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  1. Twombly, Saran, Alan Hastings, Tom Miller, Michael Cortez, Karen Abbott, Tanjona Ramiadantsoa, Julie Blackwood, and Olivia Prosper. “New Theory for Increasingly Tangled Banks.” Issues in Science and Technology 38, no. 4 (Summer 2022): 39–44. Theory has fallen out of fashion in the sciences, in favor of data collection and number crunching. But the conceptual frameworks provided by theory are essential for addressing society’s most complex and urgent problems.
    Free, publicly-accessible full text available July 1, 2023
  2. Boettiger, Carl (Ed.)
  3. Abstract

    During their lifetimes, individuals in populations pass through different states, and the notion of an occupancy time describes the amount of time an individual spends in a given set of states. Questions related to this idea were studied in a recent paper by Roth and Caswell for cases where the environmental conditions are constant. However, it is truly important to consider the case where environments are changing randomly or in directional way through time, so the transition probabilities between different states change over time, motivating the use of time-dependent stage-structured models. Using absorbing inhomogenous Markov chains and the discrete-time McKendrick–von Foerster equation, we derive explicit formulas for the occupancy time, its expectation, and its higher-order moments for stage-structured models with time-dependent transition rates. The results provide insights into the dynamics of long lived plant or animal populations where individuals transition in both directions between reproductive and non reproductive stages. We apply our approach to study a specific time-dependent model of the Southern Fulmar, and obtain insights into how the number of breeding attempts depends on external conditions that vary through time.

  4. Abstract One of the main factors that determines habitat suitability for sessile and territorial organisms is the presence or absence of another competing individual in that habitat. This type of competition arises in populations occupying patches in a metacommunity. Previous studies have looked at this process using a continuous-time modeling framework, where colonizations and extinctions occur simultaneously. However, different colonization processes may be performed by different species, which may affect the metacommunity dynamics. We address this issue by developing a discrete-time framework that describes these kinds of metacommunity interactions, and we consider different colonization dynamics. To understand potential dynamics, we consider specific functional forms that characterize the colonization and extinction processes of metapopulations competing for space as their limiting factor. We then provide a mathematical analysis of the models generated by this framework, and we compare these results to what is seen in nature and in previous models.
  5. Abstract

    Forecasting tipping points in spatially extended systems is a key area of interest to ecologists. A slowly declining spatially distributed population is an important example of an ecological system that could exhibit a cascade of tipping points. Here, we develop a spatial two-patch model with environmental stochasticity that is slowly forced through population collapse, in the presence of changing environmental conditions. We begin with a basic spatial model, then introduce a fast–slow version of the model using geometric singular perturbation theory, followed by the inclusion of stochasticity. Using the spectral density of the fluctuating subpopulation in each patch, we derive analytic expressions for candidate indicators of population extinction and evaluate their performance through a simulation study. We find that coupling and spatial heterogeneity decrease the magnitude of the proposed indicators in coupled populations relative to isolated populations. Moreover, the degree of coupling dictates the trends in summary statistics. We conclude that this theory may be applied to other contexts, including the control of invasive species.

  6. Abstract Many benthic animals begin life with a planktonic larval stage during which coastal currents may move individuals far from shore. This trait is believed to allow individuals to develop away from nearshore predators and sibling competition, based on the assumption that mortality rates are weaker offshore. However, larvae developing offshore often fail to locate suitable coastal habitats. This results in a trade-off between nearshore mortality and offshore wastage with consequences for larval delivery to adult habitats that have not been fully appreciated. We use a reaction-diffusion model to show that when the nearshore larval mortality rate is high, larval supply can vary more than 10-fold with the offshore mortality rate. If this offshore rate is weak, then larval supply is maximized by an intermediate diffusion rate or larval duration. While a low-diffusivity coastal boundary layer typically improves the larval supply by decreasing wastage, it can also reduce the larval supply by preventing individuals from exploiting low offshore mortality rates. Finally, the cross-shore structure of the mortality rate may influence the alongshore transport of larvae by determining how far offshore they reside prior to settling, and, consequently, the alongshore currents they experience. Our observations contrast with the prior argument thatmore »larval supply decreases with diffusivity and larval duration due to wastage, and challenge the widespread decision to omit cross-shore heterogeneity from studies of alongshore movement. Scenarios in which spatial variability in the mortality rate has a large effect on recruitment are important both for understanding the biological consequences of the larval stage and from a modeling perspective.« less
  7. Free, publicly-accessible full text available November 1, 2023
  8. Abstract Network-based models of epidemic spread have become increasingly popular in recent decades. Despite a rich foundation of such models, few low-dimensional systems for modeling SIS-type diseases have been proposed that manage to capture the complex dynamics induced by the network structure. We analyze one recently introduced model and derive important epidemiological quantities for the system. We derive the epidemic threshold and analyze the bifurcation that occurs, and we use asymptotic techniques to derive an approximation for the endemic equilibrium when it exists. We consider the sensitivity of this approximation to network parameters, and the implications for disease control measures are found to be in line with the results of existing studies.
  9. Chase, Jonathan (Ed.)
    Understanding mechanisms of coexistence is a central topic in ecology. Mathematical analysis of models of competition between two identical species moving at different rates of symmetric diffusion in heterogeneous environments show that the slower mover excludes the faster one. The models have not been tested empirically and lack inclusions of a component of directed movement toward favourable areas. To address these gaps, we extended previous theory by explicitly including exploitable resource dynamics and directed movement. We tested the mathematical results experimentally using laboratory populations of the nematode worm, Caenorhabditis elegans. Our results not only support the previous theory that the species diffusing at a slower rate prevails in heterogeneous environments but also reveal that moderate levels of a directed movement component on top of the diffusive movement allow species to coexist. Our results broaden the theory of species coexistence in heterogeneous space and provide empirical confirmation of the mathematical predictions.