Lecture 8 - Multiple Steady States

February 2, 2009

0. comments on multiple representation

resource depletion models

need to have the same units for the resource as the population

doesn't work with food because of metabolic losses

we'll see that in lecture 12 "coflow models" that account for energy and material linked, yet separately

solving by inspection

critical analysis of units is a key skill

tons of food (in what time period, or is it cumulative?)

poorly written axes in the journal (pore volume / mL) "/" doesn't mean divided by

equation - should have all the information

time axis (years AD, years_before_present, years_since_1950) etc

 

1. single steady state with only one solution for all starting parameters

see Lecture 6

example of birth rate vs. death rate relationships

2. multiple steady states - starting conditions (for the model) can lead to two different outcomes

what if the reproductive rate is slow at low populations - fish densitity is too low

but increases to a maximum at higher densitities

this is called the "Allee" effect, which can make decreasing population size even worse

 

Students, draw a STELLA model of this ---

did it look like this?

Do some runs with STELLA model 7-new-mss

 

3. Multiple stable states with hysteresis

background explanation

reference to recent articles on global climate change

loading up of carbon sinks, irreversibly

http://www.washingtonpost.com/wp-dyn/content/article/2009/01/26/AR2009012602037.html

bath-tub effect

http://dotearth.blogs.nytimes.com/2009/01/28/the-greenhouse-effect-and-the-bathtub-effect/?ref=science

 

4. Example of a model with phosphorus, algae and sediments in a shallow lake

focusing on the more tractable problem with P build up in shallow lakes

stocks:

• algal_P
• soluble_P
• sediment_P

fast flows:

• soluble_P --> algal_P
• release back out of sediments at high algal concentrations

slow flows:

• build up of P in sediments
• flushing of P from lake (loss of algal_P and soluble_P due to dilution)

thresholds (positive feedbacks):

• dilution of algae out of the lake will control algae at low concentrations
• high algae will increase the flow from sediments to algae

dissect out all the parts:

• soluble_P to algal_P
• dilution loss of any thing in the lake
• driving P input
• algae loss to sinking
• sediment to burial (keeps a fixed amount in the sediment)
• return from sediment is a threshold equation

5. What are the management objectives?

Is it:

• highest P load with lowest algal "pollution"

or

system resilience in the clear state

There is the danger of crossing the threshold

because of the cost of recovery, it would be very expensive