Continuum Modeling of Discrete Plant Communities: Why Does It Work and Why Is It Advantageous?

Abstract

Understanding ecosystem response to drier climates calls for modeling the dynamics of dryland plant populations, which are crucial determinants of ecosystem function, as they constitute the basal level of whole food webs. Two modeling approaches are widely used in population dynamics, individual (agent)-based models and continuum partial-differential-equation (PDE) models. The latter are advantageous in lending themselves to powerful methodologies of mathematical analysis, but the question of whether they are suitable to describe small discrete plant populations, as is often found in dryland ecosystems, has remained largely unaddressed. In this paper, we first draw attention to two aspects of plants that distinguish them from most other organisms-high phenotypic plasticity and dispersal of stress-tolerant seeds-and argue in favor of PDE modeling, where the state variables that describe population sizes are not discrete number densities, but rather continuous biomass densities. We then discuss a few examples that demonstrate the utility of PDE models in providing deep insights into landscape-scale behaviors, such as the onset of pattern forming instabilities, multiplicity of stable ecosystem states, regular and irregular, and the possible roles of front instabilities in reversing desertification. We briefly mention a few additional examples, and conclude by outlining the nature of the information we should and should not expect to gain from PDE model studies.

The Ecology of Plant Interactions: A Giant with Feet of Clay

“…First of all, we agree with (Meron et al 2019) that IBMs are mathematically less tractable than the ordinary differential equations of well-mixed demographic models or the partial differential equations of plant-continuum models. IBMs require, to a large extent, intensive computer simulations, but tools to treat them analytically also exist, e.g., moment closure methods such as the pair approximation (Matsuda et al 1992, Ellner 625 2001, Iwasa 2010), and IBMs are suitable to model plant interactions in all the remaining characteristics shown in Table 2.…”

“…IBMs require, to a large extent, intensive computer simulations, but tools to treat them analytically also exist, e.g., moment closure methods such as the pair approximation (Matsuda et al 1992, Ellner 625 2001, Iwasa 2010), and IBMs are suitable to model plant interactions in all the remaining characteristics shown in Table 2. In addition, while Meron et al (2019) defend that plantcontinuum models can function at finer spatial scales than IBM because they model biomass densities rather than individuals, we do not see a reason to state that spatially explicit IBMs cannot incorporate processes happening at finer spatial scales that the 630 individual scale, especially when they account for an explicit description of environmental conditions. Moreover, existing PDE-based models deliberately simplify many factors to keep mathematical tractability, which makes most of their results independent of the assumed net interactions.…”

“…As we argue they must be individual-based, incorporate the spatial structure of the community, and explicitly account for environmental factors. (Lotka 1924), (Jeltsch et al 2008) Individual       (Botkin et al 1972), (Shugart et al 2018) Continuum  ~    (Lefever and Lejeune 1997), (Meron et al 2019) competition for light a-a plant with initial crown size 1 intercept 1 unit of sunlight and allocate this energy in growing out, doubling its crown area, hence its solar interception, and investing 2 in growing out in the next time step, and so on. But in the case of b-an ESS, neighbors tend to overgrow the plant getting the benefit of asymmetric competition, hence the focal plant will invest 1 in growing up rather than out, reaching the canopy but 1115 not increasing its interception area.…”

“…In their review, Meron et al (2019) stand up in pro of the plant-continuum approach, which, due to its higher mathematical tractability has indeed contributed crucially to our qualitative understanding of several ecosystem-level processes in drylands, with special emphasis on desertification transitions (Rietkerk et al 2002, Bonachela et al 2015, Meron 2018, Fernandez-Oto et al 2019. Moreover, the authors defend that, unlike IBMs in which the smallest scale is that of the individual, the plant-continuum approach incorporates a higher resolution in spatial processes because it describes continuous biomass densities.…”

Species coexistence in resource-limited patterned ecosystems is facilitated by the interplay of spatial self-organisation and intraspecific competition

“…As a consequence, a number of theoretical frameworks have been developed over the last two decades [4,18,29]. Continuum approaches using partial differential equations (PDEs), in particular, have been established as a powerful tool to disentangle the complex ecohydrological dynamics in patterned vegetation [20]. However, most models do not distinguish between different plant species in their description of the ecohydrological dynamics, despite coexistence of herbaceous and woody species being commonly observed in patterned vegetation.…”