Example 1: The Battle of Trafalgar
In the battle of Trafalgar in 1805, a combined French and Spanish naval force
under Napoleon fought a British naval force under Admiral Nelson. Initially, the
French-Spanish force had 33 ships, and the British had 27 ships. During an
encounter, each side suffers a loss equal to 10% of the number of ships of the
opposing force. Fractional values are meaningful and indicate that one or more
ships are not at full capacity.
Dynamical Systems Model
Let n denote the encounter stage during the course of the battle and define
B_n = number of British ships at stage n
F_n = number of French-Spanish ships at stage n
Then the dynamical system which models the battle would be
𝐵𝑛 = 𝐵𝑛−1 − .1𝐹𝑛−1
𝐹𝑛 = 𝐹𝑛−1 − .1𝐵𝑛−1
Example 2: Predator-Prey Model
Suppose the spotted owls’ primary food source is a single prey: mice. An ecologist
wishes to predict the population levels of spotted owls and mice in a wildlife
sanctuary. Letting M_n represent the mouse population after n years and O_n the
predator owl population, the ecologist has suggested the model
𝑀𝑛+1 = 1.2 𝑀𝑛 − 0.001 𝑂𝑛 𝑀𝑛
𝑂𝑛+1 = 0.7 𝑂𝑛 + 0.002 𝑂𝑛 𝑀𝑛
The ecologist wants to know whether the two species can coexist in the habitat
and whether the outcome is sensitive to the starting populations.
To solve the system above for its equilibrium values, we want to solve the system
of equations
𝑀 = 1.2 𝑀 − 0.001 ∗ 𝑂 ∗ 𝑀
𝑂 = 0.7 ∗ 𝑂 + 0.002 ∗ 𝑂 ∗ 𝑀
Using the solve command in Mathematica, we find the equilibrium solutions are
(M,O) = (0,0) and (M,O) = (150,200). Test the following initial conditions and
observe the long-term behavior of the model:
Case 1
Case 2
Case 3
Case 4
Owls
150
150
100
10
Mice
200
300
200
20
Case 1
Case 2
Case 3
Case 4
Mice
150
150
100
10
Owls
200
300
200
20
Example 3: Voting Tendencies of the Political Parties
Consider a three-party system with Republicans, Democrats, and Independents.
Assume that in the next election, 75% of those who voted Republican again vote
Republican, 5% vote Democrat, and 20% vote Independent. Of those who voted
Democrat before, 20% vote Republican, 60% again vote Democrat, and 20% vote
Independent. Of those who voted Independent, 40% vote Republican, 50% vote
Democrat, and 10% again vote Independent. Assume that these tendencies
continue from election to election and that no additional voters enter or leave the
system. Analyze the dynamical system, equilibrium values, and sensitivity to initial
conditions.