# Laboratory 6: Lotka-Volterra, the logistic equation & Isle Royale

This is a computer-based activity using Populus software (P), followed by EcoBeaker analyses of moose populations on Isle Royale and the impact of wolves.

Introduction:

Some white-tailed deer are eating lettuces in your vegetable garden and you want to understand the interaction before you decide what to do to.

Your first step is to build a basic Lotka-Volterra Predator-Prey model for a description of the interaction using Populus:

dN/dt = r1N - CNP

and

dP/dt = gCNP – d2P

where:

N   = number of lettuce plants (prey),

P  = number of deer (predators),

r1   = the intrinsic rate of increase of lettuce (prey),

C   = the feeding rate of deer and CN is the per capita functional response of deer to lettuce density,

g    = the efficiency with which deer turn food into progeny, and gCN is the numerical response of deer (rate of new deer born as a function of prey density), and,

d2   = deer death rate.

which means that lettuce would increase without limit in the absence of deer.

This is the first of the “Continuous Predator-Prey models” in Populus.  Starting with the default parameters indicated below please answer the following questions:

Predator:       Po = 20,          d2 = 0.6,         g = 0.5

Prey:               No = 20           r1 = 0.9           C = 0.1

1. At what predator density does the prey isocline intercept?
2. At what prey density does the predator isocline intercept?
3. What is the maximum amplitude of the prey population?
4. What is the minimum amplitude of the prey population?
5. What is the maximum amplitude of the predator population?
6. What is the minimum amplitude of the predator population?
7. What is the frequency of prey peaks (N/t)?
8. What is the frequency of the predator peaks (N/t)?
9. Does the model change the amplitude or frequency of oscillations through time?

10.   Is the model any use for solving your problem?

Given that your answer to 10 is likely to be negative you decide to add more realism to the model and limit the lettuce population with intraspecific competition (you sowed the seeds too densely and foolishly forgot to thin the seedlings as your Granny always told you to do!).

So now you add density-dependent limitation of lettuces with logistic limitation to a carrying capacity as in the Populus (theta) θ–logistic predator-prey model in which:

dN/dt = r1N(1-(N/K)) - CNP

This means that in the absence of predators (P = 0) the lettuces are limited only by intraspecific competition where (1-(N/K)) is the same as (K-N)/K.

You now have the 2 extra parameters of K (the carrying capacity) and θ.  Set K to 100 and keep θ at 1 (this basically ignores the density-dependent feedback provided by θ).  All other parameters should be the same as above.  This model also includes the predator’s functional response (numbers of prey eaten at different prey densities).  Set the model to the following:

Predator:       Po = 20,          D = 0.6,          g = 0.6

Prey:               No = 20           r1 = 0.9           K = 100          θ = 1

with a type 2 functional response (C = 0.05 and h =1) and answer the following questions:

1. What shape is the prey isocline?
2. Where does the prey isocline intercept with the prey axis at P = 0?
3. Where does the predator isocline intercept the prey isocline – left or right of the hump?
4. Is the interaction stable or unstable?
5. Does the time trajectory spiral inwards or outwards?

Now change D to 0.7:

1. Does the predator isocline intercept to the right or left of the hump in the prey isocline?
2. Is the interaction stable or unstable?
3. Does the time trajectory spiral in or out?

Now change functional response to type 3 (c = 0.05, h = 1):

1. What shape is the prey isocline now?

10.   Is the new interaction stable or unstable?

Lastly, you decide to simplify the theta logistic model slightly and build your own logistically limited model of the interaction.  So you go to the Interaction Engine in Populus and add your own logistic term to the Lotka-Volterra mass action predator–prey model.

Using the Interaction Engine build the following model of the interaction:

dN1/dt = (r*N1*(K – N1)/K) – f*N2*N1

and

dN2/dt = (c*N2*N1)-(m*N2)

where:

N1      =           number of lettuce plants (prey),

N2      =           number of deer (predators),

r         =           the intrinsic rate of increase of lettuce (prey),

K        =           carrying capacity of lettuce

f         =           the feeding rate of deer,

c        =           the numerical response of deer (efficiency with which deer turn food into progeny), and,

m       =           deer death rate.

Model these continuous, differential equations using the POPULUS Interaction Engine and set the plot to N vs T, t = 100 and do not plot isoclines.  Enter the equations to build the model (uncheck N3 because you are only interested in 2 species) and set the starting parameters to:

N1     =      200

N2     =        20

r       =          0.2

m     =          0.3

K      =   1000

c       =          0.001

f        =          0.001

Run the model and turn on the gridding function .

Now that you have a functioning, resource-limited, plant-herbivore model examine how changes in r, f, and c  influence the stability and final population sizes of lettuce and deer after 100 time intervals.

To do this please fill in the following table of lettuce and deer numbers after 100 time intervals :

 Parameter Numbers of lettuce N1 No's of deer N2 r = 0.2 r = 0.4 r = 1.2 With r = 0.5 examine the following: f = 0.005 f = 0.001 f = 0.0008 With f = 0.001 examine the following: c = 0.0005 c = 0.001 c = 0.005

Using this table and your screen answer these questions :

1) What effect does the intrinsic rate of lettuce increase, r, have on the equilibrium population of lettuce plants?

2) What effect does increasing r have on deer abundance?

3) What effect does increasing r have on the rate of return to equilibrium?

4) What effect does f have on the equilibrium lettuce  population?

5) What effect does increasing f have on deer abundance?

6) What effect does increasing c have on the equilibrium lettuce  population?

7) What effect does increasing c have on the equilibrium deer population?

8) What effect does increasing c have on the stability of lettuce-deer dynamics?

In your lab report on the use of Populus to examine Lotka-Volterra predation/herbivory dynamics please include the following information:

Introduction:

Give background information on the ecological processes of predation and herbivory.  Consider functional responses and the properties of the Lotka-Volterra predation model.

Talk about our goal for the lab of modeling an interaction between plants and herbivores (white tailed deer and lettuce) and learning about the properties of the Lotka-Volterra model.

Methods:

State the software used and the models and model parameters used. State which equation parameters we varied to model the interaction.

Results:

Give the answers to the questions listed above either in paragraphs or in question and answer format.

Discussion:

State what was learned from the modeling. What properties were shown by the Lotka-Volterra model? Which parameters seemed most important for determining the stability of the plant-herbivore interaction? What are the uses and limitations of the different models?  Is modeling valuable?

Isle Royale and the dynamics of moose and wolf populations: an example of logistic population limitation, density dependence and the impact of a predator.

Work through the EcoBeaker student workbook for “Isle Royale” and while you do this keep in mind that both resource availability, as well as both intraspecific competition and predation impact moose populations on a closed island in Lake Superior.

More information about EcoBeaker software is available at the SimBiotic Software website at: EcoBeaker™: Isle Royale (101)

This popular laboratory explores basic population biology concepts including exponential and logistic growth and carrying capacity. It is based on the textbook example of a predator-prey system involving wolves and moose on an island in Lake Superior. Students start out by characterizing the growth of a colonizing population of moose in the absence of predators. Next they introduce wolves, and study the resulting predator-prey cycles. Do predators increase or decrease the health of their prey populations? Students investigate this question by sampling the energy stores of moose with and without wolves present. Finally, they try changing the plant growth rate to see how primary productivity influences population dynamics.

PRIMARY CONCEPTS: logistic & exponential population growth; carrying capacity; predator-prey cycles; energy flow