Age-structured Population Dynamics Research Articles

Demographic regulation processes in Pinna nobilis population subunits: Implications for restocking

A mass mortality event (MME) affected the endemic bivalve Pinna nobilis across the Mediterranean Sea in autumn 2016. This event affected specimens of all sizes, depth ranges and habitat types, resulting in a drastic decline in the species population close to 100%. Indeed, the consequences of a potential recovery, and therefore the viability of the populations of the species are unknown. To evaluate the likelihood of success when designing the reintroduction of the species in optimal areas, knowledge of population dynamics is essential. Here we compare two population subunits of Pinna nobilis from two areas with different environmental conditions, located in Cabrera National Park (protected area with low exposure to hydrodynamic activity) and San Telmo (unprotected area exposed to hydrodynamic activity), both in the Balearic Islands. A total of 259 individuals were tagged and monitored annually between 2011 and 2014, prior to the MME. Matrix population models were used to reduce the complexity of the age-structured population dynamics and assessing the degree of elasticity of the population. We showed that the subunit of San Telmo showed highly variable values of demographic rates, higher stochasticity and higher sensitivity to environmental variability than the population subunit of Cabrera National Park. The study shows the importance of the buffering effect of a large population as there are many adult individuals of many size classes in the population subunit of Cabrera National Park, as its age structure helps to cope with highly variable reproductive success. Therefore, these population subunits are more stable regardless of depth and would be a more suitable site for future reintroduction programs.

Modeling Population Growth under Climate Stressors Using Age-structured Matrix Models.

Climate resilience, a focus of many recent studies, has been examined from ecological, physiological, and evolutionary perspectives. However, sampling biases towards adults, males, and certain species have made establishing the link between environmental change and population-level change problematic. Here we used data from four laboratory studies, in which we administered pre- and post-natal stressors, such as suboptimal incubation temperature, heat stress, and food restriction, to zebra finches and quantified hatching success, post-hatch survival, and reproductive success, to parameterize age-structured population dynamics models with the goal of estimating the effect of the stressors on relative population growth rates. Using the same model structure, we tested the hypothesis that early life stages influence population growth rate more than later life stages. Our models suggested that stressful events during embryonic development, such as suboptimal incubation temperatures and reduced gas exchange for the embryos, have a greater total impact on population growth than post-hatch stressors, such as heat stress and food restriction. However, among life history traits, differences in hatching success and sex ratio of offspring in response to stressors changed population growth rates more than differences in any other demographic rate estimates. These results suggest that when predicting population resilience against climate change, it is critical to account for effects of climate change on all life stages, including early stages of life, and to incorporate individuals' physiology and stress tolerance that likely influence future stress responses, reproduction, and survival.

An Inverse Problem for an Age-Structured Population Dynamics Model with Migration Flows

An Inverse Problem for an Age-Structured Population Dynamics Model with Migration Flows

The optimal momentum of population growth and decline

About 50 years ago, Keyfitz (1971) asked how much further a growing human population would increase if its fertility rate were immediately to be reduced to replacement level and remain there forever. The reason for demographic momentum is an age–structure inertia due to relatively many potential parents because of past high fertility. Although nobody expects such a miraculous reduction in reproductive behavior, a gradual decline in fertility in rapidly growing populations seems inevitable. As any delay in fertility decline to a stationary level leads to an increase in the momentum, it makes sense to think about the timing and the quantum of the reduction in reproduction. More specifically, we consider an intertemporal trade-off between costly pro- and anti-natalistic measures and the demographic momentum at the end of the planning period. This paper uses the McKendrick–von Foerster partial differential equation of age–structured population dynamics to study a sketched problem in a distributed parameter control framework. Among the results obtained by applying an appropriate extension of Pontryagin’s Maximum Principle are the following: (i) monotony of adaptation efforts to net reproduction rate and convex decrease/concave increase (if initial net reproduction rate exceeds 1/is below 1); and (ii) oscillating efforts and reproduction rate if, additionally, the size of the total population does not deviate from a fixed level.

Global asymptotic stability for Gurtin-MacCamy’s population dynamics model

In this paper, we investigate the global asymptotic stability of an age-structured population dynamics model with a Ricker’s type of birth function. This model is a hyperbolic partial differential equation with a nonlinear and nonlocal boundary condition. We prove a uniform persistence result for the semiflow generated by this model. We obtain the existence of global attractors and we prove the global asymptotic stability of the positive equilibrium by using a suitable Lyapunov functional. Furthermore, we prove that our global asymptotic stability result is sharp, in the sense that Hopf bifurcation may occur as close as we want from the region global stability in the space of parameter.

Control of age-structured population dynamics with intraspecific competition in context of bioreactors

Bioreactors are used for commercial manufacturing of products or as a simplified representation of wastewater treatment plants. If it is taken into account that individual organisms from the population in the bioreactor behave differently, an additional structure such as age can be added to the population. This leads to a nonlinear, hyperbolic, integro-partial differential equation (IPDE) with non-local integral boundary conditions (BC) in the modeling. The steady-states of this system can be uniquely determined by a substitution due to the non-local BC. Using the asymptotic properties of the system, the IPDE is split into finite-dimensional dynamics and infinite dimensional internal dynamics. The internal dynamics are globally exponentially stable. Therefore, a feedforward control based on the finite dimensional dynamics is designed, augmented with a feedback control to compensate for the remaining influence of the internal dynamics on the output. Finally, the control concept is supplemented by an observer. Global exponential attractiveness of certain reference trajectories is proven. The overall control concept is validated experimentally with a bioreactor. A system parameter is identified and it is shown that by applying the control concept, the biomass can track various reference trajectories with small errors.

Spatiotemporal patterns of a structured spruce budworm diffusive model

We formulate and analyze a general reaction-diffusion equation with delay, inspired by age-structured spruce budworm population dynamics with spatial diffusion by matured individuals. The model has its particular feature for bistability due to the incorporation of a nonlinear birth function (Ricker's function) and a Holling type function of predation by birds. Here we establish some results about the global dynamics, in particular, the stability of and global Hopf bifurcation from the spatially homogeneous steady state when the maturation delay is taken as a bifurcation parameter. We also use a degree theoretical argument to identify intervals for the diffusion rate when the model system has a spatially heterogeneous steady state. Numerical experiments presented show interesting spatialtemporal patterns.

A novel approach for accurate quantification of lake residence time — Lake Kinneret as a case study

Water residence time, which is affected by increasing water demands and climate change, plays a crucial role in lakes and reservoirs since it influences many natural physical and ecological processes that eventually impact the water quality of the waterbody. Thus, accurate quantification of the water residence time and its distribution is an important tool in lake management. In this study we present a novel approach for assessing the residence time in lakes and reservoirs. The approach is based on the Leslie matrix model that was originally developed for the analysis of age-structured biological population dynamics. In this approach the water in the lake is divided into different age classes each representing the time since the “parcel” of water entered the lake and provides an overall picture of the water age structure. The traditional approach for calculating residence times, which relies only on the lake volume and annual inflow or outflow volumes thereby disregarding any previous information, is very sensitive to large interannual variation. While the proposed approach produces the fraction and volume distribution curves of all age classes within the lake for each simulated timestep. Thus, in addition to mean residence time, the fraction of young water (FYW), quantifying the “young” fraction of water in the lake can be analyzed. The same is true for any other age class of water.The approach was applied to Lake Kinneret (Sea of Galilee) historical data collected over 32 years (1987-2018) and for prediction of long-term time series based on several future scenarios (inflows and outflows). It offers a more accurate quantification of the mean residence time of water in a lake and can easily be adapted to other waterbodies. Comparison of simulation results may serve as basis for determining the lake's management policy, by controlling the inflows and outflows, that will affect both the mean residence time and the fraction of “young/old” age classes of water.

Factors influencing size-structured models’ ability to estimate natural mortality

Natural mortality (M) is crucially important for stock assessment since it strongly influences estimates of spawning stock biomass, MSY and fishing mortality. Variation in M can occur between sexes, maturity stages and temporally. Estimates of M are confounded with those of catchability, recruitment, and growth. Previous studies, which focused on age-structured population dynamics models, suggest that it would be better to estimate M rather than pre-specify it. Size-structured population dynamics models, used for species that are hard to age such as crabs, have unique characteristics such as a size-transition matrix and the possibility of a terminal molt, which could impact the ability to estimate M. Therefore, this study explores the ability of size-structured stock assessment models to estimate M for several scenarios. These scenarios include whether the operating model and estimation method have a terminal molt, whether M is constant, time-varying, or sex- and maturity stage-specific, the quality/sample size of the biomass index and the size-composition data and whether growth is estimated simultaneously with M. The results show that size-structured models can estimate time-varying, sex- and maturity stage-specific M when the estimation method mirrors the operating model. Surprisingly, terminal molt does not affect the ability to estimate M. However, estimating growth simultaneously with M has a negative impact on the ability to estimate M but a positive effect on the quality of the estimates of spawning stock biomass.

Unveiling the recovery dynamics of walleye after the invisible collapse

Walleye (Sander vitreus) populations in Alberta, Canada, collapsed by the mid-1990s and were a case study in the paper Canada’s Recreational Fisheries: The Invisible Collapse? Here we fit age-structured population dynamics models to data from a landscape-scale monitoring program to assess walleye population status and reconstruct recruitment dynamics following the invisible collapse. Assessments indicated that populations featured low Fmsyvalues of approximately 0.2–0.3 under conservative assumptions for the stock–recruitment relationship but that many populations were lightly exploited during 2000–2018. Recruitment reconstructions showed that recovery from collapse in 33/55 lakes was driven in part by large positive recruitment anomalies that occurred during 1998–2002. Additionally, 15/55 lakes demonstrated cyclic recruitment dynamics. The documented recruitment anomalies and cyclic fluctuations could be due to environmental effect(s) or cannibalism, and experimentation is likely necessary to resolve this uncertainty. These findings contribute new information on the recovery dynamics of walleye following the invisible collapse and demonstrate the effectiveness of coupling traditional fisheries science models with broad-scale monitoring data to improve understanding of population dynamics and sustainability across landscapes.

Tracking-control for age-structured population dynamics with self-competition governed by integro-PDEs

The dynamics of population systems can be observed in chemostat reactors under laboratory conditions. The modeling of these kind of systems leads to a nonlinear, hyperbolic, first-order integro-partial differential equation with integral boundary condition. Hereby, the state is the population density and the input is the restricted dilution rate. Through an appropriate choice of the steady-state input a generalized spectral analysis is carried out. By using this result and a nonlinear mapping the partial differential equation is split into an input/output-normal form and an integral delay equation. Due to the global exponential stability of the integral delay equation, a feedforward controller is designed for the input/output-normal form. This is complemented with a feedback controller to compensate the influence of the integral delay equation on the output. The global exponential attractivity of reference trajectories meeting certain requirements is proven. Simulations show the performance of the controller saturated to the input constraints in dependency of the controller settings.

Potential-Growth Indicators Revisited: Higher Generality and Wider Merit of Indication

The notion of a potential-growth indicator came to being in the field of matrix population models long ago, almost simultaneously with the pioneering Leslie model for age-structured population dynamics, although the term has been given and the theory developed only in recent years. The indicator represents an explicit function, R(L), of matrix L elements and indicates the position of the spectral radius of L relative to 1 on the real axis, thus signifying the population growth, or decline, or stabilization. Some indicators turned out to be useful in theoretical layouts and practical applications prior to calculating the spectral radius itself. The most senior (1994) and popular indicator, R0(L), is known as the net reproductive rate, and we consider two others, R1(L) and RRT(A), developed later on. All the three are different in terms of their simplicity and the level of generality, and we illustrate them with a case study of Calamagrostis epigeios, a long-rhizome perennial weed actively colonizing open spaces in the temperate zone. While the R0(L) and R1(L) fail, respectively, because of complexity and insufficient generality, the RRT(L) does succeed, justifying the merit of indication.

Trading off retained daily catch for longer seasons to maximize angler benefits in recreational fisheries management

Abstract Recreational fishing benefits associated with angling opportunity, such as fishing season duration and certainty of season duration, may outweigh trip-based benefits, such as retained catch, in measures of angler utility. We developed an age-structured population dynamics model to predict how these three beneficial attributes are affected by four recreational management parameters: minimum size limit; bag limit; inter-season quota transfer limits; and in-season adjustment of season duration. We incorporated results of stated-preference angler surveys to produce a bioeconomic model for two case studies: Atlantic Florida red snapper, and central Oregon Pacific halibut. In maximizing long-term total utility, a strong trade-off between season length and retained daily catch was observed in both fisheries, arising from increased minimum size limits which lengthened fishing seasons but reduced daily retained catch. The optimal management policy across the combination of beneficial attributes was sensitive to assumptions of the curvature of utility functions, i.e. to the degree to which anglers value those attributes with diminishing marginal returns, but generally involved satisfying the strong stated preferences for increased angling opportunity.

Null Controllability of a Four Stage and Age-Structured Population Dynamics Model

This paper is devoted to study the null controllability properties of a population dynamics model with age structuring and nonlocal boundary conditions. More precisely, we consider a four-stage model with a second derivative with respect to the age variable. The null controllability is related to the extinction of eggs, larvae, and female population. Thus, we estimate a time T to bring eggs, larvae, and female subpopulation density to zero. Our method combines fixed point theorem and Carleman estimate. We end this work with numerical illustrations.

The impact of alternative age-length sampling schemes on the performance of stock assessment methods

Stock assessments based on fitting age-structured population dynamics models using the integrated approach usually require data on the length-/age-structure of fishery removals and age-length data to estimate key population parameters such as growth rates, recruitment, natural mortality rates and selectivity. The errors in the estimates of these population parameters directly impact the ability to estimate biomass and hence recommended catch limits based on harvest control rules. The performance of a stock assessment method therefore depends in part on the quality of these “composition” data. Simulation is used to evaluate the effect of a range of effective sample sizes for the length-composition and conditional age-at-length data on the errors in estimating spawning biomass, depletion, recruitment, selected population model parameters, and key model outputs, including catch limit recommendations. Multiple operating models, including those that result in the estimation method being mis-specified, are considered, and the simulations account for the effects of life history, catch history and current stock status. As expected, the errors in estimates of management-related quantities are less for greater effective sample sizes, but effective sample size is seldom the major determinant of the magnitude of estimation errors. The effect of sample size on relative errors is neither linear nor exponential, and spatially-unbalanced sampling is more detrimental than infrequent collection of composition data but with larger sample sizes. The quantitative results are likely case-specific, as they depend on several aspects of the operating model; the framework of this paper can be applied to specific fisheries, in combination with an algorithm that links effective to actual sample sizes.

Dispersal in heterogeneous environments drives population dynamics and control of tsetse flies.

Spatio-temporally heterogeneous environments may lead to unexpected population dynamics. Knowledge is needed on local properties favouring population resilience at large scale. For pathogen vectors, such as tsetse flies transmitting human and animal African trypanosomosis, this is crucial to target management strategies. We developed a mechanistic spatio-temporal model of the age-structured population dynamics of tsetse flies, parametrized with field and laboratory data. It accounts for density- and temperature-dependence. The studied environment is heterogeneous, fragmented and dispersal is suitability-driven. We confirmed that temperature and adult mortality have a strong impact on tsetse populations. When homogeneously increasing adult mortality, control was less effective and induced faster population recovery in the coldest and temperature-stable locations, creating refuges. To optimally select locations to control, we assessed the potential impact of treating them and their contribution to the whole population. This heterogeneous control induced a similar population decrease, with more dispersed individuals. Control efficacy was no longer related to temperature. Dispersal was responsible for refuges at the interface between controlled and uncontrolled zones, where resurgence after control was very high. The early identification of refuges, which could jeopardize control efforts, is crucial. We recommend baseline data collection to characterize the ecosystem before implementing any measures.

On the existence of solution of a four-stage and age-structured population dynamics model

We consider a linear system coming from a population dynamics model with age structuring and nonlocal boundary conditions. Our population dynamics model is a four-stage model with a second derivative with respect to the age variable. Using Banach fixed point theorem, we show the existence and unicity of a non negative solution and illustrations of numerical simulations are given.

SPop: Age-structured discrete-time population dynamics model in C, Python, and R.

This article describes the sPop packages implementing the deterministic and stochastic versions of an age-structured discrete-time population dynamics model. The packages enable mechanistic modelling of a population by monitoring the age and development stage of each individual. Survival and development are included as the main effectors and they progress at a user-defined pace: follow a fixed rate, delay for a given time, or progress at an age-dependent manner. The model is implemented in C, Python, and R with a uniform design to ease usage and facilitate adoption. Early versions of the model were previously employed for investigating climate-driven population dynamics of the tiger mosquito and the chikungunya disease spread by this vector. The sPop packages presented in this article enable the use of the model in a range of applications extending from vector-borne diseases towards any age-structured population including plant and animal populations, microbial dynamics, host-pathogen interactions, infectious diseases, and other time-dependent epidemiological processes.

Age-Structured Population Dynamics with Nonlocal Diffusion.

Random diffusive age-structured population models have been studied by many researchers. Though nonlocal diffusion processes are more applicable to many biological and physical problems compared with random diffusion processes, there are very few theoretical results on age-structured population models with nonlocal diffusion. In this paper our objective is to develop basic theory for age-structured population dynamics with nonlocal diffusion. In particular, we study the semigroup of linear operators associated to an age-structured model with nonlocal diffusion and use the spectral properties of its infinitesimal generator to determine the stability of the zero steady state. It is shown that (i) the structure of the semigroup for the age-structured model with nonlocal diffusion is essentially determined by that of the semigroups for the age-structured model without diffusion and the nonlocal operator when both birth and death rates are independent of spatial variables; (ii) the asymptotic behavior can be determined by the sign of spectral bound of the infinitesimal generator when both birth and death rates are dependent on spatial variables; (iii) the weak solution and comparison principle can be established when both birth and death rates are dependent on spatial variables and time; and (iv) the above results can be generalized to an age-size structured model. In addition, we compare our results with the age-structured model with Laplacian diffusion in the first two cases (i) and (ii).

Variational approach of critical sharp front speeds in degenerate diffusion model with time delay

For the classical reaction diffusion equation, the a priori speed of fronts is determined exactly in the pioneering paper (Benguria and Depassier 1996 Commun. Math. Phys. 175 221–227) by variational characterization method. In this paper, we study the age-structured population dynamics using a degenerate diffusion equation with time delay. We show the existence and uniqueness of sharp critical fronts, where the sharp critical front is C1-smooth when the diffusion degeneracy is weaker with 1 < m < 2, and the sharp critical front is non-C1-smooth (piecewise smooth) when the diffusion degeneracy is stronger with m ⩾ 2, and the non-critical waves are C2-smooth. We give a new variational approach for the critical wave speed and investigate how the time delay affects the propagation mechanism of fronts. It is shown that the time delay slows down the critical wave speed.