In this document, you can find the summary for the course Modelling Life which is given in the second year of Biology / Life Science and Technology (old curriculum) at the University of Groningen.
For this summary, the book Modelling Life: The Mathematics
of Biological Systems is used
Modelling Life Summary
Chapter 1: Modelling, Change, and Simulation
1.1: Feedback
The prey population positively affects the number of predators, while the
predator population negatively affects the number of prey. This is a
example of a system with a negative feedback. The prey population
affects the predator population and the predator population affects the
prey population. It is difficult to predict the behavior of a feedback system
based on this kind of verbal model.
Feedback Loops
In a feedback loop, the current state of a system affects the future state of that system by changing the
inflows or outflows of the system’s components. There are two types of feedback loops: positive and
negative.
Positive Feedback
In positive feedback, a positive value of a variable leads to an increase in that variable, and a negative
value of a variable leads to a decrease (more negative) in that variable.
- Population growth: The larger the population is, the more babies are born, which makes the
population even larger. As long as resources are available, the population will keep growing.
- CO2 emissions: Carbon dioxide emissions trap heat which raises global temperatures. At higher
temperatures, soil microbes have faster metabolic rates, which means that they break down soil
organic matter faster, releasing even more CO2.
- Methane release: Large amounts of methane are trapped in Arctic permafrost and at the bottom
of the ocean. Rising temperatures cause this methane to be released, contributing to further
temperature increases.
Negative Feedback
In negative feedback, a positive value of a variable leads to a decrease in that variable, and a negative
value of a variable leads to an increase in that variable. A new variable from a given one can be defined
by choosing a reference value for the variable and then defining the new variable as the given variable
minus the reference value.
- The classic example of negative feedback is a thermostat. T0 is the set point on the thermostat. If
the current air temperature is C degrees, we define the temperature T as T = C – T0 so that values
of C above T0 produce positive values of T, and values of C below T0 produce negative values of T.
The thermostat can also control a heater, in which case a decrease in temperature causes the
heater to turn on, which raises the air temperature.
- Insulin/Glucose: Intake of glucose causes the pancreas to secrete more insulin, which then
lowers the level of glucose by helping the glucose to be metabolized in the body.
- Hormone regulation: Virtually all of the hormones of the body are under negative feedback
control by the brain and pituitary gland. This results in oscillatory behavior in many hormonal
systems.
- Gene regulation: Many genes inhibit their own transcription, resulting in oscillating gene
expression.
, - Epidemiology: Epidemiology is the study of diseases in populations. Many epidemiologists study
infectious diseases. Contact between susceptible and infected people increases the transmission
of the disease and causes the number of susceptible people to decrease. This decrease means
that there are fewer susceptible people to infect, so transmission declines.
Counterintuitive Behaviors of Feedback Systems
Most real systems consist of multiple feedback loops that interact and systems with feedback often
behave in counterintuitive (tegenstrijdig) ways. The more advanced model also predicts a
counterintuitive response: the initial response of the system to a predator-removal intervention can be a
rebound effect whereby the number of predators is increased, but transiently instead of permanently.
1.2: Functions
Function = a relationship between a set of inputs and a set of outputs, in which each input is assigned
exactly one output - never none, never more than one.
Notation For Functions
FunctionName (input) = output. A common name for functions is f, so f(X) = Y. Can also write down the
function by giving its name, the input variable, and then the expression that lets us find the value of the
output given the input.
Inputs and outputs
The set of input values that a function can accept is called its domain of the function. Real numbers are
every kind of number you can think of as representing a quantity, including whole numbers, fractions,
irrational numbers, as well as the negative values of all of these. Altogether, these numbers make up the
real numbers, abbreviated R. For convenience, R+ means the nonnegative real numbers: 0 and
everything larger. There is also a term for a set of values in which a function’s output lies, this is called
the codomain of the function. A function links each element in its domain to some element in its
codomain. Each domain element is linked to exactly one codomain element.
- function name : domain → codomain
Putting Functions Together
Composition of functions is the most natural way to combine functions. The function will be made up of
the previous two functions linked end to end.
1.3: States and State Spaces
The State of a System
State is a term for the condition of the system at a given time. The variables we use to describe the state
of a system are called state variables. The choice of state variables is determined by both the structure
of the system and how we plan to use our model. Deciding what variables to focus on is often one of the
hardest parts of building a model. State variables must be quantitative. A state variable is therefore a
quantity that describes some property of the system, such as its velocity, shark population, or blood
glucose concentration. Since a state variable can have only one value at a given time, the values of state
variables are really functions of time. Since a state variable is a function of time, we can plot this function
as a graph, with time on the horizontal axis and the state value on the vertical axis, this plot is called a
time series.
, State Space
We want to understand the system’s behavior - its changes from state to state - and why it exhibits one
pattern of behavior rather than another. The set of all conceivable state values of a system is called its
state space - the space in which the system’s state value can move.
One-Variable Systems
If a system has only one variable, its state is a real number. Thus, we can use the fundamental idea that
the real numbers exactly correspond to points on a line to say that the state space of such a system is a
line. A system’s state is represented as a point in state space, which we will sometimes call the state
point. For a variable that can be either positive or negative, the state space is just the real number line.
For a population size, the line goes from zero to infinity, excluding negative values. For a proportion, the
line goes from zero to one.
State Spaces with Multiple Variables
Starting with the one-variable case, we can define two simple operations on states:
- If X1 and X2 are two values of the state variable X, then we can add them to produce another
value of X → X3 = X1 + X2
- If X1 is any state value, we can always multiply that state value by a number. Such a number is
called a scalar.
- The rules for operating with pairs of values are similar. → (S1, T1) + (S2, T2) = (S1 + S2, T1 + T2)
- Pairs can be multiplied by a scalar. → a(S1, T1) = (aS1, aT1)
The Geometry of States
A system’s state space is often named by its variables. If X and Y are state variables, then the set of pairs
(X,Y) is the set of all states of the two-variable system. A pair of numbers is a 2-vector. The numbers
making up the vector are called its components. The space R2 is called a two-dimensional vector space.
Multiplying a vector by a scalar stretches the vector if the absolute value of the scalar is greater than 1,
and it compresses it if the absolute value of the scalar is less than 1. If the scalar is positive, then the
result is a vector in the same direction. Multiplying a vector by a negative number changes the signs of
all of the vector’s components. Geometrically, this flips the direction of the vector, in addition to
stretching it by the absolute value of the number.
State Spaces with More than Two Dimensions
If a model tracks the concentrations of three different chemical compounds, its state space is a three-
dimensional space whose axes represent the concentrations of the compounds in question. An ordered
triple specifies the position of a point in three-dimensional space. A vector with n components gives the
position of a point in n-dimensional space. The number of axes needed to represent a system’s state is
called the dimension of its state space.
An n-dimensional space as having n copies of R as axes and denote it by Rn.
• Rn = R ×···× R (n times) = {(x1,...,xn)}
1.4: Modelling Change
Change = movement through state space.
A set of hypotheses about the causes of change in a given system is called a model.
The idea is that levels are changed by flows; that is, quantities are changed by rates. A “change
equation,” in which the state variable is X and define X’ as the change in X. → X’ = [the things that
change X]
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