SIAM News Blog

Nitrogen, Litter, and Biodiversity in the North American Grasslands

By James Broda and Katherine Meyer

While it is perhaps less well-known than the carbon cycle, the nitrogen cycle is just as essential to the health of terrestrial ecosystems. In fact, scientists realized the importance of nitrogen for crop production as early as 1836 [1]. The development of the Haber-Bosch process in the early 20th century enabled the production of nitrogen fertilizers on an industrial scale. But while such fertilizers benefit crop growth, the spread of reactive nitrogen through the atmosphere and surface waters can harm ecosystems. For example, excess nitrogen has caused declines in the biodiversity of North American grasslands by disrupting the balance of resource competition among different species. The juxtaposition of cultivated land and native prairies—as in Figure 1—results in significant agricultural runoff pollution, and fossil fuel combustion exacerbates the problem by generating reactive nitrogen compounds that travel in the atmosphere and deposit on land. But on the bright side, recent efforts to reduce fossil fuel emissions and modify agricultural practices have led to localized decreases in nitrogen deposition. A natural question is as follows: Under what conditions can we expect biodiversity to bounce back?

Figure 1. Prairie grasses meet a soybean field and a line of trees outside of Hydro, a small town that straddles Caddo and Blaine counties along U.S. Route 66 in western Oklahoma. Public domain photograph courtesy of Carol M. Highsmith through the Library of Congress.
We present a mathematical approach to this question that connects recovery to the concepts of bistability and hysteresis in an ordinary differential equation (ODE) model [4]. This work is inspired by the results of long-term ecological experiments that investigated the same idea. At the University of Minnesota’s Cedar Creek Ecosystem Science Reserve, grassland biodiversity failed to recover in the 20+ years since experimental nitrogen inputs ceased — even though soil-available nitrogen returned to control levels [2]. In particular, the cool-season European grasses Elymus repens and Poa pratensis—which displaced native biodiversity during fertilization—maintained their dominance after fertilization ended. This outcome is inconsistent with the simplest versions of R resource competition theory, which predicts that nitrogen supply levels will control plant community composition [7]. Furthermore, follow-up experiments did not support an alternative explanation that involved feedback between plants and their affiliated soil organisms [5]. However, a similar study called the Rothamsted Park Grass Experiments in Hertfordshire, U.K., reached a contrasting outcome and found that grassland biodiversity recovered within a decade after cessation of nitrogen additions [6].

Ecologists conjecture that the opposing results at Cedar Creek and Park Grass might stem from differences in experimental design [6, 8]. In particular, plots at Park Grass were hayed twice a year, whereas plots at Cedar Creek were not hayed and developed dense mats of litter (dead plant biomass) near the soil surface. Litter both participates in nitrogen cycling via decomposition and forms a physical barrier for seeds, light, and seedlings. Did the accumulation of plant litter hamper biodiversity recovery at Cedar Creek? To explore this hypothesis, we developed an ODE model that synthesizes nitrogen cycling, resource competition, and litter inhibition [4].

Figure 2. Box diagram that corresponds to the ordinary differential equation model \((1-7)\). Boxes represent state variables and arrows indicate rates of change. Figure adapted from [4].
We limited the dimensions of our model by splitting all plant species into two groups: a native group correlated with high biodiversity \((P_n)\) and an exotic group correlated with low biodiversity \((P_e)\). This modeling choice is supported by observations in North American grasslands, which indicate that a greater abundance of native grasses positively correlates with both species richness and Simpson’s Diversity Index [3]. The diagram in Figure 2 portrays the relevant features of our model.

In addition to plant biomass, we kept track of inorganic soil nitrogen \(N\) and corresponding litter masses \(L_n\) and \(L_e\). The following equations describe the cycling of nitrogen between plant biomass, litter mass, and soil:

\[\frac{dP_n}{dt} = (f_n(L_n,L_e,N) - m_n - u_n)P_n + s_n \tag1\]

\[\frac{dP_e}{dt} = (f_e(L_n,L_e,N)- m_e - u_e)P_e + s_e \tag2\]

\[\frac{dL_n}{dt} = m_n P_n - k_n L_n \tag3\]

\[\frac{dL_e}{dt} = m_e P_e- k_eL_e \tag4\]

\[\frac{dN}{dt} = c - \sigma N +\gamma \Big(k_n L_n + k_e L_e +u_n P_n + u_e P_e  -f_n(L_n,L_e, N) P_n - f_e(L_n,L_e, N)P_e\Big)\tag5\]

\[\textrm{where} \qquad f_n(L_n,L_e,N)=\frac{g_n N}{h_n + N}e^{-\beta_{nn} L_n  -\beta_{en} L_e} \tag6\]

\[\textrm{and} \qquad f_e(L_n,L_e,N)=\frac{g_e N}{h_e + N} e^{-\beta_{ne} L_n-\beta_{ee} L_e}. \tag7\]

Most of the growth and decay terms are linear, but the functions \(f_i\) model nonlinear effects of soil nitrogen and litter \(L_i\) on per capita growth rates. The exponential factors in \((6)\) and \((7)\) represent litter inhibition of growth, and the parameters \(\beta_{ij}\) control the strength at which litter from group \(i\) inhibits the growth of plant group \(j\). For example, \(\beta_{en}\) corresponds to exotic litter’s inhibitory effect on native plant growth. Additional parameters of note include \(c\)—which models nitrogen inputs and is influenced by human activity—and \(s_n\) and \(s_e\), which capture the effect of seed additions by either ecosystem managers or naturally occurring plant dispersal. 

Figure 3. Plant community composition (3c) responds hysteretically to changes in nitrogen inputs (3a) that temporarily elevate available nitrogen in the soil (3b). Figure adapted from [4].
This relatively simple model allowed us to identify analytic conditions for both bistability and hysteresis. We observed bistability—namely, the presence of two stable equilibria that are alternatively dominated by native and exotic plants—over a range of parameter values. It is important to note that a necessary condition for bistability is


which captures the role of asymmetric litter inhibition in biodiversity recovery. In particular, \((8)\) suggests that bistability can result when native plants (which tend to be stronger nitrogen competitors than exotic flora) face disproportionate inhibition by exotic litter. In this case, cessation of added nitrogen may not sufficiently restore native biodiversity when the ecosystem in question gets stuck at the exotic-dominated equilibrium. Figure 3 illustrates this hysteretic scenario in a simulation of model \((1-7)\).

In addition to reproducing the qualitative behavior that researchers observed at Cedar Creek, the model also holds implications for grassland restoration strategies. Burning, haying, grazing, and other methods of biomass removal—all of which are common practices in grassland management—may support biodiversity recovery by reducing litter. Further analysis of the model also highlights the potential for the addition of native seeds to facilitate recovery. More broadly, the model suggests that combining resource competition with species-dependent inhibitory processes creates an effective recipe for bistability and hysteresis. More details about model analysis and tabulation of representative parameter values are available in [4].

James Broda presented this research during a minisymposium presentation at the 2022 SIAM Conference on Mathematics of Planet Earth (MPE22), which took place concurrently with the 2022 SIAM Annual Meeting in Pittsburgh, Pa., last summer. He received funding to attend MPE22 through a SIAM Early Career Travel Award. To learn more about Early Career Travel Awards and submit an application, visit the online page

Acknowledgments: This article describes joint work with Andrew Brettin (New York University), Mary Lou Zeeman (Bowdoin College), and Sarah Gorman, Sarah Hobbie, Forest Isbell, Richard McGehee, and María Sánchez Muñiz (all of the University of Minnesota). We are grateful for support from the American Institute of Mathematics, Cedar Creek Ecosystem Science Reserve, University of Minnesota, and Mathematics Climate Research Network. Funding from NSF grants DMS-1722578, DEB-1831944, and DMS-1645643 made this research possible.

[1] Galloway, J.N., Leach, A.M., Bleeker, A., & Erisman, J.W. (2013). A chronology of human understanding of the nitrogen cycle. Philos. Trans. R. Soc. B: Biol. Sci., 368(1621), 20130120.
[2] Isbell, F., Tilman, D., Polasky, S., Binder, S., & Hawthorne, P. (2013). Low biodiversity state persists two decades after cessation of nutrient enrichment. Ecol. Lett., 16(4), 454-460.
[3] Martin, L.M., Polley, H.W., Daneshgar, P.P., Harris, M.A., & Wilsey, B.J. (2014). Biodiversity, photosynthetic mode, and ecosystem services differ between native and novel ecosystems. Oecologia, 175(2), 687-697.
[4] Meyer, K., Broda, J., Brettin, A., Muñiz, M.S., Gorman, S., Isbell, F., … McGehee, R. (2022). Nitrogen-induced hysteresis in grassland biodiversity: A theoretical test of litter-mediated mechanisms. Preprint, arXiv:2208.12851. To appear in Am. Nat.
[5] Portales Reyes, M.C. (2020). Recovery of grassland plant communities after cessation of nutrient enrichment [Ph.D. thesis, University of Minnesota]. University of Minnesota Digital Conservancy. Retrieved from
[6] Storkey, J., Macdonald, A.J., Poulton, P.R., Scott, T., Köhler, I.H., Schnyder, H., … Crawley, M.J. (2015). Grassland biodiversity bounces back from long-term nitrogen addition. Nature, 528(7582), 401-404.
[7] Tilman, D. (1982). Resource competition and community structure. Princeton, NJ: Princeton University Press.
[8] Tilman, D., & Isbell, F. (2015). Recovery as nitrogen declines. Nature, 528(7582), 336-337.

James Broda is an assistant professor of mathematics at Washington and Lee University. He is interested in the study of both deterministic and stochastic dynamics that relate to questions of sustainability and resilience in environmental, social, and economic systems. 
Kate Meyer is an assistant professor of mathematics at Carleton College. Her research—which is motivated by applications to environmental changes and resilience—ranges from abstract mathematics to interdisciplinary collaborations. 
blog comments powered by Disqus