LS30A Midterm Week 1 Review
How can we make our simple population model better?
- By introducing a carrying capacity k - Then: X/k represents the fraction of resources that are used and (1-X/k) represents the fraction of resources that are available - The new model is X'(t)= rX(t)(1-X(t)/k). Called logistic model.
Bathtub Model
- Change equations: X'(t)= c - kX(t) - State variable: X, parameters: c and k - X(t) is the amount of water in the tub at time t - X'(t) is the rate at which the amount of water in the tub changes at time t - c is constant rate of inflow of water - kX(t) is the amount of water draining out at a rate proportional to the amount of water in the tub
Important Assumptions of the Bathtub Model
- Two things can happen: faucet pours water into tub and drain drains water out of tub - The system is the tub and the water - The rate at which k drains water is proportional to how much pressure is being put on the drain by the amount of water - Units must be consistent! X'(t) and c are in L/min, k is 1/min, and X(t) is in L
What happens when X is larger than k (for the logistic model X′=bX(1 −X/k))?
- When X is larger than k, the fraction X/k is larger than 1 and thus 1 −X/k is negative. This means that when the population is larger than the carrying capacity, X′is negative and therefore the population decreases. - If we think of X/k as the fraction of the resources used by the population X, if X/k is larger than 1, this means that the population is using more resources than what the land can sustain. Thepopulation will thus decrease.
What happens when X is smaller than k (for the logistic model X′=bX(1 −X/k))?
- When X is smaller than k, the fraction X/k is less than 1 and thus 1 −X/k is positive. Thismeans that when the population is smaller than the carrying capacity, X′ is positive and therefore the population increases. - If we think of X/k as the fraction of the resources used by the population X, if X/k is smaller than 1, this means that the population is not efficiently using the available resources. The population will thus increase.
What kind of feedback loop is the Shark and tuna model?
- negative because the two species regulate each other - If the number of sharks increases, they eat more tuna and deplete the population. However, the shark population subsequently declines as there isn't as much food. - When the shark population declines, the tuna population recovers, increasing once again, and then more sharks can be supported.
What kind of feedback loop is our simple population growth model?
- positive - As more individuals are added to the population, this leads to more births, which increases the population even more. This is thus self reinforcing.
Define a model of a system.
A set of equations that describes how the state of the system is changing over time.
Define a system.
A set of interacting things in the physical world
Simple population model
Example: rabbit population X(t) is the number of rabbits at time t X'(t) is the rate at which the rabbit population changes at time t Assumptions: 1) Rabbits are born and rabbits can die. 2) All animals are capable of giving birth, an animal's ability to give birth is constant over its lifetime, and all animals have the same likelihood of giving birth/dying. X'(t)= bX(t)- dX(t) Prediction made by this model: exponential growth, which is not really realistic and suggests that we have unlimited resources
What is the second definition of a negative feedback loop?
In a negative feedback loop: - When X is above a certain reference level, then X later decreases. - When X is below a certain reference level, then X later increases.
What is the first definition of a negative feedback loop?
In a negative feedback loop: - an increase of X leads to a later decrease of X - a decrease of X leads to a later increase of X
What is the second definition of a positive feedback loop?
In a positive feedback loop: - When X is above a certain reference level, then X increases further. - When X is below a certain reference level, then X decreases further.
What is the first definition positive feedback loop?
In a positive feedback loop: - an increase of X leads to a later increase of X - a decrease of X leads to a later decrease of X
Suppose that at t=2 the tub is empty.The faucet is running, and the drain is open as above. The model is X ′(t ) = 10 −0.2X (t ). What is the rate of change of X at t = 2?
X(2)=0 X'(2)=10-0.2*X(2) =10-0 =10
Suppose that at t = 2 there is 50 L in the tub. The faucet is running, and the drain is open as above. The model is X ′(t ) = 10 −0.2X (t ). If the capacity of the tub is 200 L, will the tub ever overflow?
X(2)=50 X'(2)= 10-0.2*X(2) =10-0.2*50 =0 No, because at t=2 the amount of water in the tub is not changing.
Write the following statements as equations. a) A is proportional to B with a proportionality constant of 2.5. b) X is proportional to Z with a proportionality constant of 3.7. c) An animal's population density, P, is proportional to body size, B, with a proportionality constant of m.
a) A=2.5B b) X=3.7Z c) P=mB
Suppose that at t = 2 the tub is empty.The faucet is running, the drain is open as above. The model is X ′(t ) = 10 −0.2X (t ). What will happen if we let the faucet running?
the amount of water will approach 50 L without quite reaching it (Tip: At what value of X(t) is X'(t)=0?)
What should happen to the number of platelets in the process of wound clotting?
C- grows really fast!
What is the basic purpose of modeling?
To predict and explain the behaviors of a system
Why do we need a second definition for feedback loops?
Some situations are not captured by the first definition. ex. Thermostat
Lotka-Volterra Model: Assumptions
Assumptions: 1) Predators represented by S(t) and preys represented by T(t) 2) Without predators, preys follow exponential growth bT(t) 3) Predators eat preys. Represented by multiplying the two states variables. S(t)T(t) 4) Only some encounters lead to the predator eating the prey. Represented by βS(t)T(t), where β represents the likelihood of a shark-tuna encounter resulting in a tuna being eaten by the shark.
Lotka-Volterra Model: Equation for the Predators and Prey
Assumptions: 1) The predator only feeds itself with this type of prey. 2) The birth rate of the predator population is directly proportional to the number of preys each predator eats. Represented by mS(t)T(t), where m is the proportionality constant for births of predators 3) Predators die at per capita rate d. => S'(t)= mβS(t)T(t) - dS(t) T'(t)= bT(t)- βS(t)T(t)
When tissue is injured, a chemical is released. This chemical causes platelets in the blood to activate. Once platelets have activated, they release a chemical which signals more platelets to activate, until the wound is clotted. What kind of feedback do the platelets and chemicals form, if any?
Positive feedback; increase of platelet activity leads to a later increase of platelet activity
You want to develop a new project of electric cars for which people change their car batteries when they are empty. If few people use your system, the cars will remain expensive, there will not be many charging stations and the project will stop. If many people are interested, the cars will become cheaper, more stations will be installed, new services will be developed and thus more people will use your system. What kind of feedback is this?
Positive feedback; when no. ppl using project is high, the cars will become cheaper, and more people will use system.
Come up with another example of a positive feedback loop and another example of a negative feedback loop.
Positive- blood clotting When you cut yourself and start bleeding, your cells release hormones that signal thrombocytes to come and stop the bleeding. Thrombocytes then release more hormones that signal even more thrombocytes to help stop bleeding until the cut is closed. Negative- blood pressure When blood pressure is too high, the brain signals the heart to slow down its pumping rate, which leads to a lower blood pressure. When the blood pressure is too low, the brain signals the heart to increase its pumping rate, which leads to a higher blood pressure.
