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# Population Modeling

Summary: analyze three simple mathematical models of a changing population as well as a model of the interactions between two populations

BIS1 COMPUTER ASSIGMENT #4

## Population Modeling

In this assignment you will analyze three simple mathematical models of a changing population as well as a model of the interactions between two populations.

Model 1 (Exponential Growth):

The first of these models involves a single population growing without any resource limitations. This model would correspond to the early stages of population growth when the population’s needs are much less than the resources available. In this model, the population size (N) is governed by a simple difference equation with a single parameter, the growth rate (r), that determines how the population would grow or decline:

 Equation for Model 1 Δ⁢N=r×N Δ N r N N t+1 = N t + r×N N t+1 = N t + r N

Model 2 (Logistic Growth):

As you will see in Part I below, the population growth predicted by Model 1 will at some point in time be unrealistic due to resource limitations. In order to correct for this, we can modify the equation to include a carrying capacity (K) that is the maximum number of individuals that the environment can sustain. The modified difference equations are as follows:

 Equation for Model 2 Δ⁢N=r×N⁢1− N K Δ N r N 1 N K N t+1 = (Nt+r)⁢Nt⁢1− Nt K N t+1 = Nt r Nt 1 Nt K

The expression for the change in population (N) above is actually quite intuitive if you think of how it behaves for the two extremes of when the population is much lower than the carrying capacity (NK) and when the population is approximately equal to the carrying capacity (PK):

ΔN=r×N1 N K r×N1-0 = r×N ( For N much less than K ) Δ N r N 1 N K r N 1-0 = r N ( For N much less than K )

ΔN=r×N1 N K r×N1-1 = r×N For N ≈ K Δ N r N 1 N K r N 1-1 = r N For N ≈ K

This shows that when there the population is much lower than the carrying capacity, the equation simply reverts to Model 1, but when the population gets close to the carrying capacity the growth drops to zero.

Model 3: “Predator-Prey” Simulation:

This model simulates the interactions between two interacting populations—one is a predator that eats the other, the prey. This model is very similar to the single population models (Models 1 & 2) described above, but this model involves a number of new parameters describing the interaction.

The equations for the Prey (N) and the Predator (P) are as follows, starting with the Prey equation

 Equation for Prey Species Δ⁢N=rprey⁢N⁢1− N K - β×N×P Δ N rprey N 1 N K - β N P Nt+1 = Nt+rprey⁢Nt⁢1− Nt K - β×N×P Nt+1 = Nt rprey Nt 1 Nt K - β N P

The first part of expression for the change in population (ΔN) above is actually just the Logistic Growth equation with the same parameters: the prey growth rate, rprey, and the prey carrying capacity, K. The second term describes the loss of prey to the predators. This term is proportion to the product of the two populations scaled by a predation factor,β, that describes the rate of loss of prey.

The equation for the predator also contains two terms:

 Equation for Predator Species ΔN = rpred⁢N×P - γ×P ΔN = rpred N P - γP Pt+1 = Pt+rpred⁢N×P - γ⁢Pt Pt+1 = Pt rpred N P - γ Pt

Equation for Predator Species

The first term indicates that the population gain of the predator population is just proportional to product of the predator and prey populations scaled by a growth rate, rpred. The inclusion of the prey population is to model the fact that the predator needs to eat the prey to survive. The second term is the spontaneous death rate of the predator population that includes a parameter, predator death rate ().

Although these equations may look quite similar to those for a single population that we modeled in the last parts, their mathematical behavior is much richer and complex because of the coupling between the two populations.

### Part I. Exponential Growth (Model 1):

In this Part you will be using the exponential model in the Excel workbook to study the effect of changing the initial population (N0) or the growth rate (r) on the growth of a population. To do this, first download the Excel workbook from the website (http://ccb.ucmerced.edu/BIS1). Save this workbook on your desktop and then double click on it to open the workbook in Excel. The workbook contains three different worksheets, containing Models 1-3. Each worksheet contains a table at the top for entering model parameters, a section where the spreadsheet calculation is performed, and a graph of the population size versus time. You will be changing values in the parameter table and observing changes in the values in the spreadsheet and/or on the graph.

Click on the Model 1 tab in the workbook and then carry out the following steps changing the indicated parameters at the top of the page.

1) Starting with the values of r=0.05 and N0=2, record the number of individuals in the population at 100 years: __________

2) Now increase the size of the initial population by 1000% to N0=20, and record the number of individuals in the population at 100 years: __________

3) Finally, return the initial population to N0=2, and increase the growth rate by 50% to r=0.075, and record the population at 100 years: ____________

4) What do your results in 1-3 above indicate about the sensitivity of population growth to changes in the initial population versus chances in the growth rate?

5) The current human growth rate is approximately r=0.019. If the world population reached 6 billion in 1997 and population growth continues at this rate, what year will the population reach 10 billion—a number sometimes cited as the carrying capacity of the earth. For convenience, assume that the population is in units of billions (so “6” means 6 billion).

### Part II. Logistic Growth (Model 2):

In this Part you will be using the logistic model in the Excel workbook to study the effect of changing the carrying capacity (K) or the growth rate (r) on the growth of a population:

1) If you start with an initial population of N0=2, a growth rate r=0.05, and a carrying capacity of K=10000, how does your population graph compare with that you got for the exponential case (Model 1) in Part I? Describe the shape of the curve and the population at year 100. How does the shape of the curve and the population at 100 years change when you reduce the values of K to 1000 or 100?

2) In some situations, the carrying capacity of an environment can drop, for example if the underground aquifers supplying water become contaminated (as is happening in many places). Model this event by setting your initial population of N0=1000, a growth rate r=0.05, and a carrying capacity of K=200. How does the final population change? How many years does this change take?

3) Using the world population growth rate of r=0.019 and starting with the world population of 6 billion in 1997 and assuming the carrying capacity of earth to be 11 billion, how long will it take the world population to reach 10 billion people? How does this compare with your estimate using the exponential model in Part I?

### Part III. Predator-Prey Interactions (Model 3):

In this part you will adjust the parameters of the predator prey model and study the dynamics of the populations in time. Note that the populations are in arbitrary units—you can think of these as 100’s, 1000’s or 1000000’s of individuals depending on the type of organism being modeled.

 Prey Predator Growth Rate 1.000 1.000 Initial Population: 0.500 0.500 Carrying Capacity(K) 1.000 N/A Predation Rate 0.800 N/A Predator death rate N/A 0.300

Observe the graph showing the change in the predator and prey populations with time. You’ll notice that both populations reach steady-state values after which they don’t change. During the oscillatory period, does the peak in the prey population occur just before, just after, or at the same time as the peak in the predator populations? Why does this make sense? Why does the predator population drop?

2) Now observe the steady-state values of N and P (i.e. the value after the oscillations have died out): N:_______ P:________ and then adjust the size of the initial populations down by a factor of 2-10 (do this separately for the prey and the predator populations) and observe the effect on the steady state populations. What changes do you see? What happens if you set the initial population to the steady state values?

3) If you increase the initial population of the predator population above some level, what happens? Is this unexpected? Do you see the same effect if you increase the initial population of the prey?

4) If you increase the death rate of the predator (perhaps by a disease being introduced), how would you expect the dynamics and steady state populations to be affected? What happens to the steady state predator and prey populations if the predator death rate is increased beyond some critical level?

5) Finally, adjust the values of the Predation Rate, which describes how effectively the predators hunt and eat the prey, from its original value of 0.8 up to .95 and down to .1, while recording the approximate steady state levels for both populations. Which steady state population is most affected by adjusting this rate?

### Part IV: Predator-Prey Phase Plots (Extra Credit—worth 2 points)

In this Part you will be using a similar model as in Part III, but you will be studying the results in terms of a “Phase Plot”. This graph plots the Predator Population on the Y-axis versus the Prey Population on the X-axis. The resulting line shows the trajectory of the population. Lines in phase planes can have many different behaviors. They can converge to a fixed point (sometimes called an attractor). Note that at this fixed point, there both populations have reached equilibrium at values that can be read from the X and Y coordinates of the fixed point. The line in phase space can diverge to infinity, but such behavior is not sensible when modeling populations. Also, the line in phase space can go into an “infinite loop”, technically known as a limit cycle in which the populations oscillate back and forth without converging to a fixed point or diverging.

There is a nifty website where you watch an animation of the phase plane for the Lotka-Volterra equations for sets of parameters that you set which is a good introduction to the concept of a phase plot: http://www.aw-bc.com/ide/Media/JavaTools/popltkvl.html. (Note: In order to view this tool on your web, you must have a Java-enable browser on your computer.) Go to this website and set the parameters to: a=1.0; b=0.2; c=0.5; and d=1.0. To start a trajectory, just click your mouse in the “phase plane”. Note that the point where you click your mouse determines the initial population of prey (X-coordinate) and predators (Y-coordinate) (You can also enter the initial populations directly into the population boxes). Watch the trajectory evolve and note the white arrows on the phase diagram. These are flow lines that show you which direction the trajectory will move from any starting point. Try starting with different initial populations—what starting conditions lead you to not have a limit cycle?

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Now try adjusting the parameters a-d to find some phase plots where no choices of the initial populations lead to a limit cycle. In the table below list some set of parameters and the dynamical behavior they lead to:

 a b c d Description of phase plot/behavior

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