Structural sensitivity and resilience in a predator–prey model with density-dependent mortality

Parameter Sensitivity of Transient Community Dynamics

AbstractTransient dynamics have always intrigued ecologists, but current rapid environmental change (inducing transients even in previously undisturbed systems) has highlighted their importance more than ever. Here, I introduce a method for analyzing the sensitivity of transient ecological dynamics to parameter perturbations. The question the method answers is: how would the community dynamics have unfolded for some time horizon had the parameters been slightly different? I apply the method to three empirically parameterized models: competition between native forbs and exotic grasses in California, a host-parasitoid system, and an experimental chemostat predator-prey model. These applications showcase the ecological insights one can gain from models using transient sensitivity analysis. First, one can find parameters and their combinations whose perturbations disproportionately affect a system. Second, one can identify particular windows of time during which the predicted deviation from the unperturbed trajectories is especially large and utilize this information for management purposes. Third, there is an inverse relationship between transient and long-term sensitivities whenever the interacting populations are ecologically similar; paradoxically, the smaller the immediate response of the system, the more extreme its long-term response will be.

Bridging Theories for Ecosystem Stability Through Structural Sensitivity Analysis of Ecological Models in Equilibrium

Ecologists are challenged by the need to bridge and synthesize different approaches and theories to obtain a coherent understanding of ecosystems in a changing world. Both food web theory and regime shift theory shine light on mechanisms that confer stability to ecosystems, but from different angles. Empirical food web models are developed to analyze how equilibria in real multi-trophic ecosystems are shaped by species interactions, and often include linear functional response terms for simple estimation of interaction strengths from observations. Models of regime shifts focus on qualitative changes of equilibrium points in a slowly changing environment, and typically include non-linear functional response terms. Currently, it is unclear how the stability of an empirical food web model, expressed as the rate of system recovery after a small perturbation, relates to the vulnerability of the ecosystem to collapse. Here, we conduct structural sensitivity analyses of classical consumer-resource models in equilibrium along an environmental gradient. Specifically, we change non-proportional interaction terms into proportional ones, while maintaining the equilibrium biomass densities and material flux rates, to analyze how alternative model formulations shape the stability properties of the equilibria. The results reveal no consistent relationship between the stability of the original models and the proportionalized versions, even though they describe the same biomass values and material flows. We use these findings to critically discuss whether stability analysis of observed equilibria by empirical food web models can provide insight into regime shift dynamics, and highlight the challenge of bridging alternative modelling approaches in ecology and beyond.

Geometric Complexity and the Information-Theoretic Comparison of Functional-Response Models

The assessment of relative model performance using information criteria like AIC and BIC has become routine among functional-response studies, reflecting trends in the broader ecological literature. Such information criteria allow comparison across diverse models because they penalize each model's fit by its parametric complexity—in terms of their number of free parameters—which allows simpler models to outperform similarly fitting models of higher parametric complexity. However, criteria like AIC and BIC do not consider an additional form of model complexity, referred to as geometric complexity, which relates specifically to the mathematical form of the model. Models of equivalent parametric complexity can differ in their geometric complexity and thereby in their ability to flexibly fit data. Here we use the Fisher Information Approximation to compare, explain, and contextualize how geometric complexity varies across a large compilation of single-prey functional-response models—including prey-, ratio-, and predator-dependent formulations—reflecting varying apparent degrees and forms of non-linearity. Because a model's geometric complexity varies with the data's underlying experimental design, we also sought to determine which designs are best at leveling the playing field among functional-response models. Our analyses illustrate (1) the large differences in geometric complexity that exist among functional-response models, (2) there is no experimental design that can minimize these differences across all models, and (3) even the qualitative nature by which some models are more or less flexible than others is reversed by changes in experimental design. Failure to appreciate model flexibility in the empirical evaluation of functional-response models may therefore lead to biased inferences for predator–prey ecology, particularly at low experimental sample sizes where its impact is strongest. We conclude by discussing the statistical and epistemological challenges that model flexibility poses for the study of functional responses as it relates to the attainment of biological truth and predictive ability.

A type IV functional response with different shapes in a predator-prey model

Group defense is a phenomenon that occurs in many predator-prey systems. Different functional responses with substantially different properties representing such a mechanism exist. Here, we develop a functional response using timescale separation. A prey-dependent catch rate represents the group defense. The resulting functional response contains a single parameter that controls whether the group defense functional response is saturating or dome-shaped. Based on that, we show that the catch rate must not increase monotonically with increasing prey density to lead to a dome-shaped functional response. We apply bifurcation analysis to show that non-monotonic group defense is usually more successful. However, we also find parameter regions in which a paradox occurs. In this case, higher group defense can give rise to a stable limit cycle, while for lower values, the predator would go extinct. The study does not only provide valuable insight on how to include functional responses representing group defense in mathematical models, but it also clarifies under which circumstances the usage of different functional responses is appropriate.

The mechanistic basis for higher-order interactions and non-additivity in competitive communities

Motivated by both analytical tractability and empirical practicality, community ecologists have long treated the species pair as the fundamental unit of study. This notwithstanding, the challenge of understanding more complex systems has repeatedly generated interest in the role of so-called higher-order interactions (HOIs) imposed by species beyond the focal pair. Here we argue that HOIs - defined as non-additive effects of density on per capita growth - are best interpreted as emergent properties of phenomenological models (e.g. Lotka-Volterra competition) rather than as distinct 'ecological processes' in their own right. Using simulations of consumer-resource models, we explore the mechanisms and system properties that give rise to HOIs in observational data. We demonstrate that HOIs emerge under all but the most restrictive of assumptions, and that incorporating non-additivity into phenomenological models improves the quantitative and qualitative accuracy of model predictions. Notably, we also observe that HOIs derive primarily from mechanisms and system properties that apply equally to single-species or pairwise systems as they do to more diverse communities. Consequently, there exists a strong mandate for further recognition of non-additive effects in both theoretical and empirical research.

Community dynamics and sensitivity to model structure: towards a probabilistic view of process-based model predictions

Statistical inference and mechanistic, process-based modelling represent two philosophically different streams of research whose primary goal is to make predictions. Here, we merge elements from both approaches to keep the theoretical power of process-based models while also considering their predictive uncertainty using Bayesian statistics. In environmental and biological sciences, the predictive uncertainty of process-based models is usually reduced to parametric uncertainty. Here, we propose a practical approach to tackle the added issue of structural sensitivity, the sensitivity of predictions to the choice between quantitatively close and biologically plausible models. In contrast to earlier studies that presented alternative predictions based on alternative models, we propose a probabilistic view of these predictions that include the uncertainty in model construction and the parametric uncertainty of each model. As a proof of concept, we apply this approach to a predator-prey system described by the classical Rosenzweig-MacArthur model, and we observe that parametric sensitivity is regularly overcome by structural sensitivity. In addition to tackling theoretical questions about model sensitivity, the proposed approach can also be extended to make probabilistic predictions based on more complex models in an operational context. Both perspectives represent important steps towards providing better model predictions in biology, and beyond.

Is structural sensitivity a problem of oversimplified biological models? Insights from nested Dynamic Energy Budget models

Many current issues in ecology require predictions made by mathematical models, which are built on somewhat arbitrary choices. Their consequences are quantified by sensitivity analysis to quantify how changes in model parameters propagate into an uncertainty in model predictions. An extension called structural sensitivity analysis deals with changes in the mathematical description of complex processes like predation. Such processes are described at the population scale by a specific mathematical function taken among similar ones, a choice that can strongly drive model predictions. However, it has only been studied in simple theoretical models. Here, we ask whether structural sensitivity is a problem of oversimplified models. We found in predator-prey models describing chemostat experiments that these models are less structurally sensitive to the choice of a specific functional response if they include mass balance resource dynamics and individual maintenance. Neglecting these processes in an ecological model (for instance by using the well-known logistic growth equation) is not only an inappropriate description of the ecological system, but also a source of more uncertain predictions.