Asymptotic analysis of an epidemic model with primary and secondary infection

For many biological systems, the behavior of interest is contained in the evolution of transients rather than in the stability of equilibria. These include systems in which perturbations and interruptions occur on a time scale much shorter than the equilibration time, and those in which any final equilibrium is sensitive to initial conditions. In this article, we examine a model of fungal root disease in a crop involving primary and secondary infection mechanisms. This system is subject to regular interruptions in the form of harvesting and sowing. Using an asymptotic approach in which certain parameter values are assumed to be small, the model can be broken down into a set of simpler subsystems respresenting recognizable biological mechanisms. These linear models can be solved to give closed-form analytical solutions for transient evolution. From this information, it is possible to construct an annual map of disease severity in the crop, and determine the parameter values under which the infection will bulk up or fade out.     

Absolute stability and dynamical stabilisation in predator-prey systems

Many ecological systems exhibit multi-year cycles. In such systems, invasions have a complicated spatiotemporal structure. In particular, it is common for unstable steady states to exist as long-term transients behind the invasion front, a phenomenon known as dynamical stabilisation. We combine absolute stability theory and computation to predict how the width of the stabilised region depends on parameter values. We develop our calculations in the context of a model for a cyclic predator-prey system, in which the invasion front and spatiotemporal oscillations of predators and prey are separated by a region in which the coexistence steady state is dynamically stabilised.     

Single species models with logistic growth and dissymmetric impulse dispersal

In this paper, two classes of single-species models with logistic growth and impulse dispersal (or migration) are studied: one model class describes dissymmetric impulsive bi-directional dispersal between two heterogeneous patches; and the other presents a new way of characterizing the aggregate migration of a natural population between two heterogeneous habitat patches, which alternates in direction periodically. In this theoretical study, some very general, weak conditions for the permanence, extinction of these systems, existence, uniqueness and global stability of positive periodic solutions are established by using analysis based on the theory of discrete dynamical systems. From this study, we observe that the dynamical behavior of populations with impulsive dispersal differs greatly from the behavior of models with continuous dispersal. Unlike models where the dispersal is continuous in time, in which the travel losses associated with dispersal make it difficult for such dispersal to evolve e.g., [25], [26], [28], in the present study it was relatively easy for impulsive dispersal to positively affect populations when realistic parameter values were used, and a rich variety of behaviors were possible. From our results, we found impulsive dispersal seems to more nicely model natural dispersal behavior of populations and may be more relevant to the investigation of such behavior in real ecological systems.     

Period 7 in a line of trees

In a line of model trees the heights tend to vary with period 7 in many cases. This is the observable effect of an interesting structural reason: in the parameter space just beyond the border of the stability range of homogeneous heights stationary solutions with long spatial periods and irregularities can be observed before simpler short periodic solutions become stable. The model can be solved mainly analytically. The models ecological significance concerns the onset of patchiness in forest systems.     

The importance of transients' dynamics in spatially extended populations

Recent theoretical works on the dynamics of metapopulations have highlighted the existence of very long transients (supertransients) with abrupt changes in behaviour which occur following perturbation of the system away from its attractor. If this phenomenon is common in natural systems, populations that do not oscillate can begin to fluctuate wildly without any change in the environmental conditions. However, the frequency of occurrence of supertransients is currently poorly understood even in model systems. Here we explore their occurrence in metapopulation models which relax the important assumption of global synchrony of events implicit in all the coupled map lattice models for which supertransients have so far been demonstrated. We find supertransients in all the models but always only for a very restricted range of parameter combinations. However, we also report for the first time another type of longer-lived transient (mesotransients) that occurs on shorter time-scales than supertransients and is found for a much wider set of conditions. We argue that these medium-term changes in the dynamics of populations can be of more ecological relevance than the long-term changes of supertransients.     

Bifurcation phenomena appearing in the Lotka-Volterra competition equations: a numerical study

Bifurcation phenomena appearing in the Lotka-Volterra competition equations with periodically varying coefficients are studied numerically. We assume sinusoidal oscillations of the coefficients and use phase differences between them as free parameters. We are mainly concerned with the case where a pair of stable and unstable positive periodic solutions exists, although one of the trivial periodic solutions is stable and the other is unstable. We obtain a very curious bifurcation diagram in which two branches of stable and unstable positive periodic solutions are connected at both ends, but are connected with no other branches. We show how this unusual diagram can be viewed as a cross-section of a multidimensional bifurcation diagram. The region in a 3-dimensional parameter space where a pair of stable and unstable positive periodic solutions exists is shown in an example, and the ecological meaning of the phase differences necessary for stable coexistence of two species is considered. Finally, a bifurcation problem with the average intrinsic growth rate as a parameter is also dealt with numerically, in relation with Cushing's result.     

Selection between multiple periodic regimes in a biochemical system: complex dynamic behaviour resolved by use of one-dimensional maps

We analyse a model biochemical system in which two autocatalytic enzyme reactions are coupled in series, in conditions where multiple stable periodic regimes coexist for the same set of parameter values. We determine how the periodic regimes are reached from different initial conditions. The structure of the attraction basins is generally simple in the case of two coexisting limit cycles (birhythmicity). This structure and the associated behaviour may, however, become highly complex. In particular, the system exhibits enhanced sensitivity to initial conditions when the boundaries of the attraction basins are fractal. In the latter case, it becomes difficult to predict the evolution towards either one of two limit cycles, a phenomenon known as final state sensitivity. We show how these complex phenomena can be explained in a unified and simple manner by means of one-dimensional return maps derived from the time evolution of the model and from fifth degree polynomial equations. We suggest experimental tests of the sensitivity to initial conditions in chemical systems presenting birhythmicity. The physiological significance of the results is discussed with respect to the sensitivity of regulatory systems admitting multiple stable biological rhythms.     

Transients: the key to long-term ecological understanding?

Ecological theory has been dominated by a focus on long-term or asymptotic behavior as a way to understand natural systems. Yet experiments are done on much shorter timescales, and the relevant timescales for ecological systems can also be relatively short. Thus, there is a mismatch between the timescales of most experiments and the timescales of many theoretical investigations. However, recent work has emphasized the importance of transient dynamics rather than long-term behavior in ecological systems, enabling the examination of forces that allow coexistence on ecological timescales. Through an examination of what leads to transients in ecological systems, a deeper appreciation of the forces leading to persistence or coexistence in ecological systems emerges, as well as a general understanding of how population levels can change through time.     

Local and global bifurcations at infinity in models of glycolytic oscillations

 We investigate two models of glycolytic oscillations. Each model consists of two coupled nonlinear ordinary differential equations. Both models are found to have a saddle point at infinity and to exhibit a saddle-node bifurcation at infinity, giving rise to a second saddle and a stable node at infinity. Depending on model parameters, a stable limit cycle may blow up to infinite period and amplitude and disappear in the bifurcation, and after the bifurcation, the stable node at infinity then attracts all trajectories. Alternatively, the stable node at infinity may coexist with either a stable sink (not at infinity) or a stable limit cycle. This limit cycle may then disappear in a heteroclinic bifurcation at infinity in which the unstable manifold from one saddle at infinity joins the stable manifold of the other saddle at infinity. These results explain prior reports for one of the models concerning parameter values for which the system does not admit any physical (bounded) behavior. Analytic results on the scaling of amplitude and period close to the bifurcations are obtained and confirmed by numerical computations. Finally, we consider more realistic modified models where all solutions are bounded and show that some of the features stemming from the bifurcations at infinity are still present. Received 4 September 1995; received in revised form 18 September 1996     

The numerical solution of stiff differential equations

This paper first discusses the conditions in which a set of differential equations should give stable solutions, starting with linear systems assuming that these do not differ greatly in this respect from non-linear systems. Methods of investigating the stability of particular systems are briefly discussed. Most real biochemical systems are known from observation to be stable, but little is known of the regions over which stability persists; moreover, models of biochemical systems may not be stable, because of inaccurate choice of parameter values.The separate problem of stability and accuracy in numerical methods of approximating the solution of systems of non-linear equations is then treated. Stress is laid on the consistently unsatisfactory results given by explicit methods for systems containing "stiff" equations, and implicit multistep methods are particularly recommended for this class of problem, which is likely to include many biochemical model systems. Finally, an iteration procedure likely to give convergence both in multistep methods and in the steady-state approach is recommended, and areas in which improvement in methods is likely to occur are outlined.     

Coupled land use and ecological models reveal emergence and feedbacks in socio‐ecological systems

Understanding the dynamics of socio‐ecological systems is crucial to the development of environmentally sustainable practices. Models of social or ecological sub‐systems have greatly enhanced such understanding, but at the risk of obscuring important feedbacks and emergent effects. Integrated modelling approaches have the potential to address this shortcoming by explicitly representing linked socio‐ecological dynamics. We developed a socio‐ecological system model by coupling an existing agent‐based model of land‐use dynamics and an individual‐based model of demography and dispersal. A hypothetical case‐study was established to simulate the interaction of crops and their pollinators in a changing agricultural landscape, initialised from a spatially random distribution of natural assets. The bi‐directional coupled model predicted larger changes in crop yield and pollinator populations than a unidirectional uncoupled version. The spatial properties of the system also differed, the coupled version revealing the emergence of spatial land‐use clusters that neither supported nor required pollinators. These findings suggest that important dynamics may be missed by uncoupled modelling approaches, but that these can be captured through the combination of currently‐available, compatible model frameworks. Such model integrations are required to further fundamental understanding of socio‐ecological dynamics and thus improve management of socio‐ecological systems.     

Model with asymptotically stable dynamics for Drosophila gap gene network

We consider a model of gap gene expression during the early development of Drosophila embryo. Parameter values in the model have been obtained by fitting to experimental patterns under an additional condition that the solution be asymptotically stable at large times. The patterns at the beginning of gastrulation in such solutions are very close to an actual attractor in the model. It is shown that such solutions are more robust to perturbations of concentrations and parameter values in the model.     

Sudden Shifts in Ecological Systems: Intermittency and Transients in the Coupled Ricker Population Model

Many real ecological systems show sudden changes in behavior, phenomena sometimes categorized as regime shifts in the literature. The relative importance of exogenous versus endogenous forces producing regime shifts is an important question. These forces’ role in generating variability over time in ecological systems has been explored using tools from dynamical systems. We use similar ideas to look at transients in simple ecological models as a way of understanding regime shifts. Based in part on the theory of crises, we carefully analyze a simple two patch spatial model and begin to understand from a mathematical point of view what produces transient behavior in ecological systems. In particular, since the tools are essentially qualitative, we are able to suggest that transient behavior should be ubiquitous in systems with overcompensatory local dynamics, and thus should be typical of many ecological systems. This work has been supported by NSF Grant EF-0434266.     

Studying the Logone floodplain,Cameroon, as a coupled human and natural system§

African floodplains are an excellent example of coupled human–natural systems because they exhibit strong interactions among multiple social, ecological, and hydrological systems. The intra-annual and interannual variations in seasonal flooding have direct and indirect impacts on ecosystems and human lives and livelihoods. Coupled human and natural system (CHANS) is a broad conceptual framework that is used to study systems in which human and natural components interact. While there are other conceptual frameworks to study social-ecological systems, the CHANS framework offers a clear way of studying the interactions, called couplings, between human and natural systems. Core features of the framework are the following: human and natural systems are analytically separated; focus is on processes within and couplings between systems; and the goal is to build an integrative, quantitative model of the coupled system. This paper explains the conceptual framework of coupled systems, using the case study of the Logone floodplain in Cameroon. We compare the CHANS framework with other frameworks that have been used to study the same floodplain, and argue for its usefulness in the study of African floodplains.     

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

Numerous formulations with the same mathematical properties can be relevant to model a biological process. Different formulations can predict different model dynamics like equilibrium vs. oscillations even if they are quantitatively close (structural sensitivity). The question we address in this paper is: does the choice of a formulation affect predictions on the number of stable states? We focus on a predator–prey model with predator competition that exhibits multiple stable states. A bifurcation analysis is realized with respect to prey carrying capacity and species body mass ratio within range of values found in food web models. Bifurcation diagrams built for two type-II functional responses are different in two ways. First, the kind of stable state (equilibrium vs. oscillations) is different for 26.0–49.4% of the parameter values, depending on the parameter space investigated. Using generalized modelling, we highlight the role of functional response slope in this difference. Secondly, the number of stable states is higher with Ivlev's functional response for 0.1–14.3% of the parameter values. These two changes interact to create different model predictions if a parameter value or a state variable is altered. In these two examples of disturbance, Holling's disc equation predicts a higher system resilience. Indeed, Ivlev's functional response predicts that disturbance may trap the system into an alternative stable state that can be escaped from only by a larger alteration (hysteresis phenomena). Two questions arise from this work: (i) how much complex ecological models can be affected by this sensitivity to model formulation? and (ii) how to deal with these uncertainties in model predictions?     

Pattern Formation,Long-Term Transients,and the Turing–Hopf Bifurcation in a Space- and Time-Discrete Predator–Prey System

Understanding of population dynamics in a fragmented habitat is an issue of considerable importance. A natural modelling framework for these systems is spatially discrete. In this paper, we consider a predator–prey system that is discrete both in space and time, and is described by a Coupled Map Lattice (CML). The prey growth is assumed to be affected by a weak Allee effect and the predator dynamics includes intra-specific competition. We first reveal the bifurcation structure of the corresponding non-spatial system. We then obtain the conditions of diffusive instability on the lattice. In order to reveal the properties of the emerging patterns, we perform extensive numerical simulations. We pay a special attention to the system properties in a vicinity of the Turing–Hopf bifurcation, which is widely regarded as a mechanism of pattern formation and spatiotemporal chaos in space-continuous systems. Counter-intuitively, we obtain that the spatial patterns arising in the CML are more typically stationary, even when the local dynamics is oscillatory. We also obtain that, for some parameter values, the system’s dynamics is dominated by long-term transients, so that the asymptotical stationary pattern arises as a sudden transition between two different patterns. Finally, we argue that our findings may have important ecological implications.     

Multistable Dynamics Mediated by Tubuloglomerular Feedback in a Model of Coupled Nephrons

To help elucidate the causes of irregular tubular flow oscillations found in the nephrons of spontaneously hypertensive rats (SHR), we have conducted a bifurcation analysis of a mathematical model of two nephrons that are coupled through their tubuloglomerular feedback (TGF) systems. This analysis was motivated by a previous modeling study which predicts that NaCl backleak from a nephron’s thick ascending limb permits multiple stable oscillatory states that are mediated by TGF (Layton et al. in Am. J. Physiol. Renal Physiol. 291:F79–F97, 2006); that prediction served as the basis for a comprehensive, multifaceted hypothesis for the emergence of irregular flow oscillations in SHR. However, in that study, we used a characteristic equation obtained via linearization from a single-nephron model, in conjunction with numerical solutions of the full, nonlinear model equations for two and three coupled nephrons. In the present study, we have derived a characteristic equation for a model of any finite number of mutually coupled nephrons having NaCl backleak. Analysis of that characteristic equation for the case of two coupled nephrons has revealed a number of parameter regions having the potential for differing stable dynamic states. Numerical solutions of the full equations for two model nephrons exhibit a variety of behaviors in these regions. Some behaviors exhibit a degree of complexity that is consistent with our hypothesis for the emergence of irregular oscillations in SHR.     

Dendritic and Synaptic Effects in Systems of Coupled Cortical Oscillators

We explore the influence of synaptic location and form on the behavior of networks of coupled cortical oscillators. First, we develop a model of two coupled somatic oscillators that includes passive dendritic cables. Using a phase model approach, we show that the synchronous solution can change from a stable solution to an unstable one as the cable lengthens and the synaptic position moves further from the soma. We confirm this prediction using a system of coupled compartmental models. We also demonstrate that when the synchronous solution becomes unstable, a bifurcation occurs and a pair of asynchronous stable solutions appear, causing a phase lag between the cells in the system. Then using a variety of coupling functions and different synaptic positions, we show that distal connections and broad synaptic time courses encourage phase lags that can be reduced, eliminated, or enhanced by the presence of active currents in the dendrite. This mechanism may appear in neural systems where proximal connections could be used to encourage synchrony, and distal connections and broad synaptic time courses could be used to produce phase lags that can be modulated by active currents.     

On the Neimark-Sacker bifurcation in a discrete predator-prey system

A two-parameter family of discrete models describing a predator-prey interaction is considered, which generalizes a model discussed by Murray, and originally due to Nicholson and Bailey, consisting of two coupled nonlinear difference equations. In contrast to the original case treated by Murray, where the two populations either die out or may display unbounded growth, the general member of this family displays a somewhat wider range of behaviour. In particular, the model has a nontrivial steady state which is stable for a certain range of parameter values, which is explicitly determined, and also undergoes a Neimark-Sacker bifurcation that produces an attracting invariant curve in some areas of the parameter space and a repelling one in others.     

Natural Shorelines Promote the Stability of Fish Communities in an Urbanized Coastal System

Habitat loss and fragmentation are leading causes of species extinctions in terrestrial, aquatic and marine systems. Along coastlines, natural habitats support high biodiversity and valuable ecosystem services but are often replaced with engineered structures for coastal protection or erosion control. We coupled high-resolution shoreline condition data with an eleven-year time series of fish community structure to examine how coastal protection structures impact community stability. Our analyses revealed that the most stable fish communities were nearest natural shorelines. Structurally complex engineered shorelines appeared to promote greater stability than simpler alternatives as communities nearest vertical walls, which are among the most prevalent structures, were most dissimilar from natural shorelines and had the lowest stability. We conclude that conserving and restoring natural habitats is essential for promoting ecological stability. However, in scenarios when natural habitats are not viable, engineered landscapes designed to mimic the complexity of natural habitats may provide similar ecological functions.