Age-structured Population Model Research Articles

An age-structured two-sex model in the space of radon measures: Well posedness

In the following paper a well-posedness of an age-structured two-sex population model in a space of Radon measures equipped with a flat metric is presented. Existence and uniqueness of measure valued solutions is proved by a regularization technique. This approach allows to obtain Lipschitz continuity of solutions with respect to time and stability estimates. Moreover, a brief discussion on a marriage function, which is the main source of a nonlinearity, is carried out and an example of the marriage function fitting into this framework is given.

Chaos in Discrete Structured Population Models

We prove analytically the existence of chaotic dynamics in some classical discrete-time age-structured population models. Our approach allows us to estimate the sensitive dependence on the initial conditions, regions of initial data with chaotic behavior, and explicit ranges of parameters for which the considered models display chaos. These properties have important implications for evaluating the influence of a chaotic regime on the predictions based on mathematical models. We illustrate through particular examples how to apply our results.

ブルーギル (Lepomis macrochirus) の個体群モデルを用いた駆除対策への示唆

ブルーギル (<i>Lepomis macrochirus</i>) の個体群モデルを用いた駆除対策への示唆

Lyapunov Functions and Global Stability for Age-Structured HIV Infection Model

We study the basic age-structured population model describing the HIV infection process, which is defined by PDEs. The model allows the production rate of viral particles and the death rate of productively infected cells to vary and depend on the infection age. By using the direct Lyapunov method and constructing suitable Lyapunov functions, dynamical properties of the age-structured model without (or with) drug treatment are established. The results show that the global asymptotic stability of the infection-free steady state and the infected steady state depends only on the basic reproductive number determined by the burst size. Further, we establish mathematically that the typical ODE and DDE (delay differential equation) models of HIV infection are equivalent to two special cases of the above PDE models.

Mixing in age-structured population models of infectious diseases

Infectious diseases are controlled by reducing pathogen replication within or transmission between hosts. Models can reliably evaluate alternative strategies for curtailing transmission, but only if interpersonal mixing is represented realistically. Compartmental modelers commonly use convex combinations of contacts within and among groups of similarly aged individuals, respectively termed preferential and proportionate mixing. Recently published face-to-face conversation and time-use studies suggest that parents and children and co-workers also mix preferentially. As indirect effects arise from the off-diagonal elements of mixing matrices, these observations are exceedingly important. Accordingly, we refined the formula published by Jacquez et al. [19] to account for these newly-observed patterns and estimated age-specific fractions of contacts with each preferred group. As the ages of contemporaries need not be identical nor those of parents and children to differ by exactly the generation time, we also estimated the variances of the Gaussian distributions with which we replaced the Kronecker delta commonly used in theoretical studies. Our formulae reproduce observed patterns and can be used, given contacts, to estimate probabilities of infection on contact, infection rates, and reproduction numbers. As examples, we illustrate these calculations for influenza based on “attack rates” from a prospective household study during the 1957 pandemic and for varicella based on cumulative incidence estimated from a cross-sectional serological survey conducted from 1988–94, together with contact rates from the several face-to-face conversation and time-use studies. Susceptibility to infection on contact generally declines with age, but may be elevated among adolescents and adults with young children.

Using an age-structured population model to define management requirements for conservation of egrets in the Murray-Darling Basin, Australia

Major modifications to river systems and their associated floodplains throughout the world have detrimental consequences for biodiversity. A reduction in the frequency of large-scale flooding events is one change that could have significant negative effects. In the Murray-Darling Basin of Australia many species of waterbirds breed in large colonies on inland floodplain systems when appropriate flooding occurs. Across the Murray-Darling Basin populations of many of these species are now at very low levels and breeding events occur much more infrequently than they did historically. I used an age-structured population model of a generic egret to assess how frequently breeding must occur to maintain populations. Using simulation I explored how sensitive results were to demographic parameter values. The results indicated that: (1) most egrets must breed every one to two years for populations to persist; and (2) the results are sensitive to adult survival, suggesting that management may also need to consider habitat availability between breeding events to ensure high adult survival. The results highlight a need to determine over what spatial scale egret populations function so that a sufficient frequency of breeding events is provided at the appropriate scale. Results were less sensitive to juvenile survival, but juvenile survival rates that average between 25 and 35% per year are probably required for persistence. Hence it is important that future research also determines whether the recent practice of managing breeding events only until fledging is achieved does not have a detrimental effect on juvenile survival. Maintaining a high frequency of flooding events that lead to successful breeding will be necessary for any species with similar life-histories.

The comparison between Homotopy Analysis Method and Optimal Homotopy Asymptotic Method for nonlinear age-structured population models

This paper presents comparison between Homotopy Analysis Method (HAM) and Optimal Homotopy Asymptotic Method (OHAM) for the solution of nonlinear age-structured population models. Three examples have been presented to illustrate and compare these methods. In OHAM the convergence region can be easily adjusted and controlled. Comparison between our solution and the exact solution shows that the both methods are effective and accurate in solving nonlinear age-structured population models with HAM being the more accurate for the same number of terms. It was also found that OHAM require more CPU time.

Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials

In this paper a numerical method for solving the nonlinear age-structured population models is presented which is based on Bernstein polynomials approximation. Operational matrices of integration, differentiation, dual and product are introduced and are utilized to reduce the age-structured population problem to the solution of algebraic equations. The method in general is easy to implement, and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.

Modeling the interaction between the exotic invasive aquatic macrophyte Myriophyllum spicatum and the native biocontrol agent Euhrychiopsis lecontei to improve augmented management programs

Mathematical modeling is used to investigate control of the invasive aquatic plant Myriophyllum spicatum L. by the native watermilfoil weevil Euhrychiopsis lecontei Dietz. An age-structured population model with discrete development class is used to separate the destructive activity of Euhrychiopsis lecontei Dietz larvae from the relatively benign activity of the other developmental stages. Empirical model behavior is evaluated and agrees with expected system characteristics. Simulation results demonstrate diminishing returns in the control impact of E. lecontei Dietz as a function of population stocking density. These results indicate that predictive modeling of this type may be used to optimize the utilization of Euhrychiopsis lecontei Dietz as a biological control agent.

Evaluating the Potential Efficacy of Invasive Lionfish (Pterois volitans) Removals

The lionfish, Pterois volitans (Linnaeus) and Pterois miles (Bennett), invasion of the Western Atlantic Ocean, Caribbean Sea and Gulf of Mexico has the potential to alter aquatic communities and represents a legitimate ecological concern. Several local removal programs have been initiated to control this invasion, but it is not known whether removal efforts can substantially reduce lionfish numbers to ameliorate these concerns. We used an age-structured population model to evaluate the potential efficacy of lionfish removal programs and identified critical data gaps for future studies. We used high and low estimates for uncertain parameters including: length at 50% vulnerability to harvest (Lvul), instantaneous natural mortality (M), and the Goodyear compensation ratio (CR). The model predicted an annual exploitation rate between 35 and 65% would be required to cause recruitment overfishing on lionfish populations for our baseline parameter estimates for M and CR (0.5 and 15). Lionfish quickly recovered from high removal rates, reaching 90% of unfished biomass six years after a 50-year simulated removal program. Quantifying lionfish natural mortality and the size-selective vulnerability to harvest are the most important knowledge gaps for future research. We suggest complete eradication of lionfish through fishing is unlikely, and substantial reduction of adult abundance will require a long-term commitment and may be feasible only in small, localized areas where annual exploitation can be intense over multiple consecutive years.

Importance of age structure in models of the response of upper trophic levels to fishing and climate change

Abstract Botsford, L. W., Holland, M. D., Samhouri, J. F., White, J. W., and Hastings, A. 2011. Importance of age structure in models of the response of upper trophic levels to fishing and climate change. – ICES Journal of Marine Science, 68: 1270–1283. There is a growing effort to use predictions of the physical state of the ocean under climate change to forecast the response of marine ecosystems. Many of these forecasts use ecosystem models rather than age-structured population models to describe upper trophic level (UTL) species. We illustrate the potential effects of climate on age-structured populations, then illustrate the ways in which ecosystem models might not depict adequately: (i) long-term changes in abundance, and (ii) variability attributable to cohort resonance. We simulated two generic species with different life histories, a short-lived semelparous species (e.g. salmon), and a long-lived iteroparous species (e.g. cod). For both species, juvenile survival was varied, first with white noise, then with the Pacific Decadal Oscillation as environmental signals. Variability in recruitment increased with fishing and became particularly sensitive to forcing at time-scales near the mean age of reproduction, consistent with the cohort resonance effect. Ecosystem models without age structure do not predict this behaviour, particularly when the ecosystem model incorrectly predicts the effective steepness of the stock–recruitment relationship, or the age structure is approximated by a stage-structured model. We suggest that ecosystem models of UTLs include full representations of age structure, fitted to available population data.

Six decades of pike and perch population dynamics in Windermere

We use six decades of catch-at-age data for perch ( Perca fluviatilis) and pike ( Esox lucius) in Windermere (UK) to estimate age- and sex-specific population sizes, natural mortalities and catchabilities in both species. Population sizes are estimated by fitting age structured population models to the catch-at-age data using standard maximum likelihood methods. We validate our methods using data simulations, and use our estimates of vital rates (natural mortality, recruitment and catahability) to address important aspects of fisheries biology. Our model indicates that strong fishery selection against male perch apparently triggered a population collapse, highlighting that sex-selective fisheries can be harmful even at a reasonable exploitation rate (here ≤30%). Recruitment (R) increased with the abundance of spawners (S) in both species, but it also responded to both abiotic and other biotic factors. In particular, increased predator (pike) abundance induced a change from compensation to depensation in the prey (perch) SR relationship, thus favouring the occurrence of an Allee effect. Our study provides reference points for the effective exploitation of pike and perch populations, and underscores the need for ecosystem-based harvesting management.

An efficient algorithm for solving nonlinear age-structured population models by combining homotopy perturbation and Padé techniques

In this study, we investigate nonlinear age-structured population models. The numerical solutions are obtained by combining homotopy perturbation method and Padé technique. Two numerical examples are presented to illustrate the strength of this efficient algorithm.

Second-order characteristic schemes in time and age for a nonlinear age-structured population model

In this paper, we analyze two new second-order characteristic schemes in time and age for an age-structured population model with nonlinear diffusion and reaction. By using the characteristic difference to approximate the transport term and the average along the characteristics to treat the nonlinear spatial diffusion and reaction terms, an implicit second-order characteristic scheme is proposed. To compute the nonlinear approximation system, an explicit second-order characteristic scheme in time and age is further proposed by using the extrapolation technique. The global existence and uniqueness of the solution of the nonlinear approximation scheme are established by using the theory of variation methods, Schauder’s fixed point theorem, and the technique of prior estimates. The optimal error estimates of second order in time and age are strictly proved for both the implicit and the explicit characteristic schemes. Numerical examples are given to illustrate the performance of the methods.

Population dynamics of splitnose rockfish ( Sebastes diploproa) in the Northeast Pacific Ocean

We developed an age-structured population model of splitnose rockfish, Sebastes diploproa, in the Northeast Pacific Ocean. Splitnose rockfish is a bycatch species that co-occurs with several commercially important species that are currently declared overfished. Bycatch species are typically not the focus of stock assessment efforts because of their limited economic importance, but they may suffer the same population declines as species with which they co-occur. To examine the dynamics of splitnose rockfish for the first time, we analyzed data from three groundfish fisheries and four research surveys conducted in the Northeast Pacific Ocean. To develop a model, we used Stock Synthesis, a statistical framework for the construction of a population dynamics models utilizing both fishery-dependent and fishery-independent data. In the model, we reconstructed the total catch of the species back to 1900, estimated the dynamics of the stock spawning output and recruitment and evaluated biomass depletion relative to the stock's unfished state, as well as sources of uncertainty in model outputs. The results indicate that the splitnose rockfish is currently not overfished even though it has experienced several periods of abrupt decline in its biomass. Revisiting age data from earlier years, monitoring fishery discard, and investigating the spatial dynamics of splitnose rockfish is important to further improve the understanding of this species’ population dynamics, and decrease uncertainty in model results.

Variational Iteration Method for Nonlinear Age-Structured Population Models Using Auxiliary Parameter

In this paper, we apply He’s variational iteration method (VIM) coupled with an auxiliary parameter which proves very effective to control the convergence region of an approximate solution. Moreover, a convenient way is considered for choosing a suitable auxiliary parameter via a residual function. The proposed algorithm is tested on some nonlinear age-structured population models, and numerical results explicitly reveal the complete reliability, efficiency, and accuracy of the suggested technique. It is observed that the approach may be implemented on other nonlinear models of physical nature.

Assessing evolutionary consequences of size-selective recreational fishing on multiple life-history traits, with an application to northern pike (Esox lucius)

Despite mounting recognition of the importance of fishing-induced evolution, methods for quantifying selection pressures on multiple adaptive traits affected by size-selective harvesting are still scarce. We study selection differentials on three life-history traits—reproductive investment, size at maturation, and growth capacity—under size-selective exploitation of northern pike (Esox lucius L.) with recreational-fishing gear. An age-structured population model is presented that accounts for the eco-evolutionary feedback arising from density-dependent and frequency-dependent selection. By introducing minimum-length limits, maximum-length limits, and combinations of such limits (resulting in harvestable-slot length limits) into the model, we examine the potential of simple management tools for mitigating selection pressures induced by recreational fishing. With regard to annual reproductive investment, we find that size-selective fishing mortality exerts relatively small positive selection differentials. By contrast, selection differentials on size at maturation are large and consistently negative. Selection differentials on growth capacity are often large and positive, but become negative when a certain range of minimum-length limits are applied. In general, the strength of selection is reduced by implementing more stringent management policies, but each life-history trait responds differently to the introduction of specific harvest regulations. Based on a simple genetic inheritance model, we examine mid- and long-term evolutionary changes of the three life-history traits and their impacts on the size spectrum and yield of pike. Fishing-induced evolution often reduces sizes and yields, but details depend on a variety of factors such as the specific regulation in place. We find no regulation that is successful in reducing to zero all selection pressures on life-history traits induced by recreational fishing. Accordingly, we must expect that inducing some degree of evolution through recreational fishing is inevitable.

Stability and Periodic Solution to a Nonlinear Partial Differential Equation Population Model with Delay

This paper studies the dynamical of a non-autonomous nonlinear age-structured partial differential equation population model with delay.The sufficient stability conditions and the existence of periodic solutions are obtained.

Spatial dynamics of a nonlocal and delayed population model in a periodic habitat

We derived an age-structured population model withnonlocal effects and time delayin a periodic habitat. The spatial dynamics of the model includingthe comparison principle, the global attractivity of spatiallyperiodic equilibrium, spreading speeds, and spatially periodictraveling wavefronts is investigated. It turns out that thespreading speed coincides with the minimal wave speed forspatially periodic travel waves.

Cohort variation, climate effects and population dynamics in a short‐lived lizard

1. Demographic theory and empirical studies indicate that cohort variation in demographic traits has substantial effects on population dynamics of long-lived vertebrates but cohort effects have been poorly investigated in short-lived species. 2. Cohort effects were quantified in the common lizard (Zootoca vivipara Jacquin 1787), a short-lived ectothermic vertebrate, for body size, reproductive traits and age-specific survival with mark-recapture data collected from 1989 to 2005 in two wetlands. We assessed cohort variation and covariation in demographic traits, tested the immediate and delayed effects of climate conditions (temperature and rainfall), and predicted consequences for population growth. 3. Most demographic traits exhibited cohort variation, but this variation was stronger for juvenile growth and survival, sub-adult survival and breeding phenology than for other traits. 4. Cohort variation was partly explained by a web of immediate and delayed effects of climate conditions. Rainfall and temperature influenced distinct life-history traits and the periods of gestation and early juvenile life were critical stages for climate effects. 5. Cohort covariation between demographic traits was usually weak, apart from a negative correlation between juvenile and sub-adult body growth suggesting compensatory responses. An age-structured population model shows that cohort variation influences population growth mainly through direct numerical effects of survival variation early in life. 6. An understanding of cohort effects is necessary to predict critical life stages and climatic determinants of population dynamics, and therefore demographic responses to future climate warming.