# Modelling in Applied Mathematics

Mind Map by katie.barclay, updated more than 1 year ago
 Created by katie.barclay about 6 years ago
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### Description

Mindmap of Dimensional Analysis, Exponential Model, Logistic Model, Equilibrium and Harvesting

## Resource summary

Modelling in Applied Mathematics
1. Dimensional Analysis
1. Mass [M], Length [L] & Time [T]
1. Can find expressions for dependence
1. Have to ensure dimensional consistency
2. Exponential Model
1. dN/dt = aN
1. Separable
1. Solves to give N = nexp(at)
2. Birth Rate (b) - Death Rate (c) = Growth Rate (a)
1. Depends on Growth Rate a and Initial Population n
1. Periodic Growth Rate - dN/dt = -acos(wt)N
2. Logistic Model
1. dN/dt = a(1 - M/N)N
1. a is linear growth rate, M is carrying capacity, n is inital population
2. Separable First Order ODE
1. Use partial fractions to solve
1. Solution; N(t) = Mn/((M-n)exp(-at) + n)
2. Depends on a, M & n
3. Equilibrium
1. Autonomous ODE
1. dy/dt = f(y)
1. Phase Space and Phase Portrait
2. Stable? Asymptotically Stable? Unstable?
1. Linearisation
1. f'(ye) < 0, a stable. f'(ye) > 0, unstable.
1. Find solutions to f(y) = 0
1. Find f'(y), then consider sign of solution
2. Harvesting
1. Constant, fixed fraction, "switch on"
1. dN/dt = a(N)N - H
1. H is constant
2. Critical Harvesting
1. Find maximum value of f(N)
1. Differentiate, solve = 0, calculate f(N) with solutions.

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