Donald L. DeAngelis

Dynamics of nutrient cycling and food webs

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EQUILIBRIUM AND NONEQUILIBRIUM CONCEPTS IN ECOLOGICAL MODELS

Mathematical models and empirical studies have revealed two potentially disruptive influences on ecosystems; (1) instabilities caused by nonlinear feedbacks and time-lags in the interactions of biological species, and (2) stochastic forcings by a fluctuating environment. Because both of these phenomena can severely affect system survival, ecol- ogists are confronted with the question of why complex ecosystems do, in fact, exist. Our study analyzes the basic themes of this research and identifies five general hypotheses that, in recent years, theoretical ecologists have built into models to increase their stability against disruptive feedback and stochasticity. To counter feedback instabilities, theoreti- cians have considered (1) functional interactions between species that act as stabilizers, (2) disturbance patterns that interrupt adverse feedback effects, and (3) the stabilizing effect of integrating small-spatial-scale systems into large landscapes. To decrease the influence of stochasticity, modelers have hypothesized (4) compensatory mechanisms operating at low population densities, and (5) the moderating effect of spatial extent and heterogeneity. We show that modeling based on these ideas can be organized in a systematic way. We also show that the stable equilibrium state should not be viewed as a fundamental property of ecological systems, but as a property that can emerge asymptotically from extrapolation to sufficiently large spatial scales.

Competition and Coexistence: The Effects of Resource Transport and Supply Rates

Classical resource competition theory can be generalized to apply to a variety of specific resource types and specific supply media (e.g., soil, water, or air). We develop a general model that relaxes the assumptions that (1) resources and organisms are sufficiently mixed that all organisms experience the same resource concentration and (2) the organisms themselves regulate the resource concentration of their shared environment. These assumptions are shown to apply to a limited subset of conditions in which the resource input rate is low and the resource transport rate in the environment is high. Under such conditions, the coexistence criteria of our general model converge with those of classical resource competition models. Such conditions may be met in some aquatic environments, but under other conditions, in which resource transport rates may be low or input fluxes high, the general model makes predictions that differ radically from those of the classical models. Specifically, our model predicts that, instead of a 1:1 ratio between limiting resources and locally coexisting species, a large number of species can coexist on a single limiting resource under steady-state conditions. Shifts from limitation by one type of resource to limitation by another type can dramatically alter the nature and intensity of competitive interactions. This phenomenon is proposed as the explanation for the ubiquitous unimodel curve of autotroph diversity along productivity gradients.

Super-individuals a simple solution for modelling large populations on an individual basis

Modelling populations on an individual-by-individual basis has proven to be a fruitful approach. Many complex patterns that are observed on the population level have been shown to arise from simple interactions between individuals. However, a major problem with these models is that the typically large number of individuals needed requires impractically large computation times. The common solution, reduction of the number of individuals in the model, can lead to loss of variation, irregular dynamics, and large sensitivity to the value of random generator seeds. As a solution to these problems, we propose to add an extra variable feature to each model individual, namely the number of real individuals it actually represents. This approach allows zooming from a real individual-by-individual model to a cohort representation or ultimately an all-animals-are-equal view without changing the model formulation. Therefore, the super-individual concept offers easy possibilities to check whether the observed behaviour is an artifact of following a limited number of individuals or of lumping individuals, and also to verify whether individual variability is indeed an essential ingredient for the observed behaviour. In addition the approach offers arbitrarily large computational advantages. As an example the super-individual approach is applied to a generic model of the dynamics of a size-distributed consumer cohort as well as to an elaborate applied simulation model of the recruitment of striped bass.

Positive feedback in natural systems

1. Introduction.- 1.1 Homeostasis.- 1.2 Positive Feedback.- 1.3 Ecological Systems with Positive Feedback.- 1.4 Generalization 1: Increasing Complexity.- 1.5 Generalization 2: Accelerating Change.- 1.6 Generalization 3: Threshold Effects.- 1.7 Generalization 4: Fragility of Complex Systems.- 1.8 Summary and Conclusions.- 2. The Mathematics of Positive Feedback.- 2.1 Graphical Analysis of a Simple Dynamic Positive Feedback System.- 2.2 A System of Two Mutualists.- 2.3 A System of Two Competitors.- 2.4 Mathematical Analysis of Positive Feedback.- 2.5 Summary and Conclusions.- 3. Physical Systems.- 3.1 The Life History of a Star.- 3.2 Geophysical Systems.- 3.3 Autocatalysis in Chemical Systems.- 3.4 Summary and Conclusions.- 4. Evolutionary Processes.- 4.1 Early Evolution of Life.- 4.2 Evolution at the Species Level.- 4.3 Coevolution.- 4.4 Summary and Conclusions.- 5. Organisms Physiology and Behaviour.- 5.1 Destructive Positive Feedback.- 5.2 Biochemical Processes in Cells and Organisms.- 5.3 Feeding and Drinking Behavior.- 5.4 Sleep.- 5.5 Movement and Motor-Sensory Relationships.- 5.6 Mind-Body Relationship.- 5.7 Summary and Conclusions.- 6. Resource Utilization by Organisms.- 6.1 Energy Allocation Tactics.- 6.2 Territorial Defense Strategies.- 6.3 Chemical Defense Strategies.- 6.4 Growth Rate Strategy.- 6.5 Summary and Conclusions.- 7. Social Behavior.- 7.1 Evolution of r- and K-strategies.- 7.2 Development of Social Strategies.- 7.3 Mating and Reproduction.- 7.4 Population Models Incorporating Sexual Reproduction.- 7.5 Small Group Dynamics.- 7.6 Castes In Insect Societies.- 7.7 Dominance Within Groups.- 7.8 Models of Group Formation and Size.- 7.9 The Schooling of Fish.- 7.10 Social Interactions and Game Theory.- 7.11 Summary and Conclusions.- 8. Mutualistic and Competitive Systems.- 8.1 Dynamics of Mutualistic Communities.- 8.2 Limits to Mutual Benefaction.- 8.3 Multi-Species Mutualism.- 8.4 Models of the Evolution of Mutualism.- 8.5 Isolation and Obligate Mutualism.- 8.6 Limited Competition.- 8.7 Summary and Conclusions.- 9. Age-Structured Populations.- 9.1 Age Structure.- 9.2 Leslie Matrices.- 9.3 Compensatory Leslie Matrices.- 9.4 Interacting Populations.- 9.5 Coexistence of Two Interacting Populations.- 9.6 Other Compensatory Models.- 9.7 Life-History Strategies.- 9.8 Intrinsic Rate of Increase.- 9.9 Reproductive Strategies.- 9.10 Summary and Conclusions.- 10. Spatially Heterogeneous Systems: Islands and Patchy Regions.- 10.1 Classical Theory of Island Biogeography.- 10.2 Island Clusters.- 10.3 Insular Reserves.- 10.4 Modeling the Patchy System.- 10.5 A Single Species in a Patchy Region.- 10.6 Time to Extinction on a Patch.- 10.7 Persistence of a Species in a Two-Patch Environment.- 10.8 Stability of a Single-Species, Two-Patch System.- 10.9 Persistence of a Species in an N-Patch Environment.- 10.10 Multi-Species, Multi-patch Systems with Competition and Mutalism.- 10.11 Persistence of a Species in a Two-Species, Two-Patch Environment.- 10.12 Persistence of a Species in an L-Species, iV-Patch Environment.- 10.13 Stability of a Two-Species, Two-Patch Model.- 10.14 Stability of an L-Species, iV-Patch Model.- 10.15 Relationship Between Reserve Design and Species Persistence.- 10.16 Summary and Conclusions.- 11. Spatially Heterogeneous Ecosystems: Pattern Formation.- 11.1 Spontaneous Emergence of Spatial Patterns.- 11.2 Diffusion Model.- 11.3 Pattern Formation Through Instability.- 11.4 Congregation of Colonial Organisms.- 11.5 Boundary Formation by Competition.- 11.6 Summary and Conclusions.- 12. Disease and Pest Outbreaks.- 12.1 Physiological Effects in the Host Species.- 12.2 Mutualistic Interactions of more than one Pathogenic Agent.- 12.3 Models of a Directly Communicated Disease or Parasite.- 12.4 Effects of Spatial Heterogeneity on Disease Outbreak Threshold Conditions.- 12.5 Design of Immunization Programs.- 12.6 Shape of the Contagion Rate Function.- 12.7 Comparison with other Spatially Heterogeneous Models.- 12.8 Host-Vector Models.- 12.9 Summary and Conclusions.- 13. The Ecosystem and Succession.- 13.1 The Ecosystem.- 13.2 Succession as a Positive Feedback Process.- 13.3 A Clementsian Model.- 13.4 Markov Chain Models.- 13.5 A Model of a Fire-Dependent System.- 13.6 Positive Feedback Loops in Ecosystems.- 13.7 Nutrient Cycling.- 13.8 Selection on the Community or Ecosystem Level.- 13.9 Summary and Conclusions.- Appendices.- Appendix A: Positive Linear Systems.- Appendix B: Stability of Positive Feedback Systems.- Appendix C: Stability of Discrete-Time Systems.- Appendix D: Positive Equilibria and Stability.- Appendix E: Comparative Statics of Positive Feedback Systems.- Appendix F: Similarity Transforms.- Appendix G: Bounds on the Roots of a Positive Linear System.- Appendix H: Relationship Between Positive Linear System Stability Criteria and the Routh-Hurwitz Criteria.- References.- Author Index.

Cannibalism and size dispersal in young-of-the-year largemouth bass: Experiment and model

Abstract A mathematical model for the dynamics of a population of young-of-the-year largemouth bass was formulated on the basis of laboratory experiments. The model was used to make a tentative test of the hypothesis that differing availabilities of alternative foods are decisive in determining the rates of cannibalism and, hence, differences in the observed behavior of the laboratory populations. It was found that slight differences in the degree of dispersal of initial sizes of individuals were probably more important in influencing the population dynamics than alternative food availability or other factors. The model was then compared with data on largemouth bass growth from pond experiments, which showed that, with some modifications of parameter values, the model is also capable of describing the young-of-the-year population dynamics in a pond.

Effect of periphyton biomass on hydraulic characteristics and nutrient cycling in streams

The effect of periphyton biomass on hydraulic characteristics and nutrient cycling was studied in laboratory streams with and without snail herbivores. Hydraulic characteristics, such as average water velocity, dispersion coefficients, and relative volume of transient storage zones (zones of stationary water), were quantified by performing short-term injections of a conservative tracer and fitting an advection-dispersion model to the conservative tracer concentration profile downstream from the injection site. Nutrient cycling was quantified by measuring two indices: (1) uptake rate of phosphorus from stream water normalized to gross primary production (GPP), a surrogate measure of total P demand, and (2) turnover rate of phosphorus in the periphyton matrix. These measures indicate the importance of internal cycling (within the periphyton matrix) in meeting the P demands of periphyton. Dense growths of filamentous diatoms and blue-green algae accumulated in the streams with no snails (high-biomass streams), whereas the periphyton communities in streams with snails consisted almost entirely of a thin layer of basal cells of Stigeoclonium sp. (low-biomass streams). Dispersion coefficients were significantly greater and transient storage zones were significantly larger in the high-biomass streams compared to the low-biomass streams. Rates of GPP-normalized P uptake from water and rates of P turnover in periphyton were significantly lower in high biomass than in low biomass periphyton communities, suggesting that a greater fraction of the P demand was met by recycling in the high biomass communities. Increases in streamwater P concentration significantly increased GPP-normalized P uptake in high biomass communities, suggesting diffusion limitation of nutrient transfer from stream water to algal cells in these communities. Our results demonstrate that accumulations of periphyton biomass can alter the hydraulic characteristics of streams, particularly by increasing transient storage zones, and can increase internal nutrient cycling. They suggest a close coupling of hydraulic characteristics and nutrient cycling processes in stream ecosystems.

Multiple nutrient limitations in ecological models

Abstract Ecological processes are often limited by more than one nutrient, for example, nitrogen and phosphorus. Therefore, mathematical representation of simultaneous limitations becomes important in many modeling and data analysis problems. This article reviews eight functional forms that have been proposed in the literature and presents a general theoretical framework that illustrates the commonalities and differences among the functions. Based on the general framework, three additional functional forms are derived. We then evaluate the functions based on their ability to fit experimental data. A total of eleven functions are fitted to eleven data sets by nonlinear least-squares. A new functional form, the Additive model, is selected as the most general function based on available data.

An individual-based approach to predicting density-dependent dynamics in smallmouth bass populations☆

Abstract The dynamics of a young-of-the-year cohort of a fish species, smallmouth bass ( Micropterus dolomieui ), were modeled using an individual-based computer simulation model. The young-of-the-year fish were simulated from the egg stage until recruitment into the yearling age class. At low initial densities of smallmouth bass in the swim-up larvae stage, recruitment to the yearling age class was roughly proportional to the initial densities of larval fish. However, when the initial densities were sufficiently high decreasing recruitment was observed as a function of further increases in larval density in the model. This occurred because high initial densities prevented all but a small fraction of the smallmouth bass from attaining large enough sizes to escape winter starvation. The addition of significant levels of size-dependent predation greatly altered this result, however. Heavy predation reduced the competition for prey among smallmouth bass and allowed a large number of smallmouth bass to grow rapidly and recruit to the yearling class.

FISH COHORT DYNAMICS: APPLICATION OF COMPLEMENTARY MODELING APPROACHES

The recruitment to the adult stock of a fish population is a function of both environmental conditions and the dynamics of juvenile fish cohorts. These dynamics can be quite complicated and involve the size structure of the cohort. Two types of models,i-state distribution models (e.g., partial differential equations) and/-state configuration models (computer simulation models following many individuals simultaneously), have been developed to study this type of question. However, these two model types have not to our knowledge previously been compared in detail. Analytical solutions are obtained for three partial differential equation models of early life-history fish cohorts. Equivalent individual-by-individual computer simulation models are also used. These two approaches can produce similar results, which suggests that one may be able to use the approaches interchangeably under many circumstances. Simple uncorrelated stochasticity in daily growth is added to the individual-by-individual models, and it is shown that this produces no significant difference from purely deterministic situations. However, when the stochasticity was temporally correlated such that a fish growing faster than the mean I d has a tendency to grow faster than the mean the next day, there can be great differences in the outcomes of the simulations.