📂Dynamics

Allee Effect in Mathematical Biology

What is the Allee Effect? ¹

The Allee effect refers to the phenomenon where the population decreases when the density is low. Mathematically, this is represented in models by setting function $a: \mathbb{R} \to \mathbb{R}$ for $N$ as an upward convex convex function.

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

Variables

• $N(t)$: Represents the number of individuals in the group at time $t$.

Examples

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

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

The decrease in population when the density is low can be seen as a situation where species that reproduce sexually cannot find a mate. For species that can reproduce on their own, it might be easier to reproduce when there are no conspecifics to compete for food and living space. However, for species that reproduce through mating, the lack of conspecifics itself can 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, rather than the entire population, can be obtained by dividing both sides by $N$. $$ {{ \dot{N} } \over { N }} = {{ r } \over { K }} ( K - N ) $$ When represented graphically, the individual growth rates appear linear as shown above. To reflect the fact that growth is hindered when population is too low, we modify the model.

Considering an upward convex function like $a$ as shown above, $$ {{ \dot{N} } \over { N }} = a(N) $$ it becomes a growth model that incorporates the Allee effect.

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