Editing Allee effect (section)

==Mathematical models==

A simple mathematical example of an Allee effect is given by the cubic growth model.
:<math> \frac{dN}{dt} = -r N \left( 1 - \frac{N}{A} \right) \left( 1 - \frac{N}{K} \right),</math>
where the population has a negative growth rate for <math> 0< N < A </math>, and 
a positive growth rate for <math> A < N < K </math> (assuming <math> 0 < A < K </math>).
This is a departure from the [[logistic function|logistic growth equation]]
:<math> \frac{dN}{dt} = r  N \left( 1- \frac{N}{K} \right)</math>
where
:''N'' = population size;
:''r'' = [[Population dynamics|intrinsic rate of increase]];
:''K'' = [[carrying capacity]];
:''A'' = critical point; and
:''dN''/''dt'' = rate of increase of the population.

After dividing both sides of the equation by the population size N, in the logistic growth the left hand side of the equation represents the per capita population growth rate, which is dependent on the population size N, and decreases with increasing ''N'' throughout the entire range of population sizes. In contrast, when there is an Allee effect the per-capita growth rate increases with increasing ''N'' over some range of population sizes [0,&nbsp;''N''].<ref>{{Google books|pQTNFYPgDdEC|Essentials of Ecology}}</ref>

The Allee effect can be explicitly modeled using birth and death rates. For instance, the equation

<math>\frac{dN}{dt} = (B(N) - D(N))N = \left( \frac{b_0N}{b_1 + N} - d - cN \right)N</math>

has a locally stable equilibrium at <math>N=0</math> when <math>b_0-d-b_1c>2\sqrt{b_1cd}</math>. Here, <math>b_0, b_1, c, d</math> are positive constants and <math>B(N)</math> and <math>D(N)</math> represent the per capita birth and death rates, respectively. This formulation is especially useful when demographic data is employed to identify parameters or when extending the model to stochastic differential equations.<ref>{{Cite journal |last=Ekanayake |first=Dinesh |last2=Croix |first2=Hunter La |last3=Ekanayake |first3=Amy |date=2024 |title=Alternative stable states and disease induced extinction |url=https://www.mmnp-journal.org/articles/mmnp/abs/2024/01/mmnp230116/mmnp230116.html |journal=Mathematical Modelling of Natural Phenomena |language=en |volume=19 |pages=18 |doi=10.1051/mmnp/2024016 |issn=0973-5348|doi-access=free }}</ref> 

Spatio-temporal models can take Allee effect into account as well. A simple example is given by the reaction-diffusion model
:<math> \frac{\partial N}{\partial t} =D \frac{\partial^2 N}{\partial x^2}+ r N \left( \frac{N}{A} - 1 \right) \left( 1 - \frac{N}{K} \right),</math>
where
: ''D'' = [[diffusion coefficient]]; 
: <math>\frac{\partial^2}{\partial x^2} ={}</math>one-dimensional [[Laplace operator]].

When a population is made up of small sub-populations additional factors to the Allee effect arise.

If the sub-populations are subject to different environmental variations (i.e. separated enough that a disaster could occur at one sub-population site without affecting the other sub-populations) but still allow individuals to travel between sub-populations, then the individual sub-populations are more likely to go extinct than the total population. In the case of a catastrophic event decreasing numbers at a sub-population, individuals from another sub-population site may be able to repopulate the area.

If all sub-populations are subject to the same environmental variations (i.e. if a disaster affected one, it would affect them all) then [[population fragmentation|fragmentation]] of the population is detrimental to the population and increases extinction risk for the total population. In this case, the species receives none of the benefits of a small sub-population (loss of the sub-population is not catastrophic to the species as a whole) and all of the disadvantages ([[inbreeding]] depression, loss of [[genetic diversity]] and increased vulnerability to environmental instability) and the population would survive better unfragmented.<ref name=Gar12 /><ref>{{Cite web |title=Minimum viable population size |vauthors=Traill LW, Brook BW, Bradshaw CJ |date=6 March 2010 |work=Ecology Theory |publisher=The Encyclopedia of Earth |url=http://www.eoearth.org/article/Minimum_viable_population_size |access-date=2012-08-12}}</ref>