# Contents

## Idea

When the population density of a species becomes very small, its members may have trouble finding mates. This leads to the Allee effect, in which the per capita population growth rate becomes smaller as the population density becomes smaller, at least when the population density is very low. This effect, also known as the underpopulation effect, can make it difficult for very small populations to survive.

## Details

Professor Martin 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

Recall that the logistic equation says

${d x \over d t} = r \,\left(1 - \frac{x}{K} \right)\, x$

where $x$, the population, is a function of time, $t$. This describes population growth that is approximately exponential with rate constant $r$ at low populations, but which slows as the population approaches the ‘carrying capacity’ $K$. To incorporate the Allee effect, we can modify the equation as follows:

${d x \over d t} = r \left(1 - \frac{x}{K} \right) \, \left(\frac{x-a}{K}\right) \, x$

where $a$ is the Allee extinction threshold, typically chosen to be much less than $K$. Now the population growth becomes zero when $P = a$ and negative for populations less than this.

### 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 Assessment - 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.

The other important thing that ecologists observed was that there was an unexpected - in the sense that ocean food chains are usually bottom up - cascade down-the whole cod food chain?, as Scheffer shows:

Overall the results of the studies of causes and consequences of the collapse of cod stocks are in line with the emerging view that marine communities may be highly nonlinear systems. Such a view suggests the need for a different look at management of marine ecosystems. It implies that sharp irreversible changes may sometimes result from gradually increasing fishing pressure and that the critical threshold will vary with climatic conditions.

Mating never occurred when density was $\le 56$ conch per hectare, and spawning never occurred at $\le 48$ conch per hectare, clearly demonstrating the operation 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 stock densities that are reduced to the point at which Allee effects begin to operate on reproductive behavior.