18
Biological models with discrete time
The most important applications, however, may be pedagogical.
The elegant body of mathematical theory pertaining to linear systems (Fourier analysis, orthogonal
functions, and so on), and its successful application to many fundamentally linear problems in the
physical sciences, tends to dominate even moderately advanced University courses in mathematics
and theoretical physics. The mathematical intuition so developed ill equips the student to confront
the bizarre behavior exhibited by the simplest of discrete nonlinear systems, such as equation (3)1 .
Yet such nonlinear systems are surely the rule, not the exception, outside the physical sciences.
I would therefore urge that people be introduced to, say, equation (3) early in their mathematical
education. This equation can be studied phenomenologically by iterating it on a calculator, or even
by hand. Its study does not involve as much conceptual sophistication as does elementary calculus.
Such study would greatly enrich the student’s intuition about nonlinear systems.
Not only in research, but also in the everyday world of politics and economics, we would all be
better off if more people realized that simple nonlinear systems do not necessarily possess simple
dynamical properties.
Robert M. May
Simple mathematical models with very complicated dynamics.
Nature, 261(5560), 1976, 459-467
A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs
of rabbits can be produced from that pair in a year if it is supposed that every month each pair
begets a new pair which from the second month on becomes productive?
Leonardo Pisano Bigollo (1170 – 1250), known as Fibonacci
Liber Abaci, 1202
18.1
Introduction
Up till now we discussed mathematical models of biological process that are characterized by continuous time; this means that at every time instant it is possible to have one elementary event, and the
parameters of our models specified the rates of the events in the system, i.e., number of events per
time unit. For population models this, for instance, means that the generations of our models are
overlapping, and birth and death events can occur at every time instant.
A number of biological systems, however, can be characterized by discrete time, which means that
there are specific time moments at which the elementary events in our system can occur, and it is not
required that at these discrete time instants only unique event happens. For example, such discrete
time is reasonable to introduce in some models of fish populations, which reproduce at specific time
moments, or for insect populations, for which quite often non-overlapping populations are what is
actually observed in reality.
Anyway, for many situations (think also about observable time series) we need a modeling tool
to describe sequences of the variables that are of interest to us, and we are naturally led to consider
discrete maps or discrete dynamical systems in the form
Nt+1 = f (Nt ),
f : U → U,
Nt ∈ U ⊆ R,
t ∈ Z,
Math 484/684: Mathematical modeling of biological processes by Artem Novozhilov
e-mail: [email protected]. Spring 2014.
1
Equation (3) is given by Xt+1 = aXt (1 − Xt )
1
(1)
where the index notation Nt emphasize the discrete character of the time variable in the system.
Sometimes, however, I will use the usual nutation N (t), when it is more convenient.
There is an equivalent notation for system (1):
N 7→ f (N ),
N ∈ U ⊆ R.
(2)
Quite often the maps that we consider are noninvertible, and in this case t ∈ Z+ = {0, 1, 2, . . .}.
If we are given an initial condition N0 then discrete dynamical system (1) defines an orbit γ(N0 ) =
{N0 , f (N0 ), f 2 (N0 ), . . .}, where I use the notation
f k := f ◦ f ◦ . . . ◦ f ◦ f,
for k times composition of function f (i.e., k times successive applications of f ). I.e., f 2 (x) = f (f (x))
and f 4 (x) = f (f (f (f (x)))).
Example 1. Consider a simple example of population growth. By definition, a relative population
growth at time moment t is defined by
rt :=
Nt+1 − Nt
,
Nt
where Nt is the population size at time t. Assuming that rt = r = const for any t, we find
Nt+1 = (1 + r)Nt = wNt ,
w := 1 + r.
This equation linear and can be easily solved explicitly:
Nt = wt N0 = (1 + r)t N0 .
Therefore, we obtain an important conclusion that our population is growing exponentially without
bounds if w > 1 (r > 0), declining to zero if 0 < w < 1 (−1 < r < 0) and stays constant if w = 1
(r = 0). It is instructive to compare these three phases of population behavior with the solutions of
the continuous time Malthus model Ṅ = mN .
In general, of course, linear models cannot be used on long time intervals, since they predict
either unbounded growth or extinction. To guarantee that the orbit is bounded, we should consider a
nonlinear model.
Example 2 (Discrete logistic equation). Consider the discrete dynamical system
)
(
Nt
Nt+1 = rNt 1 −
,
K
where r, K are positive parameters. Now the model is nonlinear, and the population cannot grow to
infinity. However, there is another drawback of this model: For Nt exceeding K, Nt+1 < 0, which
contradicts biological interpretation of the model.
Example 3 (Ricker’s equation). To make sure that the population is bounded for all t and at the same
time is nonnegative, we can consider the so called Ricker model, which is widely used for modeling
fish populations:
(
)
Nt+1 = Nt e
Here, obviously, Nt ≥ 0 for all t > 0.
2
r 1−
Nt
K
.
Example 4. Concluding this short section, consider the second epigraph to this lecture, which mathematically can be formulated as
Nt+1 = Nt + Nt−1 ,
where Nt is the number of the pairs of rabbits capable of reproduction at the t-th month. Given the
initial conditions N0 = 0, N1 = 1, it is easy to see that the solution should be given by the sequence
of Fibonacci numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . .
but the question is how to find a general solution to this equation.
Another thing to point out here is that the equation written in this example is not a discrete
dynamical system according to our definition, since the population size at t + 1 is determined through
two points before. This can be fixed by considering an additional variable
Mt = Nt−1 ,
and then
Mt+1 = Nt ,
Nt+1 = Nt + Mt ,
which is two-dimensional discrete dynamical system. In general, d-dimensional discrete dynamical
systems is defined as a map of subset U of Rd to itself:
x 7→ f (x),
18.2
x ∈ U ⊆ Rd ,
f : U → U.
Cobweb diagram
There is a simple and efficient way to get an idea of the general behavior of the orbits of (1) by looking
at the graph of f . Before describing this method, I note that point N̂ is called a fixed point of (1) if
f (N̂ ) = N̂ . Geometrically this means that fixed points are the points of the intersection of the graph
of f and the bisectrices of the first and third quadrants.
Now consider a discrete map with f shown in the figure. Let N0 be an initial point, therefore,
1
f (N )
N2
N1
N0
0
0
N0
N1
N2
N
Figure 1: Cobweb diagram
3
1
0.7
N̂
æ
N̂
æ
æ
æ
æ
æ
æ
Nt
f (N )
1
æ
æ
æ
æ
0
0
N̂
N
0
1
0
2
4
6
8
10
t
Figure 2: Cobweb diagram
N1 = f (N0 ) is the point of intersection between the graph of f and the vertical line passing through
N0 . Now we can use the diagonal Nt+1 = Nt to find the location of N1 on the N -axis: This can be
done simply by finding the intersection of the horizontal line with the ordinate N1 and the diagonal.
After this we can project to N -axis to find N1 . N2 = f (N1 ) and so on. The whole orbit is simply
is a series of reflections from the diagonal (see the figure). The picture we obtain is sometime called
cobweb diagram.
Consider now the situation as in Figure 2. The large black dots show the locations of the fixed
points, small black dot is the initial condition, and together with the cobweb diagram I present the
time series (Nt )kt=0 . A little playing with cobweb diagram and choosing different initial conditions
should convince you that the picture presented in the right panel of Fig. 2 is universal: For any initial
condition the orbit approaches the fixed point N̂ , and the convergence to this point (except for maybe
several few steps) is monotonous. It is natural to call such fixed point asymptotically stable.
Using cobweb diagram we can get an idea what kind of phenomena can be expected in discrete
dynamical systems. For example in Fig.3 one can see that asymptotically stable fixed point can
1
0.8
æ
æ
æ
æ
æ
æ
æ
æ
æ
N̂
æ
Nt
f (N )
N̂
æ
0
N̂
0
0
1
N
0
2
4
6
8
10
t
Figure 3: Cobweb diagram. Non-monotonous convergence
attract orbits in an oscillatory way. This fact alone should convince you that one dimensional discrete
dynamical system possess a richer behavior in comparison with scalar ordinary differential equations.
Due to the fact that the state space for the scalar equations is one dimensional, and orbits cannot
4
intersect, any non-monotonous behavior of solutions to ODE is prohibited.
In scalar discrete dynamical systems a periodic solutions can be observed (see Fig. 4). It can be
seen in the figure that there are two points N1 and N2 such that N2 = f (N1 ) and N1 = f (N2 ), hence
N1 = f 2 (N1 ) and N2 = f 2 (N2 ). We call the orbit {N1 , N2 } a 2-periodic orbit, or orbit with period 2.
0.9
1
N̂
æ
æ
æ
æ
æ
æ
æ
æ
æ
Nt
f (N )
æ
N̂
æ
0
N̂
0
0
1
0
2
4
6
N
8
10
t
Figure 4: Cobweb diagram. Periodic solutions
A remarkable fact is that is system (1) has 3-periodic solution then it has k-periodic solution for
any k. An example of 3-periodic solution is given in Fig. 5.
1
1
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æ æ æ æ æ æ æ
æ æ
æ
æ
æ
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æ
æ
æ
æ
æ
N̂
N̂
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Nt
f (N )
æ
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æ
N̂
0
0
1
0
10
æ
æ
æ
æ
0
æ æ æ æ æ æ æ
æ
æ
æ
20
N
æ æ æ æ æ æ
30
40
50
t
Figure 5: Cobweb diagram. 3-periodic solutions
Finally, aperiodic orbits can be observed. In Fig. 6 an example of such an orbit is given. In the top
row first 50 and 250 points respectively of the orbit are shown. Such orbits are called chaotic, and the
system itself is chaotic. We will need a few mathematical preliminaries to define what chaotic means
exactly, but here you can imagine writing down 0 every time the coordinate of the orbit in below 0.5,
and 1 when the coordinate is above 0.5. As a result you will get a sequence 00111010011011000110 . . .
You can produce a similar sequence by tossing a coin and recording 1 if it is head and 0 if it is tail.
You will get another sequence. Both sequences look (whatever this means) random. Moreover, there
is no statistical test to determine which sequence is produced by a random experiment (tossing a coin)
or by a deterministic discrete dynamical system of the form (1).
5
N̂
N̂
f (N )
1
f (N )
1
0
0
N̂
0
1
N̂
0
N
1
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Nt
N̂
0
1
N
0
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50
æ
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100
150
t
Figure 6: Cobweb diagram. Chaotic orbit
6
200
250