A guideline to study the feasibility domain of multi-trophic and changing ecological communities

Abstract

The feasibility domain of an ecological community can be described by the set of environmental abiotic and biotic conditions under which all co-occurring and interacting species in a given site and time can have positive abundances. Mathematically, the feasibility domain corresponds to the parameter space compatible with positive (feasible) solutions at equilibrium for all the state variables in a system under a given model of population dynamics. Under specific dynamics, the existence of a feasible equilibrium is a necessary condition for species persistence regardless of whether the feasible equilibrium is dynamically stable or not. Thus, the size of the feasibility domain can also be used as an indicator of the tolerance of a community to random environmental variations. This has motivated a rich research agenda to estimate the feasibility domain of ecological communities. However, these methodologies typically assume that species interactions are static, or that input and output energy flows on each trophic level are unconstrained. Yet, this is different to how communities behave in nature. Here, we present a step-by-step quantitative guideline providing illustrative examples, computational code, and mathematical proofs to study systematically the feasibility domain of ecological communities under changes of interspecific interactions and subject to different constraints on the trophic energy flows. This guideline covers multitrophic communities that can be formed by any type of interspecific interactions. Importantly, we show that the relative size of the feasibility domain can significantly change as a function of the biological information taken into consideration. We believe that the availability of these methods can allow us to increase our understanding about the limits at which ecological communities may no longer tolerate further environmental perturbations, and can facilitate a stronger integration of theoretical and empirical research.

Structural stability as a consistent predictor of phenological events

“…This range of conditions (combination of intrinsic growth rates or carrying capacities) is called the feasibility domain D F (A) [26,36]. For example, heterotrophic species should exhibit smaller negative intrinsic growth rates than the benefits they can obtain from other species in order to have a positive balance of biomass (hence feasibility) [37,38]. This feasibility domain forms an algebraic cone [19,36]:…”

Impacts of consumer–resource interaction transitions on persistence and long‐term interaction outcomes of random ecological networks

“…Second, our results might be applicable only for the parameters tested in this study. Recent theoretical works showed that only a subspace of critical parameters (such as intrinsic growth rate) led to feasible equilibriums, namely, structural stability (Song et al 2018). The BCR networks were also subject to this limitation, and it is worthy for further exploration in future studies.…”

Vulnerable species interactions are important for the stability of mutualistic networks

“…This means that weak (less frequent) links between specialists were more vulnerable than strong (more frequent) links between generalists (8). Importance was defined as the contribution of a given link to the feasibility of a network, where feasibility is a measure of a network's ability to withstand environmental variation without leading to species extinctions (15,16). Important links were those that, when removed, lowered a network's feasibility; that is, when removed, they reduced the…”

“…Feasibility is defined as the range of conditions under which all species in a community can stably coexist (16). Feasibility can therefore be thought of as the 'safe operating space' of ecological communities: it is an indicator of how much environmental stress a community can tolerate before extinction of any of its constituent species.…”

“…Importance values were expressed as ((100 * Iij) -100), such that Iij = 0 if the feasibility of the network was identical with and without the focal link. We calculated feasibility following (16). Full details of the mutualistic model and equations used can be found in (16,32,33), but we outline these briefly below.…”