560
13 Modeling Biological Systems
Chaos can arise in non-linear systems, e.g. differential equations expressing the
time derivatives of unknown functions in three or more other variables, in which
products of the unknown functions appear, such as
df (x, y, z, t)
dt
= a[f (x, y, z, t)]
3
+ · · · ,
Having products of the entities affecting the rates of those same entities is a
characteristic of feedback systems. A good example is our population model in
the previous section, in which the presence of that population causes its growth
to slow. There are many more good examples. Fluid dynamics becomes chaotic
due to the non-linear term v · ∇v in the Navier–Stokes Equation. Neuron network
models can exhibit chaos when there is feedback. The Hodgkin–Huxley neuron (see
Sect. 14.2.2) can show chaotic behavior, as can simple electronic circuits with just
one non-linear element.
Our population model above was designed so that we could express an analytic
solution. But, in general, non-linear differential equations are notorious for being
difficult to solve. Einstein’s gravitational field equations are non-linear because
localized gravitational energy has an equivalent mass, which acts back on the
strength of the gravitational field. This property alone makes Einstein’s field
equations difficult to solve, unless the system has particular simplicity or strong
symmetry.
Even though a system of non-linear differential equations may be difficult
to solve analytically, i.e. the solution in terms of known analytic functions is
difficult to find, we can ‘discretize’ the problem 2 by taking small increments for
the independent variables. We then have a system of recurrence relations for the
unknowns, indexed by the number of time increments. For example,
f i+1 = φf i + F (f i , g i ) ,
g i+1 = γ g i + G(f i , g i )
(13.6)
where φ and γ are constants and F and G are non-linear functions in f i and g i .
Now, you say: “Alright. Now we have the equations to solve the problem, at least
numerically.” Yes, but there is still the ‘butterfly effect’. You start with an initial
choice for the values f 0 , g 0 and use the Eqs. (13.6) to step through time. Great.
However, if you change the initial conditions by just a ‘little bit’, you may find that
your new solution differs widely from your first one. If you are trying to predict the
weather, your initial data will be uncertain by small amounts. That makes detailed
weather prediction over more than about a week practically impossible.
There’s another complication: Computers cannot be infinitely precise about the
value of an irrational number, or even a very large rational one. This means that
2 Actually, the problem may have originally been discrete, and had been ‘smoothed’ by a calculus
limit.
13 Modeling Biological Systems
Chaos can arise in non-linear systems, e.g. differential equations expressing the
time derivatives of unknown functions in three or more other variables, in which
products of the unknown functions appear, such as
df (x, y, z, t)
dt
= a[f (x, y, z, t)]
3
+ · · · ,
Having products of the entities affecting the rates of those same entities is a
characteristic of feedback systems. A good example is our population model in
the previous section, in which the presence of that population causes its growth
to slow. There are many more good examples. Fluid dynamics becomes chaotic
due to the non-linear term v · ∇v in the Navier–Stokes Equation. Neuron network
models can exhibit chaos when there is feedback. The Hodgkin–Huxley neuron (see
Sect. 14.2.2) can show chaotic behavior, as can simple electronic circuits with just
one non-linear element.
Our population model above was designed so that we could express an analytic
solution. But, in general, non-linear differential equations are notorious for being
difficult to solve. Einstein’s gravitational field equations are non-linear because
localized gravitational energy has an equivalent mass, which acts back on the
strength of the gravitational field. This property alone makes Einstein’s field
equations difficult to solve, unless the system has particular simplicity or strong
symmetry.
Even though a system of non-linear differential equations may be difficult
to solve analytically, i.e. the solution in terms of known analytic functions is
difficult to find, we can ‘discretize’ the problem 2 by taking small increments for
the independent variables. We then have a system of recurrence relations for the
unknowns, indexed by the number of time increments. For example,
f i+1 = φf i + F (f i , g i ) ,
g i+1 = γ g i + G(f i , g i )
(13.6)
where φ and γ are constants and F and G are non-linear functions in f i and g i .
Now, you say: “Alright. Now we have the equations to solve the problem, at least
numerically.” Yes, but there is still the ‘butterfly effect’. You start with an initial
choice for the values f 0 , g 0 and use the Eqs. (13.6) to step through time. Great.
However, if you change the initial conditions by just a ‘little bit’, you may find that
your new solution differs widely from your first one. If you are trying to predict the
weather, your initial data will be uncertain by small amounts. That makes detailed
weather prediction over more than about a week practically impossible.
There’s another complication: Computers cannot be infinitely precise about the
value of an irrational number, or even a very large rational one. This means that
2 Actually, the problem may have originally been discrete, and had been ‘smoothed’ by a calculus
limit.