Population dynamics ODE with Allee Effects

Question

From reading about Allee effects, it appears that there is a modification of the one-dimensional logistic equation to account for an allee effect given by:

$$\frac{dN(t)}{dt} = rN\left(1-\frac{N}{K} \right)\left(\frac{N}{A}-1\right)$$

Here, $r$ is the species growth rate, $K$ is the carrying capacity, and $A$ is the threshold for the population to not go extinct $(0<A<K)$.

A few questions about this model:

1) What is it in this equation that determines whether or not the allee effect is strong or weak? It seems like the parameter $A$ would determine where the threshold is but not whether or not the effect is strong/weak.

2) I'm interested in an extension to a system of $n$ interacting species. Suppose I have $n$ competing species where each one is subjected to its own allee effect. Is this a suitable modification of the above ODE for $n$ species?

$$\frac{dN_i(t)}{dt} = r_iN_i\left(1-\frac{\sum_{j=1}^n\alpha_{ij}N_j}{K_i} \right)\left(\frac{N_i}{A_i}-1\right)$$

where $1 \leq i \leq n$. In that regard, the first two terms are the usual Competitive Lotka-Volterra model, and the last term accounts for the Allee effect.

Answer 1

First, Allee effects (also positive density dependence) can be modelled in several different ways, and the equation you give is one example. The terms weak and strong Allee effects are in my experience used in a couple of different ways. Most often, strong density dependence is used to denote Allee effects where the per capita population growth rate can become negative, which means that there is a critical point in population size, below which the population will tend towards extinction. Weak Allee effects are then used to describe cases where the population growth rate is negatively affected by low population sizes, but where the per capita population growth rate cannot go below zero (so populations will grow at low population sizes). Remember that in the classic logistic model, per capita population growth rate is maximized when population size is low (limit towards N=0), while it is maximized between N=0 and N=K in models with an Allee effects (in the model includes a concept of carrying capacity).

Using this definition of weak vs strong, your model has a strong Allee effect, since per capita population growth rate will go below zero at the threshold A (see below). However, sometimes strong/weak is used to denote some sense of relative influence of the Allee effect on population dynamics, for instance if per capita population growth rate at low population sizes is decreased a lot or a little compared to the logistic model, even if both cases would correspond to a weak Allee effect in the first sense.

For comparison, here are two examples of how an Allee effect can be modelled, that shows how the per capita growth rate is affected with an weak Allee effect (see e.g. Boukal & Berec 2002 and Ferreira et al 2012) or a strong Allee effect (your model):

$$\frac{dN(t)}{dt} = \frac{r}{K}N^2\left(1-\frac{N}{K} \right) \tag{weak}\label{}$$

$$\frac{dN(t)}{dt} = rN\left(1-\frac{N}{K} \right)\left(\frac{N}{A}-1\right) \tag{strong}\label{}$$

If per capita population growth rate (dN/dtN) is plotted against population size for these two models, as well as for the standard logistic model, this is what you get:

In these examples, r=0.7, K=100 and A=20. As you can see, there is a big difference between the weak and strong case. The population growth rate (dN/dt) at different N is shown below:

As you can see, if populations using the strong Allee effect are started below A (in this case 20) the populations will decline towards extinction (since dN/dt is negative below A).

The multi-species extension including species interactions that you propose looks ok to me, for that particular formulation of the Allee effect.

Also, for a flexible way to study the Allee effect, you might be interested in this formulation (see Boukal & Berec 2002), which can produce both a strong and a weak behaviour:

$$\frac{dN(t)}{dt} = rN\left(1-\frac{N}{K} \right)\left(\frac{N}{K}-\frac{A}{K} \right)\tag{flex}\label{}$$

This function will produce a weak Allee effect if $A \leq 0$ (very weak if A is far below 0) and a strong effect is $A>0$. The weak model I give above is a special case of this formulation with $A = 0$. Per capita growth of this model is given below, for two values of A (with r = 0.5 for the logistic model and r = 0.38 for the flex models)

As a sidenote, I've also seen cases where negative density dependence at low population sizes is described as an Allee effect (so that Allee effect can be used to denote both positive and negative density dependence at low population sizes), but this is not very common. For a particular species, negative density dependence at low population sizes (in excess of what is found in the standard logistic model) could e.g. be found in situation where predators have a switching behaviour, so that they stop eating the prey when it becomes rare.