Transcript Document

Ecosystem Modeling I
Kate Hedstrom, ARSC
November 2008
Annual Phytoplankton Cycle
• Strong vertical mixing in winter,
low sun angle keep phytoplankton
numbers low
• Spring sun and reduced winds
contribute to stratitification, lead
to spring bloom
• Stratification prevents mixing
from bringing up fresh nutrients,
plants become nutrient limited,
also zooplankton eat down the
plants
Annual Cycle Continued
• In the fall, the grazing animals
have declined or gone into winter
dormancy, early storms bring in
nutrients, get a smaller fall bloom
• Winter storms and reduced sun
lead to reduced numbers of plants
in spite of ample nutrients
• We want to model these
processes to better understand
them and their interannual
variability
One Species
• The equation for one species,
growing without bounds:
• The known solution to this
ordinary differential equation:
Difference Equation
• Replacing the time derivative with
a finite change:
• Solving for the new time as a
function of the old time:
Sticking in Some Numbers
• Try b = 0.1, delta t = 1, initial N = 10
• If the units are days, we have e
times more critters after 10 days
• In the plot, green is the exact
solution while blue is the
approximate solution using a oneday timestep
Linear Plot
Log Plot
Sources of Errors
• Size of timestep relative to
timescales in the problem
• Numerical scheme
• Roundoff errors
• How do you tell which is the
trouble here?
• Since the numerical growth is also
exponential, we can adjust (tune)
the time constant to obtain the
correct solution
Two Species
• First rate equations for two species (one
prey, one predator) were written by Lotka
and Volterra during the 1920’s and 1930’s:
• Coupled, nonlinear, differential equations
• N1 is prey, N2 is predator, b, d, K1, K2 are
constants, t is time
Assumptions
• Prey will grow exponentially
without limit if no predators
• Rate of prey being eaten is
proportional to the number of prey
and the number of predators
• New predators happen
immediately after eating prey
• No other prey options
Steady Solution
• No change in time:
• Trivial solution is N1 = N2 = 0
• Any solution satisfying the
following is also steady:
Difference Equations
• We can make an approximation to
the differential equations by
assuming finite timesteps:
• These equations can be solved on
a computer
• Need initial values for N1, N2, plus
values for the constants
Numerical Solution
• One steady solution is given by
N1=1000, N2=10, d=0.1, b=0.1,
K1=0.01, K2=0.0001
• Using delta t =1, we get the
steady solution numerically
• Any perturbation from the initial
values for N1 and N2 will lead to
expanding oscillations.
Initial N1=999
Initial N1=980
Initial N1=980
• Red is ten
times
predator, blue
is prey
• Horizontal
axis is time
• Uncontionally
unstable
Why the instability?
• Invalid assumption in the
equations
– No limit to the number of prey supported
– No alternate prey
• Unstable numerical scheme
• Did you code it right?
Limit on the Prey
• Modifying the growth term for an
environment that supports up to M
prey:
• This equation is nonlinear, harder
to solve exactly
• Growth rate becomes negative if
N1 > M
One Species
Limiter in Lotka-Volterra Model
Lotka-Volterra PZ Model
• Same model as the original two
component model:
• Different constants: P=N1=75, Z=N2=10,
b=a=1, etc.
• Need a smaller timestep than one day
for such large growth rates
dt=0.1
dt=0.001
Hmmm
• The author is obviously using dt=0.1
• He then adds a Z cannibalism term
to damp the oscillations:
• This acts very much like the damper
on the prey species
My Results with Cannibalism
Planetary Orbits
Euler Timestep and Orbits
Leapfrog Step
• Second-order accurate
• Stable for orbits
• Even steps and odd
steps are decoupled
Timestepping Schemes
• Euler
– Unconditionally unstable for some classes
of problems
– Errors are linear in delta t (low order)
• Others
– There are many, many other options
– Some are higher order
– Each has its own stability properties
Back to Lotka-Volterra
• Euler step, dt=0.01
Leapfrog-Trapezoidal Step
• Second-order accurate, more stable
Large Amplitude Cycles
Conclusions
• Lotka-Volterra is cyclic, not
unstable
• Simple-minded numerical
schemes can get us in trouble
• Putting in more terms can lead to
realistic-looking results, such as
the limits on exponential growth
• More complex ecosystem models
still use Euler stepping with the
damping terms
Questions
• Timestep vs. timestepping
scheme
• Data - growth rate data, say, to
improve or disprove a model
• Error bars