Finding all multiple stable fixpoints of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>n</mml:mi></mml:math>-species Lotka–Volterra competition models

Lischke, Heike; Löffler, Thomas J.

doi:10.1016/j.tpb.2017.02.001

Cited by 13 publications

(21 citation statements)

References 49 publications

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“…The following lemmas summarize the properties of the tangent matrix at the equilibria of the system. Note that similar observations were made in different context in [19].…”

Section: 1supporting

confidence: 79%

“…While it is straightforward to find all the equilibria of (1) (it suffices to solve 2 n linear systems and determine the ones whose solutions are strictly positive, see also [19] for the efficient algorithm), finding the connections between them appears a harder task. Our aim is to give an algorithm that can be used to determine the gradient dynamics given in Theorem 8, i.e.…”

Section: Equilibria and The Local Dynamicsmentioning

confidence: 99%

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Structural stability of invasion graphs for generalized Lotka--Volterra systems

Almaraz¹,

Langa²,

Kalita³

2022

Preprint

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In this paper we study in detail the structure of the global attractor for a generalized Lotka-Volterra system with Volterra-Lyapunov stable structural matrix. We provide the full characterization of this structure and we show that it coincides with the invasion graph as recently introduced in [15]. We also study the stability of the structure with respect to the perturbation of the problem parameters. This allows us to introduce a definition of structural stability in Ecology in coherence with the classical mathematical concept where there exists a detailed geometrical structure, governing the transient and asymptotic dynamics, which is robust under perturbation.

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“…The following lemmas summarize the properties of the tangent matrix at the equilibria of the system. Note that similar observations were made in different context in [19].…”

Section: 1supporting

confidence: 79%

Section: Equilibria and The Local Dynamicsmentioning

confidence: 99%

Structural stability of invasion graphs for generalized Lotka--Volterra systems

Almaraz¹,

Langa²,

Kalita³

2022

Preprint

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“…Besides the important data requirements, however, more complex models would create new analytical challenges as simulation approaches (e.g. ODEs) might not be appropriate to efficiently characterize community dynamics and equilibrium states 51 . In this context of increasing model and data complexity, assuming a priori that species demographic parameters depend on a limited number of functional traits will likely be a critical asset to study the model behaviors and reduce the complexity of its calibration at the onset.…”

Section: Discussionmentioning

confidence: 99%

Linking functional traits and demography to model species-rich communities

et al. 2021

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It has long been anticipated that relating functional traits to species demography would be a cornerstone for achieving large-scale predictability of ecological systems. If such a relationship existed, species demography could be modeled only by measuring functional traits, transforming our ability to predict states and dynamics of species-rich communities with process-based community models. Here, we introduce a new method that links empirical functional traits with the demographic parameters of a process-based model by calibrating a transfer function through inverse modeling. As a case study, we parameterize a modified Lotka–Volterra model of a high-diversity mountain grassland with static plant community and functional trait data only. The calibrated trait–demography relationships are amenable to ecological interpretation, and lead to species abundances that fit well to the observed community structure. We conclude that our new method offers a general solution to bridge the divide between trait data and process-based models in species-rich ecosystems.

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“…A possible avenue to solve that dilemma is to create metaor hybrid models that aggregate the known mechanisms of species interactions into a simpler structure that is easier to connect to large-scale data. An example is the possibility of directly determining the equilibria of dynamic Lotka-Volterra models (Lischke & L€ offler, 2017). It incorporates the dynamic effects of species interactions and yields a very simple static model required for parameter estimation, complementing correlational approaches with dynamic add-ons (leading to hybrid models, such as KISSMIG; Nobis & Normand, 2014).…”

Section: Box 3 Dynamic Approaches To Simulate Distributions Of Interamentioning

confidence: 99%

Biotic interactions in species distribution modelling: 10 questions to guide interpretation and avoid false conclusions

Dormann

Bobrowski

Dehling

et al. 2018

Global Ecol Biogeogr

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252

202

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Aim: Recent studies increasingly use statistical methods to infer biotic interactions from cooccurrence information at a large spatial scale. However, disentangling biotic interactions from other factors that can affect co-occurrence patterns at the macroscale is a major challenge. Approach:We present a set of questions that analysts and reviewers should ask to avoid erroneously attributing species association patterns to biotic interactions. Our questions relate to the appropriateness of data and models, the causality behind a correlative signal, and the problems associated with static data from dynamic systems. We summarize caveats reported by macroecological studies of biotic interactions and examine whether conclusions on the presence of biotic interactions are supported by the modelling approaches used.Findings: Irrespective of the method used, studies that set out to test for biotic interactions find statistical associations in species' co-occurrences. Yet, when compared with our list of questions, few purported interpretations of such associations as biotic interactions hold up to scrutiny. This does not dismiss the presence or importance of biotic interactions, but it highlights the risk of too lenient interpretation of the data. Combining model results with information from experiments and functional traits that are relevant for the biotic interaction of interest might strengthen conclusions.Main conclusions: Moving from species-to community-level models, including biotic interactions among species, is of great importance for process-based understanding and forecasting ecological responses. We hope that our questions will help to improve these models and facilitate the interpretation of their results. In essence, we conclude that ecologists have to recognize that a species association pattern in joint species distribution models will be driven not only by real biotic interactions, but also by shared habitat preferences, common migration history, phylogenetic history and shared response to missing environmental drivers, which specifically need to be discussed and, if possible, integrated into models. K E Y W O R D Sbiotic interactions, communities, co-occurrence, environment, residual structure, species distribution models

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Finding all multiple stable fixpoints ofn-species Lotka–Volterra competition models

Cited by 13 publications

References 49 publications

Structural stability of invasion graphs for generalized Lotka--Volterra systems

Structural stability of invasion graphs for generalized Lotka--Volterra systems

Linking functional traits and demography to model species-rich communities

Biotic interactions in species distribution modelling: 10 questions to guide interpretation and avoid false conclusions

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