Mathematical Models In Ecology Research Articles

Inference on spatiotemporal dynamics for coupled biological populations.

Mathematical models in ecology and epidemiology must be consistent with observed data in order to generate reliable knowledge and evidence-based policy. Metapopulation systems, which consist of a network of connected sub-populations, pose technical challenges in statistical inference owing to nonlinear, stochastic interactions. Numerical difficulties encountered in conducting inference can obstruct the core scientific questions concerning the link between the mathematical models and the data. Recently, an algorithm has been proposed that enables computationally tractable likelihood-based inference for high-dimensional partially observed stochastic dynamic models of metapopulation systems. We use this algorithm to build a statistically principled data analysis workflow for metapopulation systems. Via a case study of COVID-19, we show how this workflow addresses the limitations of previous approaches. The COVID-19 pandemic provides a situation where mathematical models and their policy implications are widely visible, and we revisit an influential metapopulation model used to inform basic epidemiological understanding early in the pandemic. Our methods support self-critical data analysis, enabling us to identify and address model weaknesses, leading to a new model with substantially improved statistical fit and parameter identifiability. Our results suggest that the lockdown initiated on 23 January 2020 in China was more effective than previously thought.

Dynamics of Spruce budworms and single species competition models with bifurcation analysis

Choristoneura Fumiferana is perilous defoliators of forest lands in North America and many countries in Europe. In this study, we consider mathematical models in ecology, epidemiology and bifurcation studies; the spruce budworm model and the population model with harvesting. The study is designed based on bifurcation analysis. In particular, the results support population thresholds necessary for survival in certain cases. In a series of numerical examples, the outcomes are presented graphically to compare with bifurcation results.

Guiding Students to Understand Functional Responses: Holling's Disc Experiment Revisited

ABSTRACT Inspired by the approach first employed by C.S. Holling in his classic “disc experiment,” this article provides a sequence of learning activities that increase students' understanding of the mechanisms behind saturating effects in predator-prey scenarios. The proposed lesson is recommended for inclusion in courses that address mathematical biology or modeling from introductory to advanced levels. The featured activities include a theoretical derivation of Holling's type II functional response model, a hands-on experiment for data gathering, and tools for further exploration through individual-based computer simulations. This multi-faceted approach gives students the opportunity to gather different sets of data in order to meet multiple learning objectives. Classroom trials conducted in Spring 2019 indicate that the proposed instructional resources and activities are effective for improving students' understanding of the mechanisms in Holling's type II functional response equation, and deepen students' appreciation of authentic data-driven mathematical modeling in ecology.

An Effective Mutualism? The Role of Theoretical Studies in Ecology and Evolution.

Theoretical models often have fundamentally different goals than do empirical studies of the same topic. Models can test the logic of existing hypotheses, explore the plausibility of new hypotheses, provide expectations that can be tested with data, and address aspects of topics that are currently inaccessible empirically. Theoretical models are common in ecology and evolution and are generally well cited, but I show that many citations appearing in nontheoretical studies are general to topic and that a substantial proportion are incorrect. One potential cause of this pattern is that some functions of models are rather abstract, leading to miscommunication between theoreticians and empiricists. Such misunderstandings are often triggered by simplifying logistical assumptions that modelers make. The 2018 Vice Presidential Symposium of the American Society of Naturalists included a variety of mathematical models in ecology and evolution from across several topics. Common threads that appear in the use of the models are identified, highlighting the power of a theoretical approach and the role of the assumptions that such models make.

A cautionary note on elasticity analyses in a ternary plot using randomly generated population matrices

AbstractMatrix population models are one of the most common mathematical models in ecology, which describe the dynamics of stage‐structured populations and provide us many population statistics. One of the statistics, elasticity onto population growth rate, is frequently used and represents the degree of the relative impact of life history parameters to the population growth rate. Due to the utility of elasticities for cross‐taxonomic comparisons, Silvertown and his coauthors have published multiple papers and reported the relationship between elasticities and life forms (or life history) in multiple plant species, using a triangle map (called “ternary plot”). To understand why their elasticities are located in specific regions of the ternary plot, we constructed four archetypes of population matrices, from which we simulated 24,000 randomly generated population matrices and obtained the consequent elasticities. We found a large discrepancy when comparing our results to those in Silvertown et al.'s study (Conserv Biol 10:591–597, 1996): for our simulated matrices where rapid transitions were not allowed (e.g., trees), the elasticity distribution resulted in a line across the ternary plot. We provided the mathematical proof for this result, and found that its slope depends on matrix dimension. We also used 1230 matrices from the COMPADRE Plant Matrix Database and calculated the elasticities. Our simulated results were validated with field data from COMPADRE: two straight lines appeared in the ternary plot. Furthermore, we answered several addressed questions, such as, “Is there any special elasticity distribution in matrices with high population growth rates?” and “Why are the elasticities of natural populations concentrated in the upper half of the ternary plot?”.

Comparative Modeling of Two Special Scenarios of Extreme Dynamics in Forest Ecosystems: Psillides in Australia and Spruce Budworm Moth in Canada

Critical states of quantity remain outside the framework of classical methods of mathematical modeling in ecology. Outbreaks disrupt the idea of a species balance with the environment, the hypothesis of competition between individuals of one population and a decrease in reproduction efficiency with increasing density. In modern conditions, the extreme dynamics of populations take place more often due to the penetration of alien species into ecosystems and expansion of its areal. Explosive reproductions cause serious damage to the forest industry and can even irreversibly transform the environment. By the method of comparative analysis of situations, it is considered two scenarios of rapid, but spontaneously damped outbreaks of forest pests. For the dynamics of psyllides in the forests of Australia, a hybrid model with transition to chaotic fluctuations is proposed. Spruce budworm moth Choristoneura fumiferana in Canada is characterized by a series of damped peaks. The proposed model in the form of equations with delay describes the scenario of damped sharp oscillations, which has a more general character. A similar dynamics can be traced for invasive ctenophore Mnemiopsis leidyi in the Caspian Sea and even in the dynamics of the epidemic of the "Spanish flu" a century ago, according to mortality in the UK. The scenarios considered in new computing models are transitional processes in the interaction of the species with its biotic environment.

生態学で, 数理モデルはどのような役割を果たせるのか( 数理生態学の魅力)

生態学で, 数理モデルはどのような役割を果たせるのか( 数理生態学の魅力)

IN VIVO – IN VITRO – IN SILICO В ЭКОЛОГИИ

Обсуждается использование подхода in silico (компьютерного) в экологии для работы с эмпирико-статистическими, самоорганизующимися, имитационными и аналитическими моделями. Обсуждены особенности оценки адекватности математических моделей в экологии. Обоснован комплексный подход к моделированию («модельный штурм»).

Pools versus Queues: The Variable Dynamics of Stochastic "Steady States".

Mathematical models in ecology and epidemiology often consider populations “at equilibrium”, where in-flows, such as births, equal out-flows, such as death. For stochastic models, what is meant by equilibrium is less clear – should the population size be fixed or growing and shrinking with equal probability? Two different mechanisms to implement a stochastic steady state are considered. Under these mechanisms, both a predator-prey model and an epidemic model have vastly different outcomes, including the median population values for both predators and prey and the median levels of infection within a hospital (P < 0.001 for all comparisons). These results suggest that the question of how a stochastic steady state is modeled, and what it implies for the dynamics of the system, should be carefully considered.

Modelos matemáticos en ecología: aplicación al dilema halcón vs. paloma

Polo, V. 2013. 2013. Mathematical modeling in ecology: on the hawk vs. dove game. Ecosistemas 22(3):6-11. Doi.: 10.7818/ECOS.2013.22-3.02 Mathematical modeling is a synthetic combination of variables with dominant effects on a natural system. Thus it provides a tool to understand the meaning and the new reformulation of hypothesis in the wide range of biological systems: from the cell to the complex ecosystems. Unfortunately, because of the difficult understanding between biologist and mathematicians, this is far from appreciated to most students and professionals. But mathematical models also have some limitations. Even if they were a feasible approximation to the biological problem, then only work when they solve doubts and/or transcend the knowledge with respect to an observational or experimental approach. The goal of the present study is to describe some steps in the construction of models fitted to a biological problem. I used a known behavioral example: the hawk vs. dove behavioral dilemma firstly formulated by Maynard Smith. First, I fitted four different models to reproduce the behavior of mixed proportions of hawks and doves that confront between them by the property of a valuable resource. Second, I analyze the dynamic and the evolutionarily stable strategy in dimorphic populations. And third, I discuss the range of valuable predictions and the limitations of these models when confronted to natural populations.

Вторая национальная конференция с международным участием «Математическое моделирование в экологии. Экоматмод-2011»

Вторая национальная конференция с международным участием «Математическое моделирование в экологии. Экоматмод-2011»

The state of Darwinian theory

The year 2009 marked both the bicentenary of Charles Darwin’s birth, and the passage of 150 years since the publication of his revolutionary book, commonly referred to as The Origin of Species (Darwin 1859). As part of the national celebrations that took place in the UK, a meeting on ‘Mathematical Models in Ecology and Evolution’ was held in Bristol on September 10 and 11, with a special focus on the mathematical modelling and application of Darwin’s key theories. This special issue collects together selected papers presented at the meeting, including contributions by the four invited keynote speakers, Rob Boyd, A. W. F. Edwards, Hanna Kokko and Franjo Weissing, as well as papers received after the event. The topics of these papers span the modelling of a very wide range of Darwin’s ideas, particularly evolution through natural selection, the origin of species, sexual selection, altruism and cooperation, and pangenesis, as well as applications of Darwinian thinking to the behaviour of animals, humans, and even human societies. It is hoped that this collection of papers will provide a useful snapshot of the state-of-the-art in Darwinian theory after the last one and a half centuries, and help to identify potential directions for research over the next.

Bridging gaps between statistical and mathematical modeling in ecology

Congratulations to Heisey and colleagues (2010) for an analysis of complex ecological data within a process-oriented framework and an equally thoughtful discussion of the ongoing challenges of linking process and pattern. Their work joins a growing literature linking modern statistical methods for describing patterns in data with expanded sets of ecologically-motivated mathematical models of population dynamics. My comments build on the authors’ framework and discussion to describe a continuum that (I feel) encompasses a conceptual path between process and pattern and illustrates how hierarchical models provide a convenient area within which to explore this continuum.

An introduction to mathematical models in ecology and evolution: time and space. 2nd edn.

‘These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated’. With this rather exciting beginning to a book, Fibonacci introduced the definition of algorithms to Western Europe in his Liber Abacci (1202). It is therefore no surprise that his rigorous problem description of the growth of rabbit populations, which would later lead to the discovery of exponential growth, is the unofficial precursor of ecological modelling. It is this rigour, combined with an accessible reading, that characterizes the very approachable second edition of Gillman's book. In the challenging task of covering almost a century of work in the discipline in such a concise book, the author has succeeded in the delicate task of selecting which models are more relevant and self-explanatory, which have overarching consequences beyond the mere scope of their initial applications, and which transcend beyond the typical uncoupling between ecology and evolution. Had the reader already gone through the first edition, he or she will be especially delighted with those new subsections dedicated to evolutionary theory and stochastic processes. The introduction highlights the definition of a model beyond the obscure mathematical techniques most ecologists attribute to theoretical ecologists, paying special attention to the basics of regression. Chapters 2–4 offer an incisive description of exponential growth from discrete and continuous, to stochastic and structured populations, emphasizing the link between all four approaches and the many commonalities of the exponential behaviour in ecological and evolutionary dynamics. Density dependence and the onset of chaos in simple populations is described in Chapter 5, leading to interacting populations and harvesting models in the next chapter, which also nicely introduces spatially implicit modelling with the Nicholson–Bailey model. Finally, Chapters 7–8 give rather short insights into multiple interacting populations and spatial models. Nonetheless, the book presents some weaknesses. There is a slow progression from the exciting beginning towards more standard and superficial developments in the last two chapters which, interestingly, have not changed much from the first edition despite the enormous literature in recent years. The exclusion of the first edition's boxes explaining some key techniques relevant to students seems an unfortunate decision, since there are no appendices at all. Further, more highlighted keywords are needed to help the reader use the book also as a reference text. Other details, such as the sudden presentation of a model based on classical epidemiology without explanation of its basic assumptions point towards some lack of care in reviewing those last two chapters. However, the book, especially suitable to undergraduates in their last year, graduates interested in an introduction to modelling, and lecturers alike, is an acknowledgeable reference. Unlike some other texts on ecological modelling, it deserves the merit of being one of the few books that have actual potential to attract readers that are often frightened by the mere vision of a Greek letter. This is not a trivial point. The ever-impending need for research into complex computational algorithms and mathematical modelling requires continuous recruitment of people attracted to interesting problems in mathematical biology. If your student comes to your office with a sudden interest in ecological models, chances are you can blame Michael Gillman for it.

Philosophical Issues in Ecology: Recent Trends and Future Directions

Colyvan, M., S. Linquist, W. Grey, P. Griffiths, J. Odenbaugh, and H. P. Possingham. 2009. Philosophical issues in ecology: recent trends and future directions. Ecology and Society 14(2): 22. https://doi.org/10.5751/ES-03020-140222

Analysis of the sensitivity properties of a model of vector-borne bubonic plague.

Model sensitivity is a key to evaluation of mathematical models in ecology and evolution, especially in complex models with numerous parameters. In this paper, we use some recently developed methods for sensitivity analysis to study the parameter sensitivity of a model of vector-borne bubonic plague in a rodent population proposed by Keeling & Gilligan. The new sensitivity tools are based on a variational analysis involving the adjoint equation. The new approach provides a relatively inexpensive way to obtain derivative information about model output with respect to parameters. We use this approach to determine the sensitivity of a quantity of interest (the force of infection from rats and their fleas to humans) to various model parameters, determine a region over which linearization at a specific parameter reference point is valid, develop a global picture of the output surface, and search for maxima and minima in a given region in the parameter space.

A biologist's guide to mathematical modeling in ecology and evolution

Scientists perform experiments and gather data to gain evidence for or against various hypotheses about how the world works. This sounds straightforward, but exactly how we use this data to make quantitative statements about various hypotheses requires a bit more care. In fact, there are two related, but philosophically distinct, approaches to scientific inference. To some extent these two approaches parallel the frequency interpretation and the subjective interpretation of probability. “Classical” statistical analysis is most closely allied with the frequency interpretation, whereas “Bayesian” analysis is allied with the subjective interpretation.

Complex spatiotemporal dynamics in Lotka–Volterra ring systems

Mathematical models in ecology often need to incorporate spatial dependence to accurately model real-world systems. We consider competitive Lotka–Volterra models modified to include this spatial dependence through organization of the competing species into a one-dimensional ring by an appropriate choice of the interaction matrix. We show that these systems can exhibit complex dynamics, spatiotemporal chaos, and spontaneous symmetry breaking. A high-dimensional, spatially homogeneous, nearest-neighbor example with interaction strengths decreasing with distance is characterized including an analysis of how the dynamics of the system vary with dimension. We also show the existence of Lyapunov functions that arise from this method of including spatial dependence and how they prohibit complex dynamics for certain regions of the parameter space. We utilize these Lyapunov functions to reduce the required calculation time in brute-force searches of parameter space. A short comparison is given to derived line systems including a contrast between the eigenvalues of the two systems. .

Classification of mathematical models in ecology

This paper is devoted to the classification of mathematical models in ecology. It represents various classes of mathematical models distinguished on the base of both the characteristics of the object of investigation and research task. Each class of mathematical models is illustrated by references to particular studies. This classification aims to help ecologists, biologists and environmentalists navigate the diversity of methods and techniques in mathematical modeling and develop a standardized approach to the application of mathematics in ecological research.

Explicit mixed finite order Runge–Kutta methods

We investigate the asymptotic behavior of systems of nonlinear differential equations and introduce a family of mixed methods from combinations of explicit Runge–Kutta methods. These methods have better stability behavior than traditional Runge–Kutta methods and generally extend the range of validity of the calculated solutions. These methods also give a way of determining if the numerical solutions are real or spurious. Emphasis is put on examples coming from mathematical models in ecology.