Year-to-year Oscillations in Demography of the Strictly Biennial <Emphasis Type="Italic">Pedicularis sylvatica</Emphasis> and Effects of Experimental Disturbances

In 1998–2001, I studied disturbance effects on the population structure and dynamics of a grassland strict biennial Pedicularis sylvatica, and on the species demography (monthly dynamics of seedling recruitment in 1998 and within- and between-year survival in 1998–2000). In two Czech populations, I established three experimental disturbance regimes: (1) a gap treatment, that simulated grazing by clipping vegetation and creating small gaps, (2) a mowing treatment, where I clipped the vegetation, and (3) a no management treatment, where I left the vegetation untreated. The number of recruiting seedlings varied greatly by year, and demographic structure of populations showed significant year-to-year oscillations in mean seedling numbers, from low (3 ± 0.7 s.e. per 0.25 m² plot) to high (103 ± 20). Inversely in the same years and plots, mean adult numbers in populations oscillated from high (12 ± 2) to low (0.7 ± 0.3). Disturbance effects were only important for seedling recruitment in early census dates in all years. In 1998, most seedlings recruited in April–May in gaps in both sites, but most died before winter. Within- and between-year survival was not affected by disturbance regimes but fluctuated significantly among years. Between-year survival increased with increasing size of the overwintering bud and was higher in disturbance treatments. Since the oscillations in population structure did not significantly vary in response to experimental disturbances, population dynamics may be driven endogenously rather than by disturbance events. The weak disturbance effects on species demography may also indicate population resilience to changes in habitat quality. However, since disturbances promoted seedling recruitment, grazing or mowing regimes are strongly recommended, as they create regeneration opportunities and maintain habitat quality, meeting the species long-term conservation goals.     

Spatial scaling in a benthic population model with density-dependent disturbance.

This work investigates approaches to simplifying individual-based models in which the rate of disturbance depends on local densities. To this purpose, an individual-based model for a benthic population is developed that is both spatial and stochastic. With this model, three possible ways of approximating the dynamics of mean numbers are examined: a mean-field approximation that ignores space completely, a second-order approximation that represents spatial variation in terms of variances and covariances, and a patch-based approximation that retains information about the age structure of the patch population. Results show that space is important and that a temporal model relying on mean disturbance rates provides a poor approximation to the dynamics of mean numbers. It is possible, however, to represent relevant spatial variation with second-order moments, particularly when recruitment rates are low and/or when disturbances are large and weak. Even better approximations are obtained by retaining patch age information.     

A minimum model of prey-predator system with dormancy of predators and the paradox of enrichment

In this paper, a mathematical model of a prey-predator system is proposed to resolve the paradox of enrichment in ecosystems. The model is based on the natural strategy that a predator takes, i.e, it produces resting eggs in harsh environment. Our result gives a criterion for a functional response, which ensures that entering dormancy stabilizes the population dynamics. It is also shown that the hatching of resting eggs can stabilize the population dynamics when the switching between non-resting and resting eggs is sharp. Furthermore, the bifurcation structure of our model suggests the simultaneous existence of a stable equilibrium and a large amplitude cycle in natural enriched environments.      

Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system

We consider a continuous taxis-diffusion-reaction system of partial-differential equations describing spatiotemporal dynamics of a predator–prey system. The local kinetics of the system is defined by general Gause–Kolmogorov-type model. The predator ability to pursue the prey is modelled by the Patlak–Keller–Segel taxis model, assuming that movement velocities of predators are proportional to the gradients of specific cues emitted by prey (e.g., odour, pheromones, exometabolites). The linear stability analysis of the model showed that the non-trivial homogeneous stationary regime of the model becomes unstable with respect to small heterogeneous perturbations with increase of prey-taxis activity; an Andronov–Hopf bifurcation occurs in the system when the taxis coefficient of predator exceeds its critical bifurcation value that exists for all admissible values of model parameters. These findings generalize earlier results obtained for particular cases of the Gause–Kolmogorov-type model assuming logistic reproduction of the prey population and the Holling types I and II functional responses of the predator population. Numerical simulations with theta-logistic growth of the prey population and the Ivlev functional response of predators illustrate and support results of the analytical study.     

A basic bifurcation structure from bursting to spiking of injured nerve fibers in a two-dimensional parameter space

Two different bifurcation scenarios of firing patterns with decreasing extracellular calcium concentrations were observed in identical sciatic nerve fibers of a chronic constriction injury (CCI) model when the extracellular 4-aminopyridine concentrations were fixed at two different levels. Both processes proceeded from period-1 bursting to period-1 spiking via complex or simple processes. Multiple typical experimental examples manifested dynamics closely matching those simulated in a recently proposed 4-dimensional model to describe the nonlinear dynamics of the CCI model, which included most cases of the bifurcation scenarios. As the extracellular 4-aminopyridine concentrations is increased, the structure of the bifurcation scenario becomes more complex. The results provide a basic framework for identifying the relationships between different neural firing patterns and different bifurcation scenarios and for revealing the complex nonlinear dynamics of neural firing patterns. The potential roles of the basic bifurcation structures in identifying the information process mechanism are discussed.     

Dances with fire: Tracking metapopulation dynamics of<Emphasis Type="Italic">polygonella basiramia</Emphasis> in Florida scrub (USA)

The regional persistence of species subject to local population colonization and extinction necessarily depends on how landscape features and disturbance affect metapopulation dynamics. Here, we characterize the metapopulation structure and short-term dynamics ofPolygonella basiramia. This rare, short-lived perennial herb is endemic to Florida scrublands and lacks a seed bank. Fires create the open sand gaps within a shrub matrix that support this species but also kill established plants. Thus, persistence depends on frequent colonization of unoccupied gaps. We are monitoring population dynamics within and among 1204 gaps distributed among 19 shrub patches. Considerable subpopulation turnover is evident at the gap level with rates of gap extinction exceeding rates of colonization in the first year. Whether declines in overall abundance continue is likely to depend on patterns of disturbance and regional stochasticity in this dynamic landscape.Polygonella is more likely to occupy larger and less isolated gaps, demonstrating that landscape features and disturbance strongly affect metapopulation dynamics. BecausePolygonella basiramia displays characteristics, occupancy patterns, and turnover dynamics consistent with metapopulation theory, it represents a model system for studying plant metapopulations.     

Disturbance regimes and life-history evolution

Disturbance regimes are ecologically important, but many of their evolutionary consequences are poorly understood. A model is developed here that combines the within- and among-season dynamics of disturbances with evolutionary life-history theory. "Disturbance regime" is defined in terms of disturbance timing, frequency, predictability, and severity. The model predicts the optimal body size and time at which organisms should abandon a disturbance-prone growth habitat by maturing and moving to a disturbance-free, nongrowth habitat. The effects of both coarse-grained (those affecting the entire population synchronously) and fine-grained disturbances (those occurring in a patch dynamics setting) are explored. Several predictions are congruent with previous theory. Infrequent or temporally unpredictable disturbances should have little effect on the evolution of life-history strategies, even though they may cause high mortality. Similar to seasonal time constraints on reproduction, disturbance regimes can synchronize metamorphosis within a population, resulting in a seasonal decline in body size at maturity. Other model predictions are novel. When disturbances cause high mortality, coarse-grained disturbances have a much stronger effect on life-history strategies than fine-grained disturbances, suggesting that population structure (relative to the scale of disturbance) plays a critical evolutionary role when disturbances are severe. When within-population variance in juvenile body size is high, two consecutive seasonal declines in body size at maturity can occur, the first associated with disturbance regime and the second associated with seasonal time constraints.     

Integrodifference models for persistence in fragmented habitats

Integrodifference models of growth and dispersal are analyzed on finite domains to investigate the effects of emigration, local growth dynamics and habitat heterogeneity on population persistence. We derive the bifurcation structure for a range of population dynamics and present an approximation that allows straighforward calculation of the equilibrium populations in terms of local growth dynamics and dispersal success rates. We show how population persistence in a heterogeneous environment depends on the scale of the heterogeneity relative to the organism's characteristic dispersal distance. When organisms tend to disperse only a short distance, population persistence is dominated by local conditions in high quality patches, but when dispersal distance is relatively large, poor quality habitat exerts a greater influence.     

Stable long-period cycling and complex dynamics in a single-locus fertility model with genomic imprinting

Although long-period population size cycles and chaotic fluctuations in abundance are common in ecological models, such dynamics are uncommon in simple population-genetic models where convergence to a fixed equilibrium is most typical. When genotype-frequency cycling does occur, it is most often due to frequency-dependent selection that results from individual or species interactions. In this paper, we demonstrate that fertility selection and genomic imprinting are sufficient to generate a Hopf bifurcation and complex genotype-frequency cycling in a single-locus population-genetic model. Previous studies have shown that on its own, fertility selection can yield stable two-cycles but not long-period cycling characteristic of a Hopf bifurcation. Genomic imprinting, a molecular mechanism by which the expression of an allele depends on the sex of the donating parent, allows fitness matrices to be nonsymmetric, and this additional flexibility is crucial to the complex dynamics we observe in this fertility selection model. Additionally, we find under certain conditions that stable oscillations and a stable equilibrium point can coexist. These dynamics are characteristic of a Chenciner (generalized Hopf) bifurcation. We believe this model to be the simplest population-genetic model with such dynamics.     

Year-class coexistence in biennial plants

I extend the well known and biologically well motivated Skellam model of plant population dynamics to biennial plants. The model has two attractors: either one year class competitively excludes the other, resulting in 2-cycles with only vegetative vs only flowering plants in alternating years, or the two year classes coexist at an interior equilibrium. Contrary to earlier models, these two attractors can exist also simultaneously. I investigate the robustness of the model by including delayed flowering, a common phenomenon in plants, and provide a full numerical bifurcation analysis of the generalized model. High fecundity implies strong competition within year classes and promotes coexistence, whereas high survival results in strong competition between year classes and promotes competitive exclusion. Delayed flowering tends to stabilize the interior equilibrium, but (unlike in density-independent matrix models) the population cycles are robust with respect to some delay in flowering.     

Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission

A deterministic ordinary differential equation model for the dynamics of malaria transmission that explicitly integrates the demography and life style of the malaria vector and its interaction with the human population is developed and analyzed. The model is different from standard malaria transmission models in that the vectors involved in disease transmission are those that are questing for human blood. Model results indicate the existence of nontrivial disease free and endemic steady states, which can be driven to instability via a Hopf bifurcation as a parameter is varied in parameter space. Our model therefore captures oscillations that are known to exist in the dynamics of malaria transmission without recourse to external seasonal forcing. Additionally, our model exhibits the phenomenon of backward bifurcation. Two threshold parameters that can be used for purposes of control are identified and studied, and possible reasons why it has been difficult to eradicate malaria are advanced.     

High mortality and enhanced recovery: modelling the countervailing effects of disturbance on population dynamics

Disturbances cause high mortality in populations while simultaneously enhancing population growth by improving habitats. These countervailing effects make it difficult to predict population dynamics following disturbance events. To address this challenge, we derived a novel form of the logistic growth equation that permits time‐varying carrying capacity and growth rate. We combined this equation with concepts drawn from disturbance ecology to create a general model for population dynamics in disturbance‐prone systems. A river flooding example using three insect species (a fast life‐cycle mayfly, a slow life‐cycle dragonfly and an ostracod) found optimal tradeoffs between disturbance frequency vs. magnitude and a close fit to empirical data in 62% of cases. A savanna fire analysis identified fire frequencies of 3–4 years that maximised population size of a perennial grass. The model shows promise for predicting population dynamics after multiple disturbance events and for management of river flows and fire regimes.     

The influence of depolarization block on seizure-like activity in networks of excitatory and inhibitory neurons

The inhibitory restraint necessary to suppress aberrant activity can fail when inhibitory neurons cease to generate action potentials as they enter depolarization block. We investigate possible bifurcation structures that arise at the onset of seizure-like activity resulting from depolarization block in inhibitory neurons. Networks of conductance-based excitatory and inhibitory neurons are simulated to characterize different types of transitions to the seizure state, and a mean field model is developed to verify the generality of the observed phenomena of excitatory-inhibitory dynamics. Specifically, the inhibitory population’s activation function in the Wilson-Cowan model is modified to be non-monotonic to reflect that inhibitory neurons enter depolarization block given strong input. We find that a physiological state and a seizure state can coexist, where the seizure state is characterized by high excitatory and low inhibitory firing rate. Bifurcation analysis of the mean field model reveals that a transition to the seizure state may occur via a saddle-node bifurcation or a homoclinic bifurcation. We explain the hysteresis observed in network simulations using these two bifurcation types. We also demonstrate that extracellular potassium concentration affects the depolarization block threshold; the consequent changes in bifurcation structure enable the network to produce the tonic to clonic phase transition observed in biological epileptic networks.     

Intraspecific competition in models for vegetation patterns: Decrease in resilience to aridity and facilitation of species coexistence

Patterned vegetation is a characteristic feature of many dryland ecosystems. While plant densities on the ecosystem-wide scale are typically low, a spatial self-organisation principle leads to the occurrence of alternating patches of high biomass and patches of bare soil. Nevertheless, intraspecific competition dynamics other than competition for water over long spatial scales are commonly ignored in mathematical models for vegetation patterns. In this paper, I address the impact of local intraspecific competition on a modelling framework for banded vegetation patterns. Firstly, I show that in the context of a single-species model, neglecting local intraspecific competition leads to an overestimation of a patterned ecosystem’s resilience to increases in aridity. Secondly, in the context of a multispecies model, I argue that local intraspecific competition is a key element in the successful capture of species coexistence in model solutions representing a vegetation pattern. For both models, a detailed bifurcation analysis is presented to analyse the onset, existence and stability of patterns. Besides the strengths of local intraspecific competition, also the difference between two species has a significant impact on the bifurcation structure, providing crucial insights into the complex ecosystem dynamics. Predictions on future ecosystem dynamics presented in this paper, especially on pattern onset and pattern stability, can aid the development of conservation programs.     

The backward bifurcation in compartmental models for West Nile virus

In all of the West Nile virus (WNV) compartmental models in the literature, the basic reproduction number serves as a crucial control threshold for the eradication of the virus. However, our study suggests that backward bifurcation is a common property shared by the available compartmental models with a logistic type of growth for the population of host birds. There exists a subthreshold condition for the outbreak of the virus due to the existence of backward bifurcation. In this paper, we first review and give a comparison study of the four available compartmental models for the virus, and focus on the analysis of the model proposed by Cruz-Pacheco et al. to explore the backward bifurcation in the model. Our comparison study suggests that the mosquito population dynamics itself cannot explain the occurrence of the backward bifurcation, it is the higher mortality rate of the avian host due to the infection that determines the existence of backward bifurcation.     

Incorporating prey refuge in a prey–predator model with a Holling type I functional response: random dynamics and population outbreaks

A prey–predator discrete-time model with a Holling type I functional response is investigated by incorporating a prey refuge. It is shown that a refuge does not always stabilize prey–predator interactions. A prey refuge in some cases produces even more chaotic, random-like dynamics than without a refuge and prey population outbreaks appear. Stability analysis was performed in order to investigate the local stability of fixed points as well as the several local bifurcations they undergo. Numerical simulations such as parametric basins of attraction, bifurcation diagrams, phase plots and largest Lyapunov exponent diagrams are executed in order to illustrate the complex dynamical behavior of the system.     

Modeling density dependence and climatic disturbances in caribou: a case study from the Bathurst Island complex, Canadian High Arctic

Peary caribou Rangifer tarandus pearyi is the northernmost subspecies of Rangifer in North America and endemic to the Canadian High Arctic. Because of severe population declines following years of unfavorable winter weather with ice coating on the ground or thicker snow cover, it is believed that density-independent disturbance events are the primary driver for Peary caribou population dynamics. However, it is unclear to what extent density dependence may affect population dynamics of this species. Here, we test for different levels of density dependence in a stochastic, single-stage population model, based on available empirical information for the Bathurst Island complex (BIC) population in the Canadian High Arctic. We compare predicted densities with observed densities during 1961–2001 under various assumptions of the strength of density dependence. On the basis of our model, we found that scenarios with no or very low density dependence led to population densities far above observed densities. For average observed disturbance regimes, a carrying capacity of 0.1 caribou km⁻² generated an average caribou density similar to that estimated for the BIC population over the past four decades. With our model we also tested the potential effects of climate change-related increases in the probability and severity of disturbance years, that is unusually poor winter conditions. On the basis of our simulation results, we found that, in particular, potential increases in disturbance severity (as opposed to disturbance frequency) may pose a considerable threat to the persistence of this species.     

Smoking epidemic eradication in a eco-epidemiological dynamical model

Smoking is perceived as a major epidemic with regard to mortality. Modelling is a major tool used to obtain insight in the dynamics and possible solutions to decrease or even eradicate this epidemic. Most models on smoking consider the epidemiological context explicitly, in which smoking is regarded as an ‘infectious disease’, in which individuals ‘infect’ each other. However, the population dynamics are often ignored, while these occur at roughly the same timescale as smoking, and hence should explicitly be considered in the modelling of smoking. We present a simple but dynamical eco-epidemiological model. The model formulation consists of a resource-population dynamic part coupled to an epidemiological part resembling a SIR type model for the three compartments: non-smokers, smokers and ex-smokers. The coupling is via birth of non-smokers and death of the three classes with different death rates. The final four-dimensional system of ordinary differential equations are studied using brute force simulations for the short term dynamics and bifurcation analysis for the long-term dynamics. Due to a feed-back mechanism of the two coupling terms there is a codim-two tangent-transcritical bifurcation. This leads to bi-stability of one smoker endemic interior equilibrium and a smoker free boundary equilibrium. Changing parameters beyond the emerging tangent bifurcation leads on the short term to eradicating smoking. We consider the Netherlands in this paper for parametrization, but the modelling approach may be generally applicable.     

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.     

Multiparametric bifurcations of an epidemiological model with strong Allee effect

In this paper we completely study bifurcations of an epidemic model with five parameters introduced by Hilker et al. (Am Nat 173:72–88, 2009), which describes the joint interplay of a strong Allee effect and infectious diseases in a single population. Existence of multiple positive equilibria and all kinds of bifurcation are examined as well as related dynamical behavior. It is shown that the model undergoes a series of bifurcations such as saddle-node bifurcation, pitchfork bifurcation, Bogdanov–Takens bifurcation, degenerate Hopf bifurcation of codimension two and degenerate elliptic type Bogdanov–Takens bifurcation of codimension three. Respective bifurcation surfaces in five-dimensional parameter spaces and related dynamical behavior are obtained. These theoretical conclusions confirm their numerical simulations and conjectures by Hilker et al., and reveal some new bifurcation phenomena which are not observed in Hilker et al. (Am Nat 173:72–88, 2009). The rich and complicated dynamics exhibit that the model is very sensitive to parameter perturbations, which has important implications for disease control of endangered species.