results Phases diagram corresponding to a moderate, a slow and a fast motion

36. Evolution of the slash and burn agriculture (slow, moderate and fast motions)

37. Evolution of the slash and burn agriculture (slow, moderate and fast motions)

38. 6. Conclusions The slash and burn agriculture is practiced since remote times in the basin of the Amazon River, but this practice has recently raised concerns about its effects on deforestation and biodiversity. Considering that the natural fertility of the soil is recovered after seven years from the last harvest, we presented a model showing that, under certain conditions, it is possible to reach a sustainable, self-reproductive equilibrium in time. The model results appear to be sufficiently realistic.