About the Author

Stochastic Modeling Explores Relation Between Plants and Intermittent Rainfall and Fire

By Jillian Kunze

“Fires are events that occur once every so often, and how often that is depends on where you are on Earth,” Mara Baudena of CNR-ISAC said. “If you look at time series of rainfall, you also see this on-and-off nature.” The similarly intermittent qualities of these otherwise quite different natural events allow researchers to model fire and rainfall with similar mathematical techniques. During a minisymposium presentation at the 2022 SIAM Conference on the Life Sciences, which is taking place this week concurrently with the 2022 SIAM Annual Meeting, Baudena described the application of impulsive stochastic differential equations (SDEs) to the relation between plants, fire, and rainfall.

In this application, impulsive SDEs do not simply add a bit of white noise; the stochastic terms for parameters that relate to rainfall and fire take the form of delta-like pulses to represent these phenomena’s intermittency. Baudena detailed several applications for impulsive SDEs, starting with the influence of sporadic rainfall in drylands on plant cover. 

Figure 1. Numerical simulation of vegetation fraction versus annual rainfall under constant and intermittent rainfall. Figure courtesy of Mara Baudena [1].
Drylands—areas with arid, semi-arid, or dry-sub-humid climates—are fairly common and cover around 42 percent of the Earth’s land surface. These water-limited ecosystems are characterized by scarce, sporadic, and intermittent rainfall, as rain generally occurs in larger events and then does not reoccur for a long while. Climate change will affect these ecosystems not only by increasing temperatures, but also by altering the temporal distribution of water — there will likely be more extreme weather events and longer periods of drought. Understanding the impact of intermittent rainfall is quite important in this context. 

Baudena described a simple ordinary differential equation (ODE) model for soil water and vegetation cover in drylands. Soil moisture in a soil layer of a certain depth increases with higher rainfall infiltration rates, and both the rainfall intensity and the rainfall interarrival time are represented with stochastic, exponential distributions. The soil can then lose moisture through evaporation and transpiration. The equation for vegetation cover in drylands is connected to the soil moisture equation through transpiration, as well as the nonlinear dependence of the vegetation’s growth rate on soil moisture. 

This ODE model enabled a comparison of vegetation behavior under constant or intermittent rainfall; constant rainfall is a simpler approximation, while intermittent rainfall is a stochastic input that is more similar to a real time series that would be observed in nature. Figure 1 provides the modeled vegetation fraction versus annual rainfall, demonstrating that the average amount of vegetation in dry areas is larger when the rainfall is intermittent — certain plants are well-adapted for this kind of scenario. “It’s important to capture this intermittent rainfall in models,” Baudena said. “For example, in this model there is vegetation at values of annual rainfall where there would be no vegetation if we were to approximate this rainfall as a constant.”

Figure 2. Growth rate/water uptake versus soil moisture for a drought-resistant plant (dashed line), as well as a less drought-resistant plant with a higher optimal growth rate (solid line). Figure courtesy of Mara Baudena [2].
The talk’s focus then shifted to the way in which sporadic rainfall in drylands enables many different species of plants to coexist, even with the competition among them. Baudena was particularly interested in the effect of rainfall intermittency on the competition between two plants along an aridity gradient. These two plants have differing characteristics: one is more drought resistant, while the other is less drought resistant but has a higher optimal growth rate. 

Baudena graphed the growth rate of these two plant species (see Figure 2): the drought-resistant plant has a higher growth rate at lower levels of soil moisture, while the plant with the higher optimal growth rate fares better at higher levels of soil moisture. Figure 3 depicts the biomass versus mean average rainfall for the same plant species with three different rainfall interarrival times. Increased rainfall intermittency leads to increased coexistence of the plants due to their different strategies for water use, since they thrive at different times depending on the intermittent rainfall.

Baudena then moved to the talk’s second application of impulsive SDEs: modeling global fire ecosystems, in which the response of plants after fires is fundamental for determining the existence of different system states. “Despite fires being a threat to many human activities, they are an ancient phenomenon that have scorched the Earth almost as long as plants have been on Earth,” Baudena said. “Therefore, many plants are well-adapted to fires.” 

Figure 3. Biomass versus mean annual rainfall under different levels of rainfall intermittency for a drought-resistant plant (dashed line) and less drought-resistant plant with a higher optimal growth rate (solid line). Figure courtesy of Mara Baudena [2].

The existence of fire feedback mechanisms demonstrates how natural systems have adapted to fires. Tropical humid savannas and forests exist in the same climatic conditions, and the feedback leads to the existence of both states for the same climate. Grasses make good fuel for fires, so ecosystems with lots of grass like savannas have more frequent fires that damage any trees and prevent them from expanding. On the other hand, forests suppress fires through their humidity and by limiting grass growth.

With this in mind, Baudena and her collaborators built a simplified plant model that explores how plant characteristics can lead to the existence of different alternative ecosystem states in the presence of fires. Their model incorporated the fact that fires are influenced by climatic factors, as well as plant characteristics like flammability and fuel load. Plant responses to fires have not been researched as thoroughly as climatic factors. 

Baudena’s aim was to “look at how plant characteristics affect the existence of fire-prone ecosystems, with more than one type of ecosystem in the same kind of climate,” she said. She hoped to identify which plant communities are possible in different fire-prone ecosystems, given the plants’ characteristics. The model has two ingredients: it begins with a classical model for succession, onto which the researchers added stochastic fires that depend on the amount of vegetation cover and the flammability of the plants (see Figure 4). 

Figure 4. Feedback occurs between vegetation and stochastic fires: fires may change the type of plants in the ecosystem, which in turn affect how frequently fires occur. Figure courtesy of Mara Baudena and Marta Magnani.

“We analyzed, with sensitivity analysis, which plant characteristics are most relevant to find out the post-fire response of the dominant tree,” Baudena said. She showed several examples from different biomes that were parametrized with real data. The first example was a Mediterranean oak forest, which is well-adapted to fire and can bounce back relatively easily. However, climate change could potentially lead the system to transition to a very different biome—an open shrubland with pine trees—which would maintain more frequent fires. 

In the case of a North American boreal biome, the researchers observed cycles between two types of spruce and fir forests; the different dominant trees have different responses to fire, reflecting real data from those ecosystems. Overall, Baudena and her collaborators found that different plant communities may occur under the same climatic conditions depending on the fire resistance and post-fire response of the most competitive type of plant. “The main message is that post-fire response, which is often not included in global vegetation models that model climate change, is very important for even the qualitative outcome of a biome,” Baudena said. “So, it should be included in other types of studies.”

Acknowledgments: The speaker acknowledged Marta Magnani, Rubén Diaz-Sierra, Mart Verwijmeren, Antonello Provenzale, and Max Rietkerk.

[1] Baudena, M., Boni, G., Ferraris, L., von Hardenberg, J., & Provenzale, A. (2007). Vegetation response to rainfall intermittency in drylands: Results from a simple ecohydrological box model. Adv. Water Resour., 30(5), 1320-1328.
[2] Verwijmeren, M., Baudena, M., Wassen, M., Díaz‐Sierra, R., Smit, C., & Rietkerk, M. (2021). Intra‐seasonal rainfall variability and herbivory affect the interaction outcome of two dryland plant species. Ecosphere, 12(4), e03492.

  Jillian Kunze is the associate editor of SIAM News