Chap9: On stability and other questions Flashcards
When is a state variable in equilibrium? What’s the difference between stable and unstable equilibrium?
A system is in equilibrium, when its state variables (stocks) don’t change. In the water reservoir example, this means: dX/dt = 0, in words: in- and outflows cancel out each other. A dynamic equilibrium occurs when two opposing processes proceed at the same rate. Inflow=Outflow Stable –> If there is a disturbance in the system, the equilibrium is easily reached again. Negative loop domains. Instable–> A very small change can cause the equilibrium to collapse. Positive loop domains.
Assume that you are working in the field of nature protection. You are monitoring an endangered salamander population in a certain area over some years. In the first year you estimate the population to be 8000 individuals. In the second year, it is 4000, in the third 2000, and in the fourth year 1000 individuals. Which simple system archetype seems to describe this population decline best? Why? If the decline process continues in the same way, how many individuals will you have in the sixth year
Because the salamander population approaches zero (first quickly, later slowly), but (mathematically) never reaches zero in the four year. The amount of individuals in the sixth year is 250.
A short story: Population development in a small town. Imagine you are working in a census bureau of a small town. Your town’s population currently amounts to 10000 persons. From your database, you know that the number of births is proportional to the population size with a birth rate of 0.01/year. Similarly, the number of deaths is proportional to the population size as well. The death rate is 0.01/year. Inmigrations and emigrations are negligible. a) Draw causal-loop-diagram (CLD) for this simple system. Use the variables population, births, deaths, birth rate, and death rate. Don’t forget to mark feedback loops properly where required.
Still needs answer!
Your boss (the head of the town) wants you to use your model to forecast the population development for the next 5 years and for the next 50 years. Is your model adequate for both purposes? Why or why not?
No. It’s because in our model immigrations and emigrations are negligible. Maybe it works for the next 5 years, but it’s not practical for the next 50 years
Is feedback equivalent to stabilization? Why or why not?
‘feedback’ can be positive or negative. Positive feedback means absolute amount of stock is positively correlated with the rise in Stock and vice versa. That is the opposite of stabilization as higher stock leads to higher increments, which lead to even higher stocks Negative feedback stabilizes a system at equilibrium. As the increment of stock is negatively correlated with the absolute amount of stock, the system, i.e. the stock, finds a stabilized state after a definite time Consequently, only negative feedback is equivalent to stabilisation
Why is it important to define exactly the model boundaries?
Boundaries that are usually defined by the system observer and must determinate the inputs, outputs and feedback mechanisms to maintain an internal steady-state (called homeostasis) despite a changing external environment
What is the structural difference between exponential growth and asymptotic decay?
Both models increase and decrease very rapidly, but the decay model then level off to become asymptotic towards the x-axis. If rate is negative it leads to asymptotic decay of the population. if r is positive it leads to exponential growth. Or mathematically: N_(t+1)=N_t *(1+r)^t r < 0: asymptotic decay r > 0: exponential growth
Why do the Overshoot and Collapse system and the S-Shaped Growth system behave so differently when the growing population has reached a certain size? (Are both generic infrastructures of a growing population reaching limit)
1) Overshoot and collapse system: population exceeds usually by far its capacity in the beginning (exponential growth). Afterwards it collapses again either to zero or at least below its capacity (oscillating around capacity) 2)S-Shaped Growth – carrying capacity is not exceeded – asymptotical aproachment of the carrying capacity
Do systems whose charactarestic behavior is exponential growth tend to be long-lived or short-lived? Why?
it is not long lived because in reality there is always a boundary (a buffer), it will accelerate itself more and more and in quite a short time the exponential growth will break all limits so in reality the system will crash soon. If it is to be long lived there should be no limits allowed.
Exponential Growth S-Shaped Growth Oscillation Asymptotic decay
Exponential Growth Mouse population doubling in certain intervals of time. Human population growth.
S-Shaped Growth Increase in competition. In a fish population death rate increases with increasing population size. Carrying capacity. Population size is constrained by food availability.
Oscillation Population Cycles in a Predator-Prey System Herbivore-plant interactions (variability due parasites or bacterias, lack of ) Human impact in many ecological communities by removing predators or reducing the availability of resources.
Asymptotic decay Mouse population halving in certain intervals of time. Secondary animal extinctions following deletion of one species
