Defining and detecting structural sensitivity in biological models: developing a new framework

No sensitivity to functional forms in the Rosenzweig-MacArthur model with strong environmental stochasticity

The classic Rosenzweig-MacArthur predator-prey model has been shown to exhibit, like other coupled nonlinear ordinary differential equations (ODEs) from ecology, worrying sensitivity to model structure. This sensitivity manifests as markedly different community dynamics arising from saturating functional responses with nearly identical shapes but different mathematical expressions. Using a stochastic differential equation (SDE) version of the Rosenzweig-MacArthur model with the three functional responses considered by Fussmann & Blasius (2005), I show that such sensitivity seems to be solely a property of ODEs or stochastic systems with weak noise. SDEs with strong environmental noise have by contrast very similar fluctuations patterns, irrespective of the mathematical formula used. Although eigenvalues of linearized predator-prey models have been used as an argument for structural sensitivity, they can also be an argument against structural sensitivity. While the sign of the eigenvalues' real part is sensitive to model structure, its magnitude and the presence of imaginary parts are not, which suggests noise-driven oscillations for a broad range of carrying capacities. I then discuss multiple other ways to evaluate structural sensitivity in a stochastic setting, for predator-prey or other ecological systems.

Generalized Structural Kinetic Modeling: A Survey and Guide

Many current challenges involve understanding the complex dynamical interplay between the constituents of systems. Typically, the number of such constituents is high, but only limited data sources on them are available. Conventional dynamical models of complex systems are rarely mathematically tractable and their numerical exploration suffers both from computational and data limitations. Here we review generalized modeling, an alternative approach for formulating dynamical models to gain insights into dynamics and bifurcations of uncertain systems. We argue that this approach deals elegantly with the uncertainties that exist in real world data and enables analytical insight or highly efficient numerical investigation. We provide a survey of recent successes of generalized modeling and a guide to the application of this modeling approach in future studies such as complex integrative ecological models.

Bifurcation analysis of the predator-prey model with the Allee effect in the predator

The use of predator–prey models in theoretical ecology has a long history, and the model equations have largely evolved since the original Lotka–Volterra system towards more realistic descriptions of the processes of predation, reproduction and mortality. One important aspect is the recognition of the fact that the growth of a population can be subject to an Allee effect, where the per capita growth rate increases with the population density. Including an Allee effect has been shown to fundamentally change predator–prey dynamics and strongly impact species persistence, but previous studies mostly focused on scenarios of an Allee effect in the prey population. Here we explore a predator–prey model with an ecologically important case of the Allee effect in the predator population where it occurs in the numerical response of predator without affecting its functional response. Biologically, this can result from various scenarios such as a lack of mating partners, sperm limitation and cooperative breeding mechanisms, among others. Unlike previous studies, we consider here a generic mathematical formulation of the Allee effect without specifying a concrete parameterisation of the functional form, and analyse the possible local bifurcations in the system. Further, we explore the global bifurcation structure of the model and its possible dynamical regimes for three different concrete parameterisations of the Allee effect. The model possesses a complex bifurcation structure: there can be multiple coexistence states including two stable limit cycles. Inclusion of the Allee effect in the predator generally has a destabilising effect on the coexistence equilibrium. We also show that regardless of the parametrisation of the Allee effect, enrichment of the environment will eventually result in extinction of the predator population.

Identifying the sources of structural sensitivity in partially specified biological models

Biological systems are characterised by a high degree of uncertainty and complexity, which implies that exact mathematical equations to describe biological processes cannot generally be justified. Moreover, models can exhibit sensitivity to the precise formulations of their component functions—a property known as structural sensitivity. Structural sensitivity can be revealed and quantified by considering partially specified models with uncertain functions, but this goes beyond well-established, parameter-based sensitivity analysis, and currently presents a mathematical challenge. Here we build upon previous work in this direction by addressing the crucial question of identifying the processes which act as the major sources of model uncertainty and those which are less influential. To achieve this goal, we introduce two related concepts: (1) the gradient of structural sensitivity, accounting for errors made in specifying unknown functions, and (2) the partial degree of sensitivity with respect to each function, a global measure of the uncertainty due to possible variation of the given function while the others are kept fixed. We propose an iterative framework of experiments and analysis to inform a heuristic reduction of structural sensitivity in a model. To demonstrate the framework introduced, we investigate the sources of structural sensitivity in a tritrophic food chain model.

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.

Quantifying uncertainty in partially specified biological models: how can optimal control theory help us?

Mathematical models in biology are highly simplified representations of a complex underlying reality and there is always a high degree of uncertainty with regards to model function specification. This uncertainty becomes critical for models in which the use of different functions fitting the same dataset can yield substantially different predictions-a property known as structural sensitivity. Thus, even if the model is purely deterministic, then the uncertainty in the model functions carries through into uncertainty in model predictions, and new frameworks are required to tackle this fundamental problem. Here, we consider a framework that uses partially specified models in which some functions are not represented by a specific form. The main idea is to project infinite dimensional function space into a low-dimensional space taking into account biological constraints. The key question of how to carry out this projection has so far remained a serious mathematical challenge and hindered the use of partially specified models. Here, we propose and demonstrate a potentially powerful technique to perform such a projection by using optimal control theory to construct functions with the specified global properties. This approach opens up the prospect of a flexible and easy to use method to fulfil uncertainty analysis of biological models.

Probability bounds analysis for nonlinear population ecology models

Mathematical models in population ecology often involve parameters that are empirically determined and inherently uncertain, with probability distributions for the uncertainties not known precisely. Propagating such imprecise uncertainties rigorously through a model to determine their effect on model outputs can be a challenging problem. We illustrate here a method for the direct propagation of uncertainties represented by probability bounds though nonlinear, continuous-time, dynamic models in population ecology. This makes it possible to determine rigorous bounds on the probability that some specified outcome for a population is achieved, which can be a core problem in ecosystem modeling for risk assessment and management. Results can be obtained at a computational cost that is considerably less than that required by statistical sampling methods such as Monte Carlo analysis. The method is demonstrated using three example systems, with focus on a model of an experimental aquatic food web subject to the effects of contamination by ionic liquids, a new class of potentially important industrial chemicals.