Predator–Prey with Processing Delay

Unscaled predator–prey with discrete delay:

\[ \frac{dx(t)}{dt} = r\,x(t)\!\left(1 - \frac{x(t)}{K}\right) - m\,x(t)\,y(t) \]
Prey rule (words): natural logistic growth limited by \(K\) minus losses to predation.

\[ \frac{dy(t)}{dt} = -s\,y(t) + Y\,e^{-s\tau}\,m\,x(t-\tau)\,y(t-\tau) \]
Predator rule (words): natural deaths plus growth from prey eaten \(\tau\) time units ago; only a fraction \(e^{-s\tau}\) of predators survive that delay, and a fraction \(Y\) of prey biomass is converted to predator biomass.

Term-by-term (predator gain): \(Y\) (efficiency) × \(e^{-s\tau}\) (survival through delay) × \(m\,x(t-\tau)\,y(t-\tau)\) (past predation rate).

Idea: slice time into small steps of size \(\Delta t\). Approximate change per step = derivative × step size.

1) Pre-history for the delay. For all times \(t \le 0\), we set \(x=x_0\), \(y=y_0\). This gives values to look back to when we need \(x(t-\tau)\), \(y(t-\tau)\).

2) Convert the delay into a “look-back” index. We take \(d = \mathrm{round}(\tau/\Delta t)\). The past values \(x(t-\tau), y(t-\tau)\) are read from \(d\) steps earlier in memory.

3) Forward Euler updates each step.

\[ x_{n+1} = x_n + \Delta t\,\Big[ r\,x_n\!\left(1-\frac{x_n}{K}\right) - m\,x_n\,y_n \Big] \] \[ y_{n+1} = y_n + \Delta t\,\Big[ -s\,y_n + Y\,e^{-s\tau}\,m\,x_{n-d}\,y_{n-d} \Big] \]

4) Practical tips. Use smaller \(\Delta t\) for stability/accuracy; keep populations ≥ 0 by flooring negatives at 0; choose \(T\) big enough to see long-term behavior.