1 Time: Dynamics & FIixed Point

Question 1

What scales can biological patterns occur on?

Answer 1

From molecular to ecological

Question 2

How can biological patterns be formed?

Answer 2

Pre-pattern
Self-organization

Question 3

What system do bacteria use to replicate?

Answer 3

Biological amplification = going from one organism to many

Question 4

What is the equation for exponential growth?

Answer 4

N(t) = No x 2^t = Noe^rt

Question 5

What is r?

Answer 5

r = relative growth rate
r = ln2

Question 6

What happens when you put an exponential on a log scale?

Answer 6

It become linear

Question 7

What are the 4 phases of bacterial growth?

Answer 7

Lag
Log
Stationary
Dying

Question 8

Why is the exponential growth model not realistic?

Answer 8

If 1 E.coli divides every 20 min = take less than 2 days for colony to reach the weight of Earth

There are not enough resources to sustain the population and they could not grow heavier than the Earth because of the conservation of mass

Question 9

What are the assumptions for the exponential growth model?

Answer 9

There is no negative number of bacteria
Every bacterium has full access to all resources
There is sufficient resource to sustain growth continuously

Question 10

What is the difference in a logistic model compared to exponential?

Answer 10

Additional assumption = carrying capacity K

Question 11

Define the carrying capacity

Answer 11

For bacteria culture, this reflects the population the nutrient can support

Question 12

What shape is the logistic model?

Answer 12

N(t) = y axis
t = x axis
Sigmoid curve

Question 13

Why does logistic model saturate?

Answer 13

As population increases, proportionally less nutrients to population size
Saturation occurs when nutrients become so low and population becomes so high

Question 14

What is the equation for the logistic model?

Answer 14

N(t) = (KNoe^rt) / ((K-No) + Noe^rt)

Question 15

What are the limits for the logistic and exponential models?

Answer 15

Exponential = unbound so infinite time and growth
Logistic = bound t is infinite but N(t) can only reach K

Question 16

What does ODE stand for?

Answer 16

Ordinary Differential Equation = meaning there is only 1 independent variable (TIME)

Question 17

What is the differentiation of N(t)=Noe^rt?

Answer 17

dN/dt = rN

Question 18

What is the phase plane? ***

Question 19

What do we describe the exponential phase portrait as?

Answer 19

One-dimentional/first order linear system

Question 20

When is the phase portrait static?

Answer 20

When dN/dt = 0 the system is static because there is no change in N

This is a fixed point = N*

Question 21

What is a trajectory?

Answer 21

Series of vectors where N>1 and the length of the vector is dN/dt

This will always point away from the fixed point

Question 22

What is an unstable fixed point?

Answer 22

When the trajectory flows away from that fixed point = keeps getting bigger

Question 23

How to determine stable v unstable fixed point?

Answer 23

Look at a phase portrait = dx/dt vs x

Stable x* = negative slop through fixed point
Unstable x* = positive slop through fixed point

Question 24

What are the rules of trajectories?

Answer 24

Trajectories never cross each other in a phase portrait
Each point phase ONE and only ONE direction, which can only change when crossing a fixed point

Question 25

What does the logistic phase portrait look like?

Answer 25

Non-linear
In this case, quadratic so has two fixed points

Question 26

What is a globally stable fixed point? ***

Answer 26

A specific point in the system where, regardless of the initial conditions, the system eventually evolves or converges to that point over time