Size-structured Population Research Articles

A nonautonomous model for the impact of toxicants on size-structured aquatic populations: Well-posedness and long-term dynamics.

Mathematical models have played a crucial role in understanding and assessing the impacts of toxicants on populations. However, many existing population-toxicant interaction models are physically unstructured and represented by autonomous systems, assuming all individuals are identical and model parameters are constant over time. In this paper, we develop a nonautonomous model describing the interaction between a size-structured population and an unstructured toxicant in a polluted aquatic ecosystem. This model allows us to investigate the influence of size- and time-dependent individual vital rates (growth, reproduction, and mortality), time-varying toxicant input and degradation, and size-specific sensitivity of individuals to toxicants on population persistence. We establish the existence and uniqueness of solutions for this model using the monotone method, based on a comparison principle. We then analyze how time- and size-dependent parameters affect the long-term population dynamics. Specifically, we derive conditions on these parameters that lead to either extinction or persistence of the population. We provide a comparative analysis of numerical solutions between our size-structured model and an unstructured model with size-averaged parameters, emphasizing the significance of incorporating size structure when evaluating the effects of toxicants on populations.

Spatiotemporal variation in population dynamics of a narrow endemic, Ranunculus austro-oreganus.

Understanding how population dynamics vary in space and time is critical for understanding the basic life history and conservation needs of a species, especially for narrow endemic species whose populations are often in similar environments and therefore at increased risk of extinction under climate change. Here, we investigated the spatial and temporal variation in population dynamics of Ranunculus austro-oreganus, a perennial buttercup endemic to fragmented prairie habitat in one county in southern Oregon. We performed demographic surveys of three populations of R. austro-oreganus over 4 years (2015-2018). We used size-structured population models and life table response experiments to investigate vital rates driving spatiotemporal variation in population growth. Overall, R. austro-oreganus had positive or stable stochastic population growth rates, though individual vital rates and overall population growth varied substantially among sites and years. All populations had their greatest growth in the same year, suggesting potential synchrony associated with climate conditions. Differences in survival contributed most to spatial variation in population growth, while differences in reproduction contributed most to temporal variation in population growth. Populations of this extremely narrow endemic appear stable, with positive growth during our study window. These results suggest that populations of R. austro-oreganus are able to persist if their habitat is not eliminated by land-use change. Nonetheless, its narrow distribution and synchronous population dynamics suggest the need for continued monitoring, particularly with ongoing habitat loss and climate change.

Impacts of estuarine habitat degradation on the modeled life history of marine estuarine-dependent and resident fish species.

Shallow coastal and estuarine habitats play an essential role in the life cycles of many fish species, providing spawning, nursery, feeding, and migration areas. However, these ecologically valuable habitats are increasingly threatened by anthropogenic activities, causing substantial changes in both habitat availability and quality. Fish species use these shallow coastal habitats and estuaries during various life stages, leading to their categorization into guilds based on how and when they rely on these areas. This differential functional use of estuaries means that changes to these habitats may affect each guild differently. To understand the impact of estuarine habitat degradation on fish populations, it is therefore necessary to consider the full life cycle of fish and when they rely on these coastal habitats. Here, we use conceptual size-structured population models to study how estuarine habitat degradation affects two functionally different guilds. We use these models to predict how reduced food productivity in the estuary affects the demographic rates and population dynamics of these groups. Specifically, we model estuarine residents, which complete their entire life cycle in estuaries, and marine estuarine-dependent species, which inhabit estuaries during early life before transitioning offshore. We find that total fish biomass for both guilds decreases with decreasing food productivity. However, the density of juveniles of the marine estuarine-dependent guild can, under certain conditions, increase in the estuary. This occurs due to a shift in the population biomass distribution over different life stages and a simultaneous shift in which life stage is most limited by food. At the individual level, somatic growth of juveniles belonging to the estuarine-dependent guild decreased with lower food supply in the estuary, due to increased competition for food. The somatic growth rates of fish belonging to the resident guild were largely unaffected by low food supply, as the total fish density decreased at the same time and therefore the per-capita food availability was similar. These outcomes challenge the assumption that responses to habitat degradation are similar between fish guilds. Our study highlights the need to assess not only fish biomass but also size distributions, survival, and somatic growth rates for a comprehensive understanding of the effects of habitat degradation on fish populations. This understanding is crucial not only for estuary fish communities but also for successful conservation and management of commercially harvested offshore population components.

On hierarchical competition through reduction of individual growth

We consider a population organised hierarchically with respect to size in such a way that the growth rate of each individual depends only on the presence of larger individuals. As a concrete example one might think of a forest, in which the incidence of light on a tree (and hence how fast it grows) is affected by shading by taller trees. The classic formulation of a model for such a size-structured population employs a first order quasi-linear partial differential equation equipped with a non-local boundary condition. However, the model can also be formulated as a delay equation, more specifically a scalar renewal equation, for the population birth rate. After discussing the well-posedness of the delay formulation, we analyse how many stationary birth rates the equation can have in terms of the functional parameters of the model. In particular we show that, under reasonable and rather general assumptions, only one stationary birth rate can exist besides the trivial one (associated to the state in which there are no individuals and the population birth rate is zero). We give conditions for this non-trivial stationary birth rate to exist and analyse its stability using the principle of linearised stability for delay equations. Finally, we relate the results to the alternative, partial differential equation formulation of the model.

Stock assessment of rock lobster stocks: Past, present and future

The aim of stock assessment is to provide managers with an understanding of the biomass and age-/size-composition of a stock, stock status (ideally relative to target and reference points), and to support management decision making, often through projections and application of harvest control rules. Stock assessments are increasingly being used to form the basis for the operating models used to evaluate the ability of management strategies to achieve the goals of fisheries management. Stock assessments for rock lobsters were among the first to adopt size-structured population models as the basis for stock assessment, and population model-based stock assessments have been used extensively to evaluate management strategies for rock lobster stocks in the USA, Australia, New Zealand and South Africa. This paper identifies how closely population model-based stock assessments for rock lobsters satisfy the guidelines identified for best practices for conducting stock assessments during meetings of, for example, the Center for the Advancement of Population Assessment Methodology and papers identifying best practice guidelines for stock assessment. The paper highlights some key directions for stock assessment science for rock lobsters, including considerations related to environmental variation and climate change.

Measure-Valued Optimal Control for Size-Structured Population Models with Diffusion

Measure-Valued Optimal Control for Size-Structured Population Models with Diffusion

A discrete two time scales model of a size-structured population of parasitized trees.

The work presented a general discrete-time model of a population of trees affected by a parasite. The tree population was considered size-structured, and the parasite was represented by a single scalar variable. Parasite dynamics were assumed to act on a faster timescale than tree dynamics. The model was studied based on an associated nonlinear matrix model, in which the presence of the parasites was only reflected in the value of its parameters. For the model in all its generality, an explicit condition of viability/extinction of the parasite/tree community was found. In a simplified model with two size-classes of trees and particular forms of the vital rates, it was shown that the model undergoes a transcritical bifurcation and, likewise, a period-doubling bifurcation. It was found that, for any tree fertility rate that makes them viable without a parasite, if the parasite sufficiently reduces the survival of young trees, it can lead to the extinction of the entire community. The same cannot be assured if the parasite acts on adult trees. In situations where a high fertility rate coupled with a low survival rate of adult trees causes a non-parasitized population of trees to fluctuate, a parasite sufficiently damaging only young trees can stabilize the population. If, instead, the parasite acts on adult trees, we can find a destabilization condition on the tree population that brings them from a stable to an oscillating regime.

Biomass and sustainable yields of Eurasian perch (Perca fluviatilis) in small boreal lakes with respect to lake properties and water quality

Understanding of factors that explain variation in potential fisheries yields is essential for the ecosystem-based management of lake fisheries. We used mark-recapture and Nordic survey net sampling to obtain estimates of Eurasian perch (Perca fluviatilis) abundance in 28 small (13.3 ha ± 11.4 ha, mean ± S.D.) boreal lakes. A size-structured population model was calibrated for each lake using further individual data to derive estimates of potential yields. Principal component scores formed from physical and chemical environmental parameters and a simple score reflecting the relative photic area of the lakes were then used to explain variation in theoretical yields to identify potential environmental proxies for fisheries management purposes. The estimated mean biomass of perch in the study lakes was 52.4 ± 51.2 kg ha−1, and the maximum sustainable yield, obtained with environment-dependent recruitment size was estimated to be 9.4 ± 12.1 kg ha−1 yr−1. The estimated yields were highest in shallow lakes with good oxygen saturation and with high percentage of euphotic bottom while excess nutrients decreased yields and alkalinity was marginally predictive for catch of large individuals. Our study provides quantitative estimates of potential yields of varyingly sized perch and helps to develop environmentally informed management for small lakes of which some are surprisingly productive.

Parasite transmission in size-structured populations.

Host heterogeneity can affect parasite transmission, but determining underlying traits and incorporating them into transmission models remains challenging. Body size is easily measured and affects numerous ecological interactions, including transmission. In the snail-schistosome system, larger snails have a higher exposure to parasites but lower susceptibility to infection per parasite. We quantified the effect of size-based heterogeneity on population-level transmission by conducting transmission trials in differently size-structured snail populations and competing size-dependent transmission models. Populations with greater proportions of large snails had lower prevalence, and small snails were shielded from infection by co-occurring large conspecifics. Furthermore, a fully dependent transmission model that incorporated body size in both exposure and susceptibility outperformed other candidate models considered. Incorporating traits such as body size, which are affected by and directly affect host ecology, into transmission models could yield insights into natural dynamics and disease mitigation in many systems.

An energetic approach to the evolution of growth curve plasticity

Growth in individual body size amongst different species can to a greater or lesser extent depend on environmental factors such as resource availability. Individual growth curves can therefore be largely fixed or more plastic. Classic theory about phenotypic plasticity assumes that such plasticity has associated costs. In contrast, according to dynamic energy budget theory, maintaining a fixed growth rate in the face of variable resource availability would incur additional energetic costs. In this article, we explore the simultaneous evolution of the degree of plasticity in individual growth curves and the rate of non-plastic, environment-independent individual growth. We explore different relations between possible additional energetic costs and the degree of growth curve plasticity. To do so, we use adaptive dynamics to analyze a size-structured population model that is based on dynamic energy budget theory to account for the energetic trade-offs within an individual. We show that simultaneous evolution of the degree of growth curve plasticity and the rate of non-plastic individual growth will drive these traits to intermediate values at first. Afterwards, the degree of growth curve plasticity might evolve slowly towards extreme values depending on whether energetic costs increase or decrease with the degree of plasticity. In addition, the analysis shows that it is unlikely to encounter species in which individual growth is entirely fixed or entirely plastic, opposing general assumptions in dynamic energy budget theory.

Marine reserves promote cycles in fish populations on ecological and evolutionary time scales.

Marine reserves are considered essential for sustainable fisheries, although their effectiveness compared to traditional fisheries management is debated. The effect of marine reserves is mostly studied on short ecological time scales, whereas fisheries-induced evolution is a well-established consequence of harvesting. Using a size-structured population model for an exploited fish population of which individuals spend their early life stages in a nursery habitat, we show that marine reserves will shift the mode of population regulation from low size-selective survival late in life to low, early-life survival due to strong resource competition. This shift promotes the occurrence of rapid ecological cycles driven by density-dependent recruitment as well as much slower evolutionary cycles driven by selection for the optimal body to leave the nursery grounds, especially with larger marine reserves. The evolutionary changes increase harvesting yields in terms of total biomass but cause disproportionately large decreases in yields of larger, adult fish. Our findings highlight the importance of carefully considering the size of marine reserves and the individual life history of fish when managing eco-evolutionary marine systems to ensure both population persistence as well as stable fisheries yields.

On the asymptotic behavior of a size-structured model arising in population dynamics

We study the asymptotic behavior of a semilinear size-structured population model withdelay when the nonlinearity is small in some sense. The novelty in this work is that theoperator governing the linear part of the equation does not generate a compact semigroupunlike in the results present in literature. In such a case the spectrum does not consist whollyof eigenvalues but also has a non-trivial component called Browder’s essential spectrum. Toovercome the lack of compactness, we give a localization of Browder’s essential spectrum of theoperator governing the linear part and we use the Perron-Frobenius spectral analysis adaptedto semigroups of positive operators in Banach lattices to investigate the long time behavior ofthe system.Keywords: Perron-Frobenius, positive operators, structured population models, Browder’sessential spectrum, asymptotic behavior, semigroups of operatorsAMS Subject Classifications: 35B40, 35R10, 47D06.

Stability results for a hierarchical size-structured population model with distributed delay

In this paper we investigate a structured population model with distributed delay. Our model incorporates two different types of nonlinearities. Specifically we assume that individual growth and mortality are affected by scramble competition, while fertility is affected by contest competition. In particular, we assume that there is a hierarchical structure in the population, which affects mating success. The dynamical behavior of the model is analysed via linearisation by means of semigroup and spectral methods. In particular, we introduce a reproduction function and use it to derive linear stability criteria for our model. Further we present numerical simulations to underpin the stability results we obtained.

Does the Evolution of Ontogenetic Niche Shifts Favor Species Coexistence? An empirical test in Trinidadian streams.

A major question in ecology is how often competing species evolve to reduce competitive interactions and facilitate coexistence. One untested route for a reduction in competitive interactions is through ontogenetic changes in the trophic niche of one or more of the interacting species. In such cases, theory predicts that two species can coexist if the weaker competitor changes its resource niche to a greater degree with increased body size than the superior competitor. We tested this prediction using stable isotopes that yield information about the trophic position (δ15 N) and carbon source (δ13 C) of two coexisting fish species: Trinidadian guppies (Poecilia reticulata) and killifish (Rivulus hartii). We examined fish from locations representing three natural community types: 1) where killifish and guppies live with predators; 2) where killifish and guppies live without predators; and 3) where killifish are the only fish species. We also examined killifish from communities in which we had introduced guppies, providing a temporal sequence of the community changes following the transition from a killifish only to a killifish-guppy community. We found that killifish, which are the weaker competitor, had a much larger ontogenetic niche shift in trophic position than guppies in the community where competition is most intense (killifish-guppy only). This result is consistent with theory for size-structured populations, which predicts that these results should lead to stable coexistence of the two species. Comparisons with other communities containing guppies, killifish and predators and ones where killifish live by themselves revealed that these results are caused primarily by a loss of ontogenetic niche changes in guppies, even though they are the stronger competitor. Comparisons of these natural communities with communities in which guppies were translocated into sites containing only killifish showed that the experimental communities were intermediate between the natural killifish-guppy community and the killifish-guppy-predator community, suggesting contemporary evolution in these ontogenetic trophic differences. These results provide comparative evidence for ontogenetic niche shifts in contributing to species coexistence and comparative and experimental evidence for evolutionary or plastic changes in ontogenetic niche shifts following the formation of new communities.

Diffusive size-structured population model with time-varying diffusion rate

<p style='text-indent:20px;'>This work is devoted to the study of a size-structured population model with a time-varying diffusion rate. Due to the seasonal variation, it is natural to consider the time-varying diffusion rate. Moreover, introducing the time-varying diffusion rate makes the model more challenging and requires the results of evolution operators for the analysis. Under some assumptions on the time-dependent diffusion coefficient, the existence and uniqueness of mild solution is shown. We apply the method of characteristics and evolution operators to derive our results. Positivity and boundedness of mild solution is also shown. Some examples are provided to illustrate the theoretical findings. We also solve our model numerically to study the size and spatial dynamics of population density.</p>

Analysis of diffusive size-structured population model and optimal birth control

<p style='text-indent:20px;'>This work addresses the optimal birth control problem for invasive species in a spatial environment. We apply the method of semigroups to qualitatively analyze a size-structured population model in which individuals occupy a position in a spatial environment. With insect population in mind, we study the optimal control problem which takes fertility rate as a control variable. With the help of adjoint system, we derive optimality conditions. We obtain the optimality conditions by fixing the birth rate on three different sets. Using Ekeland's variational principle, the existence, and uniqueness of optimal birth controller to the given population model which minimizes a given cost functional is shown. A concrete example is also given to see the behaviour of population density. Outcomes of our article are new and complement the existing ones.</p>

Bifurcation Analysis of a Size-Structured Population Model

In this paper, a size-structured population model is studied. Choosing the coefficient of birth function as the varying parameter, we prove the existence of Hopf bifurcation and discuss the bifurcation properties, such as the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. The methods used in this paper include Hopf bifurcation theorem, center manifold theorem and normal form theory for the abstract Cauchy problems in a nondense domain. Numerical simulations are finally carried out to show the theoretical results.

The existence and uniqueness of solutions of a nonlinear toxin-dependent size-structured population model

In this paper, we study a toxin-mediated size-structured population model with nonlinear reproduction, growth, and mortality rates. By using the characteristic method and the contraction mapping argument, we establish the existence-uniqueness of solutions to the model. We also prove the continuous dependence of solutions on initial conditions.

Analysis of diffusive size-structured population model with stochastic perturbation

In this work, we introduce a diffusive stochastic size-struc\-tured population model. After formulating the problem in an abstract framework, we derive mild solution to the given model. We show the existence and uniqueness of the mild solution to the given model in a Hilbert space. We use the Burkholder-Davis-Gundy's lemma to prove the uniqueness of the mild solution. We also show that the system is quasi invariant in the second moment and also mean square dissipative.

Dynamical analysis on a size-structured population model of Daphnia with delayed birth process

In this paper, we investigate a size-structured population model of Daphnia coupled with an unstructured algal food source. We introduce a delay into the boundary condition of this model, where the delay describes the effect of competition for finding food or resource in short supply juvenile period. Following the semigroup theory, we transform our model into the abstract boundary delay problem and obtain the existence and uniqueness of a positive stationary solution. By means of the spectral analysis and the characteristic equation technique, the instability and linear stability results of stationary solutions are formulated in terms of the basic reproduction number ℛ(F). Some numerical simulation examples are presented to illustrate the feasibility of our main results.