Discrete Time, One-Dimensional Dynamical Systems

Created by Claudia Neuhauser Worksheet 3: Discrete Time, One-dimensional Dynamical Systems

Discrete Time, One-dimensional Dynamical Systems

Many species reproduce in distinct seasons. Populations of such species are then best modeled using discrete time models. In their simplest form where the population size at generation t+1, denoted by Nt+1, only depends on the population size at generation t, denoted by Nt, we can model this recursively as

Nt1  f (Nt ), t  0,1,2,

As the initial condition, we need to specify the population size at time 0, N0.

Exponential Growth

If the population size from one generation to the next is multiplied by a constant factor, exponential growth results. The recursive equation for this type of growth is

(1) Nt1  RNt , t  0,1,2, where R is a nonnegative constant, called the growth parameter. We assume that R  0 and that the population size at generation 0 is N0. This model is linear and can be solved explicitly, that is, we can find a function g(t) that describes the population size explicitly as a function of t, given N0. Here is how to find the function:

N1  RN0 2 N2  RN1  RRN0   R N0 2 3 N3  RN2  RR N0  R N0 ⋮ t Nt  R N0

t (2) Nt  R N0 is the solution of (1). It is an exponential function. The behavior of this function is as follows:

0 if 0  R  1  lim Nt   N0 if R  1 t    if R  1

1 Worksheet 3: Discrete Time, One-dimensional Dynamical Systems

There is a graphical method for determining long-term behavior, called cob-webbing. This uses the recursive equation. If we graph Nt+1 as a function of Nt, then the behavior can be seen from successively determining the values at generation 1,2,3,… when starting from N0.

This is illustrated in Figure 1.

(a) (b) N N t+1 N =RN t+1 t+1 t N =N N N =N t+1 t 2 t+1 t

N =RN N t+1 t 1 N 1 N 2 N N N N t N N t 0 1 1 0 Figure 1: In Figure 1(a), the parameter R is greater than 1; in Figure 1(b), it is between 0 and 1.

The value 0 is special: When R  1, it is the only value with the property that if the population size at time 0 is equal to 0, the population size will be 0 for all future times. In other words, the population size will not change when starting with N0  0 . Such a value is called a fixed point or point equilibrium. Of interest in this context is the stability of fixed points. We say that a fixed point is attracting or locally stable if the system returns to the equilibrium after a small perturbation. We call it repelling or unstable otherwise. For the exponential growth model, the equilibrium N  0 is locally stable if 0  R  1, and unstable otherwise. The equilibrium N  0 is even globally stable for 0  R  1 since no matter what the value of N0 , the population size will eventually reach N  0 . When R=1, the fixed point is indifferent.

Fixed Points and Stability

We now return to the general system

Nt1  f (Nt ), t  0,1,2, to discuss fixed points and their stability. We assume that the function f is differentiable in its domain. To find fixed points algebraically, we solve N  f (N) . Geometrically,

fixed points can be found where the graph of Nt1  f (Nt ) intersects the graph of Nt1  Nt . This is shown in Figure 2.

N N =N t+1 N =f(N ) t+1 t t+1 t

N N1 2

N N N t 1 0 Figure 2. The fixed points are indicated by the solid dots. The initial steps of the cob-webbing procedure starting from N0 are shown.

Task 1:

Investigate the stability of the fixed points in Figure 2 using the cob-webbing procedure.

Stability of a fixed point is a local property: If the system returns to the fixed point after a small perturbation, the fixed point is locally stable. An analytical criterion can be derived that is based on the idea of a small perturbation.

Criterion: A fixed point N* of Nt1  f (Nt ) is locally stable if

f N *  1

To derive this criterion, we use calculus.

Proof of the Criterion: We denote a small perturbation at time t by zt and write

Nt  N * zt

Nt1  f Nt   f N * zt 

We now linearize the function f about N  N *. We find

LN * zt   f N * f N *zt

Since the perturbation is small, we can approximate Nt1  N * zt1 by

N * zt1  f N * f N *zt

Since N*  f (N*) , this reduces to

zt1  f N *zt

But this is the recursion for exponential growth and the result follows.

Logistic Growth

We will explore a simple example of population growth with density dependence, which has received a lot of attention. The equation is given by

  Nt  Nt1  Nt 1 R1    K  where we assume that R and K are positive. The parameter R is called the growth parameter and K is called the carrying capacity. To find fixed points, we set

  N  N  N 1 R1    K 

This yields the trivial solution N  0 and N  K . To determine stability, we need to differentiate

  N  f N   N 1 R1    K 

We find (using the product rule)

2NR (3) f N   1 R  K

Since f 0  1 R  1, we conclude that N  0 is unstable. Now,

2KR f K   1 R   1 R K we conclude that N  K is locally stable if 0  R  2 .

Task 2:

Show that N  0 and N  K are fixed points of the logistic equation. Confirm that the derivative of f(N) is given by the expression in Equation (3) and confirm the stability results derived above.

Complex Dynamics

The discrete time logistic growth model exhibits complicated dynamics, which you will explore using a computer. The form of the equation can be simplified if the following transformation is used

R x  N t K(1 R) t

xt1  rxt 1 xt  where r=1+R. This is the canonical form of logistic growth. Since we assumed that R is positive, we now need to assume that r is greater than 1.

Task 3:

Derive the canonical form of the logistic growth equation and show that when r >1, two fixed points exist. Determine the stability of the two fixed points when 1 < r < 3.

Task 4:

Show that for xt to stay between 0 and 1, we need to assume that r does not exceed 4.

Periodic Orbits and Chaos

If you simulate the canonical form of logistic growth for values of r between 3 and 3.449…, xt settles into a periodic orbit of period 2, that is, for large enough times, xt will oscillate between two values. This is illustrated in the graph below.

D i s c r e t e T i m e L o g i s t i c G r o w t h w i t h P a r a m e t e r r = 3 . 2 1

0 . 7 e z

i 0 . 6 S

n o i 0 . 5 t a l u

p 0 . 4 o P

0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 G e n e r a t i o n

Figure 3: Periodic orbit with period 2.

Figure 3 was generated in Matlab using the following program:

%Discrete time logistic growth dislog.m clear fsize=15; genmax=101; popsize=zeros(1,genmax); generation=zeros(1,genmax); popsize(1)=.1; generation(1)=0; r=3.2;

6 Worksheet 3: Discrete Time, One-dimensional Dynamical Systems for j=1:genmax-1 popsize(j+1)=r*popsize(j)*(1-popsize(j)); generation(j+1)=j; end plot(generation,popsize,'ko-') xlabel('Generation','Fontsize',fsize) ylabel('Population Size','Fontsize',fsize) title(['Discrete Time Logistic Growth with Parameter r=',num2str(r)],'Fontsize',fsize) axis([0 genmax 0 1]);

A periodic orbit of period 4 appears for r between 3.449… and 3.544…. Increasing r continues to double the period (a periodic orbit of period 8 is born when r=3.544…, a periodic orbit of period 16 is born when r=3.564…, a periodic orbit of period 32 is born when r=3.567…). This period doubling occurs until r reaches a value of about 3.57 when population patterns become chaotic: there is no regular pattern, the dynamics are aperiodic, and the system shows sensitivity to initial conditions. The population dynamics seem to be random, though the rules are deterministic! Figure 4 shows the values the dynamics take on as a function of r for the cases when the dynamics exhibit fixed points, periodic orbits, and chaos as a function of r. Such a diagram is called a bifurcation diagram. To generate such a diagram, the dynamics are run for a long time and the values are plotted as a function of r.

Figure 3: The bifurcation diagram for logistic growth. Source: http://www.pha.jhu.edu/~ldb/seminar/logdiffeqn.html

Computer Lab and Homework (due ______)

 Read the worksheet and work out the tasks.

 Read the paper by R. May (1974).

In this computer lab, you will learn how to simulate populations in discrete time using a spreadsheet.

Step 1

Gause (1934) performed a number of experiments with protists in the genus Paramecium. In one of the experiments, he tracked successive population sizes of Paramecium caudatum (see Figure 5).

Figure 5: A picture of Paramecium caudatum. Source: http://protist.i.hosei.ac.jp/PDB/PCD0306/A/17.jpg

The following table contains the population size as a function of Day:

Day P. caudatum 0 2 1 4 2 8 3 9 4 14 5 21 6 57 7 94 8 142 9 175 10 189 11 217 12 199

(a) Plot population size as a function of day. (b) Assume that the population dynamics follow discrete logistic growth. Plot population size at day n+1 as a function of population size at day n and fit a function that would allow you to determine the parameters R and K.

Step 2

Investigate the behavior of the canonical logistic equation xt1  rxt 1 xt  by graphing xt as a function of t for r=2, 3.2, 3.52, and 3.8 when x0=0.3. Describe what you see.

Step 3

A hallmark of chaotic dynamics is the sensitivity to initial conditions. This means that population sizes quickly differ even if initial conditions are quite similar. Test this for r=3.8 when x0=0.19 and x0=0.2.

Step 4

If you look at the bifurcation diagram closely, you see that in the chaotic regime, there is a period-3 cycle. It occurs for values of r between 3.8284… and 3.8415…. Simulate the logistic growth model for values in this range.

Step 5

The dynamics are quite interesting when r=3.828 (i.e., right below the period-3 window). Run the simulation when x0=0.3 and r=3.828 for t=0,1,…,200 and describe what you see. The phenomenon you see (nearly period-3 interrupted by chaos) is called intermittency.

Step 6

Write up your findings and include graphs in your report. Complete the tasks and hand them in as well.

References

Gause, G.F. (1934) The struggle for existence. Hafner, New York.

May, R. (1974): Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186: 645-647.

An excellent introduction into dynamical systems is given in Strogatz, S.H. (1994) Nonlinear Dynamics and Chaos. Perseus Publishing.