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RelativeResourceManager

# RelativeResourceManager - Assignment 1(5 Due date Wednesday...

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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

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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 λ = n (0) = c 1 V 1 + c 2 V 2 + c 3 V 3 +… Solution: n (t) = c 1 λ 1 t V 1 + c 2 λ 2 t V 2 + c 3 λ 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 λ 1 t V 1 + c 2 λ 2 t V 2 + c 3 λ 3 t V 3 +… Unregulated growth of a structured population:

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n (t+1) = A . n (t) Finding the elements of the matrix Ο Ο Ο Ο Ο Ο Π Ξ Μ Μ Μ Μ Μ Μ Ν Λ = p p p p p p p p A 3 2 1 ... ... 3 3 3 3 2 3 1 2 2 3 2 2 2 1 1 ... 1 3 1 2 1 1 1 2 3 4 5 Life cycle
n (t+1) = A . n (t) Ο Ο Ο Ο Ο Ο Ο Ο Π Ξ Μ Μ Μ Μ Μ Μ Μ Μ Ν Λ = 0 0 0 0 0 0 0 ... 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ... ... 1 3 2 1 3 2 1 p p S S S S F F F F A Leslie matrix: matrix population model for age classes Fecundities Sub diagonal: survival rates Lots of zeros due to age constraints

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