January 12, 2009
try to fit to the equation
uptake_rate = maximum_rate * sugar / (half_sat_constant + sugar)
but it is really two mechanisms
diffusion at 0.02 * sugar + 10*sugar/(5 + sugar)
Constant ratio growth can't last for ever, or can it?
bacteria
other populations
"sustainable growth"
What limits growth?
by definition, resources are something that is consumed
conditions - not consumed
nutrients for plant growth
light might be a resource in a vertically structured commnity and not in another
space might might not be
temperature - not a resource, isn't decreased by action of consumers
Focus on nutrients
soluble P of N forms
in water for algae
in soil for terrestrial plants
doesn't work as well if you track carbon - because it is involved in energy and respiration
description
100 units of plant material (P content)
start to grow in a closed container that contains another 1000 units of available P
plants grow pretty fast until they run out of P
plant growth rate is a function of P
only very low levels of P limit growth rate
the total amount of plant material (yield) is limited by the total P available = 1100 units
graphical
algebraic
population + resources = 1100
new population increase = time_interval*population * Vmax * R / ( R + Km)
population at time t
R = resources at time t
Vmax is the maximum growth rate = 0.04 (new individuals per individual per time)
Km is the half-saturation constant ( the R concentration where the V = Vmax/2
delta is the time_interval (such as 1 day, 0.5 day, etc)
in any time interval, the growth rate is dependent on the nutrient concentration
when nutrients get low the growth rates slows, eventually to zero
the nutrient uptake rate is described by the Michaelis-Menten equation
Excel model
be able to say what all the equations in the first two rows of number mean and how they are constructed
STELLA model
a model within a model
description and mechanism
resource is being used by an organism
when there is a lot of resource, there is a maximum rate of use (handling)
when the resource is in a lower range, there is a near linear relationship
zero resources - zero use
graphical
iconongraphic
PacMan consumption
algebraic
v = Vmax * R / (R+ Km)
units -
v = Vmax when R>>Km
v = 0 when R = 0
v approximately Vmax*R/K when R < Km
v = Vmax/2 when R = Km
Km is constant that represents the concentration at which the velocity is 1/2 maximum
Interesting exercise
from the graph of population and resource - what is are the regions of interest in the velocity vs resource graph.
Other models for resource uptake
a. constant - uptake velocity is independent of concentration (may be over a wide range of interesting, non-zero concentrations)
draw graphs of velocity vs. R and then pop & resource vs. time
b. linear - more resource leads to faster velocity of uptake = diffusion processes
c. sigmoidal
description
the population has a maximum growth rate (rmax) and a maximum population size (carrying capacity "K")
initially the population grows exponentially at an intrinsic rate rmax
as population increases the growth rate decreases
the rate of population decreases in proportion to how close population is to the carrying capacity
graphically
draw on board
algebra
r = rmax * (K-N)/K
at each time interval r can be calculated
the term (K-N)/K goes from 1 at N near zero to 0 at N=K
Excel formulation
carrying_capacity = 1000
maximum_growth_rate = 0.2
time population logistic_rate 0 10 maximum_growth_rate*(carrying_capacity-population)/carrying_capacity 1 =pop(0) + pop(0)*log_rate(0)
