Lecture 5 -- States and Changes Geometrically PDF

# Lecture 5: States and Changes Geometrically ## The Three Driving Questions of LS 30 1. How do systems behave? - How do we think about behavior geometrically? 2. How do we model systems, especially biological ones? - What are units in models? 3. How do we use models to figure out system be...

# Lecture 5: States and Changes Geometrically ## The Three Driving Questions of LS 30 1. How do systems behave? - How do we think about behavior geometrically? 2. How do we model systems, especially biological ones? - What are units in models? 3. How do we use models to figure out system behavior? - How can we classify feedback loops and why does it matter? ## Feedback Loops - **Definition:** Feedback loops occur when a change in some quantity causes a later change in the same quantity. - Systems with feedback often behave counterintuitively, so we need **mathematical models** to study them. - Two types of feedback - **negative** and **positive** ## Negative Feedback Negative feedback occurs when an increase in a quantity causes a later decrease in that quantity, and a decrease in a quantity causes a later increase in that quantity. - **Change in some quantity - opposite change in same quantity** ## Positive Feedback Positive feedback occurs when an increase in a quantity causes a later increase in that quantity, and a decrease in a quantity causes a later decrease in that quantity. - **reinforcing** ## Identifying Feedback Loops from Models ## How To Figure Out Feedback Loops ### Equations - **T' = bT - βST** - **S' = mβST - dsS** ### Picture - Tuna affects Sharks. **1st Question:** Does the number of tuna affect the change in sharks? - Is **T** in **S' equation**? **To Answer:** Look at the change equation for **S**. Does the state variable for tuna show up in that equation? - **It does!** So **T** is in **S' equation**, so it affects sharks. - **Draw an arrow** from the tuna to the sharks ---- ### Equations - **Τ' = bΤ - βST** - **S' = mẞST - dsS** - **T affects S'** ### Picture - Tuna affects Sharks. **Next Question:** **How** ( +) or(-) effect? - Does the number of tuna affect the change in sharks? **To Answer:** Look at the change equation for **S**. What does increasing **T** do to **S'**? - **T** affects **S'** by **mẞST** - **mẞST is positive** -> **T** has a **positive** effect on **S'.** - (+) from **T** to **S**. ---- ### Equations - **T' = bT - βST** - **S' = mβST - dsS** ### Picture - Tuna affects Sharks. **Next Question:** **How** does the number of tuna affect the change in sharks? - **This is a positive causal link** **To Answer:** Look at the change equation for **S**. What does increasing **T** do to **S'**? - **S' goes up.** So I draw a + sign by my arrow ---- ### Equations - **T' = bT - βST** - **S' = mβST - dsS** - **neg-term** - **S affects, T' by -BST.** - **- BST is negative** -> **S** has **negative** effect on *T'*. - Increasing **S** has a **negative** impact on **T'**. - (this just means increasing **S makes T' smaller**) ### Picture - Tuna affects Sharks. - **this is a positive causal link** - **this is a negative causal link** **So I draw a - sign by my arrow** ---- ## This is a Feedback Loop - This gives us a loop that feeds back on itself **Start at Tuna** - increase in tuna -> **leads to increase in shark** - **go to shark** - **go back to tuna** - Ultimately decreases tuna. **This is an example of negative feedback** - **Increasing** change results in **decreasing** change in same quantity. ---- ## Another Feedback Example - The glucose/insulin case we had from lecture 1A: - **more makes less** - **Negative feedback is opposing feedback** - **regulating** - The direction of any change is opposed by the feedback loop ---- ## Positive Feedback Example - Romeo's feels towards Juliet: - State variable **R** - **R' = J** - Juliet's feels towards Romeo: - State variable **J** - **J' = R** - **Positive:** love - **Negative:** hate - **R has positive effect on J'.** - **J has positive effect on R'.** - **more makes more** - **Positive feedback is reinforcing feedback** - The direction of any change is reinforced by the feedback loop ---- ## Original RJ model ("Romeo has issues") - **J' = R** - **R' = -J** - **R has positive effect on J.** - **J has negative effect on R'.** ---- ## The Loops Can Also Get Longer - Sometimes feedback involves more than just one or two things! - Take the example of permafrost: - **higher temperature:** less permafrost - **negative effect on permafrost** - **permafrost:** traps/removes methane - **more permafrost:** less methane - negative effect on methane - **methane:** greenhouse gas inc. temperature - **more methane:** higher temperature - **positive effect on temperature** - **Positive OR Negative LOOP??** - -> shortcut... ---- ## Determining The Type Of Feedback - Take our example: - !!(permafrost: -1) - !!(temperature: +1) - !!(methane: -1) **Start at the top** - Go around the loop and multiply is out: - !!((1) * (-1) * (1) = (1)) **That's positive!** **So our loop is positive!** - **reinforcing change:** rising temps are amplified. ---- ## Positive Or Negative Feedback? - Ice at the poles reflects sunlight. When ice melts, the surrounding darker water absorbs more light, heating up the area. This causes more ice to melt. - **Greinforcing** - **A. Positive** - **B. Negative** ---- ## Positive Or Negative Feedback? - Deer ear vegetation. When there isn't much vegetation in an area, fewer deer survive. - **deer:** veg. has positive influence on deer pop - **vegetation:** deer has negative influence on vegetation - **A. Positive** - **B. Negative** ---- ## With a Partner, Come Up With Another Example Of a Feedback Loop And Say If It's Positive Or Negative ---- ## Units - **State variables:** the quantities we are modeling - Temperature, shark population, people with the flu - State variables have units. - °C, sharks/mile³, people -> do not have per time units - Give another example of a state variable and its units. - concentration of A: **mol/L** - **people/km²** - **people** ---- ## Units Of Change - **rates** will have **per time units** - Rates of change always have units of **variable/time** - °C/min, (sharks/mile³)/year, people/day -> unit of state variable/time - Give reasonable units for a rate of change of your state variable. - **people/year** - **assignments/week** - **tea leaves/bush/3 months** - **mol/L/sec** ---- ## What About Parameters? - All quantities that are not state variables are **parameters**. - Can be numbers or symbolic constants - lowercase letters - What are their units? - (people/day) = (1/year)(people) - (sharks/year) = (1/year)(sharks) - **rates of change/parameter = state var** ---- ## States And Changes Geometrically ---- ## Why? - Why does this happen? -> **higher peaks when we tried to remove them (oops)** ## Key Concepts For The Rest Of The Course - "ordered list" - The **state** of a system at time *t* is a tuple of the values of **all the system's state variables at time *t***. - Temperature of coffee - (lynx population, hare population) - (susceptible population, infected population, recovered population) - The **state space** of a system is the **collection of all possible states of that system**. - a system of elephants exists in *state space* -> might have 3 trillion elephants -> but not -5 elephants (does not) ---- ## State Space Geometrically - Hare-Lynx Space - system can exist at (20, 4) in **2D space** - **2D state space** - SIR Space - state (S, I, R ) in **3D space** - **3D state space** - Temperature Space - state T in **1D space** - **1D state space** ---- ## State Space Geometrically - Hare-Lynx Space - SIR Space - Temperature Space - **A state is a point in state space.** - **one point in space where system exists** - **Change is movement through state space.** - **moving from one state to another = change** ---- ## Which Point Represents A Community With 5 Coyotes And 3 Gila Monsters? - This system exists in a state w/ 5 coyotes and 3 gila. - **(5, 3) state** - **2D state space** ---- ## Dimension - **R Notation** - **R**^1 or **R** - **all real numbers** - **Temperature:** 1 variable - **R**^2 - **all real numbers in 2D space** - **Shark population, tuna population**: 2 var - **R**^3 - **R in 3D space** - **Susceptible, infected, recovered**: 3 var ---- ## Higher Dimensions - We aren't limited to state spaces we can visualize. - Neuron: 4 dimensions -> **R**^4 - Food webs: 10-50 dimensions (or more) -> **R**^10 - - - **R**^50 - Heart: 10 million dimensions -> **R**^10,000,000 - **R notation allots us to model when we can't visualize** ---- ## Doing Math On States - Mathematically, states like (3, 2) and (5, -1, 4) are **vectors**. - Two things we can do with **vectors** - Add: (3, 2) + (5, 8) = (8, 10) -> **add across** - Multiply by real numbers: 2(5, -1, 4) = (10, -2, 8) -> **scalar** ---- ## Doing Math On States - What is (1, 3) + (-3, 0)? - **(-2, 3)** ---- ## Doing Math On States - What is (1, 3) + (-3, 0)? - What is (3, 2, 4) + (1, 1, 2)? - **(4, 3, 6)** ---- ## Doing Math On States - What is (1, 3) + (-3, 0)? - What is (3, 2, 4) + (1, 1, 2)? - What is (2, 1, 5, 4) + (3, 2, 1, 3)? - **(5, 3, 6, 7)** ---- ## Doing Math On States - What is (1, 3) + (-3, 0)? -> **2D** - What is (3, 2, 4) + (1, 1, 2)? -> **3D** - What is (2, 1, 5, 4) + (3, 2, 1, 3)? -> **4D space** - We can do math with state spaces we can't visualize. - **We can still compute numbers in a 4D space** ---- ## States And Change - **States are points in state space.** - **Change is movement through state space.** - But what tells the state point how to move? ---- ## States And Change - **States are points in state space.** - **Change is movement through state space.** - But what tells the state point how to move? - **change equations** - **Answer: We use the model to create a set of instructions for the state point.** - **vector field gives instructions for every state in state space. (like a GPS)**