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Mathematical Models of Mycorrhizal Mutualisms

By Frederick R. Adler and Peter B. Adler

Mathematical models of ecological communities tend to focus on negative interactions, such as competition, predation, and parasitism [3]. These relationships are some of the strongest drivers of population dynamics and play a central role in maintaining diversity and shaping the course of evolution. However, these negative interactions have a positive side when they grade into so-called mutualisms, wherein the participants’ evolutionary “motivations” retain the same selfish interests—to propagate their own genes—but just so happen to align across different species.

Because plants cannot run away from problems or toward opportunities (except as seeds), survival often requires collaboration. For instance, plants acquire resources from the soil through resource collection mutualisms. Well-known examples include the nitrogen-fixing bacteria that are associated with legumes and the mycorrhizae fungi that that help inefficient plant roots extract nitrogen and phosphorus from the soil.

Defense mutualisms ward off herbivores, with examples including fungi that make plants toxic and ants that live in little homes within a plant (called domatia) and attack animals that try to take a bite from their host. Pollination mutualisms are perhaps the most familiar category, and allow plants to efficiently move their male gametes (pollen) to find female gametes on other plants. The bees that facilitate this transfer do not intend to help the plants and are instead just seeking sweet nectar and protein-rich pollen as a food source. Finally, humans take part in dispersal mutualisms when we consume fruits and excrete the seeds in favorable sites, or when we find burrs or sticky seeds on our clothing.

Figure 1. Structure of the resource-sharing model. Figure courtesy of Frederick Adler.
The benefits of mutualism are context-dependent [1]. A species of bee that does not efficiently move pollen might be beneficial to plants in the absence of superior bees but detrimental in their presence. Like bees, mycorrhizae take the sugars that plants produce via photosynthesis; they can be a net sink on plant resources if the soil is sufficiently nutrient-rich for plants to prosper on their own.

The simplest modeling techniques fail to capture this complexity. Given the well-developed and familiar Lotka-Volterra approach to competition and predation, it is sorely tempting to make the signs of both interaction coefficients be positive. Here, let \(Y\) represent the biomass of plants (we need to save \(P\) for pathogens) and \(M\) represent the biomass of mutualists:

\[\frac{dY}{dt}=(r_Y-\alpha_{YY}Y+\alpha_{YM}M)Y \\

\frac{dM}{dt}=(r_M-\alpha_{MM}M+\alpha_{MY}Y)M. \]

If both \(\alpha_{YM}\) and \(\alpha_{MY}\) are positive, each species has a beneficial effect on the other. However, this model has several problems. First, as in the simple ordinary differential equation \(dx/dt=rx^2\), this system not only blows up to infinity but does so in finite time. Second—and perhaps more importantly—the mechanistic considerations about the type of mutualism and its context dependence have nowhere to go.

We are hardly the first researchers to realize this. The long and intelligent tradition of mutualism modeling builds upon the consumer-resource approach that has remained the mainstay of ecological models [2]. Our modeling effort utilizes this background to create a framework that addresses two goals: (i) Capture the conditions for context dependence and (ii) simultaneously incorporate a mutualist and pathogen whose life histories might share many similarities.

Figure 2. Structure of the consumer-resource model with pathogens. Figure courtesy of Frederick Adler.
To embody the consumer-resource view and understand the conditions wherein resource collection mutualists are beneficial, we needed to explicitly track two resources: light for the plants that becomes carbon-based resources like sugar, and nutrients such as nitrogen (see Figure 1). Without these two resources, plants cannot coexist with their own mutualists.

We also realized that mass action cannot handle the interactions of plant roots and mycorrhizae that are stationary and physically intertwined. Instead, we developed a spatial model that scales quite differently with the interacting species’ relative biomass. Suppose that the total volume \(Y+M\) is broken into \(n\) shapes of volume \(V\), each with \(k\) faces of area \(S\) that contact their neighbors. If these shapes were independently distributed, a fraction \(\frac{2MY}{(Y+M)^2}\) of faces would fall between \(Y\) and \(M\). There are \(kn/2\) such faces where \(n=\frac{Y+M}{V}\). The total area wherein diffusion can occur is

\[A=\frac{2MY}{(Y+M)^2}\frac{k(M+Y)}{2V}S=\frac{\theta MY}{Y+M}, \tag1 \]

where \(\theta=kS/V\) can serve as a free parameter. 

Finally, we cannot treat all pathogens in exactly the same way. In line with most plant populations, our models presume that every plant has mycorrhizal associates. But if we assume that all plants have pathogens, then pathogens effectively become part of the plant physiology — much like a deleterious allele that is part of the genome. These kinds of models therefore miss the well-known ability of specialized pathogens to promote plant coexistence; when plants become rare, their pathogens also become rare, which can allow the plants to survive and their population to rebound. To capture this phenomenon, we combined our consumer-resource models with an epidemiological model that separates plants into “infected” and “uninfected” groups (see Figure 2). Due to the intimate relationship between plants and their mutualists, we must also divide the mutualists in the same way.

As predicted, the resulting model demonstrates that mutualists are beneficial to plants when resource supplies are low but detrimental when resources are sufficiently high and the plants could prosper on their own (see Figure 3a). Furthermore, plants always benefit from mutualists in the presence of pathogens. We also investigated the possibility that mutualists promote resilience, finding that mutualists do indeed buffer plant populations against changes in resource availability — though not to the same extent as pathogens. If resource availability is sufficiently low, plants can only persist when mutualists are present (see Figure 3b). 

Figure 3. Comparison of different community structures: plants only (black line), plants with mutualists (blue line), plants with pathogens (red line), and plants with both mutualists and pathogens (purple line). 3a. The effects of resource supply rate on plant biomass. 3b. The effects of an abrupt change in the resource supply rate on plant biomass. 3c. The effects of species pool size on equilibrium plant species diversity . Figure courtesy of Frederick Adler.

Ultimately, we can examine the way in which the size of the species pool affects the system. As ecological theory predicts, only a single plant species can persist in the absence of pathogens. Pathogens alone promote higher plant diversity, but diversity can be lost as more species are added and one plant type comes to dominate (see Figure 3c). When mutualists are present, their support allows the species that would be lost in their absence to survive and maintain the entire community.

Our work has several important applications. There has recently been a great deal of interest in plant-soil feedbacks, which refer to the way in which plants alter soils and make them less conducive to growth — either through resource depletion, the build-up of pathogens, or even the proliferation of mutualists [4]. These feedbacks help scientists understand the maintenance of biodiversity in plant communities, particularly in response to aspects of global shifts such as climate change, nutrient loading, and invasive species. Phenomenological models, including Lotka-Volterra models, cannot capture the resulting context dependence from these factors. We therefore need more mechanistic yet suitably simple models to understand and predict the effect of global changes on the mutualisms and parasitisms that impact plant communities.

Frederick Adler delivered a minisymposium presentation on this research at the 2022 SIAM Conference on the Life Sciences, which took place concurrently with the 2022 SIAM Annual Meeting in Pittsburgh, Pa., last year.

[1] Bronstein, J.L. (1994). Conditional outcomes in mutualistic interactions. Trends Ecol. Evol., 9(6), 214-217.
[2] Holland, J.N., & DeAngelis, D.L. (2010). A consumer-resource approach to the density-dependent population dynamics of mutualism. Ecology, 91(5), 1286-1295.
[3] May, R.M., & McLean, A.R. (Eds.) (2007). Theoretical ecology: Principles and applications. Oxford, UK: Oxford University Press.
[4] Van der Putten, W.H., Bradford, M.A., Brinkman, E.P., van de Voorde, T.F.J., & Veen, G.F. (2016). Where, when and how plant-soil feedback matters in a changing world. Funct. Ecol., 30(7), 1109-1121.

Frederick R. Adler holds a B.S. in mathematics from Harvard University and a Ph.D. in applied mathematics from Cornell University. He has spent his subsequent career becoming increasing biological and completed a postdoctoral appointment at the Center for Population Biology at the University of California, Davis. He now holds a joint position in mathematics and biology at the University of Utah, where he is currently director of the School of Biological Sciences. Adler uses mathematical modeling to investigate ecology, epidemiology, immunology, and many subfields of molecular and biomedical biology. He is also the author of two textbooks: Modeling the Dynamics of Life: Calculus and Probability for Life Scientists and Urban Ecosystems: Ecological Principles for the Built Environment.

Peter B. Adler is a professor in the Department of Wildland Resources at Utah State University. He graduated from Harvard College and earned a Ph.D. in ecology from Colorado State University in 2003. He uses long-term data sets, experiments, and statistical and simulation models to study biodiversity and the impacts of climate change on plant populations, communities, and ecosystems. He was named the Utah State University Researcher of the Year for 2023 and is a Fellow of the Ecological Society of America.
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