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Let $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined so \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] $D$ is known as the Ducci function and for $\mathbf{u} \in \mathbb{Z}_m^n$, $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ is the Ducci sequence of $\mathbf{u}$. Every Ducci sequence enters a cycle because $\mathbb{Z}_m^n$ is finite. In this paper, we aim to establish an upper bound for how long it will take for a Ducci sequence in $\mathbb{Z}_m^n$ to enter its cycle when $n$ is even.

Understanding the mountain pine beetle's dispersal patterns is critical for evaluating its threat to Canada's boreal forests. It is generally believed that higher beetle densities lead to increased long-distance dispersal due to aggregation pheromones becoming repellent at high densities, causing beetles to seek areas with less competition. However, using helicopter surveys of infested trees, along with statistical models, we find no evidence supporting a positive relationship. Instead, we observe a weak negative association between population density and dispersal at all spatial scales. A possible explanation is that at low population densities, beetles cannot successfully attack healthy trees and must travel farther to find weakened hosts. Even so, the influence of beetle density on dispersal is minor compared to the spatiotemporal variation in the overall (density-independent) scale of dispersal, as revealed by our models. This variation accounts for the MPB's erratic range expansion across western Alberta, which varied from 20 km to 220 km annually., Comment: 24 pages, 8 figures, 1 table

Understanding and predicting recreational fishing activity is important for sustainable fisheries management. However, traditional methods of measuring fishing pressure, such as surveys, can be costly and limited in both time and spatial extent. Predictive models that relate fishing activity to environmental or economic factors typically rely on historical data, which often restricts their spatial applicability due to data scarcity. In this study, high-resolution angler-generated data from an online platform and easily accessible auxiliary data were tested to predict daily boat presence and aerial counts of boats at almost 200 lakes over five years in Ontario, Canada. Lake-information website visits alone enabled predicting daily angler boat presence with 78% accuracy. While incorporating additional environmental, socio-ecological, weather and angler-generated features into machine learning models did not remarkably improve prediction performance of boat presence, they were substantial for the prediction of boat counts. Models achieved an R2 of up to 0.77 at known lakes included in the model training, but they performed poorly for unknown lakes (R2 = 0.21). The results demonstrate the value of integrating angler-generated data from online platforms into predictive models and highlight the potential of machine learning models to enhance fisheries management., Comment: 31 pages

Let $G$ be a finite group, and let $\text{Irr}(G)$ denote the set of the irreducible complex characters of $G$. An element $g\in G$ is called a vanishing element of $G$ if there exists $\chi\in\text{Irr}(G)$ such that $\chi(g)=0$ (i.e., $g$ is a zero of $\chi$) and, in this case, the conjugacy class $g^G$ of $g$ in $G$ is called a vanishing conjugacy class. In this paper we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group $G$ such that every non-linear $\chi\in\text{Irr}(G)$ vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero.

The mountain pine beetle (MPB), a destructive pest native to Western North America, has recently extended its range into Alberta, Canada. Predicting the dispersal of MPB is challenging due to their small size and complex dispersal behavior. Because of these challenges, estimates of MPB's typical dispersal distances have varied widely, ranging from 10 meters to 18 kilometers. Here, we use high-quality data from helicopter and field-crew surveys to parameterize a large number of dispersal kernels. We find that fat-tailed kernels -- those which allow for a small number of long-distance dispersal events -- consistently provide the best fit to the data. Specifically, the radially-symmetric Student's t-distribution with parameters {\nu} = 0.012 and {\rho} = 1.45 stands out as parsimonious and user-friendly; this model predicts a median dispersal distance of 60 meters, but with the 95th percentile of dispersers travelling nearly 5 kilometers. The best-fitting mathematical models have biological interpretations. The Student's t-distribution, derivable as a mixture of diffusive processes with varying settling times, is consistent with observations that most beetles fly short distances while few travel far; early-emerging beetles fly farther; and larger beetles from larger trees exhibit greater variance in flight distance. Finally, we explain why other studies have found such a wide variation in the length scale in MPB dispersal, and we demonstrate that long-distance dispersal events are critical for modelling MPB range expansion., Comment: 32 pages, 11 figures, 8 tables

Fishing is a valuable recreational activity in our society. To assess future fishing activity, identifying variables related to differences in fishing activity, such as gender or Covid-19, is helpful. We conducted a Canada-wide email survey of users of an online fishing platform and analyzed responses with a focus on gender, the impact of Covid-19, and variables directly related to fisher effort. Genders (90.1% male and 9.9% female respondents) significantly differed in demographics, socioeconomic status, and fishing skills but were similar in fishing preferences, fisher effort in terms of trip frequency, and travel distance. For almost half of the fishers, Covid-19 caused a change in trip frequency, determined by the activity level and gender of the fisher. A Bayesian network revealed that travel distance was the main determinant of trip frequency and negatively impacted the fishing activity of 61% of the fishers. Fisher effort was also directly related to fishing expertise. The study shows how online surveys and Bayesian networks can help understand the relationship between fishers' characteristics and activity and predict future fishing trends., Comment: 50 pages

This article investigates neighborhoods' sizes in the enhanced power graph (as known as the cyclic graph) associated with a finite group. In particular, we characterize finite $p$-groups with the smallest maximum size for neighborhoods of nontrivial element in its enhanced power graph., Comment: 6 pages

Large Language Models (LLMs) trained on code are revolutionizing the software development process. Increasingly, code LLMs are being integrated into software development environments to improve the productivity of human programmers, and LLM-based agents are beginning to show promise for handling complex tasks autonomously. Realizing the full potential of code LLMs requires a wide range of capabilities, including code generation, fixing bugs, explaining and documenting code, maintaining repositories, and more. In this work, we introduce the Granite series of decoder-only code models for code generative tasks, trained with code written in 116 programming languages. The Granite Code models family consists of models ranging in size from 3 to 34 billion parameters, suitable for applications ranging from complex application modernization tasks to on-device memory-constrained use cases. Evaluation on a comprehensive set of tasks demonstrates that Granite Code models consistently reaches state-of-the-art performance among available open-source code LLMs. The Granite Code model family was optimized for enterprise software development workflows and performs well across a range of coding tasks (e.g. code generation, fixing and explanation), making it a versatile all around code model. We release all our Granite Code models under an Apache 2.0 license for both research and commercial use., Comment: Corresponding Authors: Rameswar Panda, Ruchir Puri; Equal Contributors: Mayank Mishra, Matt Stallone, Gaoyuan Zhang

Forecasting the occurrence and absence of novel disease outbreaks is essential for disease management. Here, we develop a general model, with no real-world training data, that accurately forecasts outbreaks and non-outbreaks. We propose a novel framework, using a feature-based time series classification method to forecast outbreaks and non-outbreaks. We tested our methods on synthetic data from a Susceptible-Infected-Recovered model for slowly changing, noisy disease dynamics. Outbreak sequences give a transcritical bifurcation within a specified future time window, whereas non-outbreak (null bifurcation) sequences do not. We identified incipient differences in time series of infectives leading to future outbreaks and non-outbreaks. These differences are reflected in 22 statistical features and 5 early warning signal indicators. Classifier performance, given by the area under the receiver-operating curve, ranged from 0.99 for large expanding windows of training data to 0.7 for small rolling windows. Real-world performances of classifiers were tested on two empirical datasets, COVID-19 data from Singapore and SARS data from Hong Kong, with two classifiers exhibiting high accuracy. In summary, we showed that there are statistical features that distinguish outbreak and non-outbreak sequences long before outbreaks occur. We could detect these differences in synthetic and real-world data sets, well before potential outbreaks occur.

The timely detection of disease outbreaks through reliable early warning signals (EWSs) is indispensable for effective public health mitigation strategies. Nevertheless, the intricate dynamics of real-world disease spread, often influenced by diverse sources of noise and limited data in the early stages of outbreaks, pose a significant challenge in developing reliable EWSs, as the performance of existing indicators varies with extrinsic and intrinsic noises. Here, we address the challenge of modeling disease when the measurements are corrupted by additive white noise, multiplicative environmental noise, and demographic noise into a standard epidemic mathematical model. To navigate the complexities introduced by these noise sources, we employ a deep learning algorithm that provides EWS in infectious disease outbreak by training on noise-induced disease-spreading models. The indicator's effectiveness is demonstrated through its application to real-world COVID-19 cases in Edmonton and simulated time series derived from diverse disease spread models affected by noise. Notably, the indicator captures an impending transition in a time series of disease outbreaks and outperforms existing indicators. This study contributes to advancing early warning capabilities by addressing the intricate dynamics inherent in real-world disease spread, presenting a promising avenue for enhancing public health preparedness and response efforts.

Define the Ducci function $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ so \[D(x_1,x_2, ...,x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] Call $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ the Ducci sequence of $\mathbf{u}$. Because $\mathbb{Z}_m^n$ is finite, every Ducci sequence will enter a cycle. In this paper, we will prove that if $n$ is odd and $m=2^lm_1$ where $m_1$ is odd, then the longest it will take for a Ducci sequence to enter its cycle is $l$ iterations. Furthermore, we will prove the set of all tuples in a cycle for $\mathbb{Z}_m^n$ is $\{(x_1, x_2, ..., x_n) \in \mathbb{Z}_m^n \; \mid \; x_1+x_2+ \cdots +x_n \equiv 0 \; \text{mod} \; 2^l\}$.

Let $G$ be a finite group and $N_{\Omega}(G)$ be the intersection of the normalizers of all subgroups belonging to the set $\Omega(G),$ where $\Omega(G)$ is a set of all subgroups of $G$ which have some theoretical group property. In this paper, we show that $N_{\Omega}(G)= Z_{\infty}(G)$ if $\Omega(G)$ is one of the following: (i) the set of all self-normalizing subgroups of $G$; (ii) the set of all subgroups of $G$ satisfying the subnormalizer condition in $G$; (iii) the set of all pronormal subgroups of $G$; (iv) the set of all $\mathscr{H}$-subgroups of $G$; (v) the set of all weakly normal subgroups of $G$; (vi) the set of all $NE$-subgroups of $G$.

Let $G$ be a finite group. The group pseudo-algebra of $G$ is defined as the multi-set $C(G)=\{(d,m_G(d))\mid d\in{\rm Cod}(G)\},$ where $m_G(d)$ is the number of irreducible characters of with codegree $d\in {\rm Cod}(G)$. We show that there exist two finite $p$-groups with distinct orders that have the same group pseudo-algebra, providing an answer to Question 3.2 in \cite{Moreto2023}. In addition, we also discuss under what hypothesis two $p$-groups with the same group pseudo-algebra will be isomorphic.

Let $G$ be a $p$-group for some prime $p$. Let $n$ be the positive integer so that $|G:Z(G)| = p^n$. Suppose $A$ is a maximal abelian subgroup of $G$. Let $$p^l = {\rm max} \{|Z(C_G (g)):Z(G)| : g \in G \setminus Z(G)\},$$ $$p^b = {\rm max} \{|cl(g)| : g \in G \setminus Z(G) \},$$ and $p^a = |A:Z(G)|$. Then we show that $a \ge n/(b+l)$.

Reliable angler behavior data is important for effective fisheries management. Traditionally, such data is gathered through surveys, but an innovative cost-effective approach involves utilizing online platforms and smartphone applications. Previous studies have identified correlations between citizen-reported data from these applications and conventional survey information. However, it remains unclear whether conventional survey data is directly related to citizen-reported data or mainly derived from "intermediate" variables. We applied Bayesian networks to data from conventional surveys, the Angler's Atlas website, the MyCatch smartphone application, and environmental data across Alberta and Ontario, Canada, to detect probabilistic dependencies. Using Bayesian model averaging, we measured the strength of connections between variables. In Ontario, aerial boat counts were directly related to waterbody webpage views, with a 51% probability. In Alberta, creel survey-reported catch rate was directly related to citizen-reported catch rate and fishing duration, though with low probabilities (12% and 6%, respectively). Daily fishing durations were indirectly related, with air temperature and solar radiation as intermediates. These findings suggest that citizen reports can complement traditional methods for evaluating angler behavior., Comment: 18 pages, 2 tables, and 4 figures

Let $D(x_1, x_2, ..., x_n)=(x_1+x_2 \;\text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)$ where $D \in End(\mathbb{Z}_m^n)$ be the Ducci function. The sequence $\{D^k(\mathbf{u})\}_{k=0}^{\infty}$ will eventually enter a cycle. If $n=3$, we aim to establish the longest a cycle can be for a given $m$.

Let $G$ be a finite group and construct a graph $\Delta(G)$ by taking $G\setminus\{1\}$ as the vertex set of $\Delta(G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$ be the set consisting of the universal vertices of $\Delta(G)$ along the identity element. For a solvable group $G$, we present a necessary and sufficient conditon for $K(G)$ to be nontrivial. We also develop a connection between $\Delta(G)$ and $K(G)$ when $|G|$ is divisible by two distinct primes and the diameter of $\Delta(G)$ is $2$., Comment: 8 pages

Prediction of angler behaviors, such as catch rates and angler pressure, is essential to maintaining fish populations and ensuring angler satisfaction. Angler behavior can partly be tracked by online platforms and mobile phone applications that provide fishing activities reported by recreational anglers. Moreover, angler behavior is known to be driven by local site attributes. Here, the prediction of citizen-reported angler behavior was investigated by machine-learning methods using auxiliary data on the environment, socioeconomics, fisheries management objectives, and events at a freshwater body. The goal was to determine whether auxiliary data alone could predict the reported behavior. Different spatial and temporal extents and temporal resolutions were considered. Accuracy scores averaged 88% for monthly predictions at single water bodies and 86% for spatial predictions on a day in a specific region across Canada. At other resolutions and scales, the models only achieved low prediction accuracy of around 60%. The study represents a first attempt at predicting angler behavior in time and space at a large scale and establishes a foundation for potential future expansions in various directions., Comment: 36 pages, 10 figures, 4 tables (including supplementary information)

Let $G$ be a finite group. Recall that an $A$-group is a group whose Sylow subgroups are all abelian. In this paper, we investigate the upper bound on the diameter of the commuting graph of a solvable $A$-group. Assuming that the commuting graph is connected, we show when the derived length of $G$ is 2, the diameter of the commuting graph will be at most 4. In the general case, we show that the diameter of the commuting graph will be at most 6. In both cases, examples are provided to show that the upper bound of the commuting graph cannot be improved., Comment: To appear in print in the Journal of Group Theory

Let the Ducci function $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined as \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m)\] and let the Ducci sequence of $\mathbf{u}$ be the sequence $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$. %In this paper, we will prove that if $n,m$ are powers of $2$, then repeatedly applying $D$ will eventually result in $(0,0,...,0)$, as well as establish an upper bound for how many iterations it will take for this to happen. In this paper, we will provide another proof that for $n=2^k$ and $m=2^l$, that all Ducci sequences will end in $(0,0,...,0)$ and additionally prove that this will happen in at most $2^{k-1}(l+1)$ iterations of $D$.

In this paper, we investigate finite solvable tidy groups. We classify the tidy $\{ p, q \}$-groups. Combining this with a previous result, we are able to characterize the finite tidy solvable groups. Using this characterization, we bound the Fitting height of finite tidy solvable groups and we prove that the quotients of finite tidy solvable groups are tidy.

The Mountain pine beetle (MPB) is a destructive forest pest that undergoes boom-bust cycles every 20-40 years. It also has unusual life-history traits, including a chemosensory affinity for large pine trees, coordinated attacks facilitated by aggregation and anti-aggregation pheromones, and variable dispersal that increases in the absence of trees. Using an empirically calibrated model, we show that MPB's distinctive life-history characteristics are responsible for its distinctive population dynamics. Specifically, three life-history traits are necessary for episodic boom-bust dynamics: density-dependent dispersal, where beetles disperse once most of the large trees have been killed; an Allee effect, which requires a threshold number of beetles to overcome tree defenses; and a short generation time, notably in comparison with their pine tree hosts. In addition to explaining the qualitative behavior of MPB dynamics, these three ingredients explain residual tree density, the duration of outbreaks, and the average waiting time between outbreaks. The peak number of beetles, believed to be the primary factor driving range expansion, is mostly a consequence of MPB's constitutive high fecundity. However, two life history traits -- MPB's size-dependent fecundity and preference for large trees -- are responsible for 25% of the peak number of beetles. By identifying patterns across the extensive MPB literature, and by integrating data across multiple sources, we develop a highly accurate description of MPB's density-dependent interactions with lodgepole pine. We conclude with a simplified mechanistic model, distilled to two difference equations, effectively capturing the essence of MPB outbreak dynamics.

In this paper we study a broad class of non-local advection-diffusion models describing the behaviour of an arbitrary number of interacting species, each moving in response to the non-local presence of others. Our model allows for different non-local interaction kernels for each species and arbitrarily many spatial dimensions. We prove the global existence of both non-negative weak solutions in any spatial dimension and positive classical solutions in one spatial dimension. These results generalise and unify various existing results regarding existence of non-local advection-diffusion equations. We also prove that solutions can blow up in finite time when the detection radius becomes zero, i.e. when the system is local, thus showing that nonlocality is essential for the global existence of solutions. We verify our results with some numerical simulations on 2D spatial domains.

A stellar occultation of Gaia DR3 2646598228351156352 by the Centaur (2060) Chiron was observed from the South African Astronomical Observatory on 2018 November 28 UT. Here we present a positive detection of material surrounding Chiron from the 74-in telescope for this event. Additionally, a global atmosphere is ruled out at the tens of mircobar level for several possible atmospheric compositions. There are multiple 3-sigma drops in the 74-in light curve: three during immersion and two during emersion. Occulting material is located between 242-270 km from the center of the nucleus in the sky plane. Assuming the ring-plane orientation proposed for Chiron from the 2011 occultation, the flux drops are located at 352, 344, and 316 km (immersion), and 357, and 364 km (emersion) from the center, with normal optical depths of 0.26, 0.36, and 0.22 (immersion) and 0.26 and 0.18 (emersion), and equivalent widths between 0.7-1.3 km. This detection is similar to the previously proposed two-ring system and is located within the error bars of that ring-pole plane; however, the normal optical depths are less than half of the previous values, and three features are detected on immersion. These results suggest that the properties of the surrounding material have evolved between the 2011, 2018, and 2022 observations., Comment: Accepted by the Planetary Science Journal 21 Oct. 2023; 13 pages, 9 figures, 4 tables

Nonlocal interactions are ubiquitous in nature and play a central role in many biological systems. In this paper, we perform a bifurcation analysis of a widely-applicable advection-diffusion model with nonlocal advection terms describing the species movements generated by inter-species interactions. We use linear analysis to assess the stability of the constant steady state, then weakly nonlinear analysis to recover the shape and stability of non-homogeneous solutions. Since the system arises from a conservation law, the resulting amplitude equations consist of a Ginzburg-Landau equation coupled with an equation for the zero mode. In particular, this means that supercritical branches from the Ginzburg-Landau equation need not be stable. Indeed, we find that, depending on the parameters, bifurcations can be subcritical (always unstable), stable supercritical, or unstable supercritical. We show numerically that, when small amplitude patterns are unstable, the system exhibits large amplitude patterns and hysteresis, even in supercritical regimes. Finally, we construct bifurcation diagrams by combining our analysis with a previous study of the minimisers of the associated energy functional. Through this approach we reveal parameter regions in which stable small amplitude patterns coexist with strongly modulated solutions.

Efficient coverage for newly developed vaccines requires knowing which groups of individuals will accept the vaccine immediately and which will take longer to accept or never accept. Of those who may eventually accept the vaccine, there are two main types: success-based learners, basing their decisions on others' satisfaction, and myopic rationalists, attending to their own immediate perceived benefit. We used COVID-19 vaccination data to fit a mechanistic model capturing the distinct effects of the two types on the vaccination progress. We estimated that 47 percent of Americans behaved as myopic rationalist with a high variations across the jurisdictions, from 31 percent in Mississippi to 76 percent in Vermont. The proportion was correlated with the vaccination coverage, proportion of votes in favor of Democrats in 2020 presidential election, and education score., Comment: The population was narrowed to those of 12 years and older

In this note, we prove that if $G$ is solvable and ${\rm cod}(\chi)$ is a $p$-power for every nonlinear, monomial, monolithic $\chi\in {\rm Irr}(G)$ or every nonlinear, monomial, monolithic $\chi \in {\rm IBr} (G)$, then $P$ is normal in $G$, where $p$ is a prime and $P$ is a Sylow $p$-subgroup of $G$., Comment: To appear in International Journal of Group Theory

Given a discrete (resp. profinite) group $G$, we define $NCC(G)$ to be the smallest number of cyclic (resp. procyclic) subgroups of $G$ whose conjugates cover $G$. In this paper we determine all residually finite discrete groups with finite NCC and give an almost complete characterization of profinite groups with finite NCC. As a consequence we can show, for example, that for any $k\in\mathbb N$ and any prime $p>3$ the number of finite $p$-groups $P$ with more than $3$ generators and $NCC(P)=k$ is finite., Comment: 52 pages, v2: minor changes

We classify all finite groups $G$ which possesses an element $x\in G$ such that every irreducible character of $G$ takes a root of unity value at $x$.

Animal learning has interested ecologists and psychologists for over a century. Mathematical models that explain how animals store and recall information have gained attention recently. Central to this work is statistical decision theory (SDT), which relates information uptake in animals to Bayesian inference. SDT effectively explains many learning tasks in animals, but extending this theory to predict how animals will learn in changing environments still poses a challenge for ecologists. We addressed this shortcoming with a novel implementation of Bayesian Markov Chain Monte Carlo (MCMC) sampling to simulate how animals sample environmental information and learn as a result. We applied our framework to an individual-based model simulating complex foraging tasks encountered by wild animals. Simulated ``animals" learned behavioral strategies that optimized foraging returns simply by following the principles of an MCMC sampler. In these simulations, behavioral plasticity was most conducive to efficient foraging in unpredictable and uncertain environments. Our model suggests that animals prioritize highly concentrated resources even when these resources are less available overall, in line with existing knowledge on optimal foraging and ideal free distribution theory. Our innovative computational modelling framework can be applied more widely to simulate the learning of many other tasks in animals and humans., Comment: 6 figures, 1 table, 2 figures in the supplementary material

Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations (PDEs), these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection-diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy state can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. Overall, our study reveals a general picture of rich multi-stability in systems of moving and interacting species.

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., Comment: 32 pages, 5 figures

In this paper, we set $\eta (G)$ to be the number of conjugacy classes of maximal cyclic subgroups of a finite group $G$. We compute $\eta (G)$ for all metacyclic $p$-groups. We show that if $G$ is a metacyclic $p$-group of order $p^n$ that is not dihedral, generalized quaternion, or semi-dihedral, then $\eta (G) \ge n-2$, and we determine when equality holds.