Enrichment and foodchain stability: the impact of different forms of predator-prey interaction

Combining generalized modeling and specific modeling in the analysis of ecological networks

The complexity of real food webs involves uncertainty in data and in underlying ecological processes, and modeling approaches deal with these challenges differently. Generalized modeling provides a linear stability analysis without narrow specification of all processes, and conventional dynamical systems models approximate functional forms to discuss trajectories in phase space. This study compares results and ecological interpretations from both methods in four-species ecological networks at steady state. We find that a specific (dynamical systems) model only provides a subset of stability data from the generalized model, which spans many plausible dynamic scenarios, allowing for conflicting results. Nevertheless, both approaches reveal that fixed points become stable when nutrient flows to predators are fettered and even more when the basal growth rate approaches a maximum. The specific model identifies a distinct ecosystem response to bottom-up forcing, the enrichment of lower trophic levels. Enrichment stabilizes a fixed point when basal species are in a resource-deprived environment but destabilizes it if resources become more abundant. The generalized model provides less specific information since infinitely many paths of enrichment are hypothetical. Nevertheless, generalized modeling of ecological systems is a powerful technique that enables a meta analysis of these uncertain complex systems.

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.

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.

Viable supply chain model: integrating agility, resilience and sustainability perspectives-lessons from and thinking beyond the COVID-19 pandemic

Viability is the ability of a supply chain (SC) to maintain itself and survive in a changing environment through a redesign of structures and replanning of performance with long-term impacts. In this paper, we theorize a new notion-the viable supply chain (VSC). In our approach, viability is considered as an underlying SC property spanning three perspectives, i.e., agility, resilience, and sustainability. The principal ideas of the VSC model are adaptable structural SC designs for supply-demand allocations and, most importantly, establishment and control of adaptive mechanisms for transitions between the structural designs. Further, we demonstrate how the VSC components can be categorized across organizational, informational, process-functional, technological, and financial structures. Moreover, our study offers a VSC framework within an SC ecosystem. We discuss the relations between resilience and viability. Through the lens and guidance of dynamic systems theory, we illustrate the VSC model at the technical level. The VSC model can be of value for decision-makers to design SCs that can react adaptively to both positive changes (i.e., the agility angle) and be able to absorb negative disturbances, recover and survive during short-term disruptions and long-term, global shocks with societal and economical transformations (i.e., the resilience and sustainability angles). The VSC model can help firms in guiding their decisions on recovery and re-building of their SCs after global, long-term crises such as the COVID-19 pandemic. We emphasize that resilience is the central perspective in the VSC guaranteeing viability of the SCs of the future. Emerging directions in VSC research are discussed.

Stability of generalized ecological-network models

The stability of ecological networks of varying topologies and predator-prey relationships is explored by applying the concept of generalized modeling. The effects of omnivory, complexity, enrichment, number of top predators, and predatory response are discussed. The degree of omnivory plays a large role in governing web stability at steady state. Complexity as measured from connectance and network size is not a perfect indicator of stability; large, highly connected webs can be just as stable as smaller, less connected ones. Learning behavior as expressed in Holling's type III predatory response is stabilizing for food webs and provides exceptions to the paradox of enrichment for some topologies.

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.

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.

The feasibility of equilibria in large ecosystems: A primary but neglected concept in the complexity-stability debate

The consensus that complexity begets stability in ecosystems was challenged in the seventies, a result recently extended to ecologically-inspired networks. The approaches assume the existence of a feasible equilibrium, i.e. with positive abundances. However, this key assumption has not been tested. We provide analytical results complemented by simulations which show that equilibrium feasibility vanishes in species rich systems. This result leaves us in the uncomfortable situation in which the existence of a feasible equilibrium assumed in local stability criteria is far from granted. We extend our analyses by changing interaction structure and intensity, and find that feasibility and stability is warranted irrespective of species richness with weak interactions. Interestingly, we find that the dynamical behaviour of ecologically inspired architectures is very different and richer than that of unstructured systems. Our results suggest that a general understanding of ecosystem dynamics requires focusing on the interplay between interaction strength and network architecture.

Degree heterogeneity and stability of ecological networks

A classic measure of ecological stability describes the tendency of a community to return to equilibrium after small perturbations. While many advances show how the network architecture of these communities severely constrains such tendencies, one of the most fundamental properties of network structure, i.e. degree heterogeneity-the variability of the number of links associated with each species, deserves further study. Here we show that the effects of degree heterogeneity on stability vary with different types of interspecific interactions. Degree heterogeneity consistently destabilizes ecological networks with both competitive and mutualistic interactions, while its effects on networks of predator-prey interactions such as food webs depend on prey contiguity, i.e. the extent to which the species consume an unbroken sequence of prey in community niche space. Increasing degree heterogeneity tends to stabilize food webs except those with the highest prey contiguity. These findings help explain why food webs are highly but not completely interval and, more broadly, deepen our understanding of the stability of complex ecological networks.

Impact of dispersal on the stability of metapopulations

Dispersal is a key ecological process that enables local populations to form spatially extended systems called metapopulations. In the present study, we investigate how dispersal affects the linear stability of a general single-species metapopulation model. We discuss both the influence of local within-patch dynamics and the effects of various dispersal behaviours on stability. We find that positive density-dependent dispersal and positive density-dependent settlement are destabilizing dispersal behaviours while negative density-dependent dispersal and negative density-dependent settlement are stabilizing. It is also shown that dispersal has a stabilizing impact on heterogeneous metapopulations that correlates positively with the number of patches and the connectance of metapopulation networks.

Functional response of the fungus Hirsutella rhossiliensis to the nematode, Heterodera glycines

Functional response is a key index in determining the population fluctuation in predation. However, the lack of operable research system limits the studies on functional response of fungal predators. Hirsutella rhossiliensis is a dominant parasite of the soybean cyst nematode, Heterodera glycines. In a soil microcosm bioassay, we determined fungal biomass at different days within 21 days after inoculation, and parasitism rate of H. glycines by the fungus was determined. The functional response of H. rhossiliensis to H. glycines was established and found to be Holling's type III, which was influenced by mycelial densities. Meanwhile, we conducted anti-fungal analysis of metabolic fractions extracted from H. rhossiliensis to explain the potential mechanism of the intraspecific competition illustrated by functional response. The result of anti-fungal experiments indicated that the fungal predators had more complicated interaction at population level than expected, which might be regulated by self-inhibition metabolite(s). This study was the first functional response study of fungal predators in microcosm. With the increasing recognition of emerging fungal threats to animal, plant, and ecosystem health, the methodologies and hypotheses proposed in this study might inspire further research in fungal ecology.

Bifurcation analysis of models with uncertain function specification: how should we proceed?

When we investigate the bifurcation structure of models of natural phenomena, we usually assume that all model functions are mathematically specified and that the only existing uncertainty is with respect to the parameters of these functions. In this case, we can split the parameter space into domains corresponding to qualitatively similar dynamics, separated by bifurcation hypersurfaces. On the other hand, in the biological sciences, the exact shape of the model functions is often unknown, and only some qualitative properties of the functions can be specified: mathematically, we can consider that the unknown functions belong to a specific class of functions. However, the use of two different functions belonging to the same class can result in qualitatively different dynamical behaviour in the model and different types of bifurcation. In the literature, the conventional way to avoid such ambiguity is to narrow the class of unknown functions, which allows us to keep patterns of dynamical behaviour consistent for varying functions. The main shortcoming of this approach is that the restrictions on the model functions are often given by cumbersome expressions and are strictly model-dependent: biologically, they are meaningless. In this paper, we suggest a new framework (based on the ODE paradigm) which allows us to investigate deterministic biological models in which the mathematical formulation of some functions is unspecified except for some generic qualitative properties. We demonstrate that in such models, the conventional idea of revealing a concrete bifurcation structure becomes irrelevant: we can only describe bifurcations with a certain probability. We then propose a method to define the probability of a bifurcation taking place when there is uncertainty in the parameterisation in our model. As an illustrative example, we consider a generic predator-prey model where the use of different parameterisations of the logistic-type prey growth function can result in different dynamics in terms of the type of the Hopf bifurcation through which the coexistence equilibrium loses stability. Using this system, we demonstrate a framework for evaluating the probability of having a supercritical or subcritical Hopf bifurcation.

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

When we construct mathematical models to represent biological systems, there is always uncertainty with regards to the model specification--whether with respect to the parameters or to the formulation of model functions. Sometimes choosing two different functions with close shapes in a model can result in substantially different model predictions: a phenomenon known in the literature as structural sensitivity, which is a significant obstacle to improving the predictive power of biological models. In this paper, we revisit the general definition of structural sensitivity, compare several more specific definitions and discuss their usefulness for the construction and analysis of biological models. Then we propose a general approach to reveal structural sensitivity with regards to certain system properties, which considers infinite-dimensional neighbourhoods of the model functions: a far more powerful technique than the conventional approach of varying parameters for a fixed functional form. In particular, we suggest a rigorous method to unearth sensitivity with respect to the local stability of systems' equilibrium points. We present a method for specifying the neighbourhood of a general unknown function with [Formula: see text] inflection points in terms of a finite number of local function properties, and provide a rigorous proof of its completeness. Using this powerful result, we implement our method to explore sensitivity in several well-known multicomponent ecological models and demonstrate the existence of structural sensitivity in these models. Finally, we argue that structural sensitivity is an important intrinsic property of biological models, and a direct consequence of the complexity of the underlying real systems.

Diseased prey predator model with general Holling type interactions

Choice of interaction function is one of the most important parts for modelling a food chain. Many models have been proposed as a diseased-prey predator model with Holling type-I or type-II or type-III interactions, but there is no model with general Holling type interactions. In this paper, we study a diseased prey-predator model with general Holling type interactions. Local stability conditions of equilibrium points are derived. We obtain the permanence and impermanence conditions of the system. The conditions for global stability of the system are also derived. The system exhibits limit cycle, period-2, higher periodic oscillations and chaotic behaviour for different values of Holling parameters. One parameter bifurcation analysis is done with respect to general Holling parameters and infection rate. We utilize the MATCONT package to analyse the detailed bifurcation scenario as the two important interaction parameters are varied. It is interesting to note that a diseased system becomes a disease free system for proper choice of interaction functions. Our results give an idea for constructing a realistic food chain model through proper choice of general Holling parameters.

How to predict community responses to perturbations in the face of imperfect knowledge and network complexity

Recent attempts to predict the response of large food webs to perturbations have revealed that in larger systems increasingly precise information on the elements of the system is required. Thus, the effort needed for good predictions grows quickly with the system's complexity. Here, we show that not all elements need to be measured equally well, suggesting that a more efficient allocation of effort is possible. We develop an iterative technique for determining an efficient measurement strategy. In model food webs, we find that it is most important to precisely measure the mortality and predation rates of long-lived, generalist, top predators. Prioritizing the study of such species will make it easier to understand the response of complex food webs to perturbations.

Parametrization of the attainable set for a nonlinear control model of a biochemical process

In this paper, we study a three-dimensional nonlinear model of a controllable reaction [X]+[Y]+[Z]→[Z], where the reaction rate is given by a unspecified nonlinear function. A model of this type describes a variety of real-life processes in chemical kinetics and biology; in this paper our particular interests is in its application to waste water biotreatment. For this control model, we analytically study the corresponding attainable set and parameterize it by the moments of switching of piecewise constant control functions. This allows us to visualize the attainable sets using a numerical procedure. These analytical results generalize the earlier findings, which were obtained for a trilinear reaction rate (which corresponds to the law of mass action) and reported in [18,19], to the case of a general rate of reaction. These results allow to reduce the problem of constructing the optimal control to a straightforward constrained finite dimensional optimization problem.

When can we trust our model predictions? Unearthing structural sensitivity in biological systems

It is well recognized that models in the life sciences can be sensitive to small variations in their model functions, a phenomenon known as ‘structural sensitivity’. Conventionally, modellers test for sensitivity by varying parameters for a specific formulation of the model functions, but models can show structural sensitivity to the choice of functional representations used: a particularly concerning problem when system processes are too complex, or insufficiently understood, to theoretically justify specific parameterizations. Here we propose a rigorous test for the detection of structural sensitivity in a system with respect to the local stability of equilibria, the main idea being to project infinite dimensional function space onto a finite dimensional space by considering the local properties of the model functions. As an illustrative example, we use our test to demonstrate structural sensitivity in the seminal Rosenzweig–MacArthur predator–prey model, and show that the conventional parameter-based approach can fail to do so. We also consider some implications that structural sensitivity has for ecological modelling: we argue that when the model exhibits structural sensitivity but experimental results remain consistent it may indicate that there is a problem with the model construction, and that in some cases trying to find an ‘optimal’ parameterization of a model function may simply be impossible when the model exhibits structural sensitivity. Finally, we suggest that the phenomenon of structural sensitivity in biological models may help explain the irregular oscillations often observed in real ecosystems.

Complexity-stability relations in generalized food-web models with realistic parameters

We investigate the relation between complexity and stability in model food webs by evaluating the local stability of fixed points of the population dynamics using the recently developed method of generalized modeling. We first determine general conditions that lead to positive complexity-stability relations. These include (1) high resource abundance and (2) strong density-dependent mortality effects that limit consumer populations. The parameters that constitute a generalized model have clear biological meanings. In this work, emphasis is placed on using realistic values for these generalized parameters. They are derived from conventional ordinary differential equations which are commonly used to describe population dynamics and for which empirical parameter estimates exist. We find that the empirically supported generalized parameters fall in regions of the parameter space that allow for a positive relation between food-web complexity and stability.

Computational ecology as an emerging science

It has long been recognized that numerical modelling and computer simulations can be used as a powerful research tool to understand, and sometimes to predict, the tendencies and peculiarities in the dynamics of populations and ecosystems. It has been, however, much less appreciated that the context of modelling and simulations in ecology is essentially different from those that normally exist in other natural sciences. In our paper, we review the computational challenges arising in modern ecology in the spirit of computational mathematics, i.e. with our main focus on the choice and use of adequate numerical methods. Somewhat paradoxically, the complexity of ecological problems does not always require the use of complex computational methods. This paradox, however, can be easily resolved if we recall that application of sophisticated computational methods usually requires clear and unambiguous mathematical problem statement as well as clearly defined benchmark information for model validation. At the same time, many ecological problems still do not have mathematically accurate and unambiguous description, and available field data are often very noisy, and hence it can be hard to understand how the results of computations should be interpreted from the ecological viewpoint. In this scientific context, computational ecology has to deal with a new paradigm: conventional issues of numerical modelling such as convergence and stability become less important than the qualitative analysis that can be provided with the help of computational techniques. We discuss this paradigm by considering computational challenges arising in several specific ecological applications.

Towards resolving the paradox of enrichment: The impact of zooplankton vertical migrations on plankton systems stability

Eutrophication, often resulting from human activity, is a serious threat to aquatic communities. Theoretical analysis of this phenomenon, based on conceptual mathematical models, leads to controversial predictions known as Rosenzweig's paradox of enrichment. At the same time, field observations demonstrate that real plankton communities exhibit various mechanisms of self-regulation which can buffer negative effects of enrichment. In this paper, we study potential effects of zooplankton vertical migration on stability of plankton systems functioning. We consider an intrinsically unstable plankton model, which is characterized by an unlimited phytoplankton multiplication and population oscillations of increasing amplitude, and investigate whether vertical migrations of zooplankton can stabilize such a system at low plankton densities. By means of developing two different models accounting for different ecological situations, e.g. deep waters and shallow waters, we show that vertical migrations of zooplankton can result in stabilization of eutrophic plankton systems. Thus, we show that this mechanism, rarely taken into account in models of plankton dynamics, may be important for resolving the paradox of enrichment in plankton communities.

How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses

In Rosenzweig-MacArthur models of predator-prey dynamics, Allee effects in prey usually destabilize interior equilibria and can suppress or enhance limit cycles typical of the paradox of enrichment. We re-evaluate these conclusions through a complete classification of a wide range of Allee effects in prey and predator's functional response shapes. We show that abrupt and deterministic system collapses not preceded by fluctuating predator-prey dynamics occur for sufficiently steep type III functional responses and strong Allee effects (with unstable lower equilibrium in prey dynamics). This phenomenon arises as type III functional responses greatly reduce cyclic dynamics and strong Allee effects promote deterministic collapses. These collapses occur with decreasing predator mortality and/or increasing susceptibility of the prey to fall below the threshold Allee density (e.g. due to increased carrying capacity or the Allee threshold itself). On the other hand, weak Allee effects (without unstable equilibrium in prey dynamics) enlarge the range of carrying capacities for which the cycles occur if predators exhibit decelerating functional responses. We discuss the results in the light of conservation strategies, eradication of alien species, and successful introduction of biocontrol agents.

Community response to enrichment is highly sensitive to model structure

Biologists use mathematical functions to model, understand and predict nature. For most biological processes, however, the exact analytical form is not known. This is also true for one of the most basic life processes: the uptake of food or resources. We show that the use of several nearly indistinguishable functions, which can serve as phenomenological descriptors of resource uptake, may lead to alarmingly different dynamical behaviour in a simple community model. More specifically, we demonstrate that the degree of resource enrichment needed to destabilize the community dynamics depends critically on the mathematical nature of the uptake function.

Generalized models as a universal approach to the analysis of nonlinear dynamical systems

We present a universal approach to the investigation of the dynamics in generalized models. In these models the processes that are taken into account are not restricted to specific functional forms. Therefore a single generalized models can describe a class of systems which share a similar structure. Despite this generality, the proposed approach allows us to study the dynamical properties of generalized models efficiently in the framework of local bifurcation theory. The approach is based on a normalization procedure that is used to identify natural parameters of the system. The Jacobian in a steady state is then derived as a function of these parameters. The analytical computation of local bifurcations using computer algebra reveals conditions for the local asymptotic stability of steady states and provides certain insights on the global dynamics of the system. The proposed approach yields a close connection between modelling and nonlinear dynamics. We illustrate the investigation of generalized models by considering examples from three different disciplines of science: a socioeconomic model of dynastic cycles in china, a model for a coupled laser system and a general ecological food web.

Complex dynamics in a three-level trophic system with intraspecies interaction

In this paper, we present a three-level trophic food chain, including intraspecies interaction. In contrast with other analyses, we consider the effect on the third trophic level by the first-level parameters. The model shows complex, as well as, chaotic oscillations. Bifurcation diagrams show period doubling route to chaos and crises. Also from the forward and backwards sections of the bifurcation diagrams, we find hysteresis. This result implies the coexistence of attractors for the same parameter values. In particular, we consider the coexistence of a chaotic and a P1 attractors. Our results show that the regulation in the food chain is not exclusive to either a food-prey or prey-predator interaction, but to a more subtle food-prey-predator interaction, where, for some parameter values, a food-prey or a prey-predator regulation may dominate the system's dynamics. Finally, we consider the impact of the intraspecies interaction in the overall dynamics of the food chain.