Model

A transition matrix (TM) of land use/cover was integrated when evaluating the different covers in two successive periods of time (Table 1), and compares the land use/cover maps for 2000 and 2007 in México with their respective annual change rates. Once the TM was integrated, the transition probability matrix (TPM) is calculated; that is, the matrix that shows in each one of its entries the probability (p _ij ) for a formation to change from status i to j in the following evaluation period. For example, according to the information in Table 1, the probability for the forest cover in the year 2000 to change into induced grassland in the next period (2007) is 37/34,163=0.00109. The change probabilities (p _ij ) must satisfy the restriction that their sum from formation i must be equivalent to the unit, that is:

Table 1 Plant cover transition matrix (ha x 1,000) between years 2000 and 2007.  

†Rate (d) calculated as [(S ²/S ¹)(1/n)]-1; where n represents the number of years between the cover in period 1(S ¹) and period 2 (S ²). The Table was elaborated based on Series II and III from INEGI.

∑j=1mpij=1,∀i=1,2,…,m

Assuming homogeneity in the events that cause the transition, the TPM follows a first order Markovian process (^{Balzter, 2000}), so that the change probability of the land use/cover depends only on the current condition; from this, the history of the events does not have any relation with the future transition probabilities; therefore, it is implicitly assumed that the set of causes of the transition is the same throughout time. Under this assumption, it is possible to predict a future state by just knowing the current state, and the TPM through the following equation:

bt+1=Pbt

where bt is the column vector that represents the fraction of land in each one of the m categories in the period and P is the TPM of size m x m. For example, the distribution of use/ cover for period t+3 would be given by: b _t+3 = Pb _t+2 = P ² b _t+1 = P ³ b _t ; in fact it is possible to identify a stationary status, that is, the transition probability after n periods (p _j ), when n→∞i,e,πj=limn→∞⁡pj(n).

The transition probabilities (p _ij ) can be assumed to be constant (^{Balzter, 2000}), or variable, according to a set of drivers (^{Chomitz and Gray, 1996}; ^{Sandoval and Real, 2005}), with which certain adaptability is incorporated into the process. The contribution of these causes is modelled by the Markov chain modified equation to the form: b ^t-1 =P[f(t)] bt; where f(t) is a function that defines p _ij in time t based on a set of variables through a general model. This last procedure is the one used in this study to integrate the effect of socioeconomic, physical and climate variables in the definition of land use. This methodology was used successfully to evaluate the effect of temperature and density on mite populations (^{Woolhouse and Harmsen, 1987}), and in plant cover models, which include socioeconomic variables or climate conditions, and even indicators of the composition of the types of covers (^{Henderson and Wilkins, 1975}; ^{Marsden, 1983}; ^{Ludeke et al., 1990}; Chomitz and Gray, 1996; Balzter, 2000; ^{Benabdellah et al., 2003}; ^{Sandoval and Real, 2005}).

With the objective of enhancing the information to adjust the model, each State in the federation was considered as an independent unit. For this purpose, the covers at the national scale (1:250 000) were broken off by State and a TPM of the types of land use/cover was generated for each State. The transition probabilities (p _ij ) were modelled by a set of variables through a logistic model of the form:

pij=e(β0+β1X1+β2X2+β3X3+…+βnXn)1+e(β0+β1X1+β2X2+β3X3+…+βnXn)

where b represent parameters of the model and the x correspond to each one of the explicative variables of the variation in the transition probability. From construction, the pij have the restriction of ∑jpij=1 ∀i=1,2,3,…m; therefore, the set of transition probabilities of the i-nth category should be modelled simultaneously and with the restriction that the sum of probabilities be equal to the unit. This system has the following form when being analyzed:

In=Pi1/(1-pi1)=β10+β11x1+β12x2+…+β1nxn+α11y1+α12y2+α13y3+…+α1kyk

In=Pi2/(1-pi2)=β20+β21x1+β22x2+…+β2nxn+α21y1+α22y2+α23y3+…+α2kyk

In=Pij/(1-pij)=βj0+βj1x1+βj2x2+…+βjnxn+αj1y1+αj2y2+αj3y3+…+αjkyk

In=Pmm/(1-pmm)=βm0+βm1x1+βm2x2+…+βmnxn+αm1y1+αm2y2+αm3y3+…+αmkyk

1=δ+∑j=1me(βj0+βj1x1+βj2x2+βj3x3+…+βjnxn+αj1y1+αj2y2+αj3y3+…+αjk3yk)1-e(βj0+βj1x1+βj2x2+βj3x3+…+βjnxn+αj1y1+αj2y2+αj3y3+…+αjk3yk)

there ln(pij/(1-p _ij )) is the logit (lgtij) from p _ij , band x have the same notation as before, whereas the a represent parameters of the model associated to the endogenous variables (y) within the system. These variables represent variables related bi-directionally within each equation of the model, as for example the same transition probability of other formations or, in an extensive case, the same socioeconomic or accessibility conditions. The p _ij can depend on other transition probabilities within the same system, with which the errors from each equation in the model can be correlated. Therefore, the adjustment was performed with the procedure model (PROC MODEL) from SAS^®, using the adjustment variant with maximum likelihood estimators of complete information.

Based on the adjusted models and the information for year 2000, the predicted transition probabilities (p ^ij ) were calculated for each use/cover and in each State, from which the state TM was recovered for the following period (2007). Then, with the sum of the State TMs, a national TM with its respective TPM was integrated. The latter was compared statistically with the TPM calculated from Series IV (^{INEGI, 2012}), through a x² test. It must be clarified that the projection and measurement periods could be partially equivalent, given that Series II was published in the year 2000, but it was elaborated from 1994 to 1999, whereas Series III was elaborated from 2003 to 2005, and published in 2007. Therefore, the interval of projection assumed between Series II and Series III is 7 years, and from 1995 to 2002, the same that are assumed for the measurement interval between Series III and IV (^{INEGI, 2012}).

Data

The cartographic information used to estimate the transition matrices was Series II (^{INEGI, 2000}) and Series III (^{INEGI, 2007}). The breakoff of State covers was carried out based on the State cover published by INEGI (^{INEGI, 2000a}). Given that the nomenclature of covers integrates a high number of categories, they are regrouped into categories (Table 2).

The set of explicative variables of changes in transition probabilities was selected according to availability, aligning with the projection date (information up to year 2000; Table 3), and recommendations by ^{Balzter (2000)}, ^{Benabdellah et al. (2003)}, and ^{Sandoval and Real (2005)}.

Transition from forest to other formations

The predictions of transition from forest to forest (f-f), forest to shrub (f-s), and forest to other covers, among which are considered crops and urban zones (f-oc), showed satisfactory adjustments judging from the statistics for goodness of fit of the system of equations, although the transition from forest to induced grasslands (f-ig) did not. The transition from forest to natural grassland (f-ng) had low frequency, so it was considered as a false change (Table 4).

Table 4 Estimations of the model of simultaneous equations of transitions from forest to other formations. The response variable is the logit of the referred transition probability. 

The prediction of the most important transition, f-f, which measures the permanence of this cover, showed that an increase in precipitation reduces the probability of transition, that is, promotes the change from forest to other uses. The effect of temperature is: with higher maximum temperature, a lower permanence of the forest area is verified; however, the behavior is reversed when the maximum temperatures are above 26 °C. The behavior of these two variables is closely linked to the most frequent change in use of this coverage, which is directed towards agricultural crops and urban zones where lands with good precipitation and maximum temperatures that are not extreme are demanded. Therefore, in the forest localities with higher temperature and lower humidity, there will be greater conservation of the forest land, which is linked to its lower agricultural aptitude.

The Gini coefficient of the GDP distribution among municipalities was taken as proxy of the presence of poles of wealth in the State. The variable is related directly to the forest’s probability of permanence, which can be interpreted as the effect of the poles of wealth to attract population, which finally translates into lower human pressure on the forest. The accessibility, measured through the road density (km km^-2), is also related to the possibility of having infrastructure to perform other productive activities that take pressure away from the forest. The variable could be considered endogenous (i.e. the forest’s permanence can attract its exploitation and foster the construction of roads), but the form of the response variable avoids this problem.

Net migration was also highly significant to describe the process of forest conservation; the presence of a higher number of individuals who were not born in the geographic unit promotes the change from forest use to other uses. The variable can be related to two effects: that of a higher population pressure and the one related to customs in land use, although the first has been captured with the variable “density of rural population”.

The proportion of agricultural surface was significant and with an inverse relation. This variable is related to technological availability and markets associated to the agricultural activity that promote the expansion of the agricultural frontier. Also, the proportion of surface in ejidos and communities was significant and with an inverse relationship. This proportion does not represent ejidos/communities with forest, so it is not related to the higher or lower conservation that can be carried out in forest communities. In sum, all the variables used to explain the permanence of the forest were significant and with the expected sign.

The transition forest to other covers (f-oc), basically crops and urban development, is the most important because of its frequency. Albeit it is related to climate variables, its most important cause is economic variables. The climate variables show that maximum temperatures above 29.5 °C are not apt for crops, which is why the change from forest use is less attractive. The humidity also influences the availability of the terrain for agricultural use, which is why when there is higher precipitation, the probability for transition is higher. The change from forest use is also related to the standard deviation of forest disasters; with higher variation in the number of these events, there is higher probability of change from forest use, which suggests that large clearings, resulting from disasters, could be taken advantage of to open cultivation lands.

Within the economic variables, there is the per cápita GDP and accessibility. The first shows that when there is greater wealth there is higher change of forest land use up to a threshold (approximately $24 000 per year), from which the probability of use change is reduced, which is a similar effect to the trend of Kuznets environmental degradation (^{Stern et al., 1996}). The coefficient associated to the road density shows that higher accessibility is not necessarily related to a loss of temperate forests, as is thought that it happens in tropical forests (^{Chomitz, 2007}), and in this case, the model suggests that a higher accessibility reduces the pressure on the forest, presumably through the development of other economic activities and their effect on migration and the development of economic attraction poles.

In a manner analogous to the effect on f-f conservation, the net migration and the surface of agricultural crops increase the probability of transition f-oc. The variables density of rural population, proportion of rural population, and proportion of surface in ejidos and communities have an inverse behavior to the one discussed for the conservation of the forest (f-f ). In this group of sociodemographic variables, it stands out that the proportion of indigenous population increases the transition from forest to crops or urban development. The variable is not linked to the indigenous population that inhabits the forest, so its interpretation should be done with reserve.

The transition f-ig is the second most important one because of its magnitude, although it is quite limited to the States of Chiapas, Jalisco, Oaxaca and Estado de México. The transition is closely related to extreme events (fires) that apparently create large clearings, which are induced into forming grasslands, reason why it seems likely that some of these events are of anthropogenic origin, with this intention. Another variable directly related to this transition is the proportion of rural population, variable that shows that this transition is highly dependent of human pressure on the forest.

The transition f-s is not very common and its presence is higher in the States of Baja California, Coahuila, Guerrero and Querétaro. Among the important economic variables that define this transition, the ones that stand out are education (with inverse relation), road density (direct relation), net migration (inverse relation), and the proportion of surface with agricultural use. The higher road density promotes the transition surely as a result of the effect of the ill-handled change of land use. The model also shows that insofar as there is less arable surface, the transition f-s is more likely. These relationships strengthen the hypothesis that the transition f-s is linked to the abandonment of marginal cultivation lands that were originally forest (^{Chomitz, 2007}).

Transition from rainforest to other formations

The results of the adjustment of the system of equations related to the transition from rainforest to other formations showed that the prediction of transition from rainforest to crop (rf-oc), rainforest to induced grassland (rf-ig), and the permanence of rainforest (rf-rf) adjusted satisfactorily, and the adjustments were superior to those of the false transition rainforest to forest (rf-f) (Table 5).

Table 5 Estimations of the model of simultaneous equations of the transitions from rainforest to other formations. The response variable is the logit of the transition probability referred. 

The variation in the number of forest fires, the presence of economic poles, the presence of indigenous population and the consumption of firewood were the variables with greatest weight in the conservation of the rainforest (rf-rf). The temperature has less significance but is consistent with the result obtained for the transition rainforestcrop/urban zone and aligned with the presence of variations in the number of fires.

Consumption of firewood is a variable that is inversely related to the probability of the rainforest permanence. This relation evidences not only the degradation of the rainforest as a result of the extraction of firewood, but also the change of use with traditional cultivation systems (slash and burn) applied massively. The variable proportion of rural population was significant and had a positive effect, which is linked to the prior discussion. Among the economic variables, it was confirmed that inequity in the GDP, promoted by important economic centers, attracts workforce and decreases pressure on the change of rainforest use.

The transition rainforest-crop/urban zone (rfoc) depends on the temperature, the variation in disasters, and the precipitation. Results show that the transition increases in localities with higher precipitation up to the maximum limit of 1 050 mm, value from which the probability of transition is reduced. This interval is clearly related with the interval where agricultural productivity is higher; outside this interval, the desirability for land use change is reduced. The adjustment shows that when there is higher wealth, the probability of transition in a quadratic relation that resembles the Kuznets’ (^{Stern et al., 1996}) curve of environmental degradation is higher. Road density is another relevant variable that shows a direct relationship with the change in rainforest land use, in contrast with the behavior observed in temperate forests and in accordance with the one observed in other regions of the world for tropical forest (^{Chomitz, 2007}).

Other important economic variables to define the transition rainforest-crop/urban zone (rf-oc) are firewood consumption (discussed above); level of education, with a direct relation; and inequity in per cápita GDP distribution, with an inverse relation, which was discussed in prior paragraphs. Again, the variable net migration stands out with a positive relation, indicating that when there is greater presence of individuals who are not native to the zone, there is a higher impact on land use change. Another relevant variable is the proportion of the surface used for livestock production, which has an inverse relation with the transition from rainforest to crop and thus shows some substitution in land use. The contrary case occurs with the transition to induced grassland, where there is complementarity.

The higher proportion of rural population has a direct effect, which suggests the strong effect of population density on rainforest conservation. In this sense, it is observed that if the higher density is concentrated in ejidos or communities, then the effect of use change is reduced, presumably because this type of trend shows lower possibility of investment in the livestock production sector.

The transition rf-ig is reduced both with the increase in temperature and in precipitation. These variables impact directly on the reduction of livestock productivity. Other economic variables that are related to this transition are education (positive relation), inequity in per cápita GDP distribution (negative relation), road density (positive relation); all of them discussed in prior paragraphs. In this relation, it stands out that the behavior of the variable road density is quadratic, possibly because livestock production development is a secondary stage in land use when the states have greater infrastructure and better economic development. The proportion of agricultural use is an inversely related variable, which confirms a relationship of substitution between the agricultural and livestock use in the rainforest lands cleared.

The transition rainforest-forest, which is assumed to be a false relation, was considered within the system of equations because it has a high frequency and because the rainforests that are highly degraded in drier climates can originate transition formations. Nevertheless, the adjustment is the least satisfactory.

Validation of the model

The model includes the projection of all uses/ covers, but in this article, only the forest and rainforest covers are presented, as well as their transition to covers shown in Table 2. For these covers, a x² test was performed on the estimated p _ij and the p _ij obtained from Series IV (^{INEGI, 2012}). For the case of forests, the value of x²=0.0075 (a30.999) was obtained and for the case of rainforests, x²=0.0083 (a30.999). In both cases, there is no evidence to reject the hypothesis that the transition probabilities predicted are not different from those observed, so it is valid to point out that the prediction model through transition probability matrices results in predictions that are statistically similar to the values observed.