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8 recordsLinked to original sources
USGS2018

Global dynamics of a mutualism–competition model with one resource and multiple consumers

Recent simulation modeling has shown that species can coevolve toward clusters of coexisting consumers exploiting the same limiting resource or resources, with nearly identical ratios of coefficients related to growth and mortality. This paper provides a mathematical basis for such as situation; a full analysis of the global dynamics of a new model for such a class of n -dimensional consumer–resource system, in which a set of consumers with identical growth to mortality ratios compete for the same resource and in which each consumer is mutualistic with the resource. First, we study the system of one resource and two consumers. By theoretical analysis, we demonstrate the expected result that competitive exclusion of one of the consumers can occur when the growth to mortality ratios differ. However, when these ratios are identical, the outcomes are complex. Either equilibrium coexistence or mutual extinction can occur, depending on initial conditions. When there is coexistence, interaction outcomes between the consumers can transition between effective mutualism, parasitism, competition, amensalism and neutralism. We generalize to the global dynamics of a system of one resource and multiple consumers. Changes in one factor, either a parameter or initial density, can determine whether all of the consumers either coexist or go to extinction together. New results are presented showing that multiple competing consumers can coexist on a single resource when they have coevolved toward identical growth to mortality ratios. This coexistence can occur because of feedbacks created by all of the consumers providing a mutualistic service to the resource. This is biologically relevant to the persistence of pollination–mutualisms.

Journal of Mathematical Biology↗
USGS2023

Resource-driven pattern formation in consumer-resource systems with asymmetric dispersal on a plane

This paper considers resource-driven pattern formation in consumer-resource systems. Here, a planar pattern consists of many big patches, and a big patch can be regarded as combination of many patches on the plane. The consumer moves between patches asymmetrically, while the asymmetry is driven by the resource abundance. Based on experimental models with linearly-linked patches, we propose a planarly-linked-patch model with asymmetric dispersal. Using dynamical systems theory, we show global stability of equilibria in the model, and demonstrate how the resource-driven dispersal forms patterns. It is shown that appropriate asymmetry in dispersal would make the consumer persist in the system, even in sink patches. The asymmetry could also make the consumer’s total population abundance larger than that without dispersal. However, inappropriate asymmetry would make the consumer go into extinction, even in source patches. Dispersal rates are also shown to play a role in the persistence and abundance increase. Our results are consistent with experimental observations and provide new insights. Numerical simulations by the model reproduce various vegetation patterns in the real world. This work has potential applications in spatial pattern formation in biological research.

SIAM Journal on Applied Mathematics↗
USGS2023

Bridging the gap between mathematical biology and undergraduate education using applicable natural resource modeling

Mathematical biology is a wide field of study with many venues that undergraduate students can access through research. However, the topics of study for these students can be overwhelming, and many topics of study yield either only trivial results or abstract outcomes that are nonintuitive and diffcult to understand. We have used natural resource modeling, and more specifically, a partnership between academic researchers and federal scientists, as a bridge between undergraduate research and mathematical biology. Our collaboration is an interdisciplinary team that combines biology, mathematics, and statistics professors with government research scientists. As a team, we have mentored students through opportunities such as a Research Experiences for Undergraduates and other projects. In this article, we provide an overview of how we develop questions for undergraduates and outline two case studies, both of which resulted in peer reviewed journal articles. Last, we describe how we also transfer the results from these undergraduate projects to resource managers so the results may be applied to real world problems.

Letters in Biomathematics↗
USGS2024

Bayesian approaches to proxy uncertainty quantification in paleoecology: A mathematical justification and practical integration

Paleoenvironmental data are essential for reconstructing environmental conditions in the distant past, and these reconstructions strongly depend on proxies and age–depth models. Proxies are indirect measurements that substitute for variables that cannot be directly measured, such as past precipitation. Conversely, an age–depth model is a tool that correlates the observed proxy with a specific moment in time. Bayesian age–depth modelling has proved to be a powerful method for estimating sediment ages and their associated uncertainties. However, there remains considerable potential for further integration into proxy analysis. In this paper, we explore a mathematical justification and a computational approach that integrates uncertainty at the age–depth level and propagates it to the proxy scale in the form of a posterior predictive distribution. This method mitigates potential biases and errors by removing the need to assign a single age to a given proxy measurement. It allows for quantifying the likelihood that proxy data values correspond to modelled ages, thus enabling the quantification of uncertainty in both the temporal and proxy value domains. The use of Bayesian statistics in proxy analysis represents a relatively recent advancement. We aim to mathematically justify incorporating the Markov chain Monte Carlo output from age–depth models into proxy analysis and to present a novel methodology for constructing environmental reconstructions using this approach.

Journal of Agricultural, Biological and Environmen↗
USGS2024

Quantifying uncertainty when extrapolating the relationship between snorkel counts and mark-recapture estimates of juvenile salmonids

Snorkel surveys are frequently used to monitor stream-dwelling fish. Inferring local abundance from snorkel surveys is complicated by two primary factors: variable fish detection probabilities and the relative abundance of fish in habitat types below the recommended minimum depth for snorkeling. We examine these factors across three salmonid species ( Oncorhynchus spp . ), 4 years, and 113 location-years in Oregon coastal streams. We calibrate snorkel counts to mark-recapture estimates and develop mathematically explicit expressions that convert a new snorkel count into a probability density of abundance for streams containing mixed habitat types that were and were not snorkeled. Snorkelers detected 63%, 47%, and 39% of juvenile coho salmon ( Oncorhynchus kitsutch ), steelhead trout ( Oncorhynchus mykiss ), and cutthroat trout ( Oncorhynchus clarkii clarkia ) estimated by mark-recapture, respectively, but uncertainty within and among sampling units was high. Further, analytics developed here can be used to infer abundance and uncertainty for habitat types that were not snorkeled. Our quantification of uncertainty arising from using snorkel counts as a proxy for abundance will help managers balance biological risks with available resources.

Oregon↗
USGS2017

A discrete stage-structured model of California newt population dynamics during a period of drought

We introduce a mathematical model for studying the population dynamics under drought of the California newt ( Taricha torosa ), a species of special concern in the state of California. Since 2012, California has experienced a record-setting drought, and multiple studies predict drought conditions currently underway will persist and even increase in severity. Recent declines and local extinctions of California newt populations in Santa Monica Mountain streams motivate our study of the impact of drought on newt population sizes. Although newts are terrestrial salamanders, they migrate to streams each spring to breed and lay eggs. Since egg and larval stages occur in water, a precipitation deficit due to drought conditions reduces the space for newt egg-laying and the necessary habitat for larval development. To mathematically forecast newt population dynamics, we develop a nonlinear system of discrete equations that includes demographic parameters such as survival rates for newt life stages and egg production, which depend on habitat availability and rainfall. We estimate these demographic parameters using 15 years of stream survey data collected from Cold Creek in Los Angeles County, California, and our model captures the observed decline of the parameterized Cold Creek newt population. Based upon data analysis, we predict how the number of available newt egg-laying sites varies with annual precipitation. Our model allows us to make predictions about how the length and severity of drought can affect the likelihood of persistence and the time to critical endangerment of a local newt population. We predict that sustained severe drought will critically endanger the newt population but that the newt population can rebound if a drought is sufficiently short.

Journal of Theoretical Biology↗
USGS2020

Anticipating future learning affects current control decisions: A comparison between passive and active adaptive management in an epidemiological setting

Infectious disease epidemics present a difficult task for policymakers, requiring the implementation of control strategies under significant time constraints and uncertainty. Mathematical models can be used to predict the outcome of control interventions, providing useful information to policymakers in the event of such an epidemic. However, these models suffer in the early stages of an outbreak from a lack of accurate, relevant information regarding the dynamics and spread of the disease and the efficacy of control. As such, recommendations provided by these models are often incorporated in an ad hoc fashion, as and when more reliable information becomes available. In this work, we show that such trial-and-error-type approaches to management, which do not formally take into account the resolution of uncertainty and how control actions affect this, can lead to sub-optimal management outcomes. We compare three approaches to managing a theoretical epidemic: a non-adaptive management (AM) approach that does not use real-time outbreak information to adapt control, a passive AM approach that incorporates real-time information if and when it becomes available, and an active AM approach that explicitly incorporates the future resolution of uncertainty through gathering real-time information into its initial recommendations. The structured framework of active AM encourages the specification of quantifiable objectives, models of system behaviour and possible control and monitoring actions, followed by an iterative learning and control phase that is able to employ complex control optimisations and resolve system uncertainty. The result is a management framework that is able to provide dynamic, long-term projections to help policymakers meet the objectives of management. We investigate in detail the effect of different methods of incorporating up-to-date outbreak information. We find that, even in a highly simplified system, the method of incorporating new data can lead to different results that may influence initial policy decisions, with an active AM approach to management providing better information that can lead to more desirable outcomes from an epidemic.

Journal of Theoretical Biology↗