Modelling in Applied Mathematics

katie.barclay
Mind Map by katie.barclay, updated more than 1 year ago
katie.barclay
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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