Discrete-time Population Models Research Articles

Discrete three-stage population model: persistence and global stability results

A general three-stage discrete-time population model is studied. The inherent net reproductive number for this model is derived. Global stability of the origin is established provided that the inherent net reproductive number is less than one. If it is larger than one the existence of a unique positive fixed point is proved and the persistence of the system is established. Finally, for certain parameter ranges global stability of the positive fixed point is proved.

Resonance of the epidemic threshold in a periodic environment

Resonance between some natural period of an endemic disease and a seasonal periodic contact rate has been the subject of intensive study. This paper does not focus on resonance for endemic diseases but on resonance for emerging diseases. Periodicity can have an important impact on the initial growth rate and therefore on the epidemic threshold. Resonance occurs when the Euler-Lotka equation has a complex root with an imaginary part (i.e., a natural frequency) close to the angular frequency of the contact rate and a real part not too far from the Malthusian parameter. This is a kind of continuous-time analogue of work by Tuljapurkar on discrete-time population models, which in turn was motivated by the work by Coale on continuous-time demographic models with a periodic birth. We illustrate this resonance phenomenon on several simple epidemic models with contacts varying periodically on a weekly basis, and explain some surprising differences, e.g., between a periodic SEIR model with an exponentially distributed latency and the same model but with a fixed latency.

Asynchronous and Synchronous Dispersals in Spatially Discrete Population Models

This study is on the role of synchronous and asynchronous dispersals in a discrete-time single-species population model with dispersal between two patches, where predispersal dynamics are compen- satory or overcompensatory and dispersal is synchronous or asynchronous or mixed synchronous and asynchronous. It is known that single-species dispersal-linked population models behave as single- species single-patch models whenever all predispersal local dynamics are compensatory and dispersal is synchronous. However, the dynamics of the corresponding model connected by asynchronous and mixed synchronous-asynchronous dispersals depend on the dispersal rates, intrinsic growth rates, and the parameter that models the possible modes of dispersal. The species becomes extinct on at least one patch when the asynchronous dispersal rates are high, while it persists when the rates are low. In mixed synchronous-asynchronous systems, depending on the model parameters, the pioneer species either becomes extinct on all patches or persists on all patches. Overcompensatory predispersal dynamics with synchronous dispersal can lead to multiple attractors with fractal basin boundaries. However, the associated models with either asynchronous or mixed synchronous and asynchronous dispersals exhibit multiple attractors with fewer numbers of distinct attractors. That is, the long-term dynamics of synchronous dispersal-linked systems can be more sensitive to initial population sizes than that of the corresponding asynchronous and mixed synchronous-asynchronous systems. Also, synchronous, asynchronous, and mixed synchronous-asynchronous dispersals can stabilize the local patch dynamics from overcompensatory to compensatory dynamics. In our mixed synchronous-asynchronous model, the dominant mode of dispersal usually drives the dynam- ics of the full system.

On the stability analysis of a general discrete-time population model involving predation and Allee effects

This paper presents the stability analysis of equilibrium points of a general discrete-time population dynamics involving predation with and without Allee effects which occur at low population density. The mathematical analysis and numerical simulations show that the Allee effect has a stabilizing role on the local stability of the positive equilibrium points of this model.

A three-stage discrete-time population model: continuous versus seasonal reproduction

We consider a three-stage discrete-time population model with density-dependent survivorship and time-dependent reproduction. We provide stability analysis for two types of birth mechanisms: continuous and seasonal. We show that when birth is continuous there exists a unique globally stable interior equilibrium provided that the inherent net reproductive number is greater than unity. If it is less than unity, then extinction is the population's fate. We then analyze the case when birth is a function of period two and show that the unique two-cycle is globally attracting when the inherent net reproductive number is greater than unity, while if it is less than unity the population goes to extinction. The two birth types are then compared. It is shown that for low birth rates the adult average number over a one-year period is always higher when reproduction is continuous. Numerical simulations suggest that this remains true for high birth rates. Thus periodic birth rates of period two are deleterious for the three-stage population model. This is different from the results obtained for a two-stage model discussed by Ackleh and Jang (J. Diff. Equ. Appl., 13, 261–274, 2007), where it was shown that for low birth rates seasonal breeding results in higher adult averages.

Two-patch dispersal-linked compensatory-overcompensatory spatially discrete population models

We study the role of asynchronous and synchronous dispersals on discrete-time two-patch dispersal-linked population models, where the pre-dispersal local patch dynamics are of mixed compensatory and overcompensatory types. Single-species dispersal-linked models behave as single-species single-patch models whenever all pre-dispersal local patch dynamics are compensatory and dispersal is synchronous. However, the dynamics of the corresponding two-patch population model connected by asynchronous dispersal depends on the dispersal rates. The species goes extinct on at least one patch when the asynchronous dispersal rates are high, while it persists when the rates are low. We use numerical simulations to show that in both synchronous and asynchronous mixed compensatory and overcompensatory systems, symmetric and asymmetric dispersals can control and impede the onset of cyclic population oscillations via period-doubling reversal bifurcations. Also, we show that in mixed systems both asynchronous and synchronous dispersals are capable of altering the pre-dispersal local patch dynamics from overcompensatory to compensatory dynamics. Dispersal-linked population models with ‘unstructured’ overcompensatory pre-dispersal local dynamics connected by synchronous dispersal can generate multiple attractors with fractal basin boundaries. However, mixed compensatory and overcompensatory systems appear to exhibit single attractors and not coexisting (multiple) attractors.

On the mechanistic underpinning of discrete-time population models with Allee effect

The Allee effect means reduction in individual fitness at low population densities. There are many discrete-time population models with an Allee effect in the literature, but most of them are phenomenological. Recently, Geritz and Kisdi [2004. On the mechanistic underpinning of discrete-time population models with complex dynamics. J. Theor. Biol. 228, 261–269] presented a mechanistic underpinning of various discrete-time population models without an Allee effect. Their work was based on a continuous-time resource-consumer model for the dynamics within a year, from which they derived a discrete-time model for the between-year dynamics. In this article, we obtain the Allee effect by adding different mate finding mechanisms to the within-year dynamics. Further, by adding cannibalism we obtain a higher variety of models. We thus present a generator of relatively realistic, discrete-time Allee effect models that also covers some currently used phenomenological models driven more by mathematical convenience.

Local stability implies global stability in some one-dimensional discrete single-species models

We prove a criterion for the global stability of the positive equilibrium in discrete-time single-species population models of the form $x_{n+1}=x_nF(x_n)$. This allows us to demonstrate analytically (and easily) the conjecture that local stability implies global stability in some well-known models, including the Ricker difference equation and a combination of the models by Hassel and Maynard Smith. Our approach combines the use of linear fractional functions (Möbius transformations) and the Schwarzian derivative.

Signature Function for Predicting Resonant and Attenuant Population 2-cycles

Populations are either enhanced via resonant cycles or suppressed via attenuant cycles by periodic environments. We develop a signature function for predicting the response of discretely reproducing populations to 2-periodic fluctuations of both a characteristic of the environment (carrying capacity), and a characteristic of the population (inherent growth rate). Our signature function is the sign of a weighted sum of the relative strengths of the oscillations of the carrying capacity and the demographic characteristic. Periodic environments are deleterious for populations when the signature function is negative. However, positive signature functions signal favorable environments. We compute the signature functions of six classical discrete-time single species population models, and use the functions to determine regions in parameter space that are either favorable or detrimental to the populations. The two-parameter classical models include the Ricker, Beverton-Holt, Logistic, and Maynard Smith models.

On the Mechanistic Derivation of Various Discrete-Time Population Models

We present a derivation of various discrete-time population models within a single unifying mechanistic context. By systematically varying the within-year patterns of reproduction and aggression between individuals we can derive various discrete-time population models. These models include classical examples such as the Ricker model, the Beverton-Holt model, the Skellam model, the Hassell model, and others. Some of these models until now lacked a good mechanistic interpretation or have been derived in a different context. By using this mechanistic approach, the model parameters can be interpreted in terms of individual behavior.

The impact of environmental fluctuations on structured discrete time population models: Resonance, synchrony and threshold behaviour

External forcing of a discrete time ecological system does not just add variation to existing dynamics but can change the dynamics. We study the mechanisms that can bring this about, focusing on the key concepts of excitation and suppression which emerge when analysing the power spectra of the system in linear approximation. Excitation, through resonance between the system dynamics and the external forcing, is the greater the closer the system is to the boundary of the stability region. This amplification means that the extinction of populations becomes possible sooner than expected and, conversely, invasion can be significantly delayed. Suppression and the consequent redistribution of power within the spectrum proves to be a function both of the connectivity of the network graph of the system and the way that external forcing is applied to the system. It is also established that colour in stochastic forcing can have a major impact, by enhancing resonance and by greater redistribution of power. This can mean a higher risk of extinction through larger fluctuations in population numbers and a higher degree of synchrony between populations. The implications of external forcing for stage-structured species, for populations in competition and for trophic web systems are studied using the tools and concepts developed in the paper.

Attenuant cycles in periodically forced discrete-time age-structured population models

In discrete-time age-structured population models, a periodic environment is not always deleterious. We show that it is possible to have the average of the age class populations over an attracting cycle (in a periodic environment) not less than the average of the carrying capacities (in a corresponding constant environment). In our age-structured model, a periodic environment does not increase the average total biomass (no resonance). However, a periodic environment is disadvantageous for a population whenever there is no synchrony between the number of age classes and the period of the environment. As in periodically forced models without age-structure, we show that periodically forced age-structured population models support multiple attractors with complicated structures.

The frequency spectrum of structured discrete time population models: its properties and their ecological implications

Much research effort has been devoted to the study of the interaction between environmental noise and discrete time nonlinear dynamical systems. A large part of this effort has involved numerical simulation of simple unstructured models for particular ranges of parameter values. While such research is important in encouraging discussion of important ecological issues it is often unclear how general are the conclusions reached. However, by restricting attention to weak noise it is possible to obtain analytical results that hold for essentially all discrete time models and still provide considerable insight into the properties of the noise‐dynamics interface. We follow this approach, focusing on the autocorrelation properties of the population fluctuations using the power (frequency) spectrum matrix as the analytic framework. We study the relationship between the spectral peak structure and the dynamical behaviour of the system and the modulation of this relationship by its internal structure, acting as an “intrinsic” filter and by colour in the noise acting as an “extrinsic” filter. These filters redistribute “power” between frequency components in the spectrum. The analysis emphasises the importance of eigenvalues in the identification of resonance, both in the system itself and in its subsystems, and the importance of noise configuration in defining which paths are followed on the network. The analysis highlights the complexity of the inverse problem (in finding, for example, the source of long term fluctuations) and the role of factors other than colour in the persistence of populations.

Population extinction in discrete-time stochastic population models with an Allee effect

The Allee effect is often modeled as a threshold level below which there is population extinction. A minimum population size is required for population persistence. The Allee effect has been primarily studied in deterministic models. In this investigation, we develop two discrete-time stochastic population models with an Allee effect. The stochastic models are discrete-time Markov chains with demographic stochasticity. They are based on two well-known deterministic discrete-time population models. It is shown that the conditional means of the stochastic models follow the solution dynamics of the underlying deterministic models. However, near the Allee threshold value, the dynamics of the deterministic and stochastic models differ significantly. The first model is based on the well-known Ricker population model with an Allee effect. The second model is more complex. The population is subdivided according to mating status of adults and is based on a model developed for the northern spotted owl. The dynamics of the underlying deterministic models are reviewed first, then the stochastic models are formulated. The stochastic models are new formulations. Our goal is to study the probability of population extinction in the stochastic models.

International Management Strategies for a Straddling Fish Stock: A Bio-Economic Simulation Model of the Norwegian Spring-Spawning Herring Fishery

In this paper, a three-country dynamic bio-economic simulation model is presented for the spring-spawning herring fishery. The international spring-spawning herring fishery, based on potentially one of the most valuable fish stocks in the world, is currently recovering from a severe depletion of the stock and subsequent harvesting moratorium. Management of the herring fishery is complicated by its multi-nation exploitation, due to the highly migratory behaviour of the species moving between several coastal state zones (exclusive economic zones, EEZ’s) and the high seas (Ocean Loop). Based on extensive work invested on analysing both the biology of the herring stock and the fisheries economics around its exploitation we study here the profiles of different multi-agent management schemes, simulating catch levels, stock size and profit potentials of alternative management strategies. The stock dynamics are described by a linear discrete-time age-structured population model and the economics are presented by a rent maximising model with constant price of herring catch and different costs of harvesting and efficiency levels for the different national fleets. The simulations, carried out over several decades, show that the benefits of international co-operation far exceed the returns of a competitive open access fishery.

On the mechanistic underpinning of discrete-time population models with complex dynamics

We present a mechanistic underpinning for various discrete-time population models that can produce limit cycles and chaotic dynamics. Specific examples include the discrete-time logistic model and the Hassell model, which for a long time eluded convincing mechanistic interpretations, and also the Ricker- and Beverton–Holt models. We first formulate a continuous-time resource consumption model for the dynamics within a year, and from that we derive a discrete-time model for the between-year dynamics. Without influx of resources from the outside into the system, the resulting between-year dynamics is always overcompensating and hence may produce complex dynamics as well as extinction in finite time. We recover a connection between various standard types of continuous-time models for the resource dynamics within a year on the one hand and various standard types of discrete-time models for the population dynamics between years on the other. The model readily generalizes to several resource and consumer species as well as to more than two trophic levels for the within-year dynamics.

Non-linear biological responses to disturbance: consequences on population dynamics

We assessed how non-linear biological responses to environmental noise, or “noise filtering”, impact the spectra of density-dependent population dynamics, and the correlation between noise and population dynamics. The noise was assumed to affect population growth rate in a discrete-time population model by Hassell [J. Anim. Ecol. 44 (1975) 283–295] where the population growth rate was linked to the environment with an optimum type filter. When compared to unfiltered noise, the filtered noise can distort the stationary distribution of population values. The optimum type filter can make cyclic population dynamics more regular and low population values can become more frequent or rare depending on the strength of density dependence. Filtering can cause blue shifted and red shifted population dynamics and determine the strength of correlation between environmental noise and population size. In most cases, optimum type filtering makes linear correlation between population dynamics and noise weaker. The filter effect on population spectra and noise versus population correlation is sensitive to changes in population model parameters, the location where noise hits the filter, and noise colour.

Partial life cycle analysis: a model for pre‐breeding census data

Matrix population models have become popular tools in research areas as diverse as population dynamics, life history theory, wildlife management, and conservation biology. Two classes of matrix models are commonly used for demographic analysis of age‐structured populations: age‐structured (Leslie) matrix models, which require age‐specific demographic data, and partial life cycle models, which can be parameterized with partial demographic data. Partial life cycle models are easier to parameterize because data needed to estimate parameters for these models are collected much more easily than those needed to estimate age‐specific demographic parameters. Partial life cycle models also allow evaluation of the sensitivity of population growth rate to changes in ages at first and last reproduction, which cannot be done with age‐structured models. Timing of censuses relative to the birth‐pulse is an important consideration in discrete‐time population models but most existing partial life cycle models do not address this issue, nor do they allow fractional values of variables such as ages at first and last reproduction. Here, we fully develop a partial life cycle model appropriate for situations in which demographic data are collected immediately before the birth‐pulse (pre‐breeding census). Our pre‐breeding census partial life cycle model can be fully parameterized with five variables (age at maturity, age at last reproduction, juvenile survival rate, adult survival rate, and fertility), and it has some important applications even when age‐specific demographic data are available (e.g., perturbation analysis involving ages at first and last reproduction). We have extended the model to allow non‐integer values of ages at first and last reproduction, derived formulae for sensitivity analyses, and presented methods for estimating parameters for our pre‐breeding census partial life cycle model. We applied the age‐structured Leslie matrix model and our pre‐breeding census partial life cycle model to demographic data for several species of mammals. Our results suggest that dynamical properties of the age‐structured model are generally retained in our partial life cycle model, and that our pre‐breeding census partial life cycle model is an excellent proxy for the age‐structured Leslie matrix model.

The growth of general population-size-dependent branching processes year by year

We study discrete-time population models where the nearest future of an individual may depend on the individual's life-stage (age and reproduction history) and the current population size. A criterion is given for whether there is a positive probability that the population survives forever. We identify the cases when population size grows exponentially and linearly and show that in the latter population size scaled by time is asymptotically Γ-distributed.

The growth of general population-size-dependent branching processes year by year

We study discrete-time population models where the nearest future of an individual may depend on the individual's life-stage (age and reproduction history) and the current population size. A criterion is given for whether there is a positive probability that the population survives forever. We identify the cases when population size grows exponentially and linearly and show that in the latter population size scaled by time is asymptotically Γ-distributed.