Deterministic population dynamics · 2

Continuous-time logistic model

dx/dt = r_m x (1 − x)

x: population as a share of the carrying capacity K (x = N/K); it can start above 1
r_m: the largest growth rate per head (births − deaths, per unit time) when the population is tiny
(1 − x): crowding: the growth rate per head falls linearly and reaches 0 at K
t: continuous time: births and deaths happen all the time, generations overlap
Δt: the step of the Euler method in the owner's code: x_n+1 = x_n + r_mΔt x_n(1 − x_n)

Trajectory

r_m0.110 x₀0.230 t_max100 Δt0.01

The exact curve and the owner's Euler steps.

The growth rate dx/dt for every density x: where it is above zero the population grows, where it is below it shrinks.

Growth rate against density: arrows show which way x moves.

How fast the population changes along the exact curve, from x₀ to K.

How fast the population grows, over time.

Assumptions. Time is continuous; the population is large and well mixed (a mean field); no age or sex structure; births and deaths are spread evenly over time; only births and deaths change it (closed); the growth rate per head falls linearly with crowding, r = r_m(1 − N/K).
The model. Over a short time Δt, Δx ≈ r_m x(1 − x) Δt; letting Δt → 0 gives dx/dt = r_m x(1 − x).
Equilibria. x* = 0 and x* = 1 (N* = K). With f ′(x) = r_m(1 − 2x): f ′(0) = r_m > 0, so 0 is unstable; f ′(1) = −r_m < 0, so K is stable.
Exact solution. x(t) = 1 / (1 + (1/x₀ − 1) e^−r_mt): an S-curve from below, a smooth decline from above. It never crosses K.
Fastest growth is r_m/4, at x = ½, reached at t = ln((1 − x₀)/x₀)/r_m (for x₀ < ½).
No oscillations. Unlike card 1 (non-overlapping generations), the continuous model cannot overshoot: its only change of behaviour is at r_m = 0, where 0 and K swap stability.
The Euler method x_n+1 = x_n + r_mΔt x_n(1 − x_n) is card 1's map with R = 1 + r_mΔt, in the rescaled variable z = r_mΔt x/R (then z_n+1 = R z_n(1 − z_n)). A large step therefore shows overshoot, cycles and chaos that belong to the method, not to the model; keep r_mΔt well below 1.
Negative r_m. The README allows r_m < 0 (then K < 0 in its notation); the sliders keep r_m ≥ 0, where K is a real carrying capacity. If births never exceed deaths the population simply declines to 0.