<p>Created by Claudia Neuhauser Worksheet 3: Discrete Time, One-dimensional Dynamical Systems</p><p>Discrete Time, One-dimensional Dynamical Systems</p><p>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</p><p>Nt1  f (Nt ), t  0,1,2,</p><p>As the initial condition, we need to specify the population size at time 0, N0.</p><p>Exponential Growth</p><p>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</p><p>(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:</p><p>N1  RN0 2 N2  RN1  RRN0   R N0 2 3 N3  RN2  RR N0  R N0 ⋮ t Nt  R N0</p><p>Thus, </p><p> t (2) Nt  R N0 is the solution of (1). It is an exponential function. The behavior of this function is as follows:</p><p>0 if 0  R  1  lim Nt   N0 if R  1 t    if R  1</p><p>1 Worksheet 3: Discrete Time, One-dimensional Dynamical Systems</p><p>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.</p><p>This is illustrated in Figure 1.</p><p>(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</p><p>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. </p><p>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.</p><p>Fixed Points and Stability</p><p>We now return to the general system</p><p>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, </p><p>2 Worksheet 3: Discrete Time, One-dimensional Dynamical Systems</p><p> 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.</p><p>N N =N t+1 N =f(N ) t+1 t t+1 t</p><p>N N1 2</p><p>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.</p><p>Task 1: </p><p>Investigate the stability of the fixed points in Figure 2 using the cob-webbing procedure.</p><p>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. </p><p>Criterion: A fixed point N* of Nt1  f (Nt ) is locally stable if</p><p> f N *  1</p><p>To derive this criterion, we use calculus.</p><p>Proof of the Criterion: We denote a small perturbation at time t by zt and write</p><p>Nt  N * zt</p><p>3 Worksheet 3: Discrete Time, One-dimensional Dynamical Systems</p><p> and</p><p>Nt1  f Nt   f N * zt </p><p>We now linearize the function f about N  N *. We find</p><p>LN * zt   f N * f N *zt</p><p>Since the perturbation is small, we can approximate Nt1  N * zt1 by</p><p>N * zt1  f N * f N *zt</p><p>Since N*  f (N*) , this reduces to</p><p> zt1  f N *zt</p><p>But this is the recursion for exponential growth and the result follows.</p><p>Logistic Growth</p><p>We will explore a simple example of population growth with density dependence, which has received a lot of attention. The equation is given by</p><p>  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</p><p>  N  N  N 1 R1    K </p><p>This yields the trivial solution N  0 and N  K . To determine stability, we need to differentiate </p><p>  N  f N   N 1 R1    K </p><p>We find (using the product rule)</p><p>4 Worksheet 3: Discrete Time, One-dimensional Dynamical Systems</p><p>2NR (3) f N   1 R  K</p><p>Since f 0  1 R  1, we conclude that N  0 is unstable. Now,</p><p>2KR f K   1 R   1 R K we conclude that N  K is locally stable if 0  R  2 . </p><p>Task 2: </p><p>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.</p><p>Complex Dynamics</p><p>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</p><p>R x  N t K(1 R) t</p><p>Then,</p><p> 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. </p><p>Task 3:</p><p>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. </p><p>Task 4:</p><p>5 Worksheet 3: Discrete Time, One-dimensional Dynamical Systems</p><p>Show that for xt to stay between 0 and 1, we need to assume that r does not exceed 4.</p><p>Periodic Orbits and Chaos</p><p>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. </p><p>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</p><p>0 . 9</p><p>0 . 8</p><p>0 . 7 e z</p><p> i 0 . 6 S</p><p> n o i 0 . 5 t a l u</p><p> p 0 . 4 o P</p><p>0 . 3</p><p>0 . 2</p><p>0 . 1</p><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</p><p>Figure 3: Periodic orbit with period 2.</p><p>Figure 3 was generated in Matlab using the following program:</p><p>%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;</p><p>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]);</p><p>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.</p><p>Figure 3: The bifurcation diagram for logistic growth. Source: http://www.pha.jhu.edu/~ldb/seminar/logdiffeqn.html</p><p>Computer Lab and Homework (due ______)</p><p> Read the worksheet and work out the tasks. </p><p>7 Worksheet 3: Discrete Time, One-dimensional Dynamical Systems</p><p> Read the paper by R. May (1974).</p><p>In this computer lab, you will learn how to simulate populations in discrete time using a spreadsheet. </p><p>Step 1</p><p>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).</p><p>Figure 5: A picture of Paramecium caudatum. Source: http://protist.i.hosei.ac.jp/PDB/PCD0306/A/17.jpg</p><p>The following table contains the population size as a function of Day:</p><p>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</p><p>(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. </p><p>Step 2</p><p>8 Worksheet 3: Discrete Time, One-dimensional Dynamical Systems</p><p>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.</p><p>Step 3</p><p>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.</p><p>Step 4</p><p>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.</p><p>Step 5</p><p>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.</p><p>Step 6</p><p>Write up your findings and include graphs in your report. Complete the tasks and hand them in as well.</p><p>References</p><p>Gause, G.F. (1934) The struggle for existence. Hafner, New York.</p><p>May, R. (1974): Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186: 645-647.</p><p>An excellent introduction into dynamical systems is given in Strogatz, S.H. (1994) Nonlinear Dynamics and Chaos. Perseus Publishing.</p><p>9</p>