For some predominately phosphorus (P)-limited ecosystems, vegetation can be sustained under steady state conditions by atmospheric P inputs. The structure of the canopy influences deposition via the ability of the canopy to trap airborne P. This dependence suggests that a positive feedback may exist, which would have important impacts on the process of forest regeneration. One source of disturbance in the tropics is shifting cultivation. Over multiple cycles, studies have shown that shifting cultivation can lead to a large net loss of soil P, with a concomitant decline in forest biomass. In this study a model was developed to assess how shifting cultivation affects vegetation and phosphorus dynamics. This model is applied as a case study to a dry tropical forest system in the Southern Yucatan that is primarily P limited and has experienced shifting cultivation over several decades. Results show that following the second cycle, recovery would only be achieved after 100 years or longer in comparison to ∼30 years after one cycle. Examining the stable states of the system suggests that two stable states exist and that state changes as brought about by repeated disturbance can cause a shift to the other “low vegetation” stable state. Thus, for predominately P-limited ecosystems undergoing repeated disturbance, the depletion of soil P can significantly affect the long-term ability of the forest vegetation to recover. Results from this study have widespread implications, as 400 million ha of forest are affected by shifting cultivation.
 When undisturbed, ∼40% of Earth's tropical and subtropical landmass could be dominated by forest, with 42% of this area consisting of dry forest [Murphy and Lugo, 1986. However, the areas occupied by tropical dry forests have historically been attractive locations for human settlement and development, which is reflected by the significant decline in their spatial extent, by 48.5% globally and by 72% in North and Central America [Hoekstra et al., 2005; Portillo-Quintero and Sanchez-Azofeifa, 2010]. Dry tropical forests are characterized by a seasonal pattern of precipitation with the timing, frequency and duration of dry periods largely dependent on latitudinal position [Murphy and Lugo, 1986]. The gross primary productivity of dry forests is only 50–75% that of wet, tropical forests, which indicates that water availability is one of the major factors limiting to growth [Murphy and Lugo, 1986]. Apart from being water limited during the dry season, some of these forests have also been found to be phosphorus (P) limited.
 P limitations in ecosystems can emerge for many reasons [Vitousek et al., 2010], some of which include: the forests being found on highly weathered soils that are low in nonoccluded, inorganic P [Walker and Syers, 1976]; the forests being found on highly calcareous soils where P is chemically bound to calcium and clay constituents that have a high capacity for P fixation [Vitousek, 1984; Silver, 1994]; and the forests being found on soils such as sandy Spodosols that have low soil P stocks [Davidson et al., 2004]. The reasons for the P limitations described above are related to a lack of plant-available P in some of the forests located in the neotropics, which is the tropical region of the Americas that includes parts of Mexico, the Caribbean, and Central and South America [Lugo and Murphy, 1986; Campo and Vazquez-Yanes, 2004; Read and Lawrence, 2003a].
 For P-limited ecosystems, vegetation can be sustained in the long term by atmospheric inputs (e.g., rainfall, dust or fog) that contribute to the stock of plant-available soil P [Crews et al., 1995; Chadwick et al., 1999; Pett-Ridge, 2009]. The structure of the forest canopy can influence the deposition of atmospheric P via the ability of the canopy to trap airborne P, thereby promoting a positive feedback between the presence of the forest canopy and the productivity of the system [Kellman, 1979; Das et al., 2011]. This feedback is supported by certain types of atmospheric P transport mechanisms. For instance, dry deposition of P, which is associated with dust and smoke particles, can be enhanced by canopy characteristics such as leaf area index (LAI) and canopy density and structure that intercept and strip these airborne particles from air masses moving through the canopy. Wet deposition is also affected by these canopy characteristics that control the surface area available for water condensation [Lovett, 1994]. P inputs associated with dry deposition, canopy condensation and fog deposition are then transported to the forest floor via stemflow or by dripping down from the canopy. The dependence of system productivity on canopy structure suggests that an important positive feedback may exist [DeLonge et al., 2008].
 Systems affected by positive feedbacks are known for their ability to exhibit multiple stable states (i.e., a state of maximum vegetation density and a bare ground/alternate system state under disturbed conditions) [Wilson and Agnew, 1992]. When a disturbance is imposed on the system, a shift to the other stable state (i.e., the unvegetated or low-vegetation state) can take place once a threshold is passed [e.g.,Kuznetsov, 1995]. Once this shift occurs, the state dynamics remain locked in this state. More specifically, by reducing the rate of P deposition and without the use of any intervention to enhance P (i.e., fertilizers), the removal of forest vegetation may lead to a permanent shift to a state with low vegetation cover, whereby the reduced supply of P from atmospheric deposition prevents the regrowth of a dense forest canopy. This suggests that disturbance of tropical dry forests could result in state shifts in the underlying vegetation dynamics owing to the control that vegetation has on P cycling. In fact, it has been hypothesized that many stable savanna, and grassland ecosystems in the tropics originated from disturbed dry tropical forests [Murphy and Lugo, 1986].
 One important source of disturbance in tropical dry forest systems is deforestation due to shifting cultivation, which affects ∼400 million ha or 27% of the planet's arable land [Kleinman et al., 1995; Seubert et al., 1977; Giardina et al., 2000]. Shifting cultivation consists of clearing forests by slash and burn, using the land for cropping, and, leaving the land fallow during which time the forest regenerates. The number of years in a cycle is specific to the system under consideration and the length of a given cycle can vary considerably over time. This sequence of land use changes is then repeated over multiple cycles.
 Following biomass burning during the slash-and-burn phase, there is an immediate, large P input to the soil via P transported in ash and other deposited organic material as well as transformations in the soil [Lawrence and Schlesinger, 2001]. This increases the productivity of the land compared to conditions prior to the burn and can initially sustain increased crop growth and yields over the first several cycles of biomass burning, cultivation and fallow [Nye and Greenland, 1960, Seubert et al., 1977; Hughes et al., 2000; Steininger, 2000; Lawrence and Schlesinger, 2001; Lawrence, 2005; Lawrence et al., 2007]. However, during burning there is also a net loss from the system of both P stored in vegetation (smoke and airborne ash) and soil P lost to wind and water erosion [Khanna et al., 1994; Raison et al., 1985; Kauffman et al., 1993]; see Figure 1. Moreover, the export of harvested crops following cultivation and an increase in erosion and soil P leaching when the vegetation cover is removed further contribute to P losses from the system (Figure 1) [e.g., Stoorvogel et al., 1993]. These losses can be compensated by atmospheric P deposition if the fallow period is long enough. However, following multiple cycles of cultivation, the carrying capacity of the system can decrease because the initial store of available soil P has decreased substantially by up to 44% in the case of the Yucatan after 3 cycles (see Lawrence et al.  and Figure 2) and 20% in the case of Borneo after 5–6 cycles [see Lawrence, 2005]. These dynamics suggest that a decreasing periodic pattern of biomass density may emerge that is dependent on the time period over which cultivation-fallow cycles occur, the amount of available soil P, and the intensity of atmospheric deposition mechanisms that might replenish soil P.
 It is uncertain whether the cause of this decline in plant-available P is due to state changes in the coupled dynamics of vegetation and P cycling or to the relatively short duration of the fallow period [Lawrence and Schlesinger, 2001; Lawrence et al., 2007]. For instance, if the system is severely disturbed, it may cross over a critical threshold and undergo more permanent and irreversible transformations from which it could only recover with difficulty [Lawrence et al., 2007; DeLonge et al., 2008]. However, too short a fallow period, which can result from increasing population pressures, agricultural demand, a shift in labor availability or government subsidies [Lawrence et al., 2010] would truncate the time period over which P deposition is enhanced by the canopy and thus, would not fully restore the available soil P pool [Chapin et al., 2002]. For instance, in the neotropical dry forests of the Yucatan it has been estimated that biomass recovery of forests following cultivation requires a minimum of 55–95 years [Read and Lawrence, 2003b]. This suggests that if the length of the fallow period is sufficiently long, the system can recover following periods of cultivation.
 In this study, we develop a model to examine the consequences of shifting cultivation over multiple cycles (i.e., greater than three) on the system's internal (i.e., P coming from leaching, detritus or plant biomass) and external P fluxes (i.e., P coming from atmospheric sources that are external to the region) and on the productivity of the system itself. We also use this model to look at the recovery of the system from disturbances and to investigate its trajectories under different shifting cultivation scenarios. Finally, we use the model to assess whether the cause of this decline in plant-available P is associated with state changes in the coupled dynamics of vegetation and P cycling.
 In this study, we expand upon a model developed in the work of DeLonge et al. of vegetation-soil P interactions, which relates these interactions to the emergence of multiple stable states in vegetation dynamics. In contrast toDeLonge et al. , we use the model developed in this paper to look at the transient behavior of a system that is repeatedly forced by anthropogenic disturbances associated with slash and burn cultivation and examine how this affects the system's convergence rate to its attractors. We first present a model for vegetation biomass, which accounts for the control that vegetation has on soil P dynamics and for the limiting nature of available soil P on the amount of vegetation biomass that is supported. Next, we couple this model with a model of the soil P balance that relates the change in plant biomass on an annual timescale to changes in soil P concentration. An annual timescale is chosen because we are interested in examining the response of the system to disturbance and the ability of the system to recover from such disturbance over timescales ranging from 101 to 102 years. Moreover, this timescale was selected because it enables us to capture the critical P fluxes resulting from disturbance. Although we do not explicitly model the interaction between inorganic and organic P pools, we do consider available P as the sum of nonoccluded inorganic phosphorus, PI, which makes up the NaOH and bicarbonate extractable inorganic fractions and bicarbonate extractable organic phosphorus, PO (Figure 1). To account for the effect of fire intensity on internal and external P fluxes, we include a term to model P losses due to biomass burning and also consider inputs due to previously unavailable pools that were mineralized during fire.
2.1. Vegetation Dynamics
 Changes in vegetation biomass, V, can be modeled through a growth death process in which the net growth rate is expressed by the logistic equation that has been modified to include a “harvest term” as follows
where Vc is the maximum biomass of vegetation in the ecosystem, Vc − V represents the remaining net growth potential and r (t−1) regulates the temporal response of the vegetation's growth. In this equation, V is canopy biomass (Mg ha−1) and r differs between the forested and agricultural state. The Kronecker's delta function (δt,ti = 1 if t = ti and δt,ti = 0, otherwise) is used to define the times, ti, when fire occurs. The solution to equation (1) for the case study used in this paper is presented in section 3.
 To account for the limiting nature of phosphorus on vegetation, the ecosystem's carrying capacity, Vc, is expressed as in the work of DeLonge et al. ,
where Pcr is the minimum value of soil P (Ps) required for the existence of forest vegetation and Vmax is the maximum amount of vegetation supported by the system (Figure 3). While some recent studies have documented the possible existence of nitrogen limitation at the early stages of old field succession [Davidson et al., 2004], we do not consider colimitations by N in this model because of the longer timescales over which P limitation is likely to persist. This difference is also driven by the potential for biological fixation of N, which may be augmented in warm tropical soils by abundant microbial N fixation [Vitousek, 1984].
2.2. Phosphorus Dynamics
 To account for the temporal variability of soil P inputs (kg P ha−1 yr−1) and outputs (kg P ha−1 yr−1), we express the soil P (Ps) balance as
where pin is the rate of P input to the soil pool, and pout is the rate of P loss due to leaching, erosion, crop harvesting, and net plant P uptake (Figure 4). We expand upon the P model in the work of DeLonge et al.  by also including inputs due to fire, fertilizer and weathering and outputs due to fire, harvest, erosion, and leaching.
2.2.1. Phosphorus Inputs to the Soil Pool
 For an undisturbed system (i.e., not affected by logging or fire), the major long-term P inputs to the soil pool are via wet and dry atmospheric deposition. In contrast, when a system is disturbed by slash and burn, a fraction of the P stored in plant biomass is returned to the soil pool. Given that this model is run on an annual timescale, we do not explicitly account for the input of P in litterfall and only consider the net uptake of P from the soil, which is the difference between the amount that is taken up by the plant and the amount that is recycled back to the forest floor in litterfall. Thus, P inputs (pin) are expressed as
where patm is atmospheric deposition of P, which includes the deposition in rainfall, fog deposition and dry deposition of airborne dust and exogenous biogenic particles (i.e., not from biomass internal to the system), pburn is the fraction of plant P that returns to the soil locally (i.e., not transported by wind as ashes) during a fire, pfert is the input due to fertilizer during the cultivation period (when applicable), and pweath is the input of P due to weathering, which is taken as a constant. Consideration of pweath is important because nearly all the phosphorus in terrestrial ecosystems is originally derived from the weathering of calcium phosphate minerals, particularly apatite [Schlesinger, 1997].
and α expresses the strength of the vegetation deposition feedbacks and β represents deposition in the absence of a canopy [DeLonge et al., 2008] (Figure 5). Because the leaf area, which is the major control on trapping potential, reaches a saturation point with age, the amount of P deposited atmospherically approaches some constant value when the maximum value of LAI is reached.
 The solubility of P deposited atmospherically can vary widely, ranging from 7 to 100% [Mahowald et al., 2008]. For instance, phosphorus solubility derived from Saharan mineral aerosols averages 10% [Baker et al., 2006]; however, for areas affected by anthropogenic activities such as biomass burning, the soluble fraction can be greater than 75% [Mahowald et al., 2008]. Anthropogenic activities can strongly affect P solubility because anthropogenic activities can increase the acidity of the atmosphere and aerosols in the atmosphere [Nenes et al., 2011]. In turn, this can increase the solubility of phosphorus from soil-derived minerals [Nenes et al., 2011]. In fact, laboratory experiments that simulated atmospheric conditions by acidifying Saharan dust samples produced a minimum of a tenfold increase in soluble P [Nenes et al., 2011].
 The amount of P deposited locally to the soil in ash and other residual organic matter as a result of biomass burning is dependent on the amount of P stored within biomass (Pveg). Pveg is expressed using an average stoichiometric ratio (q). A single value is used despite the fact that this will underestimate Pveg initially and then overestimate Pveg as the forest matures because it will largely average out the plant P content per unit plant biomass over the lifespan of the vegetation. Thus, we do not account for the difference in species composition between the mature and secondary forest, and for changes in species composition in the secondary forest. Given the above, we express Pveg as a linear function of plant biomass as
When a burn occurs, the phosphorus stored in plant biomass, Pveg, is partly lost to the atmosphere as pyrogenic P emissions in smoke or airborne fine ashes, and the remainder is returned to the soil pool. The total amount of P that is lost atmospherically both during and immediately following a fire (pfire) is expressed as
where ω is the fraction of plant biomass that is consumed by fire and g is the fraction of Pveg that is released into the atmosphere both during and immediately following a fire. pburn is the fraction that is distributed to the soil,
Thus, pfire is a net loss from the system during combustion of biomass. This term could be increased to account for other exports from the system such as logging, fuel wood extraction, or charcoal production.
 The fraction of Pveg lost during biomass burning, as represented by the parameter g, can account for a major net loss from a slash and burn system, the magnitude of which is partially dependent on the spatial scale under consideration [Kauffman et al., 1993; Giardina et al., 2000]. Additionally, the value of g is highly variable depending on the percentage of total biomass that is consumed by fire (i.e., ω). For instance, Kauffman et al.  found for a controlled burn in a neotropical dry forest where 78% of total aboveground biomass was consumed by fire (i.e., ω = 0.78) that only 3.5% of aboveground plant P was lost (i.e., g · ω = 0.027). In contrast, during a burn where 95% of the total aboveground biomass was consumed by fire (i.e., ω = 0.95), there was a net loss of 56% of the aboveground biomass P pool (i.e., g · ω = 0.53). These losses are not only local, yet also occur at the regional scale. For instance, Mahowald et al.  observed that biomass burning in the Amazon Basin has led to a substantial net loss of P (at a rate that is about 19% of the gross deposition rate) throughout the Basin. However, it is important to note that the value of g would need to be reduced if the controlled burn occurred in an adjacent, upwind parcel. Any ash that is input from local, but nonadjacent fires is captured in patm. Because the value of b in equation (5) should be determined directly from the study area of interest, this term will capture the amount of P that is deposited locally in ash.
2.2.2. Phosphorus Outputs From the Soil Pool
 Phosphorus losses from the soil rooting zone, pout, are expressed as
where pleach is the loss due to deep leaching from the soil rooting zone, perosionis the loss of P to erosion; including the instantaneous net loss of soil P immediately following a fire (i.e., runoff minus run-on due to both water and wind transport),pnet is the net uptake of soil P by vegetation, and pcrop is the loss due to uptake by crops and removal of this crop from the system (Figure 4). In general, leaching losses of P are low because it is not desorbed easily [Chapin et al., 2002]. Because of differences in the rooting structure of younger forests versus older forests, P leaching losses do vary as a function of forest age [Lawrence et al., 2007; DeLonge, 2007]. To account for the dependency of leaching losses on the rooting structure of forests, we express (pleach) as a normalized linear function of V with respect to Vmax,
where λis a term to account for the vegetation-leaching feedback andκ represents the rate at which Ps is leached from the rooting zone in the presence of maximum vegetation cover.
 We express the net uptake rate (i.e., the net rate at which P is assimilated into forest biomass (pnet)) as being proportional to plant growth,
By expressing pnet as proportional to the vegetation growth rate and by modeling growth with the logistic equation (i.e., equation (1)) we account for the fact that nutrients are accumulated in plant biomass most rapidly during the early development of forests and more slowly as the aboveground biomass reaches a steady state [Gholz et al., 1985; Pearson et al., 1987; Reiners, 1992; Schlesinger, 1997].
 The uptake of P by crops is dependent on several factors including: (1) the type of crop and its P demand; (2) the amount of available soil P (i.e., Ps); and (3) soil characteristics such as texture, and water content [Mackay and Barber, 1985]. For this model, we assume that the primary limiting factor is available soil P [Read and Lawrence, 2003a]. Given this, we express the net uptake of P removed in crops (pcrop) as shown in Figure 6,
where Preq is the minimum amount of soil P required for crop growth in the area being considered (kg P ha−1), pmax is the maximum crop P uptake rate (kg P ha−1 yr−1), and Plim (kg P ha−1) is the amount of P that needs to be available to ensure that plant roots can access enough P to maintain their typical crop yield [Gavito and Miller, 1998]. This constraint is utilized because not all available P is accessible to the plant root owing to the slow diffusion of P ions in the soil [Schenk and Barber, 1980]. Additionally, the value of Preq is lower than Pcr in equation (2) because crops have a shallower rooting depth relative to trees. The values of Preq and Pcr will also differ depending on the plant P requirements.
 Other P losses are due to erosion because P is tightly bound to organic matter and soil minerals [Chapin et al., 2002]. Instantaneous erosion includes net losses of P adsorbed to mineral soil minus the addition due to pyromineralization (first term in equation (13)). Fire has been found to increase the amount of available P from more resistant forms of P (that are not considered in this model) in the top 10 cm of the soil [Saa et al., 1993; Kauffman et al., 1993; Lawrence and Schlesinger, 2001]. The input due to pyromineralization can offset pyrogenic losses of soil P to erosion and in some cases, can increase the amount of bioavailable soil P. Thus, although we do not explicitly model the dynamics of more resistant P pools, we do consider the transformation immediately following fire to more available forms as an input to the available pool. The amount of soil P (Ps) both lost to erosion and lost (or gained, if pyromineralization exceeds erosion) immediately following a fire owing to the combined effect of erosion and pyromineralization is expressed as
where ti are the times of fire occurrence, δt,ti is the Kronecker's delta function (δt,ti is 1 for t = ti, and is 0, otherwise), his the fraction of soil P that is instantaneously lost (or gained) from a fire (i.e., mineral soil lost/gained from the soil pool after fire-induced removal of forest cover) andυ is the time constant of soil P lost to erosion and runoff. For the system presented as a case study in this paper, the presence of seasonal wetlands suggests that during the wet season, runoff from saturation overland flow from forested areas is likely to contribute to P losses. Thus, erosion losses (Figure 4) are accounted for separately from the instantaneous losses, pfire, due to biomass burning (see equations (9)–(10)).
2.3. Equilibrium States of the System
 In order to investigate the potential for state changes, the equilibrium states of the dynamics are obtained by setting the temporal derivatives equal to zero on the left-hand side ofequations (1) and (3) for (δt,ti = 0. By setting the temporal derivative of equation (1) equal to zero, we find that V can be equal to zero or Vc, with Vc being a function of Ps (equation (2)). Similarly, by setting the temporal derivative of equation (3) equal to zero so that patm + pweath = pleach + perosion (i.e., when the system is in steady state) and rearranging these terms to solve for V, we obtain a second equation relating V to Ps. This equation is expressed as,
The equilibrium states of the system satisfy both equations and can therefore be obtained as intersections of the corresponding curves in the (Ps, V) domain. The equilibrium solutions for the case study presented in this paper are discussed in section 4 and shown in Figure 10.
2.4. Summary of the Control Vegetation Has on P Dynamics
 The rate at which vegetation in forests stores P (i.e., the net uptake rate) declines as the forest matures (i.e., when V → Vmax, pnet → 0), thereby leaving excess soil P available within a matured system to support continued productivity. For a mature forest system, the rate of atmospheric deposition is at a maximum owing to the positive feedback between canopy structure and P deposition (i.e., more canopy trapping than with bare soil or secondary forest), while the amount of soil P being leached from the rooting zone is at a minimum because of the dense rooting system of mature forests (i.e., less water percolates beyond the rooting zone). Thus, when a mature forest is present, P inputs are maximized while outputs are minimized. In contrast, removal of the forest and replacement with an agricultural system tends to maximize outputs while minimizing inputs. In fact, atmospheric deposition is reduced after the removal of the canopy, while leaching beneath the rooting zone increases, owing to both a decrease in transpiration and an increase in macropore flow along decayed or incinerated root paths. The bare soil and reduced vegetation cover also increases the susceptibility of soil P to erosion losses. Finally, a large percentage of the initial input of P to the soil from biomass burning will be lost to wind or water erosion owing to the high percentage of P contained in ash following a burn [Kauffman et al., 1993]. Other losses result when crops are harvested and the P stored in crops is transported away from agricultural fields. Thus, the removal of vegetation can alter the soil P budget from a net gaining state to a net losing state.
2.5. Model Parameterization
 To parameterize and test the model, we use vegetation biomass and soil P data from a P-limited area of the Southern Yucatan peninsular region (18°49′N, 89°23′W; mean elevation: 250 m) composed of seasonally dry tropical forests (mean annual precipitation of 890 mm yr−1) [DeLonge, 2007; Lawrence and Foster, 2002]. Available data from this area suggests that apart from water limitations during the dry season, P is the most limiting nutrient. Read and Lawrence [2003a]note that the high P-use efficiency, increasing with age, the relatively abundant nitrogen (N) and the lack of change in N use efficiency with age suggest that P is the primary limiting nutrient for this system. Additionally,Campo and Vazquez-Yanes  found that the biomass growth of young forests in the Yucatan Peninsula responded primarily to P in fertilization experiments with N and P. Although it has been found that N can become a limiting nutrient following fire because of the high N losses to volatilization [Davidson et al., 2004], for the system presented as a case study in this paper, available data suggests that both prior to and following disturbance, the system is predominately P limited. Vegetation in this area is characterized by tropical semievergreen forest in the upland areas and low seasonal wetlands (bajos) [Xuluc-Tolosa et al., 2003; DeLonge, 2007]. These two classes of vegetation are found on distinct soil types; mollisols in the upland areas and swelling clay vertisols in the lowlands [Turner et al., 2001]. This area has experienced land development since the 1960s and consists of both secondary and mature forest, mixed with fragments of land that are currently being used for agricultural purposes [Turner et al., 2001].
 Available data from this area suggests that on average, forest parcels have undergone three cultivation-fallow periods. The cultivation-fallow period consists of an ∼3 year cultivation cycle followed by a 12 year fallow period [Schmook, 2010; Klepeis et al., 2004]. Thus, under baseline conditions, a full cycle represents 15 years. For this case study, we use maize as the type of crop because it is typical of the region being considered [Klepeis, 2000]. To examine how the length of the fallow period affects these dynamics, we use parameter values for this system that are presented in Table 1 and examine the following options: reducing the cultivation period from 3 years to 1 year (i.e., one full cycle equals 13 years), increasing the length of the fallow period from 12 to 25 years (i.e., one full cycle equals 28 years), combining both of these options (i.e., one full cycle equals 26 years), and reducing the length of the fallow period from 12 to 6 years (i.e., one full cycle equals 9 years). The last option is examined because over the last 5–10 years, the study area has experienced a decrease in the length of the fallow period, from 12 years to 6 years, while the length of the cultivation period has remained largely unchanged [Schmook, 2010].
Table 1. Values of the Model Parameters Used for the Baseline Case
 We also examine the sensitivity of the model to fire using different values of g and ω that were obtained from Kauffman et al. . For the high–fire efficiency case, where 95% of the total aboveground biomass is consumed by fire, there was an instantaneous net loss of 56% of the aboveground biomass P pool. In contrast, for the low–fire efficiency case where 78% of total aboveground biomass was consumed by fire, there was an instantaneous net loss of 3.5% of the aboveground plant biomass P pool. The large losses of P immediately following a burn are also in part due to the high percentage of P that is contained in ash, which is very susceptible to entrainment and removal by air and water flow during and after the fire. Following a burn, Kauffman et al. found that 84–94% of the total aboveground P was contained in ash and 57% of this P contained in ash eroded within 17 days following a burn. We account for these losses by adjusting the parameter g. Thus, for the high-efficiency case, g was set equal to 0.79 andωequal to 0.95 and for the low-efficiency case, g was set equal to 0.53 andωequal to 0.78. Our field observations suggest that for systems undergoing multiple cycles of slash and burn, the initial burn oftentimes does not consume all of the aboveground biomass, yet consumes this residual aboveground biomass during the subsequent period of burning. For our baseline case, we use an erosion-adjusted value for g that is similar toMahowald et al.  (i.e., g = 0.60) for the first burn and distribute 15% of the total aboveground biomass P during the second cycle to account for the efficiency lag. For subsequent burns, the parameter g is set equal to 0.79 to account for the fact that the majority of vegetation is consumed by fire following the first burn (i.e., ω = 1).
3.1. Model Performance
 Results from parameterizing and applying the model to the system used as a case study suggest that the biomass for a mature forest is 142 Mg ha−1, for a 10 year old forest following one cultivation cycle is 30 Mg ha−1, for a 10 year old forest following two cultivation cycles is 18 Mg ha−1, and for a 10 year old forest following three cultivation cycles is 14 Mg ha−1 (Figure 7a). Although we use a 12 year fallow period, we compare data from above with our model results for the same aged forest and over the same number of cultivation cycles. Data suggests that the average biomass for a mature forest in the Southern Yucatan Peninsula, which has not been logged for at least 60 years, is 142 Mg ha−1 [Read and Lawrence, 2003b]. The average biomass for a 10 year old forest with one cultivation cycle is 36 Mg ha−1 (ranging from a high of 51 Mg ha−1 to a low of 30 Mg ha−1, n = 10), the average biomass for a 10 year old forest with two cultivation cycles is 21 Mg ha−1 (ranging from a high of 28 Mg ha−1 to a low of 13 Mg ha−1, n = 4), and the average biomass for a 10 year old forest with three cultivation cycles is 13 Mg ha−1 (ranging from a high of 15 Mg ha−1 to a low of 8 Mg ha−1, n = 3). These results suggest that the model accurately captures changes in biomass for this area as a function of cultivation cycle.
 Results for the soil P dynamics also suggest that the model accurately captures observed trends as a function of forest age and cultivation cycle. Results from the model indicate that the soil P content under a mature forest is 179 kg P ha−1, beneath a forest following one cultivation cycle is 171 kg P ha−1, beneath a forest following two cultivation cycles is 152 kg P ha−1, and beneath a forest following three cultivation cycles is 132 kg P ha−1 (Figure 7b). Data presented in Figure 2 and extrapolated throughout the extent of the rooting zone indicate that the average available P in the soil (0–50 cm deep) under a mature forest is 193 kg P ha−1 (ranging from a high of 207 kg P ha−1 to a low of 180 kg P ha−1, n = 3), under a forest following one cultivation cycle is 169 kg P ha−1 (ranging from a high of 226 kg P ha−1 to a low of 134 kg P ha−1, n = 10), under a forest following two cultivation cycles is 167 kg P ha−1 (ranging from a high of 197 kg P ha−1 to a low of 138 kg P ha−1, n = 4), and under a forest following three cultivation cycles is 116 kg P ha−1 (ranging from a high of 134 kg P ha−1 to a low of 98 kg P ha−1, n = 3).
3.2. Consequences of Shifting Cultivation Over Multiple Cycles Beyond the Current Cycle
 Given the ability of the model to accurately simulate observed conditions over three cultivation-fallow cycles, we then used the model to examine vegetation dynamics over subsequent cycles using the same length for the cultivation period and the same length for the fallow period (Figures 7a and 7b). Results show that the system stays in a state of low-vegetation biomass following the present cycle (i.e., the third cycle). In fact, extending the length of the fallow period during the fourth cycle, suggests that the system has difficulty recovering to its precultivation biomass within 75 years. Additionally, recovery occurs much more slowly than the growth of the precultivation mature forest. The amount of time that it takes for the forest biomass to recover to 75% of the plant biomass values typical of the mature forest increases as a function of the number of cultivation cycles (Figure 8).
3.3. Measures to Mitigate P Losses From the System
 Results from altering the length of the cultivation and/or fallow periods suggest that although vegetation biomass may differ at the end of the third cycle for each of these different sets of simulations, by the sixth cycle, the vegetation has difficulty recovering and largely converges to a bare/low vegetation state regardless of the strategy employed (Table 2). Table 2 illustrates how these different measures may affect the ability of the vegetation to recover with respect to the baseline case. Because the vegetation has difficulty recovering followed repeated deforestation in the baseline case, Table 2 illustrates that the vegetation has difficulty recovering in all cases.
Table 2. Changes in V and P Dynamics Relative to the Baseline Case as a Function of Changes in the Cultivation and/or Fallow Perioda
VA (Mg ha−1)
VB (Mg ha−1)
VC (Mg ha−1)
VD (Mg ha−1)
PA (kg ha−1)
PB (kg ha−1)
PC (kg ha−1)
PD (kg ha−1)
Superscripts are as follows: A, the option where the cultivation period is decreased from 3 years to 1 year; B, the option where the cultivation period remains the same but the fallow period is increased from 12 to 25 years; C, a combination of A and B; D, a reduction in the fallow period from 12 to 6 years while keeping the cultivation period the same.
 The efficiency of the controlled burn, as described by the amount of biomass that remains following a fire, has been cited [Hernández-Valencia and López-Hernández, 2002; Kauffman et al., 1993] as one controlling factor that determines the net loss of P from the system. We examine how changing the parameter, g and the amount of vegetation that is burned during each cycle (ω) influences the recovery of vegetation in subsequent cultivation cycles. For the case presented in Figure 9 (i.e., lower fire efficiency), there is a greater recovery of vegetation biomass relative to the baseline case during each cultivation cycle, which is sustained over four cycles. For instance, at the end of the 12 year fallow period during the fourth cycle, the lower–fire efficiency system supports 34 Mg ha−1 of biomass, whereas the higher–fire efficiency system supports 18 Mg ha−1. Following four cycles, the increased loss of P from the system is reflected in the amount of available soil P. For the system with the higher burn efficiency, available soil P at the end of the fourth cycle is 111 kg P ha−1, whereas the configuration with the lower burn efficiency has 131 kg P ha−1 at the same time. While these results show that the model is sensitive to the parameters controlling the amount of P lost during a fire, the results obtained from the model are still likely to be conservative as the model overestimates the amount of soil P available during the third cycle.
 Results from examining the P and V trajectories over additional cycles illustrate that this system is on the verge of a runaway effect: an increase in cultivation cycles leads the system into a degraded state from which it only recovers with difficulty. Similarly, extending the length of the fallow period during the second cycle, while maintaining the length of the previous fallow periods at 12 years suggests that vegetation biomass does not recover within 75 years (Figure 8). In fact, following the first cycle there is a marked change in the amount of time that it takes for the forest to recover to 75% of the precultivated state. This downward spiral in the vegetation dynamics following the first cycle illustrates the fragility of the system and further suggests that the system may undergo critical transitions during subsequent cultivation cycles.
 The potential for changes in the system state are investigated by examining the intersection of equation (1) at steady state and no slash and burn (i.e., (δt,ti = 0) with equation (14). Bistable dynamics are evident for this system and are shown by the presence of three intersections that represent three equilibrium points in a system with two stable states separated by an unstable state (Figure 10). The fully vegetated state is stable because the presence of forest vegetation enhances the amount of P trapped atmospherically and minimizes P losses due to leaching. In contrast, the low-vegetation state is stable because P losses are relatively high and the amount of P trapped atmospherically is at a minimum. If the system is at one of the two stable equilibrium points it will remain in that state unless perturbed. Conversely, if the system is not at a stable equilibrium, it will converge to one of the stable states, depending upon the initial conditions of the system with respect to the divide between the basins of attraction of the two stable states. The rate at which P is lost from the system affects the system's resilience such that each disturbance imposed on the system reduces the magnitude of disturbance required to push the system into the low-vegetation stable state. Importantly, once the low-vegetation state is reached, the dynamics would remain in that state as long as there is no intervention (i.e., the use of fertilizer), even after the cessation of the disturbance. Moreover, a shift to the low-vegetation state could negatively impact agricultural production for systems such as the one presented in this paper that do not have access to fertilizers that could be used to replenish soil P.
 The rate at which the system converges to one of these attractor basins (i.e., the set of initial conditions that leads to the system approaching a given stable state) [Strogatz, 1994] is dependent on the magnitude of P losses from the system. We examined two factors that would control the convergence rate to the attractors: altering the length of the cultivation/fallow period and examining the aboveground vegetation P losses following a burn.
 Reducing the length of the cultivation period and extending the fallow period does not seem to have the positive benefit that may be expected. This result is due to the larger net loss of P that results from a mature/older secondary forest following biomass burning relative to a younger secondary forest. The higher net loss of P from the mature forest is because more P is stored in the mature forest than the successional forest, which occurs despite a lower value of g being used for fire in the mature versus secondary forest. The decrease in soil P following the increase in the length of the fallow period reflects the increased net uptake (i.e., uptake minus the amount returned to the soil as leaf litter) and storage of P during the fallow period and subsequently, the large loss of Pvegduring biomass burning. In fact, forest regeneration requires uptake of soil P until a mature forest state is attained. During forest regeneration, losses of available soil P initially exceed the phosphorus input by deposition (up to a certain point in time) despite the effect of deposition enhancement by the forest canopy. If the forest is harvested before the soil P balance becomes positive, each cultivation cycle causes a depletion in soil resources and the longer fallow period (with respect to the baseline case) may reduce soil available P owing to the high P cost of regenerating the forest vegetation. This suggests that although a long fallow period benefits soil nutrient status via canopy-enhanced P deposition and reduced leaching losses, which aid in offsetting the initial P losses due to uptake during forest regeneration, unless losses can be reduced during biomass burning, the benefits of the long fallow period are not realized. These findings were also supported by results from decreasing the fallow period from 12 years to 6 years (Table 2).
 While these results may hold true for fallow periods less than 25 years; this would likely not be the case for extremely long fallow periods. For this system, the soil P budget for a growing forest goes from a net losing state to a net gaining state at year 25, with a 30 year old forest gaining ∼0.25 kg P ha−1 yr−1. Thus, to simply replace the loss of P from 3 years of harvest (i.e., pcrop × 3 years = 6 kg P ha−1) requires a total of 51 years of fallow. This result was obtained by calculating the annual net nutrient flux (i.e., inputs minus outputs) and determining the total number of years that it would take to replace the P lost in 3 years of harvest. Moreover, offsetting the total loss of P following the initial burn would require a fallow period of 94 years once the system attains a net gaining state. Similarly, this result was obtained by calculating the annual net nutrient flux (i.e., inputs minus outputs) once the system attains a net gaining state and determining the total number of years that it would take to replace the total loss of P following the initial burn. An input of fertilizer equal to 10 kg P ha yr−1 during the cultivation period would provide enough P to replenish soil P lost during forest regeneration and to offset losses due both to biomass burning and cultivation. Additionally, this is the minimum quantity of fertilizer that would enable the system to return to precultivation biomass in the same amount of time as the precultivated mature forest. However, for this area, adding P in fertilizer is currently not an economically viable option [Lawrence et al., 2007].
 The discussion above indicates that increasing the length of the cultivation/fallow period does not have the positive benefit that was expected, which is in part due to the high losses of P from Pveg. Increasing the fraction of vegetation that is consumed by fire also increases the amount of P (from Pveg) that is lost from the system, which sends the system into a more rapid downward spiral. Soil P dynamics for the higher-efficiency burn (Figure 9c) reflect the increased loss of P from the system and the reduced P pulse during the second cycle in comparison to the lower-efficiency burn (Figure 9d). The relative fraction of vegetation consumed by the fire influences the vegetation dynamics because this residual vegetation provides a pulse of P following the first cycle that is conserved for later uptake, which increases the amount of vegetation supported during the second cycle. However, these results are likely to underestimate the strength of removing additional vegetation cover as the loss of roughness elements (i.e., elements such as trees that create an aerodynamic resistance to flow) for higher-efficiency burns would increase the amount of P in ash that is eroded from the system. In summary, the efficiency of the burn is important because unburned residual biomass: (1) provides a later pulse of P during subsequent fallow periods; (2) reduces the amount of aboveground P that is lost from the system; and (3) reduces erosional losses of soil P, thereby sustaining higher levels of forest productivity over a longer period of time. Moreover, for any P-limited system undergoing biomass burning, conserving these losses is an important mechanism to maintaining the system's long-term productivity.
 While this paper had primarily focused on deforestation due to shifting cultivation, it is important to point out that other mechanisms of deforestation can lead to similar outcomes. For instance, in tropical forests such as the Amazon that were previously unaffected by fire, anthropogenic activities can lead to a significantly increase in the area's susceptibility to fire [Cochrane, 2003; Cochrane et al., 1999]. For P-limited systems, the increase in fire frequency would exacerbate P losses (as demonstrated in this paper) and possibly lead to state changes in the underlying P-V dynamics. Additionally, other forms of deforestation such as continued clear cutting, would also lead to significant decreases in P due to the net export of P stored in aboveground vegetation from the system.
 For systems undergoing multiple cycles of shifting cultivation, a decreasing periodic pattern of biomass density may result owing to state changes in the coupled dynamics of vegetation and P cycling. For the system used as a case study in this paper, it is unlikely that the forest will recover under its current path owing to the anthropogenic pressures that have been placed upon it. In fact, these results demonstrate that the system can undergo critical state changes and that the forest will only be able to recover with difficulty following the current cycle (i.e., the third cycle). This article lends support to the hypothesis that repeated disturbance of tropical dry forests can reduce the resilience of forest ecosystems and could ultimately lead to a highly irreversible shift without intervention (i.e., the use of fertilizer) to a degraded state with low-vegetation biomass and low soil P availability.
 This research was funded by the National Science Foundation (grant EAR0838218). We would also like to acknowledge comments provided by two anonymous reviewers, whose input greatly improved the manuscript.