# Contents

## Idea

the Wikipedia article:

The Allee effect - or underpopulation effect - is a phenomenon in biology characterized by a positive correlation between population density and the per capita population growth rate in very small populations.

## Details

Professor Scheffer states:

This is a highly important mechanism when it comes to understanding the extinction of endangered species?. If the Allee effect is strong enough, it implies that a population can go into free fall if its density goes below a certain critical level.In this case the population has two alternative stable states; one at the carrying capacity of population growth and the other at zero densities. In the previous simple growth model (logistic growth) a zero density is also an equilibrium state, as even if per capita birth rates are high , absence of parents results simply in no offspring. In that case, however, a small addition of animals will be enough o kick off a population increase that stops only when the carrying capacity is reached.

In contrast, a population with a strong Allee effect is trapped in the zero state. A small initial population number is drawn back into the zero state as long as it is below the critical density.

### Mathematical formulation

Here it is seen in the form of an extra term in the logistics equation: ${d x \over d t} = {r\over K} (K-x) x (a-x)$ where $a$ is the Allee extinction threshold.

### Effects

#### Collapse of Cod Populations

Also an example from Marten Scheffer:

Model suggest that the mechanism of exploitation can by itself cause the overexploited state to be an alternative stable state. However the lack of recovery decades after the closure of fisheries on Newfoundland cod has raised the question of whether other mechanisms may keep collapsed stock from recovering. One possibility is that a sufficiently adult stock is needed to control potential predators and competitors from their offspring. Indeed, size-structured predator-prey interactions may well lead to Allee effects (see de Roos and Persson in the References and 9.17 and 9.18 in Scheffer). from Millenium Ecosystems Assesment - Synthesis (ref.9.18):

The economic and public health costs associated with damage to ecosystem services can be substantial.

The early 1990s collapse of the Newfoundland cod

ﬁshery due to overﬁshing resulted in the loss of tens of thousands of jobs and cost at least \$2 billion in income support and retraining.

Mating never occurred when density was $\le 56$ conch per ha, and spawning never occurred at $\le 48$ conch per ha, clearly demonstrating theoperation of depensatory mechanisms. Reproductive behavior then increased rapidly to asymptotes at densities near 200 conch ha–1. Heavily exploited populations of queen conch in the Caribbean have been slow to recover despite fishery closures. Failure to recover could result from spawning stockdensities that are reduced to the point at which Allee effects begin to operate on reproductive behavior.