Population dynamics of rabbits and foxes (a) A simple Lotka–Volterra Model We ha

Population dynamics of rabbits and foxes
(a) A simple Lotka–Volterra Model
We have discussed in detail the Lotka–Volterra model for predator-prey relationships
dNprey
dt = +Rprey,oNprey(t) − γNprey(t)Npred.(t)
dNpred.
dt = γNprey(t)Npred.(t) − Rpred.,oNpred.(t)
but how does it really work in practice? Consider an example of rabbits and foxes. First,
we need to consider reasonable parameters — the following are a starting point:
• Ro,prey = 0.04
• Ro,pred = 0.2
• γ = 0.0005
• = 0.1
Assume that the time units are all in days, and that the populations are numbers of
individuals per square kilometer. What is the doubling time of rabbits without predation,
and the death rate of foxes? Explain whether or not these values are biologically reasonable.
Explain the meaning of the terms γ and and consider their values — do these make sense
(consider what conditions will lead to a balance of growth/death in each population)?
We can simulate this system using the same Forward Euler method that we used in the
first project; when we have two variables (for example, x and y), we simply use the Forward
Euler update rule for both of them. That is, at each time step, we set:
x(t + ∆t) = x(t) + dx
dt (∆t) and y(t + ∆t) = y(t) + dy
dt(∆t)
Using initial populations of 200 rabbits and 50 foxes per km2
, and a time step of 0.01
days, determine how the rabbit and fox populations will vary over one year. Plot the two
populations versus time on the same graph, as well as versus each other (on another graph).
In the latter plot use the quiver function to add velocity arrows to the plot (use a lighter
color such as gray for this). Discuss the observed behavior of the populations. What is the
range in each population?
Repeat your calculations with initial populations of 5000 rabbits and 100 foxes per km2
,
and discuss how the behavior of the system changes. Do the same for 4000 rabits and 80
foxes per km2
. What does this result tell you about the system?
Additionally, draw the phase plane graph for the interaction of these two species and
explain the graph.
1
(b) Extending the Lotka–Volterra Model
In class, we have discussed some of the limitations of the Lotka–Volterra model, and how we
might begin to address these. Consider two different models, both using the Lotka–Volterra
model (with the original set of parameters) as a starting point:
1. A model with unrestricted prey growth replaced by a logistic equation-based model.
That is, replacing α with:
A(U) = α
1 −
U
K
2. A model with restricted prey growth, as in (1), and with a predator response that
saturates at high prey density, using the Holling’s disk equation. That is, replacing α
as above, and replacing γ with:
Γ(U) = sU
1 + shU
Set K = 10, 000, s = γ and h = 0.2. Briefly explain what these values mean, in a biological
context (Hint: consider how the underlying equations behave at very low or very high
populations).
For each case, make a plot of the populations versus time and the populations versus
each other (again adding velocity arrows), in this case for a time span of at least three years.
Describe how each trajectory differs from the original Lotka–Volterra model, and from each
other, and give a suggested rationale for why these differences arise.