This year marks the 200th birthday of Charles Darwin and the 150th anniversary of the publication of On the Origin of Species, where he established that modern species are related through descent from common ancestors and proposed natural selection as the primary cause of this "descent with modification". With the advance of molecular biology and genome science biologists have been studying the evolution at the molecular level, or molecular evolution of macromolecules including DNA, RNA and proteins as well as the whole genomes. Statistical models and computational methods have been developed to infer the rates and patterns of changes in the molecular sequences during evolutionary time and to infer the evolutionary history of organisms and the macromolecules. I will show i) some observed patterns of the amino acid states on the protein sequences from different organisms, ii) introduce current models of evolutionary rates of the changes and a Markov model for protein evolution, and iii) describe a maximum likelihood approach commonly used to estimate phylogenies and reconstruct the tree of life.
Alex Georgallas,Alex Georgallas, Greg Bishop, Yousef Papadopoulos and Alan Fredeen
A Monte Carlo simulation of plant growth is presented. In the simulation the plants grow logistically but are disadvantaged if they experience shading by neighbouring plants. A normal distribution of growth rates is assigned randomly to the two-dimensional plant population. Under these conditions the plant population experiences a bifurcation resulting in a bimodal distribution of plant heights (short and tall). However, an analytical solution of the differential equations describing this growth shows no such anomalous solutions. Furthermore, similar bimodal distributions are seen in purely stochastic deposition models such as diffusion limited aggregation (DLA). Most studies of DLA focus on the fractal structure of the Brownian trees (plants), ignoring the number and their overall size distribution. This study of shading provides a link between a Monte Carlo simulation of an analytical model and stochastic models of DLA. Experimental verification of these results is currently being undertaken in trials of grass growth under controlled light.
Luju Liu,The statistical data of Tuberculosis (TB) cases shows seasonal fluctuation in many countries. A TB model incorporating seasonality is developed and analyzed. The basic reproduction ratio $R_{0}$ is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the disease always eventually disappears if $R_{0}<1$, and the disease is uniformly persistent and there exists at least one positive periodic solution if $R_{0}>1$. Numerical result indicates that there may be a unique positive periodic solution which is globally asymptotically stable if $R_{0}>1$. Parameter values of the model are estimated according to demographic and epidemiological data in China. The simulation result is in good agreement with the seasonal variation of the reported cases of active TB in China.
S. Swaminathan,Recognition is the foundation of the extreme specificity of some biological processes and is the basis of control. A mathematical model is developed to formulate the system response associated with recognition processes. Recognition is derived as one aspect of a general phenomenology of the causal relationship. A comprehensive functional context is introduced in which recognition processes occur. The model is applied to Enzyme Substrate recognition process.
Jeff Musgrave,A biological invasion is the introduction and spread of exotic organisms outside their native range.
In this thesis, the Emerald ash Borer (\textit{Agrilus planipennis}), an invasive species first discovered in Michigan in 2002, is studied. An important prediction of biological invasions is the spread rate at which a population front is moving.
In this talk I present an individual-based, probabilistic model which describes the host-searching and egg deposition behaviour of individual insects. The model is simulated on a variety of landscapes and some general conclusions about the effect of landscape connectivity on spread rates are made. The simulation model can also be expressed as a system of partial differential equations. Several versions of the PDE model are examined under different simplifying assumptions and obtained spread rates are compared with results from the simulation model. Some preliminary analysis of one of the PDE models is also presented.
Majid Jaberi Douraki,We study the non-autonomous difference equation Xn+1 = F (n, Xn , Xn-k ) - H(n, Xn ), n >eq; 0, where X-k , ..., X-1 , X0 are positive real numbers, F(n, Xn , Xn-k ) is represented by a delay rational difference equation, and H denotes the removal function of individuals during each time period as harvesting or functional responses. In particular, the periodic behavior, the at- tractivity of solutions and the stability of solutions are discussed in detail. The impact of seasonality on the behavior of solutions is also considered.
Sylvia Churcher,Available methods for calculating the biodiversity of microbial samples
The assessment of community diversity is an important and well-studied component of macro-ecology.
New environmental DNA sequencing or 'metagenomic' techniques are now being used to pose these
questions of microbial populations in many environments. Here we examine microbial
DNA sequences sampled as part of the recent Global Ocean Sampling Expedition and apply three types
of diversity measures: a single site index, and multi-site indices that either consider or ignore the degree
of phylogenetic relatedness of different taxonomic groups. Learning how to classify communities of
microbes as "similar" or "different" is the next step needed to understand how different microbes are
distributed throughout the world around us and identifying what environmental factors influence such
communities.
In this paper, we present an age-structured model of malaria transmission. We first introduce the basic reproduction ratio for this model and then show that the disease-free periodic state is globally asymptotically stable when $\mathcal{R}_0<1$, and there exists at least one positive periodic state and the disease persists when $\mathcal{R}_0>1$. We further use these analytic results to study the malaria transmission cases in KwaZulu-Natal Province, South Africa, to determine how well they represent the biological system and, consequently, how useful their predictions are. Some sensitivity analysis of $\mathcal{R}_0$ is performed, and in particular, the potential impact of climate change on seasonal transmission and populations at risk of the disease is analyzed.
Lisa Kanary Affiliation: University of New Brunswick Model of Aquatic Invasive Species Search EffectivenessWhen an aquatic invasive species is reported, a search effort follows. This study compares the efforts and results of two search methods: dive and sock lift. An experiment took place at 9 Mile Creek in Prince Edward Island in August of 2008. Four sequences of eleven mussel socks (the substrate tunicates prefer to settle upon) were chosen to deploy a set number of decoy tunicates upon. A dive and sock lift search was performed in each sequence. A model was fit to the data collected to determine which tunicates were a true positive, false positive, false negative or true nugeative. The model used combines a binomial (the probability of detecting a tunicate hidden on a sock) and poisson (the false positive probability) distribution. Our goal is to find the least proportion of false positives between the two search methods.
Chad S Gilbert Department of Engineering Mathematics & Internetworking, Dalhousie UniversitySea scallops (Placopecten magellanicus) on Georges Bank are important both ecologically and as commercial fisheries. The population is comprised of 3 distinct scallop beds, which are connected via transport of planktonic larvae spawned in the spring and fall. In order to develop sustainable management strategies, we need to quantify how the different beds and spawning times contribute to larval recruitment and how these recruitment patterns affect population dynamics.
Here, we calculate larval drift and retention using a 3D particle-tracking model, which couples seasonal currents, larval swimming, turbulent dispersion and larval development. Bed connectivity is quantified, and then analyzed using a novel Markov-chain approach. Sensitivity to variation in adult distribution, temperature-dependent growth, reproduction and mortality is assessed and the implications for management of the population are discussed.
Wendy Gentleman Dept. of Engineering Mathematics and Internetworking, DalhousieModel predictions of food-web dynamics requires characterization of the functional response, or how ingestion rates vary with prey density. The mathematical form of this dependency can significantly affect results, e.g. use of different ingestion functions in a Nutrient-Phytoplankton-Zooplankton (NPZ) model result in oscillating vs. constant densities. Contrary to previous studies, we show these results are not due to zooplankton satiation vs. non-satiation. Analysis of a predator-prey model is used to derive the necessary condition for ecological stability, which is related to food-limited clearance rates. Sensitivity studies demonstrate that zooplankton clearance rates have a strong influence on the dynamics of more complex models, including those accounting for acclimation time lags. These results are discussed in terms of practical advice to modellers who face uncertainty in choosing expressions for the functional response.
Anna Neuheimer Engineering Mathematics and InternetworkingIndividual-based models (IBMs) allow us to bridge the gap between the level at which environmental impacts occur (individuals) and the level at which observations are made (populations). Here, we present a conceptually and programmatically simple approach for building a stochastic stage-based IBM that can be used to statistically test hypotheses. A unique aspect of our model is a "Fitness" variable that characterizes individual variability, and may be prescribed a priori (e.g. genetic), or solved for dynamically based on individual history, environment, condition, etc. We demonstrate the power of our model to test hypotheses concerning environmental effects on copepod physiology through simulation of the seasonal dynamics of Calanus finmarchicus in the northwest Atlantic. We also discuss how this IBM structure can be adapted for other applications.
Shovonlal Roy Department of Oceanography, Dalhousi UniversityPhytoplankton - the unicellular primary producers in aquatic ecosystems - are prominently size-structured; different cell sizes are responsible for diverse bio-physical activities of phytoplankton species including primary production, response to light, respiration and response to change in oceanic mixed-layer. Here I develop a mathematical model to include size-effects on important parameters associated with biomass evolution of phytoplankton; and address the question as to how the diversity of size-structured biomass is maintained under changes in oceanic mixed-layer. The model is analyzed to determine conditions under which a stable size-structured biomass is achieved; and simulations are performed to investigate the effects of mixed-layer changes on the evolution of time-varying biomass.
Ammar M. Sarhan,Statistical inference of the parameters included in a competing risks model is con- sidered in this paper. It is assumed that the competing risks follow Chen distributions. The Maximum likelihood procedure and Bayes methods are used to derive point and interval estimations of the unknown parameters. The relative risks due to each cause of failure are investigated. Sets of real data are analyzed in order to:
Keywords: Exponential distribution, Weibull distribution, Generalized exponential
distribution, Maximum likelihood method, Bayes method, Biological data.
We propose a second-order Markov likelihood method for estimating animal density from binary acoustic data. The data arise from an acoustic survey consisting of a transect line with regularly-spaced listening stations. At each station, a binary variable is recorded, indicating whether vocalizations are detected or not. Our method can estimate not only density, but also the probability that an animal is vocalizing, and the range of detection. We use profile likelihood confidence intervals to quantify uncertainty in the estimates. Simulations show that our method compares favourably to conventional line transect distance sampling. Distance sampling requires measurements on the distances of individual animals from the transect, which is not easily obtained with vocalization data. The methods are illustrated using real acoustic data from whale surveys. The techniques may also be useful in surveys with non-acoustic detection of animals.
Tamara Romanuk DalhousieA central and perhaps insurmountable challenge of invasion ecology is to predict which combinations of species and habitats most effectively promote and prevent biological invasions. The Global Invasive Species Database (http://www.issg.org) suggests that there are no natural 'rules' that govern the processes of invasion that have any real predictive value because any "generalizations about invaders over too wide a taxonomic range, such as all species, or all insects, or all angiosperms, invariably [lead to] too many exceptions to be useful". Despite this well-founded pessimism, developing better models to predict when and where species will successfully invade is imperative. Here, we describe a successful exploration for general rules that govern invasion success within models of ecological networks that integrate models of food-web structure and nonlinear dynamics. The biological foundation of these models is the fundamental requirement of metabolic energy for all life and organisms' different feeding roles in consuming and providing that energy. Based on this foundation, we develop a food-web theory of species' invasions that describes how each invader's trophic function is mediated by other's species' trophic activities and the structural topology of the invaded food web. Our results suggest that general rules governing invasion success in nature emerge from computational explorations of invasions in theory. More generally, our results also suggest that tackling the challenge of predicting the properties of species and habitats that promote or inhibit invasions from a food-web perspective may aid ecologists in identifying rules that govern invasions in natural ecosystems.
Andy Foster Dept of Mathematics and Statistics, Memorial UniversityDynamical systems models are commonly used to predict species biomass and to estimate the spread of pollutants. However, such models have rarely been used to investigate the situation where biological species are subjected to moderate contamination of their environment. In this talk, I will outline the general problem of mercury contamination in ecosystems. Then I will describe a dynamical systems model for predicting methylmercury contamination levels in aquatic food webs. The model is a low-dimensional system of ODEs, relating the biomass of the fish in selected trophic levels and the concentrations of mercury in these fish populations and the environment. The research is in collaboration with Cathy McFadyen (M.Sc. in Environmental Science, MUN) and Robin Anderson (Scientist, Fisheries and Oceans Canada), in affiliation with the Collaborative Mercury Research Network. Our goal is to predict the methylmercury content in food fish in specific lakes in Labrador, under various scenarios of environmental mercury loading.
Deborah Buszard, David Iron and Brian Pellerin Dalhousie UniversityBiennial bearing, the yearly oscillation in fruit number, is a serious problem for modern apple growers who control it by removing (thinning) flowers and fruitlets. A mathematical model was developed to explain the effects of thinning on return fruit number and an optimization algorithm was used to remove biennialism. Average individual fruit weight was calculated from total yield and used to estimate fruit size distribution as the average of a Poisson process. Results suggest that yearly fruit number could be optimized to remove all biennialism and transfer the fruit weight of culls into marketable fruit sizes. Optimal fruit number and yearly crop load may differ between rootstocks and vary according to the growing conditions. Comparing data sets with and without effective thinning methods, the model appears to be heavily dependent on the thinning regime and underestimates optimal yield for unthinned trees by as much as 70%.
Sina Adl Organic agriculture cannot replace conventional agricultureIn this paper, we consider the effect of increasing the area of agricultural land under organic practices. We assumed that organic agriculture does not have effective means of pathogen control. We model pathogen dispersal with a diffusive logistic equation in which the growth/death rate is spatially heterogeneous. We find that if the ratio of the organic plots to conventional plots is below a certain threshold, the pest population is kept small. Above this threshold, the pest population in the organic plot grows rapidly. In this case, the organic plot will act as a source of pest to the surrounding regions, and will always infect organic plots as they become more closely spaced. We conclude that this will have significant impact on regional agricultural productivity and food security. We recommend that regional estimates of this threshold are necessary to manage the growth of organic agriculture.
David IronWe consider a gene regulatory network that utilizes auto- activation and cross-inhibition to establish and maintain stable boundaries of gene expression. We find that in the presence of a general activator, neither auto-activation, nor cross-inhibition alone are sufficient to maintain stable sharp boundaries of morphogen production. The minimal requirements for a self-organizing system are a coupled sys- tem of two morphogens in-which the auto-activation and cross-inhibition have Hill coeficients strictly greater than one.