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.
Highlights
Global warming and the concomitant increased frequency of intense droughts threaten the viability of plant populations and communities throughout the world [1,2,3]
That calls for a different description of the population size; rather than describing it by a discrete number density, we describe it by a biomass variable, which remains continuous even at the level of a single plant
We further demonstrate with a few examples the advantage of continuum partial differential equations (PDE) modeling over discrete individual-based models (IBM) and discuss the ecological significance of this advantage
Summary
Global warming and the concomitant increased frequency of intense droughts threaten the viability of plant populations and communities throughout the world [1,2,3]. Understanding ecosystem response to drier climates, calls for unraveling mechanisms by which plant populations tolerate water stress Such mechanisms operate at the organism level through various forms of phenotypic changes, and at higher organization levels. A powerful methodology that complements empirical studies and compensates for the time-scale limitation is the construction and study of dynamic mathematical models These models can be divided into two major groups. The application of the PDE modeling approach to plant populations in drylands should first be justified, as these populations are typically small, especially in arid regions, where the vegetation is sparse This inherent character of drylands questions the use of continuous variables to describe population sizes; variables, such as number densities, become highly discrete, and demographic noise and extinction may become important aspects of the dynamics [12,13]. We conclude the paper with a brief discussion of additional examples that demonstrate the utility of continuum PDE models in gaining mechanistic information and insights about large, landscape-scale behaviors
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