📂Dynamics

Allee Effect in Mathematical Biology

What is the Allee Effect? ¹

The Allee effect refers to a decrease in population size when the density of the population is low. Mathematically, it is expressed as a function $a: \mathbb{R} \to \mathbb{R}$ for $N$ in the model as a convex function that is convex upwards.

$$ \dot{N} = a(N) N $$

Variables

• $N(t)$: Represents the population size at point $t$.

Example

The Allee effect, for example, can be assumed by setting function $a$ as the following quadratic function.

$$ a(N) := - a_{2} N^{2} + a_{1} N - a_{0} \qquad , a_{2} ,a_{1}, a_{0} > 0 $$

A decrease in population size when the density is low can be seen as a situation where sexually reproducing species are unable to find mates. If an individual can reproduce by themselves, the absence of individuals competing for the same food and living area makes reproduction easier. However, for species that reproduce through mating, the sheer lack of conspecifics can itself be a cause of extinction.

Derivation

Logistic Growth Model: $$ \dot{N} = {{ r } \over { K }} N ( K - N ) $$

In the logistic growth model, the average growth rate of individual organisms can be obtained by dividing both sides by $N$. $$ {{ \dot{N} } \over { N }} = {{ r } \over { K }} ( K - N ) $$ When graphed, this results in a linear appearance of the individual growth rate as shown above. If the number of individuals is too small, it fails to grow, which can be adjusted to reflect by considering the convex function $a$ upwards. $$ {{ \dot{N} } \over { N }} = a(N) $$ This becomes a growth model applied with the Allee effect.

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