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Chaos in Population Dynamics

Understanding chaos in the logistic model

How does chaos emerge in deterministic models of population growth? What are the quantitative tools we need to understand and

represent chaotic behavior?

Prepared for SSAC by*David McAvity – The Evergreen State College*

© The Washington Center for Improving the Quality of Undergraduate Education. All rights reserved. *2007*

Supporting Quantitative concepts and skillsVisual Display of Data – XY scatter plotsIterationExponential and logistic growthCreating cobweb diagramsCurve fitting and trend lines

SSAC2007.QH352.DM1.1

Core Quantitative concepts and skillsModeling with difference equations

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Slides 4-5 Implement the model and create a plot of the solution.

Slides 6-7 Explore the parameter space of the logistic model – oscillations, bifurcations and chaos.

Slides 8-10 Create cobweb diagrams to visualize the approach to chaos.

Slides 11-12 Quantify chaos with the Liapunov exponent.

Slide 13 Assignment

Overview of Module

The logistic model describes the growth of a population subject to a carrying capacity which limits the total population. When the population grows discretely, as in the

case of a population of birds that breeds yearly in spring, the population dynamics can exhibit chaotic oscillations under certain conditions. In this module we

investigate what conditions lead to chaos. We will also learn a variety of quantitative methods for analyzing the route to chaos.

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The logistic model is used to describe the growth of a population in an environment with limited resources. When the population grows continuously (that is, when the births and deaths do not occur at fixed intervals) the logistic model always leads to smooth predictable growth of the population to a limiting value called the carrying capacity, which is the maximum value the environment can support. However, when the population grows discretely (for example, a population that reproduces at fixed intervals – such as once a year in spring) then the dynamics can be more complicated. The model can lead to a population that reaches a simple equilibrium, has periodic oscillations or exhibits chaotic behavior, depending on the population growth rate for the discrete time interval. In this module we will assume a time interval of one year and will explore the dynamics of the discrete logistic model with different growth rates.

Problem

The problem:•What values of the growth rate in the logistic model lead to a simple equilibrium and how does the model approach this equilibrium?•What values of the growth rate lead to periodic behavior and what type of periodicity can result? •What values of the growth rate lead to chaotic population dynamics in the logistic model?•What methods can we use to visualize and quantify the approach to chaos?

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Our first task is to create a spread sheet implementing the difference equation for the logistic model: P=rP(1-P/K), where P is the population number, r is the intrinsic growth rate and K is the carrying capacity. Recall that the difference equation for updating the population from one generation to the next is given by: Pt+1=Pt + Pt , with initial population P0 . In this example we will start with an initial population of 1, a growth rate of 0.1 per year and a carrying capacity of 100.

= cell with a number in it

= cell with a formula in it

Create a worksheet like this one, giving an iterative

solution to the logistic model.

The Logistic Model

Hint: Need help with entering formulas? Here is an example: If you have a number in Cell B2 and you want to calculate the square of it in Cell C2 then in C2 you type =B2^2.

To use absolute references type =$B$2^2 instead.

 

B C2 4 =B2 2̂

B C2 4 =$B$2 2̂

Hint: Copy the formula down for 100 cells.

Make sure you use absolute references for the parameter values.

Hint: Enter the equation for the logistic difference

equation here

B C23 P0 1

4 r (y -1 ) 0.15 K 10067 t (y) P t

8 0 19 1 1.10

10 2 1.2111 3 1.3312 4 1.4613 5 1.6014 6 1.7615 7 1.9316 8 2.1217 9 2.3318 10 2.5619 11 2.8120 12 3.0821 13 3.3822 14 3.70

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Use an XY scatter plot with points connected. Label the axes and format the graph correctly.

Create a plot of the population of the function of time as predicted by the logistic model.

Graphing the Solution

Hint: To plot a graph, select the x and y axis columns that you want to plot. Then click the chart icon and follow the instructions.

B C D E F G H I J23 P0 1.00

4 r (y -1 ) 0.105 K 100.0067 t (y) P t

8 0 1.009 1 1.10

10 2 1.2111 3 1.3312 4 1.4613 5 1.6014 6 1.7615 7 1.9316 8 2.1217 9 2.3318 10 2.5619 11 2.8120 12 3.0821 13 3.38

Logistic Model

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Now we are ready to explore how the solution to the logistic model depends on the three parameters P0, r and K.

Answer the following questions in a textbox in a new sheet named assignments.

(a) Vary the initial population P0 between 0 and 200 while keeping r=0.1 and K=100. Describe the

solution in each case. What do the solutions have in common and how do they differ?

(b) Vary the carrying capacity K between 0 and 200 while keeping r =0.1 and P0= . Describe the

the solution in each case. What do the solutions have in common and how do they differ?

(c) Vary the growth rate r, between 0 and 1 while keeping K=100 and P0 = 1. Describe the nature

of the solution in each case. What do the solutions have in common and how do they differ?

Exploring Parameter Space

B C23 P0 14 r 0.15 K 100

Logistic Model

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When the growth rate, r, varies between 0 and 1, the logistic model predicts a smooth approach to equilibrium at a population equal to the carrying capacity. However, if you consider values of the growth rate larger than 1 the model behaves in unexpected ways. For some values of r the solution overshoots the carrying capacity before approaching equilibrium, for some values it oscillates between two or more equilibria (called 2-cycles, 4-cycles, etc), and for other values the solution is chaotic.

The Route to Chaos

Logistic Model

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Logistic Model

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Answer the following question in your assignments sheet.

2. Find the range of r for which the model exhibits:

(a) oscillating approach to equilibrium.

(b) 2-cycles.

(c) 4-cycles.

(d) chaos.

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In order to understand how the different types of behavior in the logistic model come about it is useful to create a cobweb plot. A cobweb plot is a graph of Pt+1vs Pt and is used to illustrate how the sequence of values is generated by the difference equation, Pt+1=f(Pt). For the logistic model it follows from Slide 4 that f(x)=x + r x (1-x/K). In the cobweb plot we start with the initial input value x= P0, move up to the curve y=f(x) to get the value y=P1 and return to the line y=x to find the next input value x= P1.. We then repeat the process. The sequence of points to plot is (P0,0), (P0,P1), (P1,P1), (P1,P2) … See the graph below.

Copy your spreadsheet to a new one and then modified it to create the rows of points as

suggested below.

Creating a Cobweb Plot – Part 1

Cobweb Diagram

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y=f(x)y=x

The initial point is (P0, 0)

Insert a row for the point which brings the cobweb plot back to the y=x line.

Now copy the second and third rows down.

B C D23 P0 204 r 15 K 10067 t P t P t+1

8 20.00 0.009 0 20.00 36.00

10 36.00 36.0011 1 36.00 59.0412 59.04 59.0413 2 59.04 83.2214 83.22 83.2215 3 83.22 97.1916 97.19 97.1917 4 97.19 99.9218 99.92 99.92

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In order to create the full cobweb plot we need to plot the line y=x, the curve y= f(x)=x + r x (1-x/K) and the sequence of points from the previous slide.

Add new columns in your spreadsheet that will allow you to plot y=x and y= f(x)=x + r x (1-x/K). You need to first include a column of x values. These should range between 0 and 200 (or twice

the carrying capacity). When you have added your columns, plot y=x and y= f(x) and Pt+1vs Pt on the same graph. Use XY scatter plot with connected points. Use different colors for each

curve.

Creating a Cobweb Plot – Part 2

x y=x y= f(x)

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10 10 1912 12 22.5614 14 26.0416 16 29.4418 18 32.7620 20 3622 22 39.1624 24 42.2426 26 45.2428 28 48.1630 30 5132 32 53.7634 34 56.4436 36 59.0438 38 61.5640 40 6442 42 66.3644 44 68.6446 46 70.84

Cobweb Diagram

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Answer the following question in your assignments sheet.

3. Choose values of r corresponding to each of the cases below and create cobweb plots. describe the cobweb plots, qualitatively noting differences between the cases. Find values of r corresponding to transitions between two types of behavior and then note, if possible, what feature of the plot is changing at that point. Pay particular attention to how the cobweb plot behaves near the intersection between y=f(x) and y=x, and the sign of the slope of y=f(x) there.

(a) smooth approach to equilibrium

(b) oscillating approach to equilibrium

(c) 2-cycles

(d) 4-cycles

(e) chaos

Illustrating Dynamics of the Logistic Model with Cobweb Plots

As you discovered earlier, the logistic model exhibits a range of different behaviors depending on the values of the growth rate r . The cobweb diagram looks qualitatively different in each case, and illustrate why the model model changes its behavior at particular values of the growth rate.

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•Make a copy of your first spreadsheet to a new sheet. Then create an entry for the initial difference in populations .

• Add a new population column, whose initial value is P0+ . You may also want to format the columns so that they show enough decimal places to illustrate the small difference in the populations.

•Finally make a column where you calculate . This column only needs to extend about 30 time steps. You might observe that this value jumps around a bit, However you should see that it has a tendency to grow when r is in regions where the population dynamics is chaotic.

Quantifying Chaos

The word chaos is often used in imprecise ways. In this module, a chaotic system means one that exhibits sensitive dependence on initial conditions. If a model predicts two vastly different results for two populations which initially differ by only a small amount then the model is said to be chaotic. More precisely, if we define the absolute value of the difference between two populations as , then if the ratio of to the initial absolute difference diverges exponentially the system is chaotic.

B C D E2

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8 0 50 50.00001 19 1 118.75 118.75 1

10 2 57.519531 57.5195 2.7812511 3 124.71459 124.7146 1.63099912 4 39.952169 39.95222 5.07126713 5 105.9258 105.9259 7.87379814 6 88.664187 88.66402 16.3453815 7 116.3039 116.3041 18.413516 8 64.158191 64.1577 48.7353617 9 127.3957 127.3956 10.7857618 10 31.418119 31.41847 35.1266519 11 90.672745 90.67345 71.0258720 12 113.93026 113.9294 87.8604621 13 70.285611 70.28782 221.0692

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Plot / vs t. Then add an exponential trend line. Make sure you choose the option to include the equation and R2 value of trend line on the graph. Now vary the value of the growth rate r and determine the value of the Liapunov exponent for different values of r.

4. Find the range of values for r where the Liapunov is positive (indicating chaos).

The Liapunov Exponent

To determine if the ratio / grows exponentially we plot / vs time and then fit an exponential curve to the data. The exponent is called the Liapunov exponent, which is a quantitative measure of chaos. A value of less than or equal to zero implies that the system is not chaotic.

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Hint: Need help with trend lines? Right click on a data point in your graph. Select add trend line from the menu. Choose the exponential fit and then under options check the boxes for display equation on chart and display R-squared value on chart.

B C D E G H I J K L M2

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8 0 50 50.00001 19 1 118.75 118.75 1

10 2 57.519531 57.5195 2.7812511 3 124.71459 124.7146 1.63099912 4 39.952169 39.95222 5.07126713 5 105.9258 105.9259 7.87379814 6 88.664187 88.66402 16.3453815 7 116.3039 116.3041 18.413516 8 64.158191 64.1577 48.7353617 9 127.3957 127.3956 10.7857618 10 31.418119 31.41847 35.1266519 11 90.672745 90.67345 71.0258720 12 113.93026 113.9294 87.86046

Liapunov Exponent y = 1.1203e0.2939x

R 2 = 0.9526

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1. Answer the questions about the parameters of the logistic model on Slide 6.

2. Answer the questions about the range of of values of the growth rate r corresponding to different types of behavior of the logistic model on Slide 7.

3. Answer the questions about the cobweb diagrams on Slide 10.

4. Answer the question about the Liapunov exponent on Slide 12.

5. Is the the fact that the logistic model yields chaotic solutions for high values of the growth rate reasonable on biological grounds? Explain your answer.

6. The logistic model is a good enhancement to an exponential growth model because it provides a mechanism to limit the growth through the carrying capacity. It has several deficiencies, however. One is that the per capita growth rate goes below -1. Why is this unrealistic?.

7. A model that fixes this problem is the Ricker model . Show that the per capita growth rate for the Ricker model is always larger than -1.

8. Repeat the analysis of this lab on the Ricker model. In particular, give a complete account of how the behavior of the model depends on the growth rate r and compare and contrast this with the logistic model.

Write your answers to the following questions in your “Assignment” spreadsheet. When you are finished the lab submit your spreadsheet to the program drop box.

Assignment

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1. If the per capita growth rate of a population is constant would you expect the growth to be linear or exponential or something else?

2. Define the following terms:(a) Carrying capacity(b) Chaos(c) Liapunov exponent

3. Suppose a small population of birds were introduced to a lush island on which they had no natural predators. If the birds breed once annually, How would you expect the population to grow if the logistic model were an accurate model?(a) grow exponentially without limit.(b) grow for a few years until they reach a constant stable population. (c) grow for a few years and then fluctuate randomly from year to year.(d) the answer will depend on the size of the growth rate of the birds.

4. Given the difference equation xn+1=1 + 1/xn and x0=1.

(a) Find x1 and x2 .

(b) Explain in complete sentences how you would determine if the sequence generated by this equation reached a stable equilibrium, an oscillating cycle or is chaotic?

5. Given a difference equation xn+1=f(xn ), what feature of a cobweb plot indicates an equilibrium point?

Answer the following questions as best you can without using technology. Draw diagrams where needed. If you do not know an answer to a question please say so so explicitly rather than guessing or leaving the question blank. (Note to instructor: Reassure students who are taking the pretest that it is ok if they do not know these concepts.)

Pre-Post Test