Invasive Species Impact Calculator

The logistic equation describes a population that grows nearly exponentially while resources are abundant and then decelerates as it approaches the carrying capacity K of the invaded habitat. Its solution, N(t) = K divided by 1 plus ((K - N0)/N0) times e to the power minus rt, produces the S-shaped curve typical of documented invasions. The intrinsic growth rate r sets the pace: doubling time is ln(2)/r, so r = 0.15 per year gives 4.6 years and r = 0.30 gives 2.3 years. The curve is steepest at K/2, where the absolute increase reaches rK/4, and the time to reach that halfway point is ln((K - N0)/N0) divided by r. Reported r values for established invaders commonly fall between 0.1 and 1.0 per year for insects and small vertebrates, and cumulative damage here is the annual population summed over the projection and multiplied by a per-individual cost. Three simplifications limit the output. The model has no lag phase built in, yet real invasions often sit at low density for years or decades before expanding, because Allee effects such as difficulty finding mates depress growth when numbers are small; the logistic instead predicts its fastest proportional growth at the lowest densities. It is also non-spatial, tracking abundance but not the advancing front, so it cannot say how far a species will spread. And linear damage per individual ignores thresholds, where impacts such as forest mortality or fishery collapse appear abruptly above a density rather than accruing evenly.

The equation predates invasion biology by a century. Pierre-Francois Verhulst, a Belgian mathematician, published it in 1838 as a correction to Thomas Malthus unbounded exponential growth, introducing a term that brakes growth as population approaches a ceiling; he named the curve logistic in 1845. His work was largely forgotten until Raymond Pearl and Lowell Reed rediscovered the same form in 1920 while fitting United States census data. Georgii Gause gave it experimental support in his 1934 book The Struggle for Existence, fitting logistic curves to yeast and Paramecium cultures. Invasion biology as a field dates from Charles Elton, whose 1958 book The Ecology of Invasions by Animals and Plants drew together case studies and framed introduced species as a distinct ecological problem. John Skellam added the spatial dimension in a 1951 Biometrika paper, using the spread of muskrats released near Prague in 1905 to show that an invasion front advances at a roughly constant linear rate, building on Ronald Fisher 1937 wave of advance model. Economic accounting arrived much later: David Pimentel and colleagues estimated United States costs at roughly 120 billion USD per year in a 2005 paper, and the InvaCost database compiled by Christophe Diagne and co-authors put documented global costs at a minimum of 1.288 trillion USD between 1970 and 2017. The IPBES assessment released in September 2023 recorded more than 37,000 established alien species worldwide, of which over 3,500 cause documented harm.