SME
Differential Equations · Problem type
Predator and prey
Use this for “Rabbits x and foxes y, in hundreds, follow x′ = x − 0.5xy, y′ = −0.75y + 0.25xy. Where do both stay constant, and how long is one small cycle?”
Diagram
Equilibrium
(
γ
δ
,
α
β
)
=
(3,
2);
small
cycles
round
it
take
2π
÷
√
αγ
=
7.255
This is an example. Type your own values to start a new problem.
x′
=
α
x
−
β
xy
y′
=
−
γ
y
+
δ
xy
xₑ
Prey at equilibrium
worked out (not typed)
yₑ
Predators at equilibrium
worked out (not typed)
ω
Turning rate
worked out (not typed)
P
Period of small cycles
worked out (not typed)
Formulas
xₑ
=
γ
δ
3
=
0.75
0.25
yₑ
=
α
β
2
=
1
0.5
ω
=
√
α
×
γ
0.866
=
√
1 × 0.75
P
=
2
×
π
ω
7.2552
=
2
×
π
0.866
Assumptions
•
x counts the prey and y the predators. Prey grow at rate α and are eaten at βxy; predators die at rate γ and gain δxy.
•
Both stay constant where x′ = 0 and y′ = 0 with x and y above 0: x = γ ÷ δ and y = α ÷ β.
•
Linearized there, the matrix is [[0, −βγ ÷ δ], [αδ ÷ β, 0]]: T = 0 and D = αγ, a center, so small cycles turn at √(αγ).
•
x and y are in the units the rates use (here hundreds of animals), and P is in the rates’ time unit.
Step-by-step
WE KNOW
α = 1, β = 0.5, γ = 0.75, δ = 0.25
FIND
prey at equilibrium (xₑ), predators at equilibrium (yₑ), turning rate (ω), period of small cycles (P)
Step 1 · Find prey at equilibrium (xₑ)
Use xₑ = γ ÷ δ
y′
=
y(−
γ
+
δx)
is
0
for
y
above
0
only
when
δx
=
γ
.
xₑ
=
0.75
0.25
xₑ = 3
Step 2 · Find predators at equilibrium (yₑ)
Use yₑ = α ÷ β
x′
=
x(
α
−
βy)
is
0
for
x
above
0
only
when
βy
=
α
.
yₑ
=
1
0.5
yₑ = 2
Step 3 · Find turning rate (ω)
Use ω = √(α × γ)
Near
the
equilibrium
the
system
is
linear
with
T
=
0
and
D
=
αγ,
so
its
eigenvalues
are
±i
√
αγ
:
a
center
turning
at
ω
.
ω
=
√
1 × 0.75
ω
=
0.86603
Step 4 · Find period of small cycles (P)
Use P = 2 × π ÷ ω
One turn is 2π radians. Large cycles take longer, so this is the period of small ones.
P
=
2
×
π
0.86603
P
=
7.2552
Check
Put the numbers back in. Both sides should match.
3
=
0.75
0.25
✓
2
=
1
0.5
✓
0.86603
=
√
1 × 0.75
✓
7.2552
=
2
×
π
0.86603
✓
Topic check
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