Size-structured Matrix Model Research Articles

Optimizing high-dimensional stochastic forestry via reinforcement learning

In proceeding beyond the generic optimal rotation model, forest economic research has applied various specifications that aim to circumvent the problems of high dimensionality. We specify an age- and size-structured mixed-species optimal harvesting model with binary variables for harvest timing, stochastic stand growth, and stochastic prices. Reinforcement learning allows solving this high-dimensional model without simplifications. In addition to presenting new features in reservation price schedules and effects of stochasticity, our setup allows evaluating the simplifications in the existing research. We find that one- or two-dimensional models lose a high fraction of attainable economic output while the commonly applied size-structured matrix model overestimates economic profitability, yields deviations in harvest timing, including optimal rotation, and dilutes the effects of stochasticity. Reinforcement learning is found to be an efficient and promising method for detailed age- and size-structured optimization models in resource economics.

A multispecies size-structured matrix model incorporating seasonal dynamics

Multi-species size spectrum models (MSSMs) have been widely used to investigate and understand the dynamics of marine communities impacted by fishing and environmental changes to support ecosystem-based fisheries management. The continuous nature of the modelled processes makes it challenging to incorporate periodic biological processes and discontinuous life-history traits into MSSMs; therefore, a discrete multi-species model is needed. We developed a new size-structured matrix model with discrete processes to describe multi-species interactions and energy flows through predation, reproduction, metabolism, and mortality in matrix forms. A framework for assessing the population-level consequences of capital and income breeding strategies was developed, with seasonal properties. Preliminary investigations were conducted on a theoretical community comprising eight interacting species with different reproductive strategies. The utility of our model was demonstrated by showing emergent properties in the seasonal dynamics of marine communities and life-history traits such as survival, growth, and reproduction of capital and income breeders. The model enabled exploration of population dynamics caused by migration at the ecosystem level. An example application of the model in marine protected areas (MPAs), where species undertook seasonal spawning migrations, indicated that the size of MPAs may affect their potential conservation and economic benefits to fisheries. This model has the potential to unravel the relationships between drivers and seasonal dynamics and to assess the effectiveness of fisheries management strategies such as seasonal closure of fishing.

Optimal Harvesting Decision Paths When Timber and Water Have an Economic Value in Uneven Forests

We developed an uneven-aged forest economic decision-making framework that combines: (i) a size-structured matrix model, based on growth and mortality predictions of a dynamic process-based forest landscape model, (ii) an optimal control model that determines the dynamics of control and state variables, which in turn are defined by tree harvesting and forest stock, respectively, and (iii) a water yield function that depends on changes in the leaf area index (LAI), the latter being affected by forest management. This framework was used to simulate the effects of economic-driven harvesting decisions on water yields on a catchment of South-Western Swiss Alps when both timber and water benefits are considered. Water benefits are estimated as environmental prices considering current water demands for drinking, irrigation and hydropower production. We simulated optimal harvesting decisions given the initial forest structure at each 200 m × 200 m grid cells, a set of restrictions to harvesting, and specific species survival, recruitment and growth probabilities, all of which are affected by the stand’s LAI. We applied this model using different harvesting restriction levels over a period of 20 to 40-years, and accounting for single and joint timber and water benefits. The results suggested that at the environmental prices estimated at the catchment area, water benefits have a slight influence on harvesting decisions, but when water is accounted for, harvesting decisions would include more tree species and different diameter classes, which, in principle, is expected to favor more diverse forest structures.

A size-structured matrix model to simulate dynamics of marine community size spectrum.

Several types of size-based models have been developed to model the size spectra of marine communities, in which abundance scales strongly with body size, using continuous differential equations. In this study, we develop a size-structured matrix model, which can be seen as a discretization of the Mckendrick-von Foerster equation, to simulate the dynamics of marine communities. The developed model uses a series of simple body size power functions to describe the basic processes of predator–prey interactions, reproduction, metabolism, and non-predation mortality based on the principle of mass balance. Linear size spectra with slopes of approximately –1 are successfully reproduced by this model. Several examples of numerical simulations are provided to demonstrate the model’s behavior. Size spectra with cut-offs and/or waves are also found for certain parameter values. The matrix model is flexible and can be freely manipulated by users to answer different questions and is executed over a shorter numerical calculation running time than continuous models.

Colonization dynamics of a clonal pioneer plant on a glacier foreland inferred from spatially explicit and size-structured matrix models

The regional distribution of a plant species is a result of the dynamics of extinctions and colonizations in suitable habitats, especially in strongly fragmented landscapes. Here, we studied the role of spatial dynamics of the long-lived, clonal pioneer plant Geum reptans occurring on glacier forelands in the European Alps. We used demographic data from several years and sites in the Swiss Alps in combination with dispersal data to parametrize a matrix model for G. reptans to simulate extinctions, colonizations and spatial spread of established populations on glacial forelands. We used different scenarios with varying germination rates, wind and animal dispersal capabilities, and modes of spatial spread (seed-only vs clonal spread), resulting in population growth rates (λ) ranging from 1.04 to 1.20. Our results suggest that due to the low germination rate (~1%) and the very limited wind dispersal distances (99.8% of seeds are dispersed < 5 m), G. reptans has a low probability of establishing new populations and a very low spatial spread by seed dispersal alone. In contrast to the low rate of establishment, the persistence of established populations is high and even populations of only a few individuals have an extinction probability of less than 25% within 100 years. This high persistency is partly due to clonal reproduction via aboveground stolons. Clonal reproduction increases the population size and contributes considerably to the spatial spread of established populations. Our simulation results together with the known pattern of molecular diversity of G. reptans indicate that the occurrence of populations of this species in the Alps is unlikely to be a result of recent colonizations by long-distance dispersal, but rather a result of post-glacial colonizations by large migrating populations that were fragmented when glaciers retreated. Additionally, our simulations suggest that the currently observed high rates of glacial retreat might be too fast for pioneer plants, such as G. reptans, to keep up with the retreating ice and therefore might threaten existing populations.

Trade-offs between growth and maturation: the cost of reproduction for surviving environmental extremes.

Life-history trade-offs and the costs of reproduction are central concepts in evolution and ecology. Episodic climatic events such as drought and extreme temperatures provide strong selective pressures that can change the balance of these costs and trade-offs. We used size-structured matrix models parameterized from field and laboratory studies to examine the effect of periodic drought on two species of aquatic salamanders (greater siren, Siren lacertina; lesser siren, Siren intermedia) that differ in size at reproduction and maximum body size. Post-drought body size distributions of the larger species (S. lacertina) are consistent with size-dependent mortality. Smaller individuals were extirpated from the population during each drought while large animals persisted, a pattern that contrasted with that seen in several ectotherms. This appears to be largely explained by estivation proficiency and a positive relationship between body size and estivation potential. Increased body size, however, may come at the cost of fecundity and maturation rate compared to a closely related congener. The cost of somatic allocation in this case may manifest itself via reduced per-capita competitive ability, which (at least in simulation studies) allows the smaller, fast-maturing species to outcompete the larger, slow-maturing species when drought is minimal or nonexistent.

Economics of harvesting uneven-aged forest stands in Fennoscandia

This study analyzes the optimal harvesting of uneven-aged Norway spruce (Picea abies (L.) Karst.), Scots pine (Pinus sylvestris L.), and birch (Betula pendula Roth. and B. pubescens Ehrh.) stands. The analysis is based on an economic description of uneven-aged forestry using a size-structured transition matrix model and a single-tree model. The optimization problem is solved in its general dynamic form using gradient-based interior point methods. Increasing the harvesting interval decreases the annual volume yield. Assuming natural regeneration, this suggests that volume yield is maximized by uneven-aged rather than even-aged management. The present value of stumpage revenues is maximized after saw timber and pulpwood prices, interest rate, and a 15-year harvesting interval are included. The economically optimal solution with a 3% interest rate produces an annual yield of 1.9, 6.2, and 3.1 cubic meters for Scots pine, Norway spruce, and birch, respectively. Both the optimal volume yield and net present value maximization solutions converge to unique species- and site-type-specific steady states with constant harvests. The transition matrix model typically used in optimization studies is computationally less demanding than the single-tree model, but the differences in optimal solutions are more remarkable than earlier studies suggest.

The impact of seed extraction on the population dynamics of Pinus maximartinezii

Pinus maximartinezii is a rare, endemic, threatened species known from a single small population in the state of Zacatecas, Mexico. Among the pine species that produce edible nuts, it produces one of the largest and most nutritious seeds. The seeds of P. maximartinezii have historically been used for human consumption. The cones are harvested directly from the trees, and the seeds are sold illegally in local, national and international markets. However, the effects of seed extraction must be thoroughly evaluated to determine the potential impacts on population stability. To assess the impact of different rates of seed harvesting on the demography of this species, a 2-yr study of population dynamics was conducted in three 0.1-ha plots. A 9 × 9 size-structured matrix model was used to simulate changes in population growth over time in conjunction with increasing stepwise reductions in fecundity. The population growth rate (λ) of P. maximartinezii was 1.1175, with a 95% confidence interval (CI) from 1.1008 to 1.1321, and it was relatively insensitive to changes in fecundity and growth. Under a seed extraction intensity of 99%, λ decreased to 1.0241, with a CI from 1.0177 to 1.0361. Elasticity analysis was then performed to identify the combined effects of proportional changes in fecundities and the largest stasis elements on λ. The results suggest that a sound conservation strategy should focus on improving the survival of juveniles and adults during their first reproductive events and on the largest adults, as well as on protecting the habitat of this threatened endemic species.

Modeling population growth and site specific control of the invasive Lantana camara L. (Verbenaceae) under differing fire regimes

AbstractIt is at the population level that an invasion either fails or succeeds. Lantana camara L. (Verbenaceae) is a weed of great significance in Queensland Australia and globally but its whole life‐history ecology is poorly known. Here we used 3 years of field data across four land use types (farm, hoop pine plantation and two open eucalyptus forests, including one with a triennial fire regime) to parameterise the weed's vital rates and develop size‐structured matrix models. Lantanacamara in its re‐colonization phase, as observed in the recently cleared hoop pine plantation, was projected to increase more rapidly (annual growth rate, λ = 3.80) than at the other three sites (λ 1.88–2.71). Elasticity analyses indicated that growth contributed more (64.6 %) to λ than fecundity (18.5 %) or survival (15.5 %), while across size groups, the contribution was of the order: juvenile (19–27 %) ≥ seed (17–28 %) ≥ seedling (16–25 %) &gt; small adult (4–26 %) ≥ medium adult (7–20 %) &gt; large adult (0–20 %). From a control perspective it is difficult to determine a single weak point in the life cycle of lantana that might be exploited to reduce growth below a sustaining rate. The triennial fire regime applied did not alter the population elasticity structure nor resulted in local control of the weed. However, simulations showed that, except for the farm population, periodic burning could work within 4–10 years for control of the weed, but fire frequency should increase to at least once every 2 years. For the farm, site‐specific control may be achieved by 15 years if the biennial fire frequency is tempered with increased burning intensity.

Re‐evaluating the effect of harvesting regimes on Nile crocodiles using an integral projection model

Crocodile populations are size-structured, and for populations that are subject to harvesting, removal is typically size selective. For this reason, size-structured matrix models are typically used to analyse the dynamics of crocodile populations. The boundaries between the size classes used to classify individuals in these models are typically chosen arbitrarily. This is problematic because results can depend upon the number and width of size classes. The recent development of continuous character population models termed integral projection models (IPM) has removed the need to arbitrarily classify individuals. These models are yet to be applied to harvested animal populations. Using information obtained from the literature, we develop an IPM for crocodiles. We use perturbation analyses to investigate how altering size-specific demographic rates influences the population growth rate and the strength of selection on snout to vent length. We find that perturbations can lead to complex responses. Sensitivity analysis to population growth and fertility selection reveals that the smallest animals and the sizes of early breeding individuals and their eggs may have more influence on these population biology parameters than previously thought. Although our model is relatively simple, our results show that IPM can be used to gain theoretical insight into the possible consequences of altering size-specific demographic rates on the population and evolutionary ecology of harvested populations.

Relative effects of environment and direct species interactions on the population growth rate of an exotic ascidian

The success of exotic species can be influenced by both the abiotic environment and species interactions. Many studies have demonstrated significant effects of either type of factor on aspects of exotic success, but few have considered their relative effects on population growth rate, a more holistic measure of success. To quantify the relative effects of environment and direct competition on an exotic ascidian, Botrylloides violaceus, I manipulated direct contact interactions at four sites with different abiotic environments and tracked individual colonies over 3 years. I tested site and contact treatment effects on survival, growth and fecundity, and then conducted a life table response experiment on a periodic, size-structured population matrix model to test their effects on population growth rate. Both site and contact interaction were important to explaining variation in survival and growth. Contact interactions decreased the survival and growth of larger colonies but unexpectedly increased the survival of small colonies at some sites, which led to relatively weaker and spatially variable effects on overall population growth rates. Site effects on population growth rates were an order of magnitude larger than contact effects, and site variation in winter vital rates made the largest contributions to changes in population growth rate. The results of this study suggest that the abiotic environment plays a larger role in the success of B. violaceus. Thus, environmental variables, such as temperature and salinity, could be used to predict this exotic species' success under different environmental scenarios, including global climate change.

Predicting the population dynamics of three understory broad-leaved evergreen species under the influence of Sika deer in primary and secondary forests of mid-western Japan

We examined the population dynamics of three broad-leaved tree species with different susceptibilities to deer predation. Simulation analysis was conducted using a size-structured matrix model for a primary forest plot (PP) and a secondary forest plot (SP) with 56% and 12% evergreen conifer composition in the canopy, respectively. In both plots, populations of Neolitsea sericea, a species that is susceptible to deer predation, initially declined significantly but eventually leveled off. The number of small stems decreased, while that of larger stems increased, indicating that the population dynamics of N. sericea are strongly affected by browsing pressure and that the number of large trees is important for population maintenance. When we examined two deer-resistant species, Pieris japonica and Illicium anisatum, the population of P. japonica increased in the SP and decreased in the PP, whereas that of I. anisatum increased in both plots, likely because mortality tends to increase in persistently dark environments. No significant difference was observed between the present and predicted size distributions of resistant species in the PP. Competition for resources is expected to intensify in the SP as a result of the predicted increase in large stems of the resistant species I. anisatum. Therefore, a specific conservation and management strategy for tree species should be considered for each forest type under the influence of Sika deer.

Restoration of threatened red gorgonian populations: An experimental and modelling approach

The increasing disturbances affecting marine communities highlight the need to examine restoration measures that can be added to other conservation efforts for threatened populations. The main goal of this study was to examine the usefulness of ecological restoration in the management of gorgonian populations damaged by diving activity in intensively visited marine protected areas (MPAs). We used field experiments as well as simulations from size-structured matrices to assess the utility of transplantation of living fragments from damaged colonies to increase the viability of threatened populations. Despite results showing that technique failure caused the loss of 40% of transplants, well-attached transplants achieved survival rates (80%) similar to those of natural colonies. Surprisingly, environmental conditions (light level and presence of algae) did not have a significant effect on the mortality of the transplants, but did affect methodological failure rates (37% of transplants were lost in the photophilous treatments in contrast to the 25% lost in sciaphilous treatment). The simulations showed a substantial increase in the annual population growth rate ( λ) only when transplantation was performed every two years and under the most demanding conditions (recovering 75% of the detached colonies and obtaining 3 fragments from each one). Predictions from the size-structured matrix model suggest severe limitations of this technique at larger spatial scales. However, our study confirms the feasibility of this restoration measure to contribute to the recovery of populations in MPAs affected by local disturbances. The experimental and modelling approaches developed here may provide useful guidelines for future studies on the restoration of marine populations.

Elasticity Analyses Of Size-Based Red And White Abalone Matrix Models: Management And Conservation

Prospective elasticity analyses have been used to aid in the management of fished species and the conservation of endangered species. Elasticities were examined for deterministic size-based matrix models of red abalone, Haliotis rufescens, and white abalone, H. sorenseni, to evaluate which size classes influenced population growth (lambda) the most. In the red abalone matrix, growth transitions were determined from a tag recapture study and grouped into nine size classes. In the white abalone matrix, abalone growth was determined from a laboratory study and grouped into five size classes. Survivorship was estimated from tag recapture data for red abalone using a Jolly-Seber model with size as a covariate and used for both red and white abalone. Reproduction estimates for both models used averages of the number of mature eggs produced by female red and white abalone in each size class from four-year reproduction studies. Population growth rate (lambda) was set to 1.0, and the first-year survival (larval survival through to the first size class) was estimated by iteration. Survival elasticities were higher than fecundity elasticities in both the red and white matrix models. The sizes classes with the greatest survival elasticities, and therefore the most influence on population growth in the model, were the sublegal red abalone (150-178 mm) and the largest white abalone size class (140-175 mm). For red abalone, the existing minimum legal size (178 mm) protects the size class the model suggests is critical to population growth. Implementation of education programs for novice divers coupled with renewed enforcement may serve to minimize incidental mortality of the critical size class. For white abalone, conservation efforts directed at restoring adults may have more of an impact on population growth than efforts focusing on juveniles. Our work is an example of how prospective elasticity analyses of size-structured matrix models can be used to quantitatively evaluate research priorities, fishery management strategies, and conservation options.

Dynamical response to fishing varies with compensatory mechanism: An abalone population model

Compensatory density-dependent processes in fish stocks are critical for the sustainability of fisheries, yet the implications of different mechanisms of density dependence in populations are seldom considered. We examine a size-structured matrix model of an abalone population subject to compensatory density-dependent processes at various life-stages. Matrix elements governing transitions amongst seven length classes are derived from growth, fecundity and mortality data for abalone populations in Australia, employing a probabilistic Gompertz growth model that accurately represents tag-recapture data for these populations. The resulting transition matrix is then modified to take account of density-dependent fecundity, survival and growth processes at various life stages. Equilibrium states of these models are obtained in the absence of fishing. Hypothetical populations at these equilibria are then subjected to fishing pressure and their responses examined. We find that density-dependent fecundity, survival and growth are each sufficient to stabilize these populations under harvesting. Density-dependent fecundity and survival lead to altered population structures after short time delays, while density-dependent growth stabilizes populations over longer time scales. Some forms of density-dependent growth lead to a false recovery from the initial fishing impact, followed by a decline to a lower stable level. The varied response to fishing under different compensatory mechanisms implies that simple stock–recruitment models may not capture key aspects of the dynamics of fishery stocks, nor predict their response to fishing correctly. Since density-dependent growth has been demonstrated for many species, its implications for fisheries deserve further theoretical investigation.

A size-structured model for a stock-recovery program for an exploited endemic fisheries resource

The size of harvests temporarily decreases following stock-recovery programs for overexploited resources. The process of stock recovery is quite uncertain, because of unknown life-history parameters, changes in environmental conditions, and human impact. To project future stock recovery, we considered a size-structured matrix model of a stock-recovery program, in which the minimum body size in the first harvest is increased. We applied some common concepts, including yield per recruitment, spawning per recruitment, and reproductive value, to a size-structured matrix model that incorporated economic discounting. The size-structured model predicted the following characteristics in stock-recovery processes: several years must pass for the harvest to increase, and the time-lag between the inception of the management program and the increase in the catch is caused by intergenerational momentum in demography. We also investigated the effects on the recovery process of temporal environmental fluctuations. We applied our model to a stock-recovery program of the corbiculid Seta clam ( Corbicula sandai ) in Lake Biwa, Japan.

A size-structured model for a stock-recovery program for an exploited endemic fisheries resource

The size of harvests temporarily decreases following stock-recovery programs for overexploited resources. The process of stock recovery is quite uncertain, because of unknown life-history parameters, changes in environmental conditions, and human impact. To project future stock recovery, we considered a size-structured matrix model of a stock-recovery program, in which the minimum body size in the first harvest is increased. We applied some common concepts, including yield per recruitment, spawning per recruitment, and reproductive value, to a size-structured matrix model that incorporated economic discounting. The size-structured model predicted the following characteristics in stock-recovery processes: several years must pass for the harvest to increase, and the time-lag between the inception of the management program and the increase in the catch is caused by intergenerational momentum in demography. We also investigated the effects on the recovery process of temporal environmental fluctuations. We applied our model to a stock-recovery program of the corbiculid Seta clam ( Corbicula sandai) in Lake Biwa, Japan.

ESTIMATING BIOCONTROL AGENT IMPACT WITH MATRIX MODELS:CARDUUS NUTANSIN NEW ZEALAND

We demonstrate the use of matrix models to assess the impact of biological control and other pest management strategies with a case study of Carduus nutans (nodding thistle) in New Zealand. Modeling can facilitate the choice of the best control agents, improve evaluation of biocontrol attempts, and assist in the design of integrated pest management plans. Field data for ∼8000 mapped C. nutans plants at two sites in New Zealand were used to develop size-structured matrix models. The matrix models confirm that both populations of C. nutans were increasing in number, as is expected of a noxious weed in its invasion phase. Elasticity analysis indicated that seed/seedling and small-plant/seed transitions were more crucial to population growth than rosette survival rates. However, simulations of attack by the biocontrol agent Rhinocyllus conicus (nodding thistle receptacle weevil) showed that seed losses of ∼69% would be required to make the populations decrease in size, far more than the observed losses of 30–...

Population Level Consequences of Toxicological Influences on Individual Growth and Reproduction inLumbricus rubellus(Lumbricidae, Oligochaeta)

The effects of increased environmental concentrations of copper on the population dynamics ofLumbricus rubellusare investigated. A size-structured matrix model is used to translate sublethal effects on individual growth and reproduction into their population dynamical consequences. Laboratory data on growth and reproduction under different, sublethal conditions of copper stress are used to parameterize the model. An estimate for the critical threshold concentration of copper (critical in a sense that the population growth rate at this concentration equals zero), obtained from the model analysis, agrees well with observations on field populations ofL. rubellus.

The Influence of Rainfall and Grazing on the Demography of Some African Savanna Grasses: A Matrix Modelling Approach

Size-structured matrix models were constructed for the perennial grasses of two African savannas (Aristida bipartita, Bothriochloa insculpta, Digitaria eriantha, Heteropogon contortus, Setaria incrassata and Themeda triandra) separately for each of four 1-year transitions under light or heavy grazing. The asymptotic population growth rate varied between years in relation to rainfall trends, but there were few marked differences between grazing treatments. Aristida had the highest asymptotic growth rate (λ 1 = 1.624) over a single transition period, while the heavily grazed population of Heteropogon had the lowest (λ 1 = 0.552). Most species declined in the final dry year. The stable stage distribution was generally a reversed «J» (...)