Non-linear dynamics of marine ecosystem models

Thesis (Ph.D.) University of Alaska Fairbanks, 2004Despite a rapid trend towards more realistic Nutrient-Phytoplankton-Zooplankton (NPZ) models, in which zooplankton are presented with multiple nutritional resources, investigations into the fundamental dynamics of these newer models have been limited. The objective of this dissertation was to explore the dynamical behavior of such NPZ models parameterized for the coastal Gulf of Alaska. With alternative stationary forcing regimes and zooplankton grazing functions, the dynamics of one-dimensional NPZ models were investigated for a range of specific predation rates (h) and predation exponents (q), which together define the form of the predation (model closure) function. Oscillations in state variables are shown to be an intrinsic property of the NPZ models, not dependent on variable seasonal forcing for their existence. Increasing mixed layer diffusivity or reducing mixed layer depth increased model excitability; it is hypothesized that this is due to the resultant increase in flux of utilizable nutrient. Model behavior was also strongly influenced by the form of both the grazing and predation functions. For all of the grazing functions implemented, Hopf bifurcations, where the form of the solution transitioned between steady equilibrium and periodic limit cycles, persisted across the q-h parameter space. Regardless of the values of h and q, with some forms of the grazing function steady equilibrium solutions that simultaneously comprised non-zero concentrations for all model components could not be found. The inclusion of sinking detritus in the model had important implications for the composition and excitability of model solutions, generally increasing the region of q-h space for which oscillatory solutions were found. Therefore, in order to correctly simulate the depth-explicit concentrations of model components, or to have an accurate understanding of the potential excitability of the system, inclusion of this component is valuable. This dissertation highlights the importance of understanding the potential impact that choice of functional response may have on the intrinsic oscillatory nature of a model prior to interpreting results from coupled bio-physical simulations. As we come to rely more on ecosystem models as a tool to interpret marine ecosystem functionality it will be important to improve our understanding of the non-linear behavior inherent in these models.General introduction -- Development of an NPZ model with multiple prey types -- Development of an NPZ model with multiple prey types -- Linear stability analysis of an NPZ model with multiple prey types -- Non-linear dynamics of a pelagic ecosystem model with multiple predator and prey types -- The role of detritus in NPZ model dynamics -- Discussion and conclusions

Investigations into a plankton population model: Mortality and its importance in climate change scenarios

The potential for marine plankton ecosystems to influence climate by the production of dimethylsulphide (DMS) has been an important topic of recent research into climate change. Several General Circulation Models, used to predict climate change, have or are being modified to include interactions of ecosystems with climate. Climate change necessitates that parameters within ecosystem models must change during long-term simulations, especially mortality parameters that increase as organisms are pushed toward the boundaries of their thermal tolerance. Changing mortality parameters can have profound influences on ecosystem model dynamics. There is therefore a pressing need to understand the influence of varying mortality parameters on the long-term behaviour of ecosystem models. This work examines the sensitivity of a model of DMS production by marine ecosystems to variations in three linear mortality coefficients. Significant differences in behaviour are observed, and we note the importance of these results in formulating ecosystem models for application in simulations of climate change

Chaos to Order: Role of Toxin Producing Phytoplankton in Aquatic Systems

Toxin producing phytoplankton (TPP) plays an important role in aquatic systems. To observe the role of TPP, we consider a three species food chain model consisting of TPP-zooplankton-fish population. The similar type of model considered by Upadhyay et al. [1] for terrestrial ecosystem and obtained chaotic dynamics in some region of parametric space. We modify their models by taking into account the toxin liberation process of TPP population and represented as aquatic systems. We consider Holling type I, type II and type III functional forms for this process. We observe that increasing the strength of toxic substance change the state from chaos to order. Our conclusion is that TPP has a stabilizing contribution in aquatic systems and may be used as a bio-control mechanism

Chaos to order: role of toxin producing phytoplankton in aquatic system

Toxin producing phytoplankton (TPP) plays an important role in aquatic systems. To observe the role of TPP, we consider a three species food chain model consisting of TPP-zooplankton-fish population. The similar type of model considered by Upadhyay et al. [1] for terrestrial ecosystem and obtained chaotic dynamics in some region of parametric space. We modify their models by taking into account the toxin liberation process of TPP population and represented as aquatic systems. We consider Holling type I, type II and type III functional forms for this process. We observe that increasing the strength of toxic substance change the state from chaos to order. Our conclusion is that TPP has a stabilizing contribution in aquatic systems and may be used as a bio-control mechanism

Generalized modeling of complex dynamical systems: an application to the stability of ecological networks

Dissertation (Ph.D.) University of Alaska Fairbanks, 2023Understanding the stability of food webs is crucial for resource sustainability and conservation of ecosystems, especially in the context of climate change. Specific models describe the biomass flow in food webs by a set of ordinary differential equations that require the explicit reconstruction of a mathematical expression for each of the interactions or processes, like predator-prey interactions, primary production, and mortality. Although specific models are rich with time-evolution information, limited access for empirical observation of these typically immense systems induce uncertainty in the data and approximations in the corresponding models that can threaten robustness or relevance of results. Generalized models can produce the stability of all the equilibria of specific models that have the same vague structure and bypasses the requirement to specify every function by evoking a normalizing transformation. The analysis is subsequently computationally efficient and can be used to study large food webs with a great number of replicates. Often generalized ecological network studies confine the scope to a small subset of the variable dynamical scenarios, but this limits the interpretations that can be inferred. With this in mind, we develop a deterministic food-web generator that can be used to compare large food webs that differ by only a single link and maintain an expansive dynamical scope. We found behavior that indicates the existence of critical links and a grander theory on topological equivalence. We explicitly show how we can create hypothetical paths the system may traverse upon enrichment of lower trophic levels using the expanded dynamical scope. Generalized modeling is unable to produce evolution solutions among other things, but it has an unlimiting access to the stability of equilibria while specific models provide only a subset of stability data. Generalized modeling is a relatively new method and its relation to specific model outcomes/results is not clearly understood. Specific models can inform generalized modeling studies on properties like coexistence of fixed points or actually occurring relative weighting of flows between ecosystem members. We combined the methods and demonstrated the validity of the abstract technique of generalized modeling in emphasis to its usefulness/power for the analysis of network stability. The specific model provided a unifying explanation to a conglomerate of related microcosm experiments that showed conflicting results on enrichment and implied stabilization upon the hampering of predatory efficiency. We identified the conditions by which enrichment is stabilizing to a steady state when basal species are in a resource-deprived environment but destabilizing if resources become more abundant. A prevalent issue in ecology involves discrepancies between simulation and empirical observation about food-web stability such as how intuition says enrichment or complexity in some way are favorable to stability but mathematical models find it predominately the opposite. A common rationalization for these discrepancies includes discourse on reductionistic versus holistic rhetoric. The idea being that as models become better representations of ecosystems that capture more intricacies and detail, they will help to resolve the issue. We constructed over a million food webs that reveal positive effects on fixed-point stability from the incorporation of more realistic ecosystem features that include species specialization, habitat modularity, and predator's prey preferences. Arctic warming is a portent to changes in species composition and ecological theory predicts the existence of key ecosystem members that have extraordinary influence on overall ecosystem function or the state of the system. Motivated by sea-ice loss and northward expansion of species distributions, ice-obligate species are removed from the food webs and southern competitors are introduced. Although it is common understanding that apex predators can enhance biodiversity, we find the presence of "super killers" significantly destabilizes food webs. Ecosystems have immense complexity with thousands of species, but ecosystem models condense and consider only a few species that are of the most interest or abundance, neglecting the many weak interactions comprising the larger ecosystem. Considering this, we suppose a food web is subsumed by a larger phantom ecological network that represents hypothetically rare species or predator-prey relationships. Each link from the phantom network contributes a variably weak perturbation, but collectively, induce a net positive effect on the average stability of the food webs, considerably so near the optimal perturbation strength.Chapter 1: introduction -- 1.1. Ecological background -- 1.2. Dynamical systems -- 1.2.1. Example: one-dimensional flows -- 1.2.2. Example: two-dimensional flows -- 1.3. Lotka-volterra competition -- 1.3.1. Specific model -- 1.3.2. Conventional nondimensionalization -- 1.3.3. Normalizing nondimensionalization -- 1.3.4. Generalized model -- 1.3.5. Connecting the generalized model to the specific model -- 1.4. A different view of the system: combining processes -- 1.5. Synopsis -- 1.6. References. Chapter 2: stability of generalized ecological network models -- 2.1. Abstract -- 2.2. Abstract extension -- 2.3. Introduction -- 2.4. Model -- V2.4.1. Food web topology -- 2.4.2. Food web dynamics -- 2.4.3. Generalized modeling -- 2.5. Sensitivity to topology -- 2.6. Predatory response and stability -- 2.7. Paradox of enrichment -- 2.8. Omnivory: food chains to food webs -- 2.9. Complexity-stability debate and the ratio of intermediate to top predators -- 2.10. Conclusion -- 2.11. References. Chapter 3: Combining generalized modeling and specific modeling in the analysis of ecological networks -- 3.1. Abstract -- 3.2. Abstract extension -- 3.3. Introduction -- 3.4. Model -- 3.4.1. Generalized model -- 3.4.2. Specific models -- 3.5. Four-species food web and analysis -- 3.5.1. Specific model and steady states -- 3.5.2. Generalized parameters -- 3.5.3. Generalized versus specific model: parameter scenarios -- 3.6. Interface of specific and generalized model -- 3.7. Robustness of stability for various basal production -- 3.8. Enrichment stabilizes and destabilizes -- 3.9. Adding the omnivorous link stabilizes -- 3.10. Coexisting fixed points -- 3.11. Feeding nonlinearities and stability -- 3.12. Weighting of links in steady state -- 3.13. Conclusion -- 3.14. References. Chapter 4: How realistic features affect the stability of an Arctic marine food web model -- 4.1. Introduction -- 4.2. The Beaufort Sea food web -- 4.3. Food web dynamics -- 4.3.1. Generalized ecological model -- 4.3.2. Generalized parameters for the base model -- 4.3.3. Measuring stability -- 4.4. Impact of refined species characteristics on stability -- 4.4.1. Adjusting the base model -- 4.4.2. Results -- 4.5. Impact of species introduction and removal on stability -- 4.6. Impact of background species on stability -- 4.6.1. Adjusting the model -- 4.6.2. Results -- 4.7. Conclusion -- 4.8. References. Chapter 5: Conclusion -- References

Phase tipping: how cyclic ecosystems respond to contemporary climate

We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator–prey cycle to extinction in two paradigmatic predator–prey models with an Allee effect. Our analysis of these examples uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping, where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors

Rhythms and Evolution: Effects of Timing on Survival

The evolution of metabolism regulation is an intertwined process, where different strategies are constantly being developed towards a cognitive ability to perceive and respond to an environment. Organisms depend on an orchestration of a complex set of chemical reactions: maintaining homeostasis with a changing environment, while simultaneously sending material and energetic resources to where they are needed. The success of an organism requires efficient metabolic regulation, highlighting the connection between evolution, population dynamics and the underlying biochemistry. In this work, I represent organisms as coupled information-processing networks, that is, gene-regulatory networks receiving signals from the environment and acting on chemical reactions, eventually affecting material flows. I discuss the mechanisms through which metabolism control is improved during evolution and how the nonlinearities of competition influence this solution-searching process. The propagation of the populations through the resulting landscapes generally point to the role of the rhythm of cell division as an essential phenotypic feature driving evolution. Subsequently, as it naturally follows, different representations of organisms as oscillators are constructed to indicate more precisely how the interplay between competition, maturation timing and cell-division synchronisation affects the expected evolutionary outcomes, not always leading to the \"survival of the fastest\"

Phase transitions in fluids and biological systems

In this thesis, I consider systems from two seemingly different fields: fluid dynamics and microbial ecology. In these systems, the unifying features are the existences of global non-equilibrium steady states. I consider generic and statistical models for transitions between these global states, and I relate the model results with experimental data. A theme of this thesis is that these rather simple, minimal models are able to capture a lot of functional detail about complex dynamical systems. In Part I, I consider the transition between laminar and turbulent flow. I find that quantitative and qualitative features of pipe flow experiments, the superexponential lifetime and the splitting of turbulent puffs, and the growth rate of turbulent slugs, can all be explained by a coarse-grained, phenomenological model in the directed percolation universality class. To relate this critical phenomena approach closer to the fluid dynamics, I consider the transition to turbulence in the Burgers equation, a simplified model for Navier-Stokes equations. Via a transformation to a model of directed polymers in a random medium, I find that the transition to Burgers turbulence may also be in the directed percolation universality class. This evidence implies that the turbulent-to-laminar transition is statistical in nature and does not depend on details of the Navier-Stokes equations describing the fluid flow. In Part II, I consider the disparate subject of microbial ecology where the complex interactions within microbial ecosystems produce observable patterns in microbe abundance, diversity and genotype. In order to be able to study these patterns, I develop a bioinformatics pipeline to multiply align and quickly cluster large microbial metagenomics datasets. I also develop a novel metric that quantifies the degree of interactions underlying the assembly of a microbial ecosystem, particularly the transition between neutral (random) and niche (deterministic) assembly. I apply this metric to 16S rRNA metagenomic studies of 6 vertebrate gastrointestinal microbiomes and find that they assembled through a highly non-neutral process. I then consider a phase transition that may occur in nutrient-poor environments such as ocean surface waters. In these systems, I find that the experimentally observed genome streamlining, specialization and opportunism may well be generic statistical phenomena