Invasive Species: Early vs Delayed Response

Outbreak

Growth rate r 0.18 /day

Removal rate E 40 /day

Response delay 10 days

Playback speed 2.0×

Status

Day

0.0

Population

Cumulative effort

Break-even (E/r)

–

Outcome

Idle — press Start Outbreak

Population vs. day (log scale). Dashed line = break-even population E/r; removal beats growth only below it.

Eradication cost vs. response delay, computed instantly for the current r/E. Green = eradicable; red = growth permanently outruns removal. Marker tracks the delay slider.

How it works

Before removal starts, the population grows unchecked: dP/dt = rP, so P(t) = P₀·e^(rt). Once removal begins at rate E, the dynamics become dP/dt = rP − E, an unstable linear ODE with equilibrium at P* = E/r.

P(t) = (P(t₀) − E/r)·e^(r(t−t₀)) + E/r

If the population at the moment removal starts is below E/r, the term (P−E/r) is negative and shrinks further from zero every day — population is driven to exactly zero in finite time. Eradication.
If it's above E/r, the same term is positive and grows without bound — the same removal effort E now merely subtracts a constant while growth compounds, so the population keeps climbing forever. Control, never eradication.
Because P(delay) = P₀·e^(r·delay) grows exponentially with delay, there is a sharp critical delay beyond which no amount of *waiting* can be recovered by the *same* effort rate — only a larger E would work, and cumulative cost near that threshold explodes.

Real-world relevance: this is why invasive-species agencies prioritize rapid response over response size — a small crew acting in week one can do what an army cannot do in year three.

Invasive species Native vegetation Lake / water

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