Modelling in Applied Mathematics

Modelling in Applied Mathematics

1. Dimensional Analysis

   • Mass [M], Length [L] & Time [T]
   • Can find expressions for dependence
   • Have to ensure dimensional consistency

2. Exponential Model

   • dN/dt = aN
     • Separable
     • Solves to give N = nexp(at)
   • 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

3. Logistic Model

   • dN/dt = a(1 - M/N)N
     • 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)
   • Depends on a, M & n

4. Equilibrium

   • Autonomous ODE
     • dy/dt = f(y)
       • Phase Space and Phase Portrait
   • 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

5. Harvesting

   • Constant, fixed fraction, "switch on"
   • dN/dt = a(N)N - H
     • H is constant
   • Critical Harvesting
     • Find maximum value of f(N)
       • Differentiate, solve = 0, calculate f(N) with solutions.