arXiv subjects

Mark A. Lewis

Publications and source records attributed to Mark A. Lewis.

At least 37 records · Page 2Linked to original sources

arXiv2022

Calculating the timing and probability of arrival for sea lice dispersing between salmon farms

Sea lice are a threat to the health of both wild and farmed salmon and an economic burden for salmon farms. With a free living larval stage, sea lice can disperse tens of kilometers in the ocean between salmon farms, leading to connected sea lice populations that are difficult to control in isolation. In this paper we develop a simple analytical model for the dispersal of sea lice between two salmon farms. From the model we calculate the arrival time distribution of sea lice dispersing between farms, as well as the level of cross-infection of sea lice. We also use numerical flows from a hydrodynamic model, coupled with a particle tracking model, to directly calculate the arrival time of sea lice dispersing between two farms in the Broughton Archipelago, BC, in order to fit our analytical model and find realistic parameter estimates. Using the parametrized analytical model we show that there is often an intermediate inter-farm spacing that maximizes the level of cross infection between farms, and that increased temperatures will lead to increased levels of cross infection.

q-bio.PE↗

arXiv2022

Detecting seasonal episodic-like spatiotemporal memory patterns using animal movement modelling

1. Spatial memory plays a role in the way animals perceive their environments, resulting in memory-informed movement patterns that are observable to ecologists. Developing mathematical techniques to understand how animals use memory in their environments allows for an increased understanding of animal cognition. 2. Here we describe a model that accounts for the memory of seasonal or ephemeral qualities of an animal's environment. The model captures multiple behaviors at once by allowing for resource selection in the present time as well as long-distance navigations to previously visited locations within an animal's home range. 3. We performed a set of analyses on simulated data to test our model, determining that it can provide informative results from as little as one year of discrete-time location data. We also show that the accuracy of model selection and parameter estimation increases with more location data. 4. This model has potential to identify a specific mechanism in which animals use memory to optimize their foraging, by revisiting temporally and predictably variable resources at consistent time lags.

q-bio.QM↗

arXiv2022

A Fisher-KPP model with a nonlocal weighted free boundary: analysis of how habitat boundaries expand, balance or shrink

In this paper, we propose a novel free boundary problem to model the movement of single species with a range boundary. The spatial movement and birth/death processes of the species found within the range boundary are assumed to be governed by the classic Fisher-KPP reaction-diffusion equation, while the movement of a free boundary describing the range limit is assumed to be influenced by the weighted total population inside the range boundary and is described by an integro-differential equation. Our free boundary equation is a generalization of the classical Stefan problem that allows for nonlocal influences on the boundary movement so that range expansion and shrinkage are both possible. In this paper we prove that the new model is well posed and possesses steady state. We show that the spreading speed of the range boundary is smaller than that for the equivalent problem with a Stefan condition. This implies that the nonlocal effect of the weighted total population on the boundary movement slows down the spreading speed of the population. While the classical Stefan condition categorizes asymptotic behavior via a spreading-vanishing dichotomy, the new model extends this dichotomy to a spreading-balancing-vanishing trichotomy. We specifically analyze how habitat boundaries expand, balance or shrink. When the model is extended to have two free boundaries, we observe the steady state scenario, asymmetric shifts, or even boundaries moving synchronously in the same direction. These are newly discovered phenomena in the free boundary problems for animal movement.

math.AP↗

arXiv2021

Coupling the socio-economic and ecological dynamics of cyanobacteria: single lake and network dynamics

In recent decades freshwater lakes have seen an increase in human presence. A common byproduct of this human presence is eutrophication, which readily results in harmful cyanobacteria blooms. In this work we propose a model that couples the socio-economic and ecological dynamics related to cyanobacteria systems. The socio-economic dynamics considers the choices a human population makes regarding whether or not to mitigate their pollution levels. These choices are based on various costs related to social ostracism, social norms, environmental concern and financial burden. The coupled model exhibits bistable dynamics, with one stable state corresponding to high mitigation efforts and low CB abundance, and the other to low mitigation efforts and high CB abundance. Furthermore, we consider social interactions among a network of lakes and present dynamic outcomes pertaining to various associated costs and social situations. In each case we show the potential for regime shifts between levels of cooperation and CB abundance. Social ostracism and pressure are shown to be driving factors in causing such regime shifts.

physics.soc-ph↗

arXiv2021

Local and Global Existence for Non-local Multi-Species Advection-Diffusion Models

Non-local advection is a key process in a range of biological systems, from cells within individuals to the movement of whole organisms. Consequently, in recent years, there has been increasing attention on modelling non-local advection mathematically. These often take the form of partial differential equations, with integral terms modelling the non-locality. One common formalism is the aggregation-diffusion equation, a class of advection diffusion models with non-local advection. This was originally used to model a single population, but has recently been extended to the multi-species case to model the way organisms may alter their movement in the presence of coexistent species. Here we prove existence theorems for a class of non-local multi-species advection-diffusion models, with an arbitrary number of co-existent species. We prove global existence for models in n=1 spatial dimension and local existence for n>1. We describe an efficient spectral method for numerically solving these models and provide example simulation output. Overall, this helps provide a solid mathematical foundation for studying the effect of inter-species interactions on movement and space use.

math.AP↗

arXiv2021

A framework for studying transients in marine metapopulations

Transient dynamics can often differ drastically from the asymptotic dynamics of systems. In this paper we provide a unifying framework for analysing transient dynamics in marine metapopulations, from the choice of norms to the addition of stage structure. We use the $\ell_1$ norm, because of its biological interpretation, to extend the transient metrics of reactivity and attenuation to marine metapopulations, and use examples to compare these metrics under the more commonly used $\ell_2$ norm. We then connect the reactivity and attenuation of marine metapopulations to the source-sink distribution of habitat patches and demonstrate how to meaningfully measure reactivity when metapopulations are stage-structured.

q-bio.PE↗

arXiv2020

A hybrid gravity and route choice model to assess vector traffic in large-scale road networks

Human traffic along roads can be a major vector for infectious diseases and invasive species. Though most road traffic is local, a small number of long-distance trips can suffice to move an invasion or disease front forward. Therefore, understanding how many agents travel over long distances and which routes they choose is key to successful management of diseases and invasions. Stochastic gravity models have been used to estimate the distribution of trips between origins and destinations of agents. However, in large-scale systems it is hard to collect the data required to fit these models, as the number of long-distance travellers is small, and origins and destinations can have multiple access points. Therefore, gravity models often provide only relative measures of the agent flow. Furthermore, gravity models yield no insights into which roads agents use. We resolve these issues by combining a stochastic gravity model with a stochastic route choice model. Our hybrid model can be fitted to survey data collected at roads that are used by many long-distance travellers. This decreases the sampling effort, allows us to obtain absolute predictions of both vector pressure and pathways, and permits rigorous model validation. After introducing our approach in general terms, we demonstrate its benefits by applying it to the potential invasion of zebra and quagga mussels (Dreissena spp.) to the Canadian province British Columbia. The model yields an R-squared value of 0.73 for variance-corrected agent counts at survey locations.

physics.soc-ph↗

arXiv2020

A robust and efficient algorithm to find profile likelihood confidence intervals

Profile likelihood confidence intervals are a robust alternative to Wald's method if the asymptotic properties of the maximum likelihood estimator are not met. However, the constrained optimization problem defining profile likelihood confidence intervals can be difficult to solve in these situations, because the likelihood function may exhibit unfavorable properties. As a result, existing methods may be inefficient and yield misleading results. In this paper, we address this problem by computing profile likelihood confidence intervals via a trust-region approach, where steps computed based on local approximations are constrained to regions where these approximations are sufficiently precise. As our algorithm also accounts for numerical issues arising if the likelihood function is strongly non-linear or parameters are not estimable, the method is applicable in many scenarios where earlier approaches are shown to be unreliable. To demonstrate its potential in applications, we apply our algorithm to benchmark problems and compare it with 6 existing approaches to compute profile likelihood confidence intervals. Our algorithm consistently achieved higher success rates than any competitor while also being among the quickest methods. As our algorithm can be applied to compute both confidence intervals of parameters and model predictions, it is useful in a wide range of scenarios.

stat.CO↗

arXiv2020

Managing aquatic invasions: optimal locations and operating times for watercraft inspection stations

Aquatic invasive species (AIS) cause significant ecological and economic damages around the world. A major spread mechanism for AIS is traffic of boaters transporting their watercraft from invaded to uninvaded waterbodies. To inhibit the spread of AIS, several Canadian provinces and American states set up watercraft inspection stations at roadsides, where potentially infested boats are screened for AIS and, if necessary, decontaminated. However, since budgets for AIS control are limited, watercraft inspection stations can only be operated at specific locations and daytimes. Though theoretical studies provide managers with general guidelines for AIS management, more specific results are needed to determine when and where watercraft inspections would be most effective. This is the subject of this paper. We show how linear integer programming techniques can be used to optimize watercraft inspection policies under budget constraints. We introduce our approach as a general framework and apply it to the prevention of the spread of zebra and quagga mussels (Dreissena spp.) to the Canadian province British Columbia. We consider a variety of scenarios and show how variations in budget constraints, propagule sources, and model uncertainty affect the optimal policy. Based on these results, we identify simple, generally applicable principles for optimal AIS management.

q-bio.QM↗

arXiv2019

Analysis of propagation for impulsive reaction-diffusion models

We study a hybrid impulsive reaction-advection-diffusion model given by a reaction-advection-diffusion equation composed with a discrete-time map in space dimension $n\in\mathbb N$. The reaction-advection-diffusion equation takes the form \begin{equation*}\label{} u^{(m)}_t = \text{div}(A\nabla u^{(m)}-q u^{(m)}) + f(u^{(m)}) \quad \text{for} \ \ (x,t)\in\mathbb R^n \times (0,1] , \end{equation*} for some function $f$, a drift $q$ and a diffusion matrix $A$. When the discrete-time map is local in space we use $N_m(x)$ to denote the density of population at a point $x$ at the beginning of reproductive season in the $m$th year and when the map is nonlocal we use $u_m(x)$. The local discrete-time map is \begin{eqnarray*}\label{}\left\{ \begin{array}{lcl} u^{(m)}(x,0) = g(N_m(x)) \quad \text{for} \ \ x\in \mathbb R^n , \\ N_{m+1}(x):=u^{(m)}(x,1) \quad \text{for} \ \ x\in \mathbb R^n , \end{array}\right. \end{eqnarray*} for some function $g$. The nonlocal discrete time map is \begin{eqnarray*}\label{}\left\{ \begin{array}{lcl} u^{(m)}(x,0) = u_{m}(x) \quad \text{for} \ \ x\in \mathbb R^n , \\ \label{mainb2} u_{m+1}(x) := g\left(\int_{\mathbb R^n} K(x-y)u^{(m)}(y,1) dy\right) \quad \text{for} \ \ x\in \mathbb R^n, \end{array}\right. \end{eqnarray*} when $K$ is a nonnegative normalized kernel. SEE THE ARTICLE FOR COMPLETE ABSTRACT.

math.AP↗

arXiv2019

Second-generation stoichiometric mathematical model to predict methane emissions from oil sands tailings

Microbial metabolism of fugitive hydrocarbons produces greenhouse gas (GHG) emissions from oil sands tailings ponds (OSTP) and end pit lakes (EPL) that retain semisolid wastes from surface mining of oil sands ores. Predicting GHG production, particularly methane (CH4), would help oil sands operators mitigate tailings emissions and would assist regulators evaluating the trajectory of reclamation scenarios. Using empirical datasets from laboratory incubation of OSTP sediments with pertinent hydrocarbons, we developed a stoichiometric model for CH4 generation by indigenous microbes. This model improved on previous first-approximation models by considering long-term biodegradation kinetics for 18 relevant hydrocarbons from three different oil sands operations, lag times, nutrient limitations, and microbial growth and death rates. Laboratory measurements were used to estimate model parameter values and to validate the new model. Goodness of fit analysis showed that the stoichiometric model predicted CH4 production well; normalized mean square error analysis revealed that it surpassed previous models. Comparison of model predictions with field measurements of CH4 emissions further validated the new model. Importantly, the model also identified parameters that are currently lacking but are needed to enable future robust modeling of CH4 production from OSTP and EPL in situ.

q-bio.QM↗

arXiv2019

Spatial memory and taxis-driven pattern formation in model ecosystems

Mathematical models of spatial population dynamics typically focus on the interplay between dispersal events and birth/death processes. However, for many animal communities, significant arrangement in space can occur on shorter timescales, where births and deaths are negligible. This phenomenon is particularly prevalent in populations of larger, vertebrate animals who often reproduce only once per year or less. To understand spatial arrangements of animal communities on such timescales, we use a class of diffusion-taxis equations for modelling inter-population movement responses between $N \geq 2$ populations. These systems of equations incorporate the effect on animal movement of both the current presence of other populations and the memory of past presence encoded either in the environment or in the minds of animals. We give general criteria for the spontaneous formation of both stationary and oscillatory patterns, via linear pattern formation analysis. For $N=2$, we classify completely the pattern formation properties using a combination of linear analysis and non-linear energy functionals. In this case, the only patterns that can occur asymptotically in time are stationary. However, for $N \geq 3$, oscillatory patterns can occur asymptotically, giving rise to a sequence of period-doubling bifurcations leading to patterns with no obvious regularity, a hallmark of chaos. Our study highlights the importance of understanding between-population animal movement for understanding spatial species distributions, something that is typically ignored in species distribution modelling, and so develops a new paradigm for spatial population dynamics.

q-bio.PE↗

arXiv2016

Climate change and integrodifference equations in a stochastic environment

Climate change impacts population distributions, forcing some species to migrate poleward if they are to survive and keep up with the suitable habitat that is shifting with the temperature isoclines. Previous studies have analyzed whether populations have the capacity to keep up with shifting temperature isoclines, and have mathematically determined the combination of growth and dispersal that is needed to achieve this. However, the rate of isocline movement can be highly variable, with much uncertainty associated with yearly shifts. The same is true for population growth rates. Growth rates can be variable and uncertain, even within suitable habitats for growth. In this paper we reanalyze the question of population persistence in the context of the uncertainty and variability in isocline shifts and rates of growth. Specifically, we employ a stochastic integrodifference equation model on a patch of suitable habitat that shifts poleward at a random rate. We derive a metric describing the asymptotic growth rate of the linearized operator of the stochastic model and yielding a threshold criterion for population persistence. We demonstrate that the variability in the yearly shift and in the growth rate has a significant negative effect on the persistence in the sense that it decreases the threshold criterion for population persistence. Mathematically, we show how the persistence metric can be connected to the principal eigenvalue problem for a related integral operator, at least for the case where isocline shifting speed is deterministic. Analysis of dynamics for the case where the dispersal kernel is Gaussian leads to the existence of a critical shifting speed. Finally we illustrate our different results for butterfly population using numerical simulations and demonstrate how increased variances in isocline shifts and growth rates translate into decreased likelihoods of persistence.

math.AP↗

arXiv2016

State-space models' dirty little secrets: even simple linear Gaussian models can have estimation problems

State-space models (SSMs) are increasingly used in ecology to model time-series such as animal movement paths and population dynamics. This type of hierarchical model is often structured to account for two levels of variability: biological stochasticity and measurement error. SSMs are flexible. They can model linear and nonlinear processes using a variety of statistical distributions. Recent ecological SSMs are often complex, with a large number of parameters to estimate. Through a simulation study, we show that even simple linear Gaussian SSMs can suffer from parameter- and state-estimation problems. We demonstrate that these problems occur primarily when measurement error is larger than biological stochasticity, the condition that often drives ecologists to use SSMs. Using an animal movement example, we show how these estimation problems can affect ecological inference. Biased parameter estimates of a SSM describing the movement of polar bears (\textit{Ursus maritimus}) result in overestimating their energy expenditure. We suggest potential solutions, but show that it often remains difficult to estimate parameters. While SSMs are powerful tools, they can give misleading results and we urge ecologists to assess whether the parameters can be estimated accurately before drawing ecological conclusions from their results.

q-bio.QM↗

arXiv2016

A unifying framework for quantifying the nature of animal interactions

Collective phenomena, whereby agent-agent interactions determine spatial patterns, are ubiquitous in the animal kingdom. On the other hand, movement and space use are also greatly influenced by the interactions between animals and their environment. Despite both types of interaction fundamentally influencing animal behaviour, there has hitherto been no unifying framework for the models proposed in both areas. Here, we construct a general method for inferring population-level spatial patterns from underlying individual movement and interaction processes, a key ingredient in building a statistical mechanics for ecological systems. We show that resource selection functions, as well as several examples of collective motion models, arise as special cases of our framework, thus bringing together resource selection analysis and collective animal behaviour into a single theory. In particular, we focus on combining the various mechanistic models of territorial interactions in the literature with step selection functions, by incorporate interactions into the step selection framework and demonstrating how to derive territorial patterns from the resulting models. We demonstrate the efficacy of our model by application to a population of insectivore birds in the Amazon rainforest.

q-bio.QM↗

arXiv2015

Integrated step selection analysis: bridging the gap between resource selection and animal movement

A resource selection function is a model of the likelihood that an available spatial unit will be used by an animal, given its resource value. But how do we appropriately define availability? Step-selection analysis deals with this problem at the scale of the observed positional data, by matching each used step (connecting two consecutive observed positions of the animal) with a set of available steps randomly sampled from a distribution of observed steps or their characteristics. Here we present a simple extension to this approach, termed integrated step-selection analysis (iSSA), which relaxes the implicit assumption that observed movement attributes (i.e. velocities and their temporal autocorrelations) are independent of resource selection. Instead, iSSA relies on simultaneously estimating movement and resource-selection parameters, thus allowing simple likelihood-based inference of resource selection within a mechanistic movement model. We provide theoretical underpinning of iSSA, as well as practical guidelines to its implementation. Using computer simulations, we evaluate the inferential and predictive capacity of iSSA compared to currently used methods. Our work demonstrates the utility of iSSA as a general, flexible and user-friendly approach for both evaluating a variety of ecological hypotheses, and predicting future ecological patterns.

q-bio.QM↗

arXiv2015

Robustness of movement models: can models bridge the gap between temporal scales of data sets and behavioural processes?

Discrete-time random walks and their extensions are common tools for analyzing animal movement data. In these analyses, resolution of temporal discretization is a critical feature. Ideally, a model both mirrors the relevant temporal scale of the biological process of interest and matches the data sampling rate. Challenges arise when resolution of data is too coarse due to technological constraints, or when we wish to extrapolate results or compare results obtained from data with different resolutions. Drawing loosely on the concept of robustness in statistics, we propose a rigorous mathematical framework for studying movement models' robustness against changes in temporal resolution. In this framework, we define varying levels of robustness as formal model properties, focusing on random walk models with spatially-explicit component. With the new framework, we can investigate whether models can validly be applied to data across varying temporal resolutions and how we can account for these different resolutions in statistical inference results. We apply the new framework to movement-based resource selection models, demonstrating both analytical and numerical calculations, as well as a Monte Carlo simulation approach. While exact robustness is rare, the concept of approximate robustness provides a promising new direction for analyzing movement models.

q-bio.QM↗

arXiv2015

Differentiating the Lévy walk from a composite correlated random walk

1. Understanding how to find targets with very limited information is a topic of interest in many disciplines. In ecology, such research has often focused on the development of two movement models: i) the Lévy walk and; ii) the composite correlated random walk and its associated area-restricted search behaviour. Although the processes underlying these models differ, they can produce similar movement patterns. Due to this similarity and because of their disparate formulation, current methods cannot reliably differentiate between these two models. 2. Here, we present a method that differentiates between the two models. It consists of likelihood functions, including one for a hidden Markov model, and associated statistical measures that assess the relative support for and absolute fit of each model. 3. Using a simulation study, we show that our method can differentiate between the two search models over a range of parameter values. Using the movement data of two polar bears (\textit{Ursus maritimus}), we show that the method can be applied to complex, real-world movement paths. 4. By providing the means to differentiate between the two most prominent search models in the literature, and a framework that could be extended to include other models, we facilitate further research into the strategies animals use to find resources.

q-bio.QM↗