MA354
February 15, 2012
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 described the interaction of two
predator populations. We will systematically vary model parameters describing growth
rates and the competitive interactions in order to understand their roles in the model. Also,
we will survey parameter space for the types of long-time outcomes that are possible for
the 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 and C) 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). We will study
how varying the other parameters affects the solution to the dynamical system.
1. Investigating the role of k3.
a. 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.
b. 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.
2. What are the roles of other parameters? Summarize your results succinctly with one
or more graphs. Plan on presenting your results to the class.
Group 1: What is the role of the initial population size? Vary these two parameters in
systematic ways to discover their roles.
Group 2: What is the role of the growth rate of the owls? Vary this parameter in a
systematic way to discover its functional role. Summarize your results succinctly with
one or more graphs. Is there a growth rate that would lead to a steady population of owls
and hawks?
Group 3: What is the role of the growth rate of the hawks? Vary this parameter in a
systematic way to discover its functional role. Is there a growth rate that would lead to a
steady population of owls and hawks?
Group 4: What is the role of k4? Vary this parameter in a systematic way to discover its
functional role. Is there a value that would lead to a steady population of owls and hawks?
Part C: Saving the Owl Population
1. Return to our initial model parameters:
Initial Size:
Growth Rate:
Competitive Interaction:
Owls
Hawks
W0 = 100;
H0 = 100;
k1 = 0;
k 2= 0;
k3 = 0.005
k4 = 0;
Activists are concerned with preserving the 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 to maintain a stable owl population?
2. Now consider that the hawk population is negatively affected by competition from the
owls:
Initial Size:
Growth Rate:
Competitive Interaction:
Owls
Hawks
W0 = 100;
H0 = 100;
k1 = 0;
k 2= 0;
k3 = 0.005
k4 = 0.001;
How many owls and/or hawks would you recommend adding each year? (Assume that
activists would like a stable population of both species.)
3. Now consider that the hawk population is negatively affected by competition from the
owls, but the extent of competition is initially unknown since there are so few owls.
Initial Size:
Growth Rate:
Competitive Interaction:
Owls
Hawks
W0 = 20;
H0 = 100;
k1 = 0;
k 2= 0;
k3 = 0.005
k4 = ?
Draft a plan for owl rehabilitation that includes checking on the hawk population and
amending the owl and hawk rehabilitation over time. Then describe how this
rehabilitation would play out for several scenarios for different values of k4.