Discrete vs continuous models

Biological quantities change through time. A mathematical model must decide when that change is represented: only at separate time points, or continuously through time. This leads to discrete-time and continuous-time models.

One biological example

Suppose a population starts with \(N_0=100\) individuals and grows at 10% per day. We can represent this same growth idea in two different ways.

Discrete time: change one step at a time

If the population is updated once each day, 10% of the current population is added during each step. Therefore

\[ N_{n+1}=N_n+0.1N_n. \]

Collecting \(N_n\) gives

\[ N_{n+1}=1.1N_n. \]

Starting from \(N_0=100\), repeated application gives

\[ N_1=110,\qquad N_2=121,\qquad N_3=133.1,\ldots \]

Because the same multiplication occurs at every step, after \(n\) steps

\[ \boxed{N_n=N_0(1.1)^n.} \]

How the function was obtained. The biological rule “increase the current population by 10% each day” first gives the recurrence \(N_{n+1}=1.1N_n\). Repeating that rule produces the explicit function \(N_n=N_0(1.1)^n\).

Continuous time: change at every instant

Now suppose growth occurs continuously rather than only at daily updates. We describe the instantaneous rate of change of population by

\[ \frac{dN}{dt}=rN. \]

This equation says that the rate of population growth at any time is proportional to the population currently present. The constant \(r\) is the continuous per-capita growth rate.

To obtain the population as a function of time, separate the variables:

\[ \frac{1}{N}\,dN=r\,dt. \]

Integrating gives

\[ \ln N=rt+C. \]

Exponentiating,

\[ N=Ae^{rt}. \]

At \(t=0\), \(N(0)=N_0\), so \(A=N_0\). Hence

\[ \boxed{N(t)=N_0e^{rt}.} \]

How the function was obtained. In the continuous model we first state the biological rule as a differential equation, \(dN/dt=rN\). Solving that equation gives the continuous function \(N(t)=N_0e^{rt}\).

What is the visual difference?

Discrete-time model

The model specifies population values at \(n=0,1,2,\ldots\). The connecting line only helps the eye; the model itself is defined at the marked steps.

Continuous-time model

The function \(N(t)\) has a value at every time, including times between whole days.

They are related, but the rates are not identical

A 10% increase per discrete day gives the multiplier \(1.1\). If we want a continuous exponential model that has exactly the same values at whole days, we require

\[ e^r=1.1, \]

\[ r=\ln(1.1)\approx0.0953\text{ day}^{-1}. \]

Then

\[ N(t)=100e^{0.0953t} \]

passes through the same population values as \(N_n=100(1.1)^n\) at \(t=n\). The continuous curve additionally describes what happens between those times.

Discrete and continuous state

Time and state are separate choices. A population count such as

\[ I\in\{0,1,2,\ldots,N\} \]

is a discrete state, whereas a concentration \(C\geq0\) is often represented as continuous. A model can therefore have continuous time but discrete state, as in a continuous-time Markov chain.

Feature	Discrete	Continuous
Time	Separate steps \(n=0,1,2,\ldots\)	Every time \(t\)
Typical equation	\(N_{n+1}=f(N_n)\)	\(dN/dt=f(N)\)
State	Separated values	Any value in an interval

Key idea. A discrete-time model gives a rule for moving from one time step to the next. A continuous-time model gives an instantaneous rate of change; solving the differential equation produces a function defined for every time. Neither approach is automatically better—the appropriate choice depends on how the biological process is naturally represented.