2967663
Description
Mind Map by katie.barclay, updated more than 1 year ago
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Created by katie.barclay
almost 9 years ago
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Modelling in Applied Mathematics
- Dimensional Analysis
- Mass [M], Length [L] & Time [T]
- Can find expressions for dependence
- Have to ensure dimensional consistency
- Mass [M], Length [L] & Time [T]
- Exponential Model
- dN/dt = aN
- Separable
- Solves to give N = nexp(at)
- Separable
- Birth Rate (b) - Death Rate (c) = Growth Rate (a)
- Depends on Growth Rate a and Initial Population n
- Periodic Growth Rate - dN/dt = -acos(wt)N
- dN/dt = aN
- Logistic Model
- dN/dt = a(1 - M/N)N
- a is linear growth rate, M is carrying capacity, n is inital population
- a is linear growth rate, M is carrying capacity, n is inital population
- Separable First Order ODE
- Use partial fractions to solve
- Solution; N(t) = Mn/((M-n)exp(-at) + n)
- Solution; N(t) = Mn/((M-n)exp(-at) + n)
- Depends on a, M & n
- dN/dt = a(1 - M/N)N
- Equilibrium
- Autonomous ODE
- dy/dt = f(y)
- Phase Space and Phase Portrait
- Phase Space and Phase Portrait
- dy/dt = f(y)
- Stable? Asymptotically Stable? Unstable?
- Linearisation
- f'(ye) < 0, a stable. f'(ye) > 0, unstable.
- Find solutions to f(y) = 0
- Find f'(y), then consider sign of solution
- Find f'(y), then consider sign of solution
- f'(ye) < 0, a stable. f'(ye) > 0, unstable.
- Autonomous ODE
- Harvesting
- Constant, fixed fraction, "switch on"
- dN/dt = a(N)N - H
- H is constant
- H is constant
- Critical Harvesting
- Find maximum value of f(N)
- Differentiate, solve = 0, calculate f(N) with solutions.
- Differentiate, solve = 0, calculate f(N) with solutions.
- Find maximum value of f(N)
- Constant, fixed fraction, "switch on"
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