The reliability of approximate reduction techniques in population models with two time scales

As a result of the complexity inherent in some natural systems, mathematical models employed in ecology are often governed by a large number of variables. For instance, in the study of population dynamics we often find multiregional models for structured populations in which individuals are classified regarding their age and their spatial location. Dealing with such structured populations leads to high dimensional models. Moreover, in many instances the dynamics of the system is controlled by processes whose time scales are very different from each other. For example, in multiregional models migration is often a fast process in comparison to the growth of the population.Approximate reduction techniques take advantage of the presence of different time scales in a system to introduce approximations that allow one to transform the original system into a simpler low dimensional system. In this way, the dynamics of the original system can be approximated in terms of that of the reduced system. This work deals with the study of that approximation. In particular, we work with a non-autonomous discrete time model previously presented in the literature and obtain different bounds for the error we incur when we describe the dynamics of the original system in terms of the reduced one.The results are illustrated by some numerical simulations corresponding to the reduction of a Leslie type model for a population structured in two age classes and living in a two patch system.     

A discrete model with density dependent fast migration

The aim of this work is to develop an approximate aggregation method for certain non-linear discrete models. Approximate aggregation consists in describing the dynamics of a general system involving many coupled variables by means of the dynamics of a reduced system with a few global variables. We present discrete models with two different time scales, the slow one considered to be linear and the fast one non-linear because of its transition matrix depends on the global variables. In our discrete model the time unit is chosen to be the one associated to the slow dynamics, and then we approximate the effect of fast dynamics by using a sufficiently large power of its corresponding transition matrix. In a previous work the same system is treated in the case of fast dynamics considered to be linear, conservative in the global variables and inducing a stable frequency distribution of the state variables. A similar non-linear model has also been studied which uses as time unit the one associated to the fast dynamics and has the non-linearity in the slow part of the system. In the present work we transform the system to make the global variables explicit, and we justify the quick derivation of the aggregated system. The local asymptotic behaviour of the aggregated system entails that of the general system under certain conditions, for instance, if the aggregated system has a stable hyperbolic fixed point then the general system has one too. The method is applied to aggregate a multiregional Leslie model with density dependent migration rates.     

Reduction of Supercritical Multiregional Stochastic Models with Fast Migration

In this work we study the behavior of a time discrete multiregional stochastic model for a population structured in age classes and spread out in different spatial patches between which individuals can migrate. The dynamics of the population is controlled both by reproduction-survival and by migration. These processes take place at different time scales in the sense of the latter being much faster than the former. We incorporate the effect of demographic stochasticity into the population, which results in both dynamics being modelled by multitype Bienaymé–Galton–Watson branching processes. We present a multitype global model that incorporates the effect of both processes and, making use of the existence of different time scales for demography and migration, build a reduced model in which the variables correspond to the total population in each age class. We extend previous results that relate the behavior of the original and the reduced model showing that, given a large enough separation of time scales between demography and migration, we can obtain information about the behavior of the multitype global model through the study of the simpler reduced model. We concentrate on the case where the two systems are supercritical and therefore the expected number of individuals grows to infinity, and show that we can approximate the asymptotic structure of the population vector and the asymptotic population size of the original system through the study of the reduced model.     

Linear Discrete Population Models with Two Time Scales in Fast Changing Environments I: Autonomous Case

In this work we consider a structured population with groups and subgroups of individuals. The intra-group dynamics is assumed to be fast in comparison with the inter-group dynamics. We study linear discrete models where the slow dynamics is represented by a single matrix and the fast dynamics is described by means of the first k terms of a converging sequence of different matrices. The number k can be interpreted as the ratio between the two time scales.The aim of this work is to extend aggregation techniques to the case of fast changing environments. The main idea of aggregation is to build up a new system, with lower dimension, that summarizes the information concerning the fast process. This "aggregated" system provides essential information on the original one. It is shown that the asymptotic behavior of the original system can be approximated by the asymptotic behavior of the aggregated system when the ratio between the two time scales is large enough.We present an example of an age structured population in a patchy environment. The migration process is assumed to be fast in comparison with the demographic process. Numerical simulations illustrate that the asymptotic growth rate and the stable age distribution of the population in the original and the aggregated systems are getting closer as the ratio k increases.     

A density dependent model describing Salmo trutta population dynamics in an arborescent river network. Effects of dams and channelling

Our aim is to model the Salmo trutta population dynamics (three age-classes) in an arborescent river network (four levels, 15 patches), by considering both migrations (fast time scale) and demography (slow time scale). We study how the environmental management can influence the global population dynamics. We present a general model coupling both a linear discrete model for constant migrations and a non-linear density-dependent Leslie model for the demography, with (15 × 3) difference equations (15 patches, three age-classes). The variable aggregation method applied to discrete time models allows us to aggregate the previous model into a new one with only three equations. We assume fecundity and survival gradients with respect to the river network levels. The Salmo trutta whole population tends towards an equilibrium state depending on the environmental structure, and we show that dams have a stronger influence than channelling on this equilibrium.     

Approximate reduction of multiregional models with environmental stochasticity

In this work we extend previous results regarding the use of approximate aggregation techniques to simplify the study of discrete time models for populations that live in an environment that changes randomly with time. Approximate aggregation techniques allow one to transform a complex system involving many coupled variables and in which there are processes with different time scales, by a simpler reduced model with a fewer number of 'global' variables, in such a way that the dynamics of the former can be approximated by that of the latter. We present the reduction of a stochastic multiregional model in which the population, structured by age and spatial location, lives in a random environment and in which migration is fast with respect to demography. However, the technique works in much more general settings as, for example, those of stage-structured populations living in a multipatch environment. By manipulating the original system and appropriately defining the global variables we obtain a simpler system. The paper concentrates on establishing relationships between the original and the reduced systems for a given separation of time scales between the two processes. In particular, we relate the original state variables and the global variables and, in the case the pattern of temporal variation is Markovian, we relate the presence of strong stochastic ergodicity for the original and reduced systems. Moreover, we relate different measures of asymptotic population growth for these systems.     

Linear discrete population models with two time scales in fast changing environments II: non-autonomous case

As the result of the complexity inherent in nature, mathematical models employed in ecology are often governed by a large number of variables. For instance, in the study of population dynamics we often deal with models for structured populations in which individuals are classified regarding their age, size, activity or location, and this structuring of the population leads to high dimensional systems. In many instances, the dynamics of the system is controlled by processes whose time scales are very different from each other. Aggregation techniques take advantage of this situation to build a low dimensional reduced system from which behavior we can approximate the dynamics of the complex original system.In this work we extend aggregation techniques to the case of time dependent discrete population models with two time scales where both the fast and the slow processes are allowed to change at their own characteristic time scale, generalizing the results of previous studies. We propose a non-autonomous model with two time scales, construct an aggregated model and give relationship between the variables governing the original and the reduced systems. We also explore how the properties of strong and weak ergodicity, regarding the capacity of the system to forget initial conditions, of the original system can be studied in terms of the reduced system.     

Linear discrete models with different time scales

Aggregation of variables allows to approximate a large scale dynamical system (the micro-system) involving many variables into a reduced system (the macro-system) described by a few number of global variables. Approximate aggregation can be performed when different time scales are involved in the dynamics of the micro-system. Perturbation methods enable to approximate the large micro-system by a macro-system going on at a slow time scale. Aggregation has been performed for systems of ordinary differential equations in which time is a continuous variable. In this contribution, we extend aggregation methods to time-discrete models of population dynamics. Time discrete micro-models with two time scales are presented. We use perturbation methods to obtain a slow macro-model. The asymptotic behaviours of the micro and macro-systems are characterized by the main eigenvalues and the associated eigenvectors. We compare the asymptotic behaviours of both systems which are shown to be similar to a certain order.     

Population Dynamics Modelling in an Hierarchical Arborescent River Network: An Attempt with Salmo trutta

The balance between births and deaths in an age-structured population is strongly influenced by the spatial distribution of sub-populations. Our aim was to describe the demographic process of a fish population in an hierarchical dendritic river network, by taking into account the possible movements of individuals. We tried also to quantify the effect of river network changes (damming or channelling) on the global fish population dynamics. The Salmo trutta life pattern was taken as an example for.We proposed a model which includes the demographic and the migration processes, considering migration fast compared to demography. The population was divided into three age-classes and subdivided into fifteen spatial patches, thus having 45 state variables. Both processes were described by means of constant transfer coefficients, so we were dealing with a linear system of difference equations. The discrete case of the variable aggregation method allowed the study of the system through the dominant elements of a much simpler linear system with only three global variables: the total number of individuals in each age-class.From biological hypothesis on demographic and migratory parameters, we showed that the global population dynamics of fishes is well characterized in the reference river network, and that dams could have stronger effects on the global dynamics than channelling.     

Timing regulation in a network reduced from voltage-gated equations to a one-dimensional map

 We discuss a method by which the dynamics of a network of neurons, coupled by mutual inhibition, can be reduced to a one-dimensional map. This network consists of a pair of neurons, one of which is an endogenous burster, and the other excitable but not bursting in the absence of phasic input. The latter cell has more than one slow process. The reduction uses the standard separation of slow/fast processes; it also uses information about how the dynamics on the slow manifold evolve after a finite amount of slow time. From this reduction we obtain a one-dimensional map dependent on the parameters of the original biophysical equations. In some parameter regimes, one can deduce that the original equations have solutions in which the active phase of the originally excitable cell is constant from burst to burst, while in other parameter regimes it is not. The existence or absence of this kind of regulation corresponds to qualitatively different dynamics in the one-dimensional map. The computations associated with the reduction and the analysis of the dynamics includes the use of coordinates that parameterize by time along trajectories, and “singular Poincaré maps” that combine information about flows along a slow manifold with information about jumps between branches of the slow manifold. Received: 19 May 1997 / Revised version: 6 April 1998     

Methods of aggregation of variables in population dynamics

In ecology, we are faced with modelling complex systems involving many variables corresponding to interacting populations structured in different compartmental classes, ages and spatial patches. Models that incorporate such a variety of aspects would lead to systems of equations with many variables and parameters. Mathematical analysis of these models would, in general, be impossible. In many real cases, the dynamics of the system corresponds to two or more time scales. For example, individual decisions can be rapid in comparison to growth of the populations. In that case, it is possible to perform aggregation methods that allow one to build a reduced model that governs the dynamics of a lower dimensional system, at a slow time scale. In this article, we present a review of aggregation methods for time continuous systems as well as for discrete models. We also present applications in population dynamics. A first example concerns a continuous time model of a single population distributed on a system of two connected patches (a logistic source and a sink), by fast migration. It is shown that under a certain condition, the total equilibrium population can be larger than the carrying capacity of the logistic source. A second example concerns a discrete model of a population distributed on two patches, still a source and a sink, connected by fast migration. The use of aggregation methods permits us to conclude that density-dependent migration can stabilize the total population.     

Effect of Hawk-Dove Game on the Dynamics of Two Competing Species

Outcomes of interspecific competition, and especially the possibility of coexistence, have been extensively studied in theoretical ecology because of their implications in community assemblages. During the last decades, the influence of different time scales through the local/regional dynamics of animal communities has received an increasing attention. Nevertheless, different time scales involved in interspecific competition can result form other processes than spatial dynamics. Here, we envision and analyze a new theoretical framework that couples a game theory approach for competition with a demographic model. We take advantage of these two time scales to derive a reduced model governing the total densities of the two populations and we study how these two time scales interfere and influence outcomes of species competition. We find that a competition process occurring on a faster time scale than demography yields a “priority effect” where the first species introduced outcompetes the other one. We then confirm previous findings stipulating that species coexistence is favored by large difference in time scales because the extinction/recolonization process. Our results then highlight that an integration of demographic and competition time scales at both local and regional levels is mandatory to explain communities assemblages and should become a research priority.     

Analysis of models for crustacean stretch receptors

 The mechanisms underlying the diverse responses to step current stimuli of models [Edman et al. (1987) J Physiol (Lond) 384: 649–669] of lobster slowly adapting stretch receptor organs (SAO) and fast-adapting stretch receptor organs (FAO) are analyzed. In response to a step current, the models display three distinct types of firing reflecting the level of adaptation to the stimulation. Low-amplitude currents evoke transient firing containing one to several action potentials before the system stabilizes to a resting state. Conversely, high-amplitude stimulations induce a high frequency transient burst that can last several seconds before the model returns to its quiescent state. In the SAO model, the transition between the two regimes is characterized by a sustained pacemaker firing at an intermediate stimulation amplitude. The FAO model does not exhibit such a maintained firing; rather, the duration of the transient firing increases at first with the stimulus intensity, goes through a maximum and then decreases at larger intensities. Both models comprise seven variables representing the membrane potential, the sodium fast activation, fast inactivation, slow inactivation, the potassium fast activation, slow inactivation gating variables, and the intra cellular sodium concentration. To elucidate the mechanisms of the firing adaptations, the seven-variable model for the lobster stretch receptor neuron is first reduced to a three-dimensional system by regrouping variables with similar time scales. More precisely, we substituted the membrane potential V for the sodium fast activation equivalent potential V _m , the potassium fast inactivation V _n for the sodium fast inactivation V _h , and the sodium slow inactivation V _l for the potassium slow inactivation V _r . Comparison of the responses of the reduced models to those of the original models revealed that the main behaviors of the system were preserved in the reduction process. We classified the different types of responses of the reduced SAO and FAO models to constant current stimulation. We analyzed the transient and stationary responses of the reduced models by constructing bifurcation diagrams representing the qualitatively distinct dynamics of the models and the transitions between them. These revealed that (1) the transient firings prior to reaching the stationary state can be accounted for by the sodium slow inactivation evolving more slowly than the other two variables, so that the changes during the transient firings reflect the bifurcations that the two-dimensional system undergoes when the sodium slow inactivation, considered as a parameter, is varied; and (2) the stationary behaviors of the models are captured by the standard bifurcations of a two-dimensional system formed by the membrane potential and the potassium fast inactivation. We found that each type of firing and the transitions between them is due to the interplay between essentially three variables: two fast ones accounting for the action potential generation and the post-discharge refractoriness, and a third slow one representing the adaptation. Received: 28 February 2000 / Accepted in revised form: 4 October 2000     

Approximate reduction of linear population models governed by stochastic differential equations: application to multiregional models

In this work we develop approximate aggregation techniques in the context of slow-fast linear population models governed by stochastic differential equations and apply the results to the treatment of populations with spatial heterogeneity. Approximate aggregation techniques allow one to transform a complex system involving many coupled variables and in which there are processes with different time scales, by a simpler reduced model with a fewer number of ‘global’ variables, in such a way that the dynamics of the former can be approximated by that of the latter. In our model we contemplate a linear fast deterministic process together with a linear slow process in which the parameters are affected by additive noise, and give conditions for the solutions corresponding to positive initial conditions to remain positive for all times. By letting the fast process reach equilibrium we build a reduced system with a lesser number of variables, and provide results relating the asymptotic behaviour of the first- and second-order moments of the population vector for the original and the reduced system. The general technique is illustrated by analysing a multiregional stochastic system in which dispersal is deterministic and the rate growth of the populations in each patch is affected by additive noise.     

A review on spatial aggregation methods involving several time scales

This article is a review of spatial aggregation of variables for time continuous models. Two cases are considered. The first case corresponds to a discrete space, i.e. a set of discrete patches connected by migrations, which are assumed to be fast with respect to local interactions. The mathematical model is a set of coupled ordinary differential equations (O.D.E.). The spatial aggregation allows one to derive a global model governing the time variation of the total numbers of individuals of all patches in the long term. The second case considers a continuous space and is a set of partial differential equations (P.D.E.). In that case, we also assume that diffusion is fast in comparison with local interactions. The spatial aggregation allows us again to obtain an O.D.E. governing the total population density, which is obtained by integration all over the spatial domain, at the slow time scale. These aggregations of variables are based on time scales separation methods which have been presented largely elsewhere and we recall the main results. We illustrate the methods by examples in population dynamics and prey–predator models.     

Interaction of fast and slow dynamics in endocrine control systems with an application to β-cell dynamics

Endocrine dynamics spans a wide range of time scales, from rapid responses to physiological challenges to with slow responses that adapt the system to the demands placed on it. We outline a non-linear averaging procedure to extract the slower dynamics in a way that accounts properly for the non-linear dynamics of the faster time scale and is applicable to a hierarchy of more than two time scales, although we restrict our discussion to two scales for the sake of clarity. The procedure is exact if the slow time scale is infinitely slow (the dimensionless ε-quantity is the period of the fast time scale fluctuation times an upper bound to the slow time scale rate of change). However, even for an imperfect separation of time scales we find that this construction provides an excellent approximation for the slow-time dynamics at considerably reduced computational cost. Besides the computation advantage, the averaged equation provided a qualitative insight into the interaction of the time scales. We demonstrate the procedure and its advantages by applying the theory to the model described by Toli? et al. [I.M. Toli?, E. Mosekilde, J. Sturis, Modeling the insulin–glucose feedback system: the significance of pulsatile insulin secretion, J. Theor. Biol. 207 (2000) 361–375.] for ultradian dynamics of the glucose–insulin homeostasis feedback system, extended to include β-cell dynamics. We find that the dynamics of the β-cell mass are dependent not only on the glycemic load (amount of glucose administered to the system), but also on the way this load is applied (i.e. three meals daily versus constant infusion), effects that are lost in the inappropriate methods used by the earlier authors. Furthermore, we find that the loss of the protection against apoptosis conferred by insulin that occurs at elevated levels of insulin has a functional role in keeping the β-cell mass in check without compromising regulatory function. We also find that replenishment of β-cells from a rapidly proliferating pool of cells, as opposed to the slow turn-over which characterises fully differentiated β-cells, is essential to the prevention of type 1 diabetes.     

The Temporal Structure of Behaviour and Sleep Homeostasis

The amount and architecture of vigilance states are governed by two distinct processes, which occur at different time scales. The first, a slow one, is related to a wake/sleep dependent homeostatic Process S, which occurs on a time scale of hours, and is reflected in the dynamics of NREM sleep EEG slow-wave activity. The second, a fast one, is manifested in a regular alternation of two sleep states – NREM and REM sleep, which occur, in rodents, on a time scale of ∼5–10 minutes. Neither the mechanisms underlying the time constants of these two processes – the slow one and the fast one, nor their functional significance are understood. Notably, both processes are primarily apparent during sleep, while their potential manifestation during wakefulness is obscured by ongoing behaviour. Here, we find, in mice provided with running wheels, that the two sleep processes become clearly apparent also during waking at the level of behavior and brain activity. Specifically, the slow process was manifested in the total duration of waking periods starting from dark onset, while the fast process was apparent in a regular occurrence of running bouts during the waking periods. The dynamics of both processes were stable within individual animals, but showed large interindividual variability. Importantly, the two processes were not independent: the periodic structure of waking behaviour (fast process) appeared to be a strong predictor of the capacity to sustain continuous wakefulness (slow process). The data indicate that the temporal organization of vigilance states on both the fast and the slow time scales may arise from a common neurophysiologic mechanism.     

Non-local concepts and models in biology

In this paper, we consider local and non-local spatially explicit mathematical models for biological phenomena. We show that, when rate differences between fast and slow local dynamics are great enough, non-local models are adequate simplifications of local models. Non-local models thus avoid describing fast processes in mechanistic detail, instead describing the effects of fast processes on slower ones. As a consequence, non-local models are helpful to biologists because they describe biological systems on scales that are convenient to observation, data collection, and insight. We illustrate these arguments by comparing local and non-local models for the aggregation of hypothetical organisms, and we support theoretical ideas with concrete examples from cell biology and animal behavior.