Behaviour of simple population models under ecological processes

The two most popular and extensively-used discrete models of population growth display the generic bifurcation structure of a hierarchy of period-doubling sequence to chaos with increasing growth rates. In this paper we show that these two models, though they belong to a general class of one-dimensional maps, show very different dynamics when important ecological processes such as immigration and emigration/depletion, are considered. It is important that ecologists recognize the differences between these models before using them to describe their data—or develop optimization strategies—based on these models.     

Models for Bounded Systems with Continuous Dynamics

Summary .  Models for natural nonlinear processes, such as population dynamics, have been given much attention in applied mathematics. For example, species competition has been extensively modeled by differential equations. Often, the scientist has preferred to model the underlying dynamical processes (i.e., theoretical mechanisms) in continuous time. It is of both scientific and mathematical interest to implement such models in a statistical framework to quantify uncertainty associated with the models in the presence of observations. That is, given discrete observations arising from the underlying continuous process, the unobserved process can be formally described while accounting for multiple sources of uncertainty (e.g., measurement error, model choice, and inherent stochasticity of process parameters). In addition to continuity, natural processes are often bounded; specifically, they tend to have nonnegative support. Various techniques have been implemented to accommodate nonnegative processes, but such techniques are often limited or overly compromising. This article offers an alternative to common differential modeling practices by using a bias-corrected truncated normal distribution to model the observations and latent process, both having bounded support. Parameters of an underlying continuous process are characterized in a Bayesian hierarchical context, utilizing a fourth-order Runge–Kutta approximation.     

Pair-level approximations to the spatio-temporal dynamics of epidemics on asymmetric contact networks

The process of infection during an epidemic can be envisaged as being transmitted via a network of routes represented by a contact network. Most differential equation models of epidemics are mean-field models. These contain none of the underlying spatial structure of the contact network. By extending the mean-field models to pair-level, some of the spatial structure can be contained in the model. Some networks of transmission such as river or transportation networks are clearly asymmetric, whereas others such as airborne infection can be regarded as symmetric. Pair-level models have been developed to describe symmetric contact networks. Here we report on work to develop a pair-level model that is also applicable to asymmetric contact networks. The procedure for closing the model at the level of pairs is discussed in detail. The model is compared against stochastic simulations of epidemics on asymmetric contact networks and against the predictions of the symmetric model on the same networks. DEFRA funded project FC1153     

On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations

The expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented.     

A two-stage model for the SIR outbreak: accounting for the discrete and stochastic nature of the epidemic at the initial contamination stage

The evolution of an infectious disease outbreak in an isolated population is split into two stages: a stochastic Markov process describing the initial contamination and a linked deterministic dynamical system with random initial conditions for the continued development of the outbreak. The initial contamination stage is well approximated by the randomized SI (susceptible/infected) model. We obtain the probability density function for the early behavior of the epidemic. This provides an appropriate distribution for the initial conditions with which to describe the subsequent deterministic evolution of the system. We apply the method of matching asymptotic expansions to link the two stages. This allows us to estimate the standard deviation of the number of infectives in the developed outbreak, and the statistical characteristics of the outbreak time. The potential trajectories caused by the stochastic nature of the contamination stage show greatest divergence at the initial and fade-out stages and coincide most tightly just after the peak of the epidemic. The time to the peak of the outbreak is not strongly dependent on the initial trajectory.     

On the analysis of competition at the level of the individual plant

Summary The extent to which some measure of local crowding can account for the performance of individual plants is examined with reference to populations of two species of annual plant. Only a relatively small proportion of the variation in individual plant yield could be accounted for by measures of local crowding. These included the number of close neighbours, an estimate of the area available to each plant and competitive pressure. A multiple regression that took account of both emergence time and local crowding increased the proportion of variance that could be accounted for up to 50%. Computer simulations of the growth of indivudual plants in monoculture were then caried out in order to determine whether the unexplained variation resulted from fundamental flaws in the models or from unaccounted for sources of variation in the field. The results from the simulations again indicated that only a relatively low proportion of the variation in individual plant yield could be accounted for by emergence time and local density, even though these were known to be the only variables present. These findings are discussed in relation to the relative importance of one-sided and two-sided competition, and the complex cross-correlations that occur between individuals in plant populations. These two factors will make it very difficult for field workers to determine accurately what factors determine individual plant yield and in particular to predict the effects of local crowding on the performance of individual plants.     

A computational approach to studying ageing at the individual level

The ageing process is actively regulated throughout an organism''s life, but studying the rate of ageing in individuals is difficult with conventional methods. Consequently, ageing studies typically make biological inference based on population mortality rates, which often do not accurately reflect the probabilities of death at the individual level. To study the relationship between individual and population mortality rates, we integrated in vivo switch experiments with in silico stochastic simulations to elucidate how carefully designed experiments allow key aspects of individual ageing to be deduced from group mortality measurements. As our case study, we used the recent report demonstrating that pheromones of the opposite sex decrease lifespan in Drosophila melanogaster by reversibly increasing population mortality rates. We showed that the population mortality reversal following pheromone removal was almost surely occurring in individuals, albeit more slowly than suggested by population measures. Furthermore, heterogeneity among individuals due to the inherent stochasticity of behavioural interactions skewed population mortality rates in middle-age away from the individual-level trajectories of which they are comprised. This article exemplifies how computational models function as important predictive tools for designing wet-laboratory experiments to use population mortality rates to understand how genetic and environmental manipulations affect ageing in the individual.     

On the formulation and analysis of general deterministic structured population models I. Linear Theory

—We define a linear physiologically structured population model by two rules, one for reproduction and one for “movement” and survival. We use these ingredients to give a constructive definition of next-population-state operators. For the autonomous case we define the basic reproduction ratio R ₀ and the Malthusian parameter r and we compute the resolvent in terms of the Laplace transform of the ingredients. A key feature of our approach is that unbounded operators are avoided throughout. This will facilitate the treatment of nonlinear models as a next step. Received 26 July 1996; received in revised form 3 September 1997     

New species in evolving networks--stochastic theory of sensitive networks and applications on the metaphorical level

In this paper we develop a theory to describe stochastic influences on the fate of new species with non-linear growth rates in evolutionary processes. We develop a theoretical framework based on notions of species, network, innovation, competition, survival and fitness. We introduce a stochastic picture describing the role of fluctuations in the survival of new species in non-linear systems. In particular we consider the fate of new species with non-linear growth. As an application of the general model framework we consider the fate of 'rare species' in early biological evolution. We show that hypercycle systems do not represent the end of the evolutionary process as they may evolve further in small niches. This has implications for different types of applications ranging from biological systems on one level to socio-technological systems on a more metaphoric level.     

Bridging the gap between theory and data in ecological models

In September 2011, the University of Essex, UK, hosted an interdisciplinary conference, Mathematical and Theoretical Ecology 2011 (MATE 2011), with the theme of ‘Linking models with ecological processes’. The aim of the meeting was to create discussion and debate between modellers and empiricists working in ecology. A wide range of topics were discussed at the meeting including evolutionary and community models of ecosystem structure, epidemiological models, non-linear models of population dynamics, spatiotemporal models, individual and collective movement behaviour, and applications of ecological models to engineering problems. In this introductory article, we provide a report of the MATE 2011 meeting, and briefly review the most recent relevant research in the fields of mathematical and theoretical ecology. We introduce and summarise the eight contributed articles that were selected for this special issue. The diverse range of topics and the wide range of mathematical, statistical and computational tools used illustrate the broad appeal and depth of research in the rich field of mathematical and theoretical ecology.     

Estimating the effect of continual disturbances on discrete-time population models

In this paper, methods for estimating the effect of continual (but bounded) disturbances on ecosystems modelled by difference equations are discussed. The approach adopted is to estimate the region of state space (called a reachable set) which can be reached by the disturbed system from an initial healthy state in a given time period. Liapunov stability methods for estimating these reachable sets are presented and applied to two specific population models.     

Tuberculosis models with fast and slow dynamics: the role of close and casual contacts

Models that incorporate local and individual interactions are introduced in the context of the transmission dynamics of tuberculosis (TB). The multi-level contact structure implicitly assumes that individuals are at risk of infection from close contacts in generalized household (clusters) as well as from casual (random) contacts in the general population. Epidemiological time scales are used to reduce the dimensionality of the model and singular perturbation methods are used to corroborate the results of time-scale approximations. The concept and impact of optimal average cluster or generalized household size on TB dynamics is discussed. We also discuss the potential impact of our results on the spread of TB.     

Evaluation of models for the evolution of protein sequences and functions under structural constraint

In the field of evolutionary structural genomics, methods are needed to evaluate why genomes evolved to contain the fold distributions that are observed. In order to study the effects of population dynamics in the evolved genomes we need fast and accurate evolutionary models which can analyze the effects of selection, drift and fixation of a protein sequence in a population that are grounded by physical parameters governing the folding and binding properties of the sequence. In this study, various knowledge-based, force field, and statistical methods for protein folding have been evaluated with four different folds: SH2 domains, SH3 domains, Globin-like, and Flavodoxin-like, to evaluate the speed and accuracy of the energy functions. Similarly, knowledge-based and force field methods have been used to predict ligand binding specificity in SH2 domain. To demonstrate the applicability of these methods, the dynamics of evolution of new binding capabilities by an SH2 domain is demonstrated.     

Realistic population dynamics in epidemiological models: the impact of population decline on the dynamics of childhood infectious diseases. Measles in Italy as an example

Most contributions in the field of mathematical modelling of childhood infectious diseases transmission dynamics have focused on stationary or exponentially growing populations. In this paper an epidemiological model with realistic demography is used to investigate the impact of the non-equilibrium conditions typical of the transition to sustained below replacement fertility (BRF) recently observed in a number of western countries, upon the transmission dynamics of measles. The results depend on the manner we model the relation between the (changing) age distribution of the population and contacts. Under some circumstances the transitional ageing phase typical of BRF populations might complexly interact with epidemiological variables leading to (i) a substantial reduction in the amount of vaccination effort required for eliminating the disease; (ii) a significant magnification of the perverse impact of vaccination in terms of the burden of severe age related morbidity.     

On the uniqueness of the positive solution of an integral equation which appears in epidemiological models

 In this paper, we show that the positive solution of a non-linear integral equation which appears in classical SIR epidemiological models is unique. The demonstration of this fact is necessary to justify the correctness of any approximate or numerical solution. The SIR epidemiological model is used only for simplicity. In fact, the methods used can be easily extended to prove the existence and uniqueness of the more involved integral equations that appear when more biological realities are considered. Thus the inclusion of a latent class (SLIR models) and models incorporating variability in the infectiousness with duration of the infection and spatial distribution lead to integral equations to which the results derived in this paper apply immediately. Received: 7 May 1999     

Persistent solutions for age-dependent pair-formation models

Conditions on the vital rates and the mating function are derived which imply existence or nonexistence of exponentially growing persistent age-distributions for age-dependent pair-formation models.     

Agent-based models to investigate sound impact on marine animals: bridging the gap between effects on individual behaviour and population level consequences

Marine ambient sound levels have risen due to noisy human activities, such as shipping, fishing, seismic surveys and piling for windfarms. Marine mammals and fishes are two prominent taxonomic groups that are exposed to this noise pollution, which may experience detrimental effects at the population level. Acoustic effects on individual behaviour such as deterrence, disturbance, distraction and masking of biologically relevant sounds, can be translated energetically to changes in vital rates (growth, maturation, reproduction and survival) in a population consequences of acoustic disturbance (PCAD) approach. However, we typically neglect spatial variation in species distributions and noise pollution, while abiotic factors like temperature, bathymetry and currents, as well as habitat quality in terms of feeding or hiding opportunities, will also have a geographically variable impact on potential consequences. We here address the conceptual integration of agent based models (ABM) into the PCAD framework, as a suitable theoretical tool with high potential for the exploration of these spatial factors and their modifying role in noise impact assessment studies. We review five ABM case studies, including investigations into: 1) effects of movement strategy on the impact of explosions in harbour porpoise; 2) effects of disturbance sensitivity on pile driving impact on migrating cod; 3) impact of seismic survey sounds on Atlantic mackerel distribution and movement; 4) population-level impact of mitigation of harbour porpoise bycatch with pingers; and 5) population effects of alternative windfarm construction scenarios in harbour porpoise. We discuss similarities and differences among these studies in sound and species mapping approaches and we evaluate model realism and pattern validation. We believe that ABMs are a valuable tool for integrating spatial information into ecological impact studies that investigate acoustic disturbance, for any type of sound source, and for both marine mammals and fish.     

Following the initiation and development of individual leaf primordia at the level of the shoot apical meristem: the case of distichous phyllotaxis in Begonia

BACKGROUND AND AIMS: By using the technique of replicas of a developing apex it is possible to obtain a direct measure of phyllotactic parameters (plastochrone and platochronic ratio) involved in the initiation of two successive primordia at the level of the SAM. The goal of this study is to compare, in a real time setting, the value of phyllotactic parameters in distichous systems using Begonia as a case study, with the value of the same parameters in spiral phyllotactic systems. METHODS: To determine the real-time sequence of events at the level of the SAM, replicas were made of the developing apex at different intervals using previously described techniques. Impression moulds were made at 24-h intervals. The following phyllotactic parameters were measured: plastochrone, angle of divergence, plastochrone ratio and ratio between the diameter of the leaf and the apex. RESULTS: The time between the appearance of two successive leaves is 15-20 d. The average value of the plastochrone ratio (R) is 1.3, and the ratio of the leaf to the diameter of the apex (Gamma) is 2.5. The angle of divergence varies from 165 masculine to 180 masculine. The speed of advection of the primordium from the apex, varies from 0.28 to 0.37 microm d(-1). CONCLUSIONS: The speed of advection of primordia in Begonia is lower than that of Anagalis. This is not in accordance with theoretical simulations that predict the opposite. In Begonia, the plastochrone ratio does not reflect the real time of appearance of two successive primordia. The time separating the appearance of two primordia is not directly related to the distance of these two primordia from the centre of the apex but is related instead to the enlargement of leaves.