Department

Life SciencesCourse Code

LIFESCI 30AProfessor

Conley William JosephStudy Guide

FinalThis

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LIFESCI 30A

FINAL EXAM

STUDY GUIDE

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LS30A Lecture 2

28 September 2016

State of a dynamical system and its state variables – status/current conditions

(i.e. with pituitary gland: current level of a hormone, concentration of GNRH in

the blood, level of testosterone, etc)

or

(Insulin release, amount of glucose in the blood, level of insulin in the

bloodstream)

Steps to set up a mathematical model:

1. Identify the quantities you want to keep track of in the system (*quantities that are

changing with time).

a. Things are always changing over time (t), so time is not considered as a

changing variable

2. Assign a variable to each one of them called state variables

a. Of a population: T = number of Tuna; S = number of sharks

b. State variables are changing over time, so the variables S & T are

considered functions of time (i.e. T(t) is the number of Tuna at time t, year

t, day t, etc. S(t) = number of Sharks at time t, year t, day t, etc)

c. Usually, we’ll omit (t). Just remember that state variables are functions of

time

d. State variables must always have units

Example:

3-species food chain:

o R = # of rabbits

o F = # foxes

o L = # of Mountain Lions

Hypothalamus-Pituitary-Gonad models:

o H = Level (concentration in blood) of GnRH

o P = level of FSH and LH

o G = level of estrogen and progesterone

What is the state of a system?

Not just a property of a system as a whole, but at a specific time. The state is

also changing over time.

In the shark-tuna example. If right now there are 507 tuna and 38 sharks, we’d

say the state of the system is (507 tuna, 38 sharks).

Order matters

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Take the values of all the state variables and put them into a list

Definition: The state of a system (at a particular time t) is the collection of the values of

each of the state variables put into a “list,” more accurately called a vector

If the state is 507 tuna and 38 sharks, we write this state as

_ _

| 507 | with or without the units *Note: In the textbook, vectors

| | next to it. are written like (507, 38). I will

|_ 38 _| always write them in brackets.

The state of the H-P-G system at some time might be (10, 4, 3.792)

This might mean 10 mg/dL of GnRH, 4 mg/dL FSH + LH, etc.

State Space of the sytem – The set of all possible states a system could ever be in

Example: The shark-tuna system:

All possible values of T? Any real # > 0 or # = 0

All possible values of S? Any real # > 0 or # = 0

We will represent this state space graphically as

State space for H-P-G model is a 3D space. It makes a cube. States are not just a

collection of numbers, but points in a state space.

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