ecosystem modeling i kate hedstrom, arsc november 2008
TRANSCRIPT
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 one-day timestep
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=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
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
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
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
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