Biology 377
Predation Models
Today's exercise is intended to familiarize you with the basics of predatory
prey models based upon modifications of the Lotka-Volterra models. The
basic Lotka-Volterra model includes no density dependence in the prey population,
and a very simple Type I functional response on the part of the predators.
We will examine the effect on the dynamics of predator prey systems when
both of these assumptions are removed. We will again use the program called
Populus.
Modified Lotka-Volterra Predation Models
At the main menu choose "Multi-Species Interactions". On the multi-species
interactions menu choose "Theta-Logistic Predatory-Prey". This will get
you into the model that we will use this week. The theta-logistic is similar to the classic Lotka-Volterra predator prey model, but it incorporates both different predator behaviors (functional responses) and different kinds of density dependence in the prey. If you have a type-1 functional response ( more on functional responses here) and a large value for K, the dynamics approach the classical lotka-volterra model.
The Lotka-Volterra model of predation, as presented by Populus, is
as follows:
where N = density of the prey, P = density of the predator, r = prey's
intrinsic rate of increase, K = prey's carrying capacity, g = predator's
reproductive rate, D = predator's threshold food consumption for reproduction,
and f = the predator's functional response, which can take on any of the
following forms:
where C is a parameter that can be thought of as capture success, and
h is a parameter that describes the extent to which the predator's functional
response saturates.
Type I Functional Response
Exercise #1 - The Effects of Density Dependence
Choose the Type I functional response and set the rest of the parameters
as follows: t = 100, r = 1, K = 100, theta
= 1, D = 1, g = 1, and C = .02,N = 20, P = 15. Make a run by striking the
enter key, and then examine the results. Do predator and prey coexist?
Is there a stable equilibrium density or do the populations exhibit continuous
fluctuations? Examine the trajectory as it crosses each of the isoclines.
Is the behavior of the trajectory consistent with the concept of isoclines?
Now, vary the value of the parameter theta,
which modifies the way in which the density dependent term (N/K) affects
the growth of the prey. When theta is large the
effect of density on prey population growth is minimal. When theta
is 1, the prey population grows following a logistic patterns (discounting
for the predator, of course). Set the value of theta
to 10, and make another run. What is the effect of removing density
dependence on the prey? How have the shapes and positions of the isoclines
changed? How has the trajectory changed. Did this increase or decrease
the qualitative stability of the interaction?
Exercise #2 - Shifting the Predator Isocline
Be sure that theta is reset to 1 so
that density dependence is back in force, and then do a series of runs
in which you vary the value of D. This will shift the predator isocline
left and right. What affect does this have on the stability of the equilibrium
point? What effect does this have on the rate at which equilibrium is approached?
Try shifting the initial densities of predator and prey. Does this
have any effect on the patterns that you see?
Exercise #3 - Changing Reproductive Parameters
Be sure to reset the parameters to the basic values (t = 100, r = 1, K
= 100, theta = 1, D = 1, g = 1, and
C = .02, N = 20, P = 15). Now vary the parameter s, which describes the
rate at which the predator population can grow when prey are available.
Making these changes is equivalent to changing the kind of predator prey
interaction. For example, high predator growth rate relative to prey would
be characteristic of diseases, whereas high prey growth rate of prey relative
predator would be characteristic of vertebrate systems, such as owls and
mice. What effect does relative predator growth rate have on the periodicity
and magnitude of cycles?
Type II Functional Response
Exercise #4 - Basic effects of the Type II functional response
Reset the parameters of the model for a Type II functional response as
follows: t = 100, r = 1, K = 100, theta
= 1, D = 1, g = 10, and C = .15, h = .9, N = 20, P = 15. Make a run with
these parameters. How has the shape of the prey isocline changed? What
effect do you think the humped shaped isocline will have on the stability
of the interaction?
Exercise #5 - Position of the Predator Isocline relative to the Prey Isocline
Vary the value of C from around 0.1 to about 1.0. What does this do
to the position of the predator isocline relative to the hump of the prey
isocline? What is the correlation between the position at which the isoclines
cross and the stability of the interaction? For what conditions is the
equilibrium stable? For what conditions are there limit cycles around an
unstable equilibrium? For what conditions does the predator go extinct?
Exercise #6 - Productivity of the Prey
Set the parameters back to standard values (t = 100, r = 1, K = 100, theta
= 1, D = 1, g = 10, and C = .15, h = .9, N = 20, P = 15). Now do a series
of runs in which you vary the value of K from 50 to 1000. How does the
producitivity of the environment, as measured by K, affect the dynamics
of the interaction. What lessons might this provide for the consequences
of enriching an ecosystem?
Exercise #7 - Predator Growth Rates
Once again, explore the effect of changing the predator's growth rate parameter,
g. Set C = .22, so that the predator isocline is about in the middle of
the prey isocline. Now vary the value of g, and see what effect this has
on the dynamics of the interaction. Does the value of g affect the periodicity
of the cycles, or the lag between predator and prey peaks?
Type III Functional Response
Change the parameters of the model so that the Type III functional response
is chosen and following parameter values are set: t = 100, r = 1, K = 100,
q = 1, D = 1, g = 10, and C =
1, h = .9, N = 20, P = 15. Do a quick run with the values and inspect the
isoclines. Notice that the prey isocline has a very different shape
when there is a Type III functional response. Effectively, there a
low density refuge for the prey. How would you characterize the pattern
of population dynamics characterized by this set of conditions? Now
vary the value of C to progressively lower values, which will shift the
predator isocline to the right. Unfortunately, the isocline graphs produced
by Populus become unreadable under these conditions, but trust me, the
shape of the prey isocline is remaining constant. As the predator isocline
shifts to the right, how do the population dynamics change?
General Questions
Overall, how would you characterize the effects that Type I, Type II
and Type III functional responses have on the dynamics and stability of
these predator prey interactions?
Thanks to John Addicott