A. Yu. Morozov

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.

Evolution of virulence driven by predator-prey interaction: Possible consequences for population dynamics

The evolution of pathogen virulence in natural populations has conventionally been considered as a result of selection caused by the interactions of the host with its pathogen(s). The host population, however, is generally embedded in complex trophic interactions with other populations in the community, in particular, intensive predation on the infected host can increase its mortality, and this can affect the course of virulence evolution. Reciprocally, in the long run, the evolution of virulence within an infected host can affect the patterns of population dynamics of a predator consuming the host (e.g. resulting in large amplitude oscillations, causing a severe drop in the population size, etc.). Surprisingly, neither the effect of predation on the evolution of virulence within a host, nor the influence of the evolution of virulence upon the consumers dynamics has been addressed in the literature yet. In this paper, we consider a classical S-I ecoepidemiological model in which the infected host is consumed by a predator. We are particularly interested in the evolutionarily stable virulence of the pathogen in the model and its dependence upon ecologically relevant parameters. We show that predation can prominently shift the evolutionarily stable virulence towards more severe strains as compared to the same system without predation. We demonstrate that the evolution of virulence can result in a succession of dynamical regimes and can even lead to the extinction of the predator in the long run. The presence of a predator can indirectly affect the evolution within its prey since the evolutionarily stable virulence becomes a function of the prey growth rate, which would not be the case in a predator-free system. We find that the evolutionarily stable virulence largely depends on the carrying capacity K of the prey in a non-monotonous way. The model also predicts that in an eutrophic environment the shift of virulence towards evolutionarily stable benign strains can cause demographically stochastic evolutionary suicide, resulting in the extinction of both species, thus artificially maintaining severe strains of pathogen can enhance the persistence of both species.

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

Bifurcation Analysis of Models with Uncertain Function Specification: How Should We Proceed?

n

Towards a correct description of zooplankton feeding in models: Taking into account food-mediated unsynchronized vertical migration

n 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.

Long-term transients and complex dynamics of a stage-structured population with time delay and the Allee effect.

It is well known that predation/harvesting on a species subjected to an infectious disease can affect both the infection prevalence and the population dynamics. In this paper, I model predator–prey–pathogen interactions in the case where the presence of a predator indirectly affects the transmission rate of the infection in its prey. I call this phenomenon the predator-dependent disease transmission. Such a scenario can arise, for example, as a consequence of anti-predator defence behaviour, debilitating the immune system of the prey. Although being well documented, the predator-dependent disease transmission has rarely been taken into account in ecoepidemiological models. Mathematically, I consider a classical S-I-P ecoepidemiological model in which the infected and/or the healthy host can be consumed by a predator where the coefficient in the mass action transmission term is predator-dependent. Investigation of the model shows that including such a predator-dependent disease transmission can have important consequences for shaping predator–prey–pathogen interactions. In particular, this can enhance the survival of the predator, restricted in a system with a predator-independent disease transmission. I demonstrate the emergence of a disease-mediated strong Allee effect for the predator population. I also show that in the system with predator-dependent disease transmission, the predator can indirectly promote epidemics of highly virulent infectious diseases, which would die out in a predator-free system. Finally, I argue that taking into account predator-dependent disease transmission can have a destabilizing effect in a eutrophic environment, which can potentially cause the extinction of both species. I also show that including the predator-dependent disease transmission may increase the infection prevalence, and this fact will question the ‘keeping herds healthy’ hypothesis concerning the management of wildlife infections by natural predators.

Spatio-temporal horizontal plankton patterns caused by biological invasion in a two-species model of plankton dynamics allowing for the Allee effect

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.

Tri-trophic Plankton Models Revised: Importance of Space, Food Web Structure and Functional Response Parametrisation

Understanding how patterns and travelling waves form in chemical and biological reaction–diffusion models is an area which has been widely researched, yet is still experiencing fast development. Surprisingly enough, we still do not have a clear understanding about all possible types of dynamical regimes in classical reaction–diffusion models, such as Lotka–Volterra competition models with spatial dependence. In this study, we demonstrate some new types of wave propagation and pattern formation in a classical three species cyclic competition model with spatial diffusion, which have been so far missed in the literature. These new patterns are characterized by a high regularity in space, but are different from patterns previously known to exist in reaction–diffusion models, and may have important applications in improving our understanding of biological pattern formation and invasion theory. Finding these new patterns is made technically possible by using an automatic adaptive finite element method driven by a novel a posteriori error estimate which is proved to provide a reliable bound for the error of the numerical method. We demonstrate how this numerical framework allows us to easily explore the dynamical patterns in both two and three spatial dimensions.