PIMS lectures
Title: Population persistence in a source-sink metapopulation
Speaker: Julien Arino, University of Manitoba
Date and time:
22 Nov 2018,
3:00pm -
4:20pm
Location: Meet and Greet in DTB A514 then talk in CLE A202
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Please join us for a meet and greet at 3 pm in DTB A514, refreshments will be served. The talk will take place at 3:30 pm after the meet and greet.
ABSTRACT:
The problem of source-sink dynamics in mathematical ecology originated with a 1969 paper of Levins. The idea is that a species lives in an environment that is heterogeneous: some locations are favourable to the species persistence (sources), while others are unfavourable and can only be populated if there is an inflow of individuals to these locations (sinks). This setup has been studied extensively in the case of so-called Levins-type metapopulation models, which count the number of locations (patches) in various states and couple them implicitly. The case of metapopulation models with explicit movement, where individuals move between locations, is less known.
In this talk, I will discuss the solution to a simple problem set in the latter context: is there a critical number of the number of patches that are sources such that the population persists in the entire system when the number of sources is above the threshold? An existence result is provided that relies heavily on matrix analysis, illustrating the power of linear algebra in this type of large scale problem. In a particular case, the principle of equitable partitions allows us to obtain an explicit form for the threshold. This is joint work with Nicolas Bajeux (University of Manitoba) and Steve Kirkland (University of Manitoba).
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Title: Kemeny's Constant for Markov Chains
Speaker: Stephen Kirkland, University of Manitoba
Date and time:
18 Oct 2018,
3:30pm -
4:20pm
Location: Clearihue A308
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There will be refreshments at a 3:00 pm reception prior to the talk.
Abstract:
Markov chains are a much-studied class of stochastic processes, and it is well- known that if the transition matrix A associated with a Markov chain possesses a certain property (called primitivity), then the long-term behaviour of the Markov chain is described by a particular eigenvector of A, known as the stationary distribution vector. Rather less well-known is Kemeny’s constant for a Markov chain, which can be interpreted in terms of the expected number of time steps taken to arrive at a randomly chosen state, starting from initial state i. In particular, if Kemeny’s constant is small, then we can think of the Markov chain as possessing good mixing properties.
In this talk, we will give a short overview of Kemeny’s constant, and discuss some results dealing with the problem of minimising Kemeny’s constant over transition matrices that are subject to various constraints. We will also describe some results showing that Kemeny’s constant can exhibit surprising behaviour when the transition matrix is perturbed. Throughout, the techniques used rely on matrix theory and graph theory.
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