Traveling waves in a simple population model involving growth and death

The structure of solutions to a simple spatially dependent population model involving growth and death is investigated. Two forms of motility of the population are considered: (1) random motion only modeled by a Fickian law, and (2) a directed component of motion (chemotaxis), included in addition to the random motion. Under certain growth conditions a traveling wave of constant speed is approached. This speed can be increased by the addition of the chemotaxis with a corresponding increase in the asymptotic population. Development of initial conditions into a wave is illustrated numerically.     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

A cross-diffusion model of forest boundary dynamics

A simple mathematical model of mono-species forest with two age classes which takes into account seed production and dispersal is presented in the paper. This reaction — diffusion type model is then reduced by means of an asymptotic procedure to a lower dimensional reaction — cross-diffusion model. The existence of standing and travelling wave front solutions corresponding to the forest boundary is shown for the later model. On the basis of the analysis, possible changes in forest boundary dynamics caused by antropogenic impacts are discussed.     

Traveling waves in visual cortex

Electrode recordings and imaging studies have revealed that localized visual stimuli elicit waves of activity that travel across primary visual cortex. Traveling waves are present also during spontaneous activity, but they can be greatly reduced by widespread and intensive visual stimulation. In this Review, we summarize the evidence in favor of these traveling waves. We suggest that their substrate may lie in long-range horizontal connections and that their functional role may involve the integration of information over large regions of space.     

Density-structured models for plant population dynamics

Density-structured models are structured population models in which the state variable is the proportion of populations or sites in a small number of discrete density states. Although such models have rarely been used, they have the advantage that they are straightforward to parameterize, make few assumptions about population dynamics, and permit rapid data collection using coarse density assessment. In this article, we highlight their use in relating population dynamics to environmental variation and their robustness to measurement error. We show that density-structured models are able to accurately represent population dynamics under a wide range of conditions. We look at the effects of including a persistent seedbank and describe numerical approximations for the mean and variance of population size. For simulated data, we determine the extent to which the underlying continuous process may be inferred from density-structured data. Finally, we discuss issues of parameter estimation and applications for which these types of models may be useful.     

Direction of regeneration waves in grid-based models for forest dynamics

Progressing waves of regeneration are observed in forest ecosystems such as Shimagare fir forests. The patterns generated by lattice models for forest dynamics often show similar waves of disturbance and recovery. This paper introduces a method to detect and quantify the directional movement of these waves. The method is based only on the disturbance times of the sites and allows to distinguish three types of wave patterns: patterns with global direction, patterns with local direction, and patterns without direction. We apply this to several grid-based models for forest dynamics which are evaluated analytically or by simulation. The results reveal a clear distinction of the models which earlier studies were not able to detect.     

The dynamics of density dependent population models

Summary The dynamics of density-dependent population models can be extraordinarily complex as numerous authors have displayed in numerical simulations. Here we commence a theoretical analysis of the mathematical mechanisms underlying this complexity from the viewpoint of modern dynamical systems theory. After discussing the chaotic behavior of one-dimensional difference equations we proceed to illustrate the general theory on a density-dependent Leslie model with two age classes. The pattern of bifurcations away from the equilibrium point is investigated and the existence of a strange attractor is demonstrated — i.e. an attracting limit set which is neither an equilibrium nor a limit cycle. Near the strange attractor the system exhibits essentially random behavior. An approach to the statistical analysis of the dynamics in the chaotic regime is suggested. We then generalize our conclusions to higher dimensions and continuous models (e.g. the nonlinear von Foerster equation).Supported by NSF Grant No. BMS 74-21240.     

Stability results for delayed-recruitment models in population dynamics

This paper relates the stability properties of a class of delay-difference equations to those of an associated scalar difference equation. Simple but powerful conditions for testing global stability are presented which are independent of the length of the time delay involved. For models which do not have globally stable equilibria, estimates of stability regions are obtained. Some well known baleen whale models are used to illustrate the results.     

Traveling waves in a chemotactic model

A model for chemotaxis in a bacteria-substrate mixture introduced by Keller and Segel, which is described by nonlinear partial differential equations, is studied analytically. The existence of traveling waves is shown for the system in which the substrate diffusion is taken into account and the chemotactic coefficient is greater than the motility one, and the instability of traveling waves is discussed.     

Adaptive dynamics for physiologically structured population models

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309–338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579–612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka–Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.      

Dynamical and statistical models of vertebrate population dynamics

Population dynamics methodology now powerfully combines discrete time models (with constant parameters, density dependence, random environment, and/or demographic stochasticity) and capture-recapture models for estimating demographic parameters. Vertebrate population dynamics has strongly benefited from this progress: survival estimates have been revised upwards, trade-offs between life history traits have been demonstrated, analyses of population viability and management are more and more realistic. Promising developments concern random effects, multistate and integrated models. Some biological questions (density dependence, links between individual and population levels, and diversification of life histories) can now be efficiently attacked.     

Markov population processes as models of primate social and population dynamics

Thanks to recently developed theory of Markov population processes, models of how an individual primate migrates from one casual social group to another or from one breeding troop to another can now deal exactly with transition rates which depend nonlinearly on the sizes of both the group (or troop) left and the group (or troop) entered. Examples of such models presented here are consistent with existing observations of primate social and population dynamics and are more plausible as explanations of these data than previous linear models.     

Complex discrete dynamics from simple continuous population models

Nonoverlapping generations have been classically modelled as difference equations in order to account for the discrete nature of reproductive events. However, other events such as resource consumption or mortality are continuous and take place in the within-generation time. We have realistically assumed a hybrid ODE bidimensional model of resources and consumers with discrete events for reproduction. Numerical and analytical approaches showed that the resulting dynamics resembles a Ricker map, including the doubling route to chaos. Stochastic simulations with a handling-time parameter for indirect competition of juveniles may affect the qualitative behaviour of the model.     

Traveling waves of infection in the hantavirus epidemics

Traveling waves are analyzed in a model of the hantavirus infection in deer mice. The existence of two kinds of wave phenomena is predicted. An environmental parameter governs a transition between two regimes of propagation. In one of them the front of infection lags behind at a constant rate. In the other, fronts of susceptible and infected mice travel at the same speed, separated by a constant delay. The dependence of the delay on system parameters is analyzed numerically and through a piecewise linearization.     

Traveling electrical waves in cortex: insights from phase dynamics and speculation on a computational role

The theory of coupled phase oscillators provides a framework to understand the emergent properties of networks of neuronal oscillators. When the architecture of the network is dominated by short-range connections, the pattern of electrical output is predicted to correspond to traveling plane and rotating waves, in addition to synchronized output. We argue that this theory provides the foundation for understanding the traveling electrical waves that are observed across olfactory, visual, and visuomotor areas of cortex in a variety of species. The waves are typically present during periods outside of stimulation, while synchronous activity typically dominates in the presence of a strong stimulus. We suggest that the continuum of phase shifts during epochs with traveling waves provides a means to scan the incoming sensory stream for novel features. Experiments to test our theoretical approach are presented.     

Incorporating movement into models of grey seal population dynamics

1. One of the most difficult problems in developing spatially explicit models of population dynamics is the validation and parameterization of the movement process. We show how movement models derived from capture-recapture analysis can be improved by incorporating them into a spatially explicit metapopulation model that is fitted to a time series of abundance data. 2. We applied multisite capture-recapture analysis techniques to photo-identification data collected from female grey seals at the four main breeding colonies in the North Sea between 1999 and 2001. The best-fitting movement models were then incorporated into state-space metapopulation models that explicitly accounted for demographic and observational stochasticity. 3. These metapopulation models were fitted to a 20-year time series of pup production data for each colony using a Bayesian approach. The best-fitting model, based on the Akaike Information Criterion (AIC), had only a single movement parameter, whose confidence interval was 82% less than that obtained from the capture-recapture study, but there was some support for a model that included an effect of distance between colonies. 4. The state-space modelling provided improved estimates of other demographic parameters. 5. The incorporation of movement, and the way in which it was modelled, affected both local and regional dynamics. These differences were most evident as colonies approached their carrying capacities, suggesting that our ability to discriminate between models should improve as the length of the grey seal time series increases.     

Rapidly converging numerical algorithms for models of population dynamics

We propose algorithms for the approximation of the age distributions of populations modeled by the McKendrick-von Foerster and the Gurtin-MacCamy systems both in one- and two-sex versions. For the one-sex model methods of second and fourth order are proposed. For the two-sex model a second order method is described. In each case the convergence is demonstrated. Several numerical examples are given.     

The Krylov-Bogoliubov-Mitropolskii method applied to models of population dynamics

In this study the possibility of applying the asymptotic method of Krylov-Bogoliubov-Mitropolskii to problems of population dynamics is shown. Especially a general Volterra-Gause-Witt type model for prey-predator interaction is investigated. A discussion on the results obtained is given for the general model and for a particular case as well.     

The dynamics of population models with distributed maturation periods

An integro-differential equation for the dynamics of a subpopulation of adults in a closed system where only the adults compete and where there is a distribution of maturation periods is described. We show how the careful choice of a general weighting function based on the gamma distribution with a shift in origin enables us to characterize adequately some observed maturation-period distributions, and also makes local stability and numerical analyses straightforward. Using these results we examine the progression in the behavior of the distributed-delay model as the distribution is narrowed toward the limit of a discrete delay. We conclude that while local stability properties approach those of the limiting equation very rapidly, the persistent fluctuation behavior converges more slowly, with the dominant period and maximum amplitude being least affected by the details of the distribution, and the fine structure of solutions being most sensitive. Finally, we examine the consequences for population modeling, and using several examples of insect populations, conclude that although quite often a full maturation-period distribution should be incorporated in a given model, in many cases a discrete-delay approximation will suffice.