Predator–Prey with Delay (Forward Euler)

Parameters

Prey growth r

1.00

Carrying capacity K

1.0

Attack rate m

1.00

Predator death s

0.020

Conversion Y

0.60

Delay τ (time)

0.0

Initial conditions & numerics

Prey x(≤0)

0.10

Predator y(≤0)

0.10

Time step Δt

0.0100

Duration T

1200

Numerics: Forward Euler updates: x_n+1 = x_n + Δt · [r x_n(1 − x_n/K) − m x_n y_n], y_n+1 = y_n + Δt · [−s y_n + Y e^−sτ m x_n−d y_n−d], with delay index d = round(τ/Δt). History for t ≤ 0 set to x = x₀, y = y₀. Values are floored at 0.

Tip: If things blow up or oscillate too wildly numerically, reduce Δt.

Time series

Prey x(t) Predator y(t)

Phase plane (x vs y)

Trajectory