Stead State and Feedbacks

1. Steady state

• input = output
• could be very fast flows in both directions
• important to understand that at steady state
  • net flow through system decreases to zero
  • individual flows do not decrease

example model diagram for discussion

 

2. Positive feedback

• flow into a stock increases with higher stock concentration
• example population growth
  • higher population
  • more births
  • even though births per 1000 people could be constant

3. Negative feedback

• flow into a stock decreases with higher stock concentration
• example crowding inhibition of growth rate
  • more population means more crowding
  • growth rate slows down, fewer births

4. Stock depletion flow control

• when stock runs out, the flow stops
• maybe as stock decreases flow rate decreases
• example: flow out of a bucket with a hole in the bottom

5. Combinations of feedbacks control flows

• simple set of feedback controls
  • more deer means more fawns are born - increase in deer leads to increase in births
    • even if birth_rate (fawns per year per female deer) is constant
    • because birth_rate * population = birth
  • more deer mean more deaths
  • this system will increase to a set level - steady state at the carrying capacity

• multiple controls
  • food level may lead to higher births when there is more food and fewer births when food is lower
  • death rate may be higher when there is low food (starvation and disease)
  • "well behaved" regulation can lead to "tuning" of the population level to the food
  • reach new steady states that are appropriate for the amount of resources available

• BUT, there is not guarantee that these systems are always well behaved,
  • low food may weaken the population and the loss to deaths is much larger
  • this would result in a much lower average population or maybe even a boom and bust type behavior under conditions of food limitation

6. Thresholds

• the range in which a new feedback may "kick in" may lead to new operating steady state
• each new level is a balance between
  • positive feedbacks that accelerate the change
  • negative feedbacks that resist the change
• often thresholds are the consequence of saturating the ability of the negative feedback to resist the change
  • leads to "runaway" positive feedback until a new steady state is reached
• Examples:
  • methane leakage from frozen soils (more methane leads to higher temperature that can lead to drammatic methane increases)
  • tropical forests promote "green" water flow and maintainence of the forest, loss of forest decreases rain water cycling, which leads to loss of forest, which leads to less availalbe water and further forest loss