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.     

Stability of discrete age-structured and aggregated delay-difference population models

Sufficiency conditions for local stability are derived for a class of density dependent Leslie matrix models. Four of the recruitment functions in common use in fisheries management are then considered. In two of these oscillating instability can never occur (Beverton and Holt and Cushing forms). In the other two (Deriso-Schnute and Shepherd forms) undamped oscillations are possible within the region of parameter space described here. An algorithm is developed for calculating necessary and sufficient local stability conditions for a simplified form of the general age-structured model. The complete spectrum of stability states (monotonic stability; monotonic instability; oscillating-stable; oscillating-unstable) and the bifurcation periods are given for selected examples of this model. The examples cover a large portion of the parameter space of interest in resource management. It is shown that in perfectly deterministic systems which are observed with error, oscillating instabilities may be missed, and such systems could be erroneously assumed to be stable.     

Stability in cyclic epidemic models

A general formulation for a family of cyclic epidemic models with density-dependent feedback mechanisms and removed classes is presented. A parameter, , related to the basic reproductive rate determines the asymptotic behaviour of solutions of the model. It is shown that if <1 the trivial solution is globally stable, and if >1 it is conditionally stable. The results are applied to a set of differential equations that has been used to model the life cycle of a parasite that has two hosts.     

Equilibrium solutions of age-specific population dynamics of several species

Summary A mathematical model describing the dynamics of a population consisting of several species is studied. The interactions in the population are assumed to be age-specific. Using an evolution equation approach, sufficient conditions for well-posedness in L ¹ of the dynamics and for existence as well as for stability of equilibrium solutions are given.     

Relationships among recent models for insect population dynamics with variable rates of development

A classification scheme for those population models which allow variation in development rates is proposed, based on two ways of modifying standard age-structured models. The resulting classes of models are termed development index models and sojourn time models. General formulations for the two classes of models are developed from two basic balance equations, and numerous specific models from the literature are shown to fit into the scheme. Concepts from competing risks theory are shown to be important in understanding the interplay between mortality and maturation. Relationships among the classes are investigated both for the most general forms of the models and for the simpler forms often used. The scheme can provide guidance in developing appropriate insect population models for specific modelling situations.Contribution 3878871     

Stability analysis for models of diseases without immunity

Summary A cyclic, constant parameter epidemiological model is described for a closed population divided into susceptible, exposed and infectious classes. Distributed delays are introduced and the model is formulated as two coupled Volterra integral equations. The delays do not change the general nature of thresholds or asymptotic stability; in all cases considered the disease either dies out, or approaches an endemic steady state.This work was partially supported by NIH Grant AI 13233 and NSERC Grant A-4645     

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.     

Stability in a class of cyclic epidemic models with delay

A detailed analysis of a general class of SIRS epidemic models is given. Sufficient conditions are derived which guarantee the global stability of the endemic equilibrium solution. Further conditions are found which ensure instability for the equilibrium. Finally, the dependence of the stability on the contact number and the ratio of the mean length of infection to the mean removed time is considered.     

A mathematical model for the dynamics of malaria in mosquitoes feeding on a heterogeneous host population

We describe and develop a difference equation model for the dynamics of malaria in a mosquito population feeding on, infecting and getting infected from a heterogeneous population of hosts. Using the force of infection from different classes of humans to mosquitoes as parameters, we evaluate a number of entomological parameters, indicating malaria transmission levels, which can be compared to field data. By assigning different types of vector control interventions to different classes of humans and by evaluating the corresponding levels of malaria transmission, we can compare the effectiveness of these interventions. We show a numerical example of the effects of increasing coverage of insecticide-treated bed nets in a human population where the predominant malaria vector is Anopheles gambiae.     

One- and multi-locus multi-allele selection models in a random environment

Summary We deduce conditions for stochastic local stability of general perturbed linear stochastic difference equations widely applicable in population genetics. The findings are adapted to evaluate the stability properties of equilibria in classical one- and multi-locus multi-allele selection models influenced by random temporal variation in selection intensities. As an example of some conclusions and biological interpretations we analyse a special one-locus multi-allele model in more detail.This work was supported in part by Stiftung Volkswagenwerk.     

Developmental variability and stability in continuous-time host-parasitoid models

Insect host-parasitoid systems are often modeled using delay-differential equations, with a fixed development time for the juvenile host and parasitoid stages. We explore here the effects of distributed development on the stability of these systems, for a random parasitism model incorporating an invulnerable host stage, and a negative binomial model that displays generation cycles. A shifted gamma distribution was used to model the distribution of development time for both host and parasitoid stages, using the range of parameter values suggested by a literature survey. For the random parasitism model, the addition of biologically plausible levels of developmental variability could potentially double the area of stable parameter space beyond that generated by the invulnerable host stage. Only variability in host development time was stabilizing in this model. For the negative binomial model, development variability reduced the likelihood of generation cycles, and variability in host and parasitoid was equally stabilizing. One source of stability in these models may be aggregation of risk, because hosts with varying development times have different vulnerabilities. High levels of variability in development time occur in many insects and so could be a common source of stability in host-parasitoid systems.     

Behaviour of simple population models under ecological processes

The two most popular and extensively-used discrete models of population growth display the generic bifurcation structure of a hierarchy of period-doubling sequence to chaos with increasing growth rates. In this paper we show that these two models, though they belong to a general class of one-dimensional maps, show very different dynamics when important ecological processes such as immigration and emigration/depletion, are considered. It is important that ecologists recognize the differences between these models before using them to describe their data—or develop optimization strategies—based on these models.     

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.     

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.     

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.     

Structured population dynamics: continuous size and discontinuous stage structures

A nonlinear stochastic model for the dynamics of a population with either a continuous size structure or a discontinuous stage structure is formulated in the Eulerian formalism. It takes into account dispersion effects due to stochastic variability of the development process of the individuals. The discrete equations of the numerical approximation are derived, and an analysis of the existence and stability of the equilibrium states is performed. An application to a copepod population is illustrated; numerical results of Eulerian and Lagrangian models are compared.      

ODE models for oncolytic virus dynamics

Replicating oncolytic viruses are able to infect and lyse cancer cells and spread through the tumor, while leaving normal cells largely unharmed. This makes them potentially useful in cancer therapy, and a variety of viruses have shown promising results in clinical trials. Nevertheless, consistent success remains elusive and the correlates of success have been the subject of investigation, both from an experimental and a mathematical point of view. Mathematical modeling of oncolytic virus therapy is often limited by the fact that the predicted dynamics depend strongly on particular mathematical terms in the model, the nature of which remains uncertain. We aim to address this issue in the context of ODE modeling, by formulating a general computational framework that is independent of particular mathematical expressions. By analyzing this framework, we find some new insights into the conditions for successful virus therapy. We find that depending on our assumptions about the virus spread, there can be two distinct types of dynamics. In models of the first type (the “fast spread” models), we predict that the viruses can eliminate the tumor if the viral replication rate is sufficiently high. The second type of models is characterized by a suboptimal spread (the “slow spread” models). For such models, the simulated treatment may fail, even for very high viral replication rates. Our methodology can be used to study the dynamics of many biological systems, and thus has implications beyond the study of virus therapy of cancers.     

Separable models for age-structured population genetics

This paper is concerned with the applications of nonlinear age-dependent dynamics to population genetics. Age-structured models are formulated for a single autosomal locus with an arbitrary number of alleles. The following cases are considered: a) haploid populations with selection and mutation; b) monoecious diploid populations with or without mutation reproducing by self-fertilization or by two types of random mating. The diploid models do not deal with selection. For these cases the genic and genotypic frequencies evolve towards time-persistent forms, whether the total population size tends towards exponential growth or not.     

Stability properties of pulse vaccination strategy in SEIR epidemic model

The problem of the applicability of the pulse vaccination strategy (PVS) for the stable eradication of some relevant general class of infectious diseases is analyzed in terms of study of local asymptotic stability (LAS) and global asymptotic stability (GAS) of the periodic eradication solution for the SEIR epidemic model in which is included the PVS. Demographic variations due or not to diseased-related fatalities are also considered. Due to the non-triviality of the Floquet's matrix associate to the studied model, the LAS is studied numerically and in this way it is found a simple approximate (but analytical) sufficient criterion which is an extension of the LAS constraint for the stability of the trivial equilibrium in SEIR model without vaccination. The numerical simulations also seem to suggest that the PVS is slightly more efficient than the continuous vaccination strategy. Analytically, the GAS of the eradication solutions is studied and it is demonstrated that the above criteria for the LAS guarantee also the GAS.     

Stability in a two host epidemic model

An epidemic model is derived for a two host infectious disease. It is shown that if a non-trivial equilibrium solution exists, it is globally stable. This result is also proved for a similar one host model.