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Unformatted text preview: Assignment 1 (5%) Due date: Wednesday Feb 1 1. Unregulated growth Matrix population models 2. Spatial dynamics: immigration/ emigration 3. Stability in Continuous / Discrete time models Late policy: 10% per day; not accepted after Monday Feb 7 Unregulated growth of a structured population: Population with stage or age structure Matrix projection model: n (t): vector of population sizes per class A : population projection matrix Eigenvectors and eigenvalues Vector decomposition along eigenvectors: Equation: n (t+1) = A . n (t) Solution: n (t) = A t n (0) e e e V AV = n (0) = c 1 V 1 + c 2 V 2 + c 3 V 3 + Solution: n (t) = c 1 1 t V 1 + c 2 2 t V 2 + c 3 3 t V 3 + Unregulated growth of a structured population: Population with stage or age structure Matrix projection model: n (t): vector of population sizes per class A : population projection matrix Eigenvectors and eigenvalues Vector decomposition along eigenvectors: Equation: n (t+1) = A . n (t) Solution: n (t) = A t n (0) e e e V AV = n (0) = c 1 V 1 + c 2 V 2 + c 3 V 3 + Solution: n (t) = c 1 1 t V 1 + c 2 2 t V 2 + c 3 3 t V 3 + Long term behaviour: Asymptotic exponential growth rate Dominant eigenvalue (largest one in absolute value) Asymptotic population structure Dominant eigenvector Equation: n (t+1) = A . n (t) Solution: n (t) = c 1 1 t V 1 + c 2 2 t V 2 + c 3 3 t V 3 + Unregulated growth of a structured population: n (t+1) = A . n (t) Finding the elements of the matrix...
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 Winter '12
 SLEIMAN
 Organic chemistry

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