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The well known linear chain trick (LCT) allows modellers to derive mean field ODEs that assume gamma (Erlang) distributed passage times, by transitioning individuals sequentially through a chain of sub-states. The time spent in these sub-states is the sum of k exponentially distributed random variables, and is thus gamma distributed. The generalized linear chain trick (GLCT) extends this technique to the broader phase-type family of distributions, which includes exponential, Erlang, hypoexponential, and Coxian distributions. Phase-type distributions are the family of matrix exponential distributions on that represent the absorption time distributions for finite-state, continuous time Markov chains (CTMCs). Here we review CTMCs and phase-type distributions, then illustrate how to use the GLCT to efficiently build ODE models from underlying stochastic model assumptions. We introduce two novel model families by using the GLCT to generalize the Rosenzweig-MacArthur predator-prey model, and the SEIR model. We illustrate the kinds of complexity that can be captured by such models through multiple examples. We also show the benefits of using a GLCT-based model formulation to speed up the computation of numerical solutions to such models. These results highlight the intuitive nature, and utility, of using the GLCT to derive ODE models from first principles.

Most of earth's biodiversity is comprised of interactions among species, yet it is unclear what causes variation in interaction diversity across space and time. We define interaction diversity as the richness and relative abundance of interactions linking species together at scales from localized, measurable webs to entire ecosystems. Large-scale patterns suggest that two basic components of interaction diversity differ substantially and predictably between different ecosystems: overall taxonomic diversity and host specificity of consumers. Understanding how these factors influence interaction diversity, and quantifying the causes and effects of variation in interaction diversity are important goals for community ecology. While previous studies have examined the effects of sampling bias and consumer specialization on determining patterns of ecological networks, these studies were restricted to two trophic levels and did not incorporate realistic variation in species diversity and consumer diet breadth. Here, we developed a food web model to generate tri-trophic ecological networks, and evaluated specific hypotheses about how the diversity of trophic interactions and species diversity are related under different scenarios of species richness, taxonomic abundance, and consumer diet breadth. We investigated the accumulation of species and interactions and found that interactions accumulate more quickly; thus, the accumulation of novel interactions may require less sampling effort than sampling species in order to get reliable estimates of either type of diversity. Mean consumer diet breadth influenced the correlation between species and interaction diversity significantly more than variation in both species richness and taxonomic abundance. However, this effect of diet breadth on interaction diversity is conditional on the number of observed interactions included in the models. The results presented here will help develop realistic predictions of the relationships between consumer diet breadth, interaction diversity, and species diversity within multi-trophic communities, which is critical for the conservation of biodiversity in this period of accelerated global change.

In this paper we generalize the Linear Chain Trick (LCT; aka the Gamma Chain Trick) to help provide modelers more flexibility to incorporate appropriate dwell time assumptions into mean field ODEs, and help clarify connections between individual-level stochastic model assumptions and the structure of corresponding mean field ODEs. The LCT is a technique used to construct mean field ODE models from continuous-time stochastic state transition models where the time an individual spends in a given state (i.e., the dwell time) is Erlang distributed (i.e., gamma distributed with integer shape parameter). Despite the LCT’s widespread use, we lack general theory to facilitate the easy application of this technique, especially for complex models. Modelers must therefore choose between constructing ODE models using heuristics with oversimplified dwell time assumptions, using time consuming derivations from first principles, or to instead use non-ODE models (like integro-differential or delay differential equations) which can be cumbersome to derive and analyze. Here, we provide analytical results that enable modelers to more efficiently construct ODE models using the LCT or related extensions. Specifically, we provide (1) novel LCT extensions for various scenarios found in applications, including conditional dwell time distributions; (2) formulations of these LCT extensions that bypass the need to derive ODEs from integral equations; and (3) a novel Generalized Linear Chain Trick (GLCT) framework that extends the LCT to a much broader set of possible dwell time distribution assumptions, including the flexible phase-type distributions which can approximate distributions on \\[{\\mathbb {R}}^+\\] and can be fit to data.

Terrestrial tri-trophic interactions account for a large part of biodiversity, with approximately 75% represented in plant-insect-parasitoid interactions. Herbivore diet breadth is an important factor mediating these tri-trophic interactions, as specialisation can influence how herbivore fitness is affected by plant traits. We investigated how phytochemistry, herbivore immunity, and herbivore diet breadth mediate plant-caterpillar-parasitoid interactions on the tropical plant genus (Piperaceae) at La Selva Biological station in Costa Rica and at Yanayacu Biological Station in Ecuador. We collected larval stages of one generalist species, , (Lepidoptera: Hesperiidae) and 4 specialist species in the genus (Lepidoptera: Geometridae) from 15 different species of , reared them on host leaf material, and assayed phenoloxidase activity as a measure of potential larval immunity. We combined these data with parasitism and caterpillar species diet breadth calculated from a 19-year database, as well as established values of phytochemical diversity calculated for each plant species, in order to test specific hypotheses about how these variables are related. We found that phytochemical diversity was an important predictor for herbivore immunity, herbivore parasitism, and diet breadth for specialist caterpillars, but that the direction and magnitude of these relationships differed between sites. In Costa Rica, specialist herbivore immune function was negatively associated with the phytochemical diversity of the host plants, and rates of parasitism decreased with higher immune function. The same was true for Ecuador with the exception that there was a positive association between immune function and phytochemical diversity. Furthermore, phytochemical diversity did not affect herbivore immunity and parasitism for the more generalised herbivore. Results also indicated that small differences in herbivore diet breadth are an important factor mediating herbivore immunity and parasitism success for at both sites. These patterns contribute to a growing body of literature that demonstrate strong cascading effects of phytochemistry on higher trophic levels that are dependent on herbivore specialisation and that can vary in space and time. Investigating the interface between herbivore immunity, plant chemical defence, and parasitoids is an important facet of tri-trophic interactions that can help to explain the enormous amount of biodiversity found in the tropics.

Phytochemical variation among plant species is one of the most fascinating and perplexing features of the natural world and has implications for both human health and the functioning of ecosystems. A key area of research on phytochemical variation has focused on insects that feed on plants and the enormous diversity of plant-derived compounds that reduce or deter damage by insects. Empirical studies on the ecology and evolution of these chemically mediated plant–insect interactions have been guided by a long history of theoretical development. However, until recently, such theory was substantially limited by inadequate data, a situation that is rapidly changing as ecologists partner with chemists utilizing the latest technological advances. In this Review, we aim to facilitate the union of ecological theory with modern chemistry by discussing important theoretical frameworks for studying chemical ecology and outlining the steps by which hypotheses on insect–phytochemical interactions can be advanced using current methodologies and statistical approaches. We highlight unique approaches to isolation, synthesis, spectroscopy, metabolomics and genomics relevant to chemical ecology and describe future areas for research that will bring an unprecedented understanding of phytochemical variation. The union of theory in chemical ecology with modern methods in chemistry has enhanced our understanding of phytochemical variation among and within plants. This Review outlines these theoretical frameworks and approaches for hypothesis testing, with a focus on chemically mediated plant–insect interactions.

Emerging infectious disease is a key factor in the loss of amphibian diversity. In particular, the disease chytridiomycosis has caused severe declines around the world. The lethal fungal pathogen that causes chytridiomycosis, Batrachochytrium dendrobatidis ( Bd ), has affected amphibians in many different environments. One primary question for researchers grappling with disease-induced losses of amphibian biodiversity is what abiotic factors drive Bd pathogenicity in different environments. To study environmental influences on Bd pathogenicity, we quantified responses of Bd phenotypic traits (e.g., viability, zoospore densities, growth rates, and carrying capacities) over a range of environmental temperatures to generate thermal performance curves. We selected multiple Bd isolates that belong to a single genetic lineage but that were collected across a latitudinal gradient. For the population viability, we found that the isolates had similar thermal optima at 21°C, but there was considerable variation among the isolates in maximum viability at that temperature. Additionally, we found the densities of infectious zoospores varied among isolates across all temperatures. Our results suggest that temperatures across geographic point of origin (latitude) may explain some of the variation in Bd viability through vertical shifts in maximal performance. However, the same pattern was not evident for other reproductive parameters (zoospore densities, growth rates, fecundity), underscoring the importance of measuring multiple traits to understand variation in pathogen responses to environmental conditions. We suggest that variation among Bd genetic variants due to environmental factors may be an important determinant of disease dynamics for amphibians across a range of diverse environments.

Seasonal epizootics of vector-borne pathogens infecting multiple species are ecologically complex and difficult to forecast. Pathogen transmission potential within the host community is determined by the relative abilities of host species to maintain and transmit the pathogen and by ecological factors influencing contact rates between hosts and vectors. Increasing evidence of strong feeding preferences by a number of vectors suggests that the host community experienced by the pathogen may be very different from the local host community. We developed an empirically informed transmission model for West Nile virus (WNV) in four sites using one vector species (Culex pipiens) and preferred and non-preferred avian hosts. We measured strong feeding preferences for American robins (Turdus migratorius) by Cx. pipiens, quantified as the proportion of Cx. pipiens blood meals from robins in relation to their abundance (feeding index). The model accurately predicted WNV prevalence in Cx. pipiens at three of four sites. Sensitivity analysis revealed feeding preference was the most influential parameter on intensity and timing of peak WNV infection in Cx. pipiens and a threshold feeding index for transmission was identified. Our findings indicate host preference-induced contact heterogeneity is a key mediator of vector-borne pathogen epizootics in multi-species host communities, and should be incorporated into multi-host transmission models.

We explore pathogen virulence evolution during the spatial expansion of an infectious disease epidemic in the presence of a novel host movement trade-off, using a simple, spatially explicit mathematical model. This work is motivated by empirical observations of the Mycoplasma gallisepticum invasion into North American house finch (Haemorhous mexicanus) populations; however, our results likely have important applications to other emerging infectious diseases in mobile hosts. We assume that infection reduces host movement and survival and that across pathogen strains the severity of these reductions increases with pathogen infectiousness. Assuming these trade-offs between pathogen virulence (host mortality), pathogen transmission, and host movement, we find that pathogen virulence levels near the epidemic front (that maximize wave speed) are lower than those that have a short-term growth rate advantage or that ultimately prevail (i.e., are evolutionarily stable) near the epicenter and where infection becomes endemic (i.e., that maximize the pathogen basic reproductive ratio). We predict that, under these trade-offs, less virulent pathogen strains will dominate the periphery of an epidemic and that more virulent strains will increase in frequency after invasion where disease is endemic. These results have important implications for observing and interpreting spatiotemporal epidemic data and may help explain transient virulence dynamics of emerging infectious diseases.

Nonhost species can strongly affect the timing and progression of epidemics. One central interaction—between hosts, their resources, and parasites—remains surprisingly underdeveloped from a theoretical perspective. Furthermore, key epidemiological traits that govern disease spread are known to depend on resource density. We tackle both issues here using models that fuse consumer–resource and epidemiological theory. Motivated by recent studies of a phytoplankton–zooplankton–fungus system, we derive and analyze a family of dynamic models for parasite spread among consumers in which transmission depends on consumer (host) and resource densities. These models yield four key insights. First, host–resource cycling can lower mean host density and inhibit parasite invasion. Second, host–resource cycling can create Allee effects (bistability) if parasites increase mean host density by reducing the amplitude of host–resource cycles. Third, parasites can stabilize host–resource cycles; however, host–resource cycling can also cause disease cycling. Fourth, resource dependence of epidemiological traits helps to govern the relative dominance of these different behaviors. However, these resource dependencies largely have quantitative rather than qualitative effects on these three-species dynamics. Given the extent of these results, host–resource–parasite interactions should become more fundamental components of the burgeoning theory for the community ecology of infectious diseases.

Ordinary differential equations (ODE) models have a wide variety of applications in the fields of mathematics, statistics, and the sciences. Though they are widely used, these models are sometimes viewed as inflexible with respect to the incorporation of time delays. The Generalized Linear Chain Trick (GLCT) serves as a way for modelers to incorporate much more flexible delay or dwell time distribution assumptions than the usual exponential and Erlang distributions. In this paper we demonstrate how the GLCT can be used to generate new ODE models by generalizing or approximating existing models to yield much more general ODEs with phase-type distributed delays or dwell times.