Stage-structured Population Dynamics Research Articles

Stability and Bifurcation Analysis of a Spatially Size–Stage-Structured Model with Resting Phase

In this paper, we consider a spatially size–stage-structured population dynamics model with resting phase. The primary objective of this model is to study size structure, stage structure, resting phase and spatial location simultaneously in a single population system. At first, we reformulate the problem as an abstract nondensely defined Cauchy problem. Then, taking advantage of the integrated semigroup and bifurcation theories, we investigate the stability and Hopf bifurcation at the positive equilibrium of the model. Finally, numerical simulations are presented as evidence to support our analytical results. A discussion of related problems is also presented briefly.

A nonlinear model for stage-structured population dynamics with nonlocal density-dependent regulation: An application to the fall armyworm moth

The assessment and the management of the risks linked to insect pests can be supported by the use of physiologically-based demographic models. These models are useful in population ecology to simulate the dynamics of stage-structured populations, by means of functions (e.g., development, mortality and fecundity rate functions) realistically representing the nonlinear individuals physiological responses to environmental forcing variables. Since density-dependent responses are important regulating factors in population dynamics, we propose a nonlinear physiologically-based Kolmogorov model describing the dynamics of a stage-structured population in which a time-dependent mortality rate is coupled with a nonlocal density-dependent term. We prove existence and uniqueness of the solution for this resulting highly nonlinear partial differential equation. Then, the equation is discretized by finite volumes in space and semi-implicit backward Euler scheme in time. The model is applied for simulating the population dynamics of the fall armyworm moth (Spodoptera frugiperda), a highly invasive pest threatening agriculture worldwide.

A temperature-dependent mathematical model of malaria transmission with stage-structured mosquito population dynamics

Abstract In this paper, we formulate a temperature-dependent model for malaria transmission dynamics which includes immature stages of mosquitoes. The model is constructed by using ordinary differential equations with some parameters which are periodic functions. Two thresholds dynamics associated to the model have been derived: the vector reproduction ratio ℛ v and the basic reproduction ratio ℛ0. Through a rigorous analysis via theories and methods of dynamical systems, we prove that the global behavior of the model depends strongly on these two parameters. More precisely, we show that if ℛ v is greater than one and ℛ0 is less than one then, the disease-free periodic equilibrium is globally attractive. If ℛ v is greater than one and ℛ0 is greater than one, the disease remains persistent and the system admits at least one positive periodic solution. Finally, using the reported monthly mean temperature for Burkina Faso, numerical simulations are carried out to illustrate our mathematical results.

The mango tree – blossom gall midge system: toward in silico assessment of management practices

Mango (Mangifera indica L.), a major fruit production in tropical and subtropical regions, is facing many production constraints. Mango yield is irregular across years, fruit quality is heterogeneous at harvest, and mango tree exhibits phenological asynchronisms within and between trees that result in long periods with phenological stages susceptible to pests and diseases. Among them, the mango blossom gall midge (BGM, Procontarinia mangiferae Felt) is a major pest of mango which can cause significant yield losses by damaging mango inflorescences. A mango-BGM model is currently developed from experimental data. Its final objective will be in silico assessment of BGM management levers (e.g., manipulation of mango phenology to synchronize flowering and soil mulching during flowering, used as physical barrier to break the BGM life-cycle). The mango-BGM model simulates the dynamics of inflorescence and BGM populations of an orchard at a daily time-step during the period of mango flowering. The orchard is structured into three zones (or patches) according to applied mulching treatments. In a first approach, the model is defined at the patch scale and considers: i) age-structured inflorescence population dynamics within each patch, accounting for natural development and BGM-induced mortality of inflorescences; ii) stage-structured BGM population dynamics within each patch, differentiating the effect of mulching treatments on the BGM life-cycle for each patch; and iii) orchard colonization and movements of BGM adults between patches, both driven by inflorescence abundance in each patch. The model was parameterized from existing knowledge and experimental data collected in Reunion Island on 'Cogshall' cultivar. In a further step, virtual experiments will be performed with the model to assess the effects of management levers on mango flowering and subsequent fruit yield, according to exogenous pest pressure. The on-going modeling approach and preliminary results are presented and discussed.

Disappointing Survival Forecast for a Local Population of Androsace albana in a Random Environment

A local population of Andropace albana, a short-lived perennial plant, has been monitored during 10 years on permanent plots laid down in an alpine lichen heath in 2009. We summarize the outcome of monitoring as a non-autonomous matrix model of stage-structured population dynamics. The model originates from a life cycle graph constructed earlier for the stages of ontogenesis and consists of 9 annual “projection” matrices that are calibrated in a unique way from the observation data. Five of the 9 matrices have their dominant eigenvalues greater than 1, i.e., give favorable forecasts for the local population survival, while the rest four have those values less than 1, i.e., give the negative forecasts. To make the resulting prediction, we apply an original concept of the pattern-geometric averaging of given nonnegative matrices and obtain the dominant eigenvalue, λ1(G9), of the average matrix G9 markedly less than 1, indicating the population decline in the long term. The traditional method to forecast the local population is to estimate λS, the stochastic growth rate of the population in a random environment formed by a random choice from the same 9 annual matrices. Assuming the choice to be independent and equiprobable, we obtain the negative result as well, yet with higher quantitative values of λS. We associate these higher values with the artificial assumption of equal choice probability when forming the random sequence of annual matrices, each of which is indirectly reflecting the habitat conditions that have influenced the growth and development of plants during the year prior the calibration moment. This motivates the task to construct a more adequate model for choosing annual matrices, in which the probability of choice would be related to the dynamics of the real habitat factors for a given local population.

Complex interaction of resource availability, life-history and demography determines the dynamics and stability of stage-structured populations

The dynamics of stage-structured populations facing stage-specific variability in resource availability and/or demographic factors like unequal sex-ratios, remains poorly understood. We addressed these issues using a stage-structured individual-based model that incorporates life-history parameters common to many holometabolous insects. The model was calibrated using time series data from a 49-generation experiment on laboratory populations of Drosophila melanogaster, subjected to four different combinations of larval and adult nutritional levels. The model was able to capture multiple qualitative and quantitative aspects of the empirical time series across three independent studies. We then simulated the model to explore the interaction of various life-history parameters and nutritional levels in determining population stability. In all nutritional regimes, constancy stability of the populations was reduced upon increasing egg-hatchability, critical mass, and proportion of body resource allocated to female fecundity. However, the effects of increasing sensitivity of female-fecundity to adult density on constancy stability varied across nutrition regimes. The effects of unequal sex-ratio and sex-specific culling were greatly influenced by fecundity but not by levels of juvenile nutrition. Finally, we investigated the implications of some of these insights on the efficiency of the widely-used pest control method, the Sterile Insect Technique (SIT). We show that increasing the amount of juvenile food had no effects on SIT efficiency when the density-independent fecundity is low, but reduces SIT efficiency when the density-independent fecundity is high.

Modelling the effects of spatial heterogeneity and temporal variation in extinction probability on mosquito populations.

Spatial synchrony plays an important role in dictating the dynamics of spatial and stage-structured populations. Here we argue that, unlike the Moran effect where spatial synchrony is driven by exogenous factors, spatial correlation in intrinsic/local-scale processes can affect the level of spatial synchrony among distinct sub-populations, and therefore the persistence of the entire population. To explore this mechanism, we modelled the consequences of spatial heterogeneity in aquatic habitat quality, and that of temporal variation in local extinction probability, on the persistence of stage-structured mosquito populations. As a model system, we used two widely distributed mosquito species, Aedes albopictus and Culex pipiens, both key vectors of a range of infectious diseases. Spatial heterogeneity in aquatic habitat quality led to increased population persistence, and this pattern was more pronounced at intermediate dispersal rates, and in the long-dispersing species (C.pipiens). The highest regional persistence was obtained at high dispersal rates. This is probably because dispersal, in our model, did not carry any additional costs. Population persistence of both species was negatively correlated with increased temporal variation in local extinction probability. These differences were stronger in the short-dispersing species (A.albopictus), especially at intermediate dispersal rates. The dispersal of A.albopictus adults in each time step was limited to the nearest habitat patches, weakening the positive effect of spatial heterogeneity in aquatic habitat quality on population persistence. In contrast, C.pipiens adults could disperse into more remote sub-populations, resulting in much higher recolonization rates. Hence, the negative effect of temporal variation in local extinction probability on patch occupancy disappeared at intermediate dispersal rates. We suggest that effectively controlling these two mosquito species requires making few spatially synchronized control efforts (i.e., generating high temporal variation in local extinction probability), rather than many asynchronized local control efforts. Finally, our model can be easily fitted to other organisms characterized by complex life cycles, and it can be also used to examine alternative scenarios, including the effect of spatial configuration of local habitat patches and dispersal kernel shape on population persistence.

Local population of Eritrichium caucasicum as an object of mathematical modelling. I. Life cycle graph and a nonautonomous matrix model

For the plant species, which is considered a short-lived perennial, we have composed a scale of ontogenetic stages and the life cycle graph (LCG) according to annual observations on permanent sample plots in an Alpine lichen heathland during the 2009–2014 period. The LCG that reflects seed reproduction has been reduced to the one that avoids the stage of soil seed bank, yet preserves the arcs of annual recruitment. The corresponding matrix model of stage-structured population dynamics has four stages: juvenile plants (including seedlings), vegetative, generative, and ‘terminally generative’ (the plants die after seed production). Model calibration reduces to directly calculating the rates of transition between stages and those of delays within stages from the data of only one time step, while keeping the two reproduction rates uncertain, yet confined to the quantitative bounds of observed recruitment. This has enabled us to determine a feasible range for the dominant eigenvalue of the model matrix, i.e., the quantitative bounds for the measure of how the local population adapts to its environment, at each of the five time steps, resulting in a formally nonautonomous model. To obtain “age-specific parameters” from a stage-classified model, we have applied the technique that constructs a virtual absorbing Markov chain and calculates its fundamental matrix. In a nonautonomous model, the estimates of life expectancy also depend on the time of observation (that fixes certain environmental conditions), and vary from two to nearly seven years. The estimates reveal how specifically short lives the short-lived perennial, while their range motivates the task to average the model matrices over the whole period of observation. The model indicates that Eritrichium caucasicum plants spend the most part of their life span in the adult vegetative stage under each of the environment conditions observed, thus revealing the space holder strategy by C. Korner (2003), or the delayed-development strategy by L.A. Zhukova (1995). We discuss the prospects of model experiments with a logically nonautonomous model to forecast the long-term dynamics of E. caucasicum, should a scenario of climate changes be given.

Evolution of Thermal Reaction Norms in Seasonally Varying Environments.

Thermal reaction norms of ectotherms exhibit a distinctive latitudinal pattern: the temperature at which performance is maximized coincides with the mean habitat temperature in tropical ectotherms but exceeds the mean temperature in temperate ectotherms. We hypothesize, on the basis of Jensen's inequality, that this pattern is driven by latitudinal variation in seasonal temperature fluctuations. We test this hypothesis with an eco-evolutionary model that integrates the quantitative genetics of reaction norm evolution with stage-structured population dynamics, which we parameterize with data from insects. We find that thermal optima of temperate and Mediterranean species evolve to exceed the mean habitat temperature if seasonal fluctuations are strong, while the thermal optimum of tropical species evolves to coincide with the mean habitat temperature if fluctuations are weak. Importantly, ecological dynamics can impose a constraint on reaction norm evolution. Tropical species cannot tolerate an increase in seasonal fluctuations at the high mean habitat temperature it experiences, while the temperate species cannot tolerate a reduction in seasonal fluctuations if the mean temperature is higher. In both cases, stochastic extinction during periods of low abundances precludes adaptation to a novel thermal environment. Our findings suggest a potential directionality in colonization success. Tropical ectotherms, because of their high thermal optima, can successfully colonize temperate habitats, while temperate ectotherms, because of their low optima, are less successful in colonizing tropical habitats.

Analyse d'un modèle décrivant la dynamique d'une population structurée en stade utilisant le jeu faucon-colombe

The purpose of this paper is to investigate the effects of conflicting tactics of resource acquisition on stage structured population dynamics. We present a population subdivided into two distinct stages (immature and mature). We assume that immature individual survival is density dependent. We also assume that mature individuals acquire resources required to survive and reproduce by using two contrasted behavioral tactics (hawk versus dove). Mature individual survival thus is assumed to depend on the average cost of fights while individual fecundity depends on the average gain in the competition to access the resource. Our model includes two parts: a fast part that describes the encounters and fights involves a game dynamic model based upon the replicator equations, and a slow part that describes the long-term effects of conflicting tactics on the population dynamics. The existence of two time scales let us investigate the complete system from a reduced one, which describes the dynamics of the total immature and mature densities at the slow time scale. Our analysis shows that an increase in resource value may decrease total population density, because it promotes individual (i.e. selfish) behavior. Our results may therefore find practical implications in animal conservation or biological control for instance. Le but de cet article est d'étudier les effets de comportements agressifs dans l'acquisition de ressources sur la dynamique d'une population structurée en deux classes d'âge. Nous considérons une population divisée en deux sous populations distinctes (immatures et adultes matures). Nous supposons que la survie individuelle de la population immature est densité dépendante. Nous supposons également que les individus matures sont en compétition pour acquérir les ressources nécessaires pour leur survie et leur reproduction. Les adultes utilisent deux tactiques comportementales (faucon et colombe). Lors de confrontations entre adultes, la survie d'un individu mature est supposée être affectée par le coût moyen des combats alors que la fécondité dépend du gain moyen obtenu en accédant à la ressource. Notre modèle comprend deux parties : une partie rapide qui décrit les rencontres et les combats basée sur les équations du réplicateur, et une partie lente qui décrit les effets à long terme des tactiques conflictuelles sur la dynamique de la population. L'existence de deux échelles de temps nous permet d'étudier le système complet à partir d'un système réduit, qui décrit la dynamique des densités totales des immatures et des adultes à l'échelle de temps lente. Notre analyse montre que le taux de croissance global de la population dépend de la valeur de la ressource et du coût des combats entre adultes. Nos résultats trouvent des implications pratiques dans la conservation des espèces et le contrôle biologique.

Estimating growth within size-structured fishery stock assessments: What is the state of the art and what does the future look like?

Most growth studies have considered the relationship between age and growth. Such relationships are essential to age-structured fishery stock assessments. In contrast, assessments based on stage-structured population dynamics models require information on the probability of animals moving from one stage to each of the other stages at each time-step. Size-structured population dynamics models are a special case of stage-structured population dynamics models in which each class represents a unique set of sizes. These models form the basis of assessments for many valuable, hard to age, marine species, including crabs, abalone, lobsters and prawns. Growth within these models is governed by a size-transition matrix. The values for the parameters of a size-transition matrix can either be estimated externally to the assessment, generally utilizing data from tag-recapture experiments, or the estimation of growth can be integrated within the assessment model. This paper reviews the approaches used to construct size-transition matrices, including the underlying structural formulation, statistical estimation framework, and the consequences of error when specifying these matrices on the ability to estimate population size and manage populations sustainably.

Projection matrices revisited: a potential-growth indicator and the merit of indication

The mathematics of matrix models for age- and/or stage-structured population dynamics substantiates the use of the dominant eigenvalue λ 1 of the projection matrix L as a measure of the growth potential, or of adaptation, for a given species population in modern plant or animal demography. The calibration of L = T +F on the “identified-individuals-of-unknown-parents” kind of empirical data determines precisely the transition matrix T, but admits arbitrariness in the estimation of the fertility matrix F. We propose an adaptation principle that reduces calibration to the maximization of λ 1(L) under the fixed T and constraints on F ensuing from the data and expert knowledge. A theorem has been proved on the existence and uniqueness of the maximizing solution for projection matrices of a general pattern. A conjugated maximization problem for a “potential-growth indicator” under the same constraints has appeared to be a linear-programming problem with a ready solution, the solution testing whether the data and knowledge are compatible with the population growth observed.

Complex Transient Dynamics of Stage-Structured Populations in Response to Environmental Changes

Stage structures of populations can have a profound influence on their dynamics. However, not much is known about the transient dynamics that follow a disturbance in such systems. Here we combined chemostat experiments with dynamical modeling to study the response of the phytoplankton species Chlorella vulgaris to press perturbations. From an initially stable steady state, we altered either the concentration or dilution rate of a growth-limiting resource. This disturbance induced a complex transient response-characterized by the possible onset of oscillations-before population numbers relaxed to a new steady state. Thus, cell numbers could initially change in the opposite direction of the long-term change. We present quantitative indexes to characterize the transients and to show that the dynamic response is dependent on the degree of synchronization among life stages, which itself depends on the state of the population before perturbation. That is, we show how identical future steady states can be approached via different transients depending on the initial population structure. Our experimental results are supported by a size-structured model that accounts for interplay between cell-cycle and population-level processes and that includes resource-dependent variability in cell size. Our results should be relevant to other populations with a stage structure including organisms of higher order.

Complexity in matrix population models: Polyvariant ontogeny and reproductive uncertainty

Linear matrix models of stage-structured population dynamics are widely used in plant and animal demography as a tool to evaluate the growth potential of a population in a given environment. The potential is identified with λ1, the dominant eigenvalue of the projection matrix, which is compiled of stage-specific transition and fertility rates. Advanced botanical studies reveal polyvariant ontogeny in perennial plants, i.e., multiple different versions of individual development within a local population of a single species. This phenomenon complicates any standard, successive-stage, life cycle graph to a digraph defined on a 2D lattice in the age and stage dimensions, the pattern of projection matrix becoming more complex too. In a kind of experimental design, the transition rates can be calculated directly from the data for two successive time moments, but the age-stage-specific rates of reproduction still remain uncertain, adding more complexity to the calibration problem. Simple additional assumptions could technically eliminate the uncertainty, but they contravene the biology of a species in which polyvariant ontogeny is considered to be the major mechanism of adaptation. Given the data and expert constraints, the calibration can be reduced instead to a nonlinear maximization problem, yet with linear constraints. I prove that it has a unique solution to be attained at a vertex of the constraint polyhedral. To facilitate searching for the solution in practice, I use the net reproductive rate R0, a well-known indicator for the principal property of λ1 to be greater or less than 1. The method is exemplified with the calibration of a projection matrix in an age-stage-structured model (published elsewhere) for Calamagrostis canescens, a perennial herbaceous species with a complex (multivariant) life cycle that features unlimited growth when colonizing open areas.

Identifying non-invasible habitats for marine copepods using temperature-dependent R 0

If a non-indigenous species is to thrive and become invasive it must first persist under its new set of environmental conditions. Net reproductive rate (R 0) represents the average number of female offspring produced by a female over its lifetime, and has been used as a metric of population persistence. We modeled R 0 as a function of ambient water temperature (T) for the invasive marine calanoid copepod Pseudodiaptomus marinus, which is introduced to west coast of North America from East Asia by ship ballast water. The model was based on temperature-dependent stage-structured population dynamics given by a system of ordinary differential equations. We proposed a methodology to identify habitats that are non-invasible for P. marinus using the threshold of R 0(T) < 1 in order to identify potentially invasible habitats. We parameterized the model using published data on P. marinus and applied R 0(T) to identify the range of non-invasible habitats in a global scale based on sea surface temperature data. The model predictions matched the field evidence of species occurrences well.

Nonnegative matrices as a tool to model population dynamics: Classical models and contemporary expansions

Matrix models of age-and/or stage-structured population dynamics rest upon the Perron-Frobenius theorem for nonnegative matrices, and the life cycle graph for individuals of a given biological species plays a major role in model construction and analysis. A summary of classical results in the theory of matrix models for population dynamics is presented, and generalizations are proposed, which have been motivated by a need to account for an additional structure, i.e., to classify individuals not only by age, but also by an additional (discrete) characteristic: size, physiological status, stage of development, etc.

Convexity in projection matrices: Projection to a calibration problem

Convexity, as a fundamental property of sets and functions defined on convex sets, plays an important role in many mathematical and applied disciplines, including extremal and optimal-control problems. We prove the set of all feasible projection matrices in a general class of matrix models for stage-structured population dynamics to be convex and the dominant eigenvalue ( λ 1) of any projection 2 × 2 matrix to be either a convex, or a concave function on a simplex of the matrix first-row entries (i.e., stage-specific reproduction rates). The latter is also conjectured for the general n × n case. Though looking far from practical needs of matrix population models, this mathematical result has appeared to be quite useful in solving a practical problem to calibrate the projection matrix, i.e., to estimate all the stage-specific vital rates, from empirical data. The data from monitoring of individual life histories of marked plants on permanent sample plots during successive years enable direct calculation of the stage-specific survival and ontogenetic transition rates, but the rates of reproduction do remain uncertain as far as the parent plants can hardly be determined for the (not yet marked!) recruitment. To overcome this insurmountable uncertainty in data and to conclude the model calibration, we use a general principle that was proposed by Yuri M. Svirezhev in late 70s of the 20th century and applied by his students and followers in a number of applications, namely, to substitute an adaptation conjecture (“empirical generalization”) for the lack of data/information. We assume the unobservable stage-specific reproduction rates to be such that they furnish the maximum value to λ 1 under certain constraints which ensue from available data and knowledge. The calibration problem hereafter reduces to constraint maximization of λ 1 over a bounded convex set of unknown variables (stage-specific reproduction rates), the problem having a unique solution. This solution appears to differ from those proposed by a software package for ecological modelling (RAMAS ® EcoLab), which were based on artificial assumptions of uniformity or singularity in the distribution of the reproduction rates among the stages. Expert knowledge, on the contrary, suggests a certain hierarchy among the rates, and whether the hierarchy admits the population growth can be tested vs. data by means of the former results concerning the potential-growth indicator function.

Use of individual-based models for population parameters estimation

A method for parameters estimation of stage-specific mortality and fecundity rate functions in poikilotherm organisms, and in particular for arthropod structured population, is proposed. The application of this method requires three types of information: stage-frequency data of a sampled population, development rate function and time evolution of forcing variables affecting the rate functions. By means of an individual-based model (a microscopic model) the number of eggs produced by the adults is generated starting from the number of individuals collected at each sampling time. Using a compartmental model (a macroscopic model) a stage-structured population dynamics is described and compared with observations. Non-linear regression methods based on least square principle are used to estimate the optimal parameters of the mortality and fecundity rate functions combining microscopic and macroscopic models. As a case study, the parameter estimation of the temperature-dependent mortality function of olives fruit fly Bactrocera oleae is presented.

Modelling Pseudocalanus elongatus stage-structured population dynamics embedded in a water column ecosystem model for the northern North Sea

This paper outlines an approach to couple a structured zooplankton population model with state variables for eggs, nauplii, two copepodites stages and adults adapted to Pseudocalanus elongatus into the complex marine ecosystem model ECOHAM2 with 13 state variables resolving the carbon and nitrogen cycle. Different temperature and food scenarios derived from laboratory culture studies were examined to improve the process parameterisation for copepod stage dependent development processes. To study annual cycles under realistic weather and hydrographic conditions, the coupled ecosystem–zooplankton model is applied to a water column in the northern North Sea. The main ecosystem state variables were validated against observed monthly mean values. Then vertical profiles of selected state variables were compared to the physical forcing to study differences between zooplankton as one biomass state variable or partitioned into five population state variables. Simulated generation times are more affected by temperature than food conditions except during the spring phytoplankton bloom. Up to six generations within the annual cycle can be discerned in the simulation.

Robustness: Predicting the effects of life history perturbations on stage-structured population dynamics

Matrix-based models lie at the core of many applications across the physical, engineering and life sciences. In ecology, matrix models arise naturally via population projection matrices (PPM). The eigendata of PPMs provide detailed quantitative and qualitative information on the dynamic behaviour of model populations, especially their asymptotic rates of growth or decline. A fundamental task in modern ecology is to assess the effect that perturbations to life-cycle transition rates of individuals have on such eigendata. The prevailing assessment tools in ecological applications of PPMs are direct matrix simulations of eigendata and linearised extrapolations to the typically non-linear relationship between perturbation magnitude and the resulting matrix eigenvalues. In recent years, mathematical systems theory has developed an analytical framework, called ‘Robustness Analysis and Robust Control’, encompassing also algorithms and numerical tools. This framework provides a systematic and precise approach to studying perturbations and uncertainty in systems represented by matrices. Here we lay down the foundations and concepts for a ‘robustness’ inspired approach to predictive analyses in population ecology. We treat a number of application-specific perturbation problems and show how they can be formulated and analysed using these robustness methodologies.