Continuoustime models

1
Continuous-time models

• (An absolutely too short presentation of the
  ridiculously simplified nothing-but-the-bare-essen
  tials content for the purposes of the
  Introduction to Ecological modelling)

2
Continous time

• Resource consumption
• individuals need and may take up resources
  constantly
• competition
• growth, biomass accumulation
• nutrient and other element cycles
• Reproduction
• some species can reproduce at any time of a year
  (or other time step)
• overlapping generations
• Also other reasons for use analytical solutions,
  mathematical beauty, familiarity to phycisists,
  etc.

3
Analytical methods

• Phase plane a graph where state variables are
  plotted against each other
• Zero growth isoclines graphs in the phase plane
  where dN/dt  0
• can be derived with pen and paper
• can tell a lot of the system behavior
• More sophisticated methods based on the analysis
  of the Jacobian matrix, etc.

4
Logistic growth
K
Integration
Differentiation
N(t)
N(0)
Time, t
0
5
But ...

• ... most differential equations can not be solved
  explicitly, and ...
• ... solutions therefore require numerical
  methods, ...
• ... which are approximations, ...
• ..., but usually a sufficiently good method can
  be found

6
Numerical integration in Matlab

• One m-file contains the population model as a
  function
• Another m-file contains an integration command
  (e.g., ode45 or ode23) with arguments
• population model (the function)
• initial conditions
• simulation time span
• Execution of the latter file runs the simulation