MA354
Project 3: Analysis of a System of Difference Equations
Competitive Hunter Model
Objective
The objective of this project is to analyze a dynamical system defined by a system of
difference equations using Mathematica. The system describes the interaction of two
predator populations.
Competitive Hunter Model
Populations of owls (W) and hawks (H) over time are summarized by the following
coupled difference equations:
Wn 1  1  k1  Wn  k3 Wn H n
H n 1  1  k2  H n  k4 Wn H n
where k1 and k2 are the growth rates of the owls and hawks respectively, and k3 and k4 are
the strength of their competitive interactions.
Model Parameters (Six Total)
W0 = initial owl population ∈ 𝑅 +
k1 = growth rate of owls ∈ 𝑅 +
k2 = growth rate of hawks ∈ 𝑅 +
k3 = (detrimental) competitive effect of hawk/owl interactions upon owls ∈ 𝑅 +
k4 = (detrimental) competitive effect of hawk/owl interactions upon hawks ∈ 𝑅 +
Model Response Variables
Owl and hawk populations over time.
How can we characterize/summarize the solution of the dynamical system for a particular
set of parameters?
For example, we could choose to describe:
 Owl and hawk populations at a fixed time interval after initial conditions (e.g. t= 100).
 Long term trends: monotonically increasing? steady state? oscillatory? zero?
 Both populations will have a stable positive population? Only 1? None?
Project 3 Instructions: Turn in a Mathematica Notebook titled Project3_name.xls in
which the following questions (Part A, B, C and D) are addressed.
Competitive
Hunters
Wn 1  1  k1  Wn  k3 Wn H n
H n 1  1  k2  H n  k4 Wn H n
Part A: Developing the Model
Develop and study the dynamical system for a simple set of parameters.
1. A template is provided for defining a dynamical system describing non-interacting
owl and hawk populations in Mathematica. Modify the template to model the
interacting population of owls and hawks.
(*Non-Interactive Population Model - Spotted Owls and Hawks*)
(*Defining the System*)
Owls[0]=Owls0;
Hawks[0]=Hawks0;
Owls[n_]:=Owls[n]=(1+k1)*Owls[n-1];
Hawks[n_]:=Hawks[n]=(1+k2)*Hawks[n-1];
(*Defining the Parameters*)
Owls0=100;Hawks0=100;
k1=0.1; k2=0.2;
(*Plotting the Population Sizes verses Time*)
W1=Table[Owls[n],{n,0,10}];
H1=Table[Hawks[n],{n,0,10}];
ListPlot[{W1,H1},JoinedTrue]
2. Define and plot the system for the following parameters:
Initial Size:
Growth Rate:
Competitive Interaction:
Owls
Hawks
W0 = 100
H0 = 100
k1 = 0
k2 = 0
k3 = 0.005
k4 = 0
The owls and hawks have the same initial population and the same zero growth rate.
The owls and hawks differ only in their respective competitive interaction.
i.
ii.
iii.
iv.
In terms of the competitive effect of the interaction on the hawks, and the
competitive effect of the interaction on the owls, explain what it means if k3 is
positive and k4 is 0.
What is the long term outcome of the dynamical system for these parameters?
Write down an equation in terms of k3 = 0.005 that describes the number of
owls that are lost per time step
Figure out how to apply “H” and “W” markers for the hawk and owl
populations respectively and how to have the origin (0,0) show on the plot.
Part B: No Competitive Interaction of the Owls on the Hawks
For now, we will continue to focus on studying the dynamical system in the simpler case
when the hawk population is unaffected by the owl population (k4 = 0).
Investigating the role of k3.
1. Vary k3 systematically from 0.001 to 0.005. Figure out how to plot the
population of owls and hawks for different values of k3 on the same graph.
2. What happens when k3=0.015? Explain in detail why this is not a biologically
relevant outcome. Modify the model so this won’t happen.
Part C: Case With Dual Competitive Interaction
1. Create a model for the population of owls and hawks in Mathematica for the
following parameters:
Owls
Hawks
Initial Size:
Growth Rate:
Competitive Interaction:
W0 = 100
H0 = 100
k1 = 0.2
k2 = 0.3
k3 = 0.001
k4 = 0.002
In class, we found the equilibrium values {WE, HE} for this system:
2. On a graph, show the trajectories of the owl and hawk populations at equilibrium.
Then begin to vary the size of the owl and hawk populations by one animal (±1 owl,
±1 hawk) at a time to discover the effect on the long-term outcome.
Part D: Resource Management “Saving the Owl Population”
Owls
Hawks
Initial Size:
W0 = 20
H0 = 160
Growth Rate:
k1 = 0.2
k2 = N/A
k3 = 0.001
k4 = N?A
Competitive Interaction:
1. Again consider the case of dual competition between the hawks and the owls. In a
hypothetical scenario, local hawks are part of a larger national hawk population and
their population size remains constant at 160 hawks. Activists are concerned with
preserving the local owl population. An animal nursery is able to breed and release a
small number of owls per year. Modify your model to include the release of a small
number of owls per year. Would this measure help? How many owls would you
recommending adding each year?
2. Now consider that the hawk population is not fixed, but also negatively affected by
competition from the owls. Consider that the extent of competition is currently
unknown since there are so few owls, but that once the owl population recovers we
can determine their effect on the hawks.
Owls
Hawks
Initial Size:
W0 = 20
H0 = 160
Growth Rate:
k1 = 0.2
k2= 0.3
k3 = 0.001
k4 = ?
Competitive Interaction:
Draft a plan for owl rehabilitation that includes checking on the hawk population and
amending the owl rehabilitation over time. Describe how this rehabilitation would play
out for several scenarios for different values of k4.