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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:

$$frac{dN(t)}{dt} = rNleft(1-frac{N}{K} ight)left(frac{N}{A}-1 ight)$$

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 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?

$$frac{dN_i(t)}{dt} = r_iN_ileft(1-frac{sum_{j=1}^nalpha_{ij}N_j}{K_i} ight)left(frac{N_i}{A_i}-1 ight)$$

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.

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

**Allee effects are in my experience used in a couple of different ways. Most often,**

*strong**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):

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

$$frac{dN(t)}{dt} = rNleft(1-frac{N}{K} ight)left(frac{N}{A}-1 ight) ag{strong}label{}$$

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:

$$frac{dN(t)}{dt} = rNleft(1-frac{N}{K} ight)left(frac{N}{K}-frac{A}{K} ight) ag{flex}label{}$$

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.

## Abstract

We take a well-known dynamic model of an isolated, unstructured population and modify this to include a factor that allows for a reduction in fitness due to declining population sizes, often termed an Allee effect. Analysis of the behaviour of this model is carried out on two fronts – determining the equilibrium values and examining the stability of these equilibria. Our results point to the stabilising effect on population dynamics of the Allee effect and an unexpected increase in stability with increased competition due to the interaction between competitive and Allee effects.

Author to whom correspondence should be addressed. E-mails: [email protected] [email protected]

## Population Dynamics with Threshold Effects Give Rise to a Diverse Family of Allee Effects

The Allee effect describes populations that deviate from logistic growth models and arises in applications including ecology and cell biology. A common justification for incorporating Allee effects into population models is that the population in question has altered growth mechanisms at some critical density, often referred to as a threshold effect. Despite the ubiquitous nature of threshold effects arising in various biological applications, the explicit link between local threshold effects and global Allee effects has not been considered. In this work, we examine a continuum population model that incorporates threshold effects in the local growth mechanisms. We show that this model gives rise to a diverse family of Allee effects, and we provide a comprehensive analysis of which choices of local growth mechanisms give rise to specific Allee effects. Calibrating this model to a recent set of experimental data describing the growth of a population of cancer cells provides an interpretation of the threshold population density and growth mechanisms associated with the population.

**Keywords:** Logistic growth Per-capita growth rate Population dynamics Population models.

### Figures

Comparison of typical logistic growth,…

Comparison of typical logistic growth, Weak Allee, and Strong Allee models. The mathematical…

Schematic for the Binary Switch…

Schematic for the Binary Switch Model. Individuals in a population **a** can sense…

Bifurcation diagram of the Binary…

Bifurcation diagram of the Binary Switch Model, shown in (6), for Case 1…

Bifurcation diagram of the Binary…

Bifurcation diagram of the Binary Switch Model, shown in (8), for Case 2…

Bifurcation diagram of the Binary…

Bifurcation diagram of the Binary Switch Model for Case 3, shown in (11),…

Bifurcation diagram of the Binary…

Bifurcation diagram of the Binary Switch Model for Case 3, shown in (11),…

Population density of U87 glioblastoma…

Population density of U87 glioblastoma cells compared to the calibrated Binary Switch Model.…

## Mathematical model

Describing the Allee effect in a mathematical model is more challenging. We will use the form (1) for the logistic equation in cases where the coefficients are constant, but we want to consider the situations when the intrinsic growth rate *r* changes sign (concerning time, or in spatial models for space).

The assumptions that increase population density lead to decreases in the birth rate and increase the death rate may not always be valid.

Allee (1931) observed that many animals engage in social behavior such as cooperative hunting or group defense that can cause their birth rate to increase or their death rate to decrease with population density [1]. Also, the rate of predation may decrease with prey density in some cases, as discussed by Ludwig in [31] (see recent studies on predation under the Allee effect in [42] and references therein). In the presence of such effects, which are typically known as Allee effects, the model (1) will take a more general pattern

where *f*(*t*, *x*, *u*) may be increasing for some values of *u* and decreasing for others. A simple case of a model with an Allee effect is

where, (r > 0) and (0< M < K) . The model (3) implies that the density of *u* will decrease if (0< u < M) or (u>K) but increase if (M< u < K) . Introducing diffusion in this model is more challenging and effective in spatial heterogeneity.

To demonstrate the model with diffusion and the Allee effect, it is essential to introduce the following short description. Mathematically the growth rate per capita function *f*(*t*, *x*, *u*) will not have maximum value at (u=0) . If *f*(*t*, *x*, *u*) is negative when *u* is small, we define such a growth pattern has a strong Allee effect or mandatory extinction. If *f*(*t*, *x*, *u*) is decreasing but still positive for the very low density of *u*, then this growth has a weak Allee effect or extinction-survival situation. Considering diffusion in ecology model has open a new era.

Let us now consider the general reaction-diffusion mathematical model with homogeneous Neumann boundary conditions

xin Omega , abla ucdot n =0,&<>t>0,

Here *d* is the diffusion rate. The habitat (Omega) is a bounded region in (*n* is the outward normal vector. The functions *u*(*t*, *x*) represent the population density of the species and it’s migration rate is positive, (d > 0) . We assume that all the functions are in the class of (C^ <1+eta >(

The similar type model (4) was studied by Shi and Shivaji in [16] for weak Allee effects with global bifurcation analysis. They considered the following functions as a reaction term:

where the Allee effect is due to a type-II functional response.

It is important to note that the function *f*(*t*, *x*, *u*) in (4) will be negative or positive such that the Allee effect will be seen depending on the choice of species mating difficulty or, low growth rate or high death rate as well as competition, etc. There are several kinds of growth function with the Allee Effect has been discovered since 1931. There are numerous example of *f*(*t*, *x*, *u*) with Allee effects, for examples:

In 1988, Leah Edelstein-Keshet use a quadratic polynomial of the form [32]

The negative quadratic Eq. (5) describes the per capita growth rate under the Allee effect. Here the function *r*(*x*) is the intrinsic growth rate of species and bounded. Also (a,b>0) , and depending on the choice of *a* there could be unconditional stability extinction-stability situation and unavoidable extinction which we defined as no Allee effect (similar to logistic growth) weak Allee effect and strong Allee effect. We will now define an important function for further studies.

### Definition 1

A sparse function is a function (f: Omega
ightarrow *f* is also in the class of (C^<1+alpha >(Omega ),<alpha <1) .

Biologically sparse means thinly scattered (species are set or planted here and there or, not being dense or close together), and any species of the population having this property is known as a sparse population. This idea leads to a sparsity function, which gives biological meaning to a mathematical expression.

The next two growth function with Allee effects are a variation of (5) and are obtained by modifying the Verhulst logistic model

The function (6) was broadly discussed in [33, 34]. Here *M*(*x*) is a sparse function, defined as (M(x)<K(x)) , with *K*(*x*) being the carrying capacity. And by choice of different *M*(*x*), different type of Allee effects occurs. Similarly (7) was discussed in [35].

Another choice of *f*(*t*, *x*, *u*) shown in (8), which is more appealing choice for Allee effect than the previous Eqs. (6) and (7) and is broadly discussed in [36, 37].

Here *M*(*x*) has the same interpretation as Eq. (6) and (7). Also by choice of constant *C*, (8) is a captivating choice, since the dynamics approaches logistic growth for large *u*. Model (8) has already been used with (Cequiv 0) in competition, predation and meta-population models ([36, 38] and references therein). Therefore, considering model (8) with (C>0) yields a more flexible model with variable extinction rates.

The problem (8) with (C=sup limits _

Now choosing (b=-frac

In 1996, Takeuchi proposed and discussed this form (9) of Allee effect with (a, b, C > 0) .

The per capita growth function (10) proposed by Jacobs in 1984 combines features of the heuristic approach with biological reasoning (see [40])

where (alpha , eta , omega , gamma >0) . The function defined in (10) treats the Allee effect and negative density dependence separately and may combine them in a wide range of biologically plausible relationships. On the other hand, the model is quite complicated, and consequently, much work has not been studied.

Since each functions (5)–(10) are biologically meaningfull and generalized, see [41] for more details, here we mainly consider the function (7) with our diffusion model (4). Using Eq. (7) is more appropriate to this paper since the main concept of this paper is to show both strong and weak Allee effect in a heterogeneous environment with and without diffusion. The sparsity function here gives the demographic Allee effect and made extinction of species within a short period in the presence of diffusion. The following sections will cover the analytic discussions and numerical approximations for further possibilities of this kind of Allee effect. By considering suitable sparse function and parameters the function of the other could give similar results and also broadly discussed and established in the references provided. Numerically we also consider the cases where functions (6) and (8) appears.

Let us now consider the following mathematical model of (4) with *f*(*t*, *x*, *u*) as defined in (7). Then we get the following regular (classical) diffusion model with Allee effect:

xin Omega , abla ucdot n =0,&<>t>0 ,

For spatial positive functions *M*(*x*) and *K*(*x*), it is assumed that (0<M(x)<K(x)) for any (xin *M*(*x*) is the sparsity function and (r(x)>0) is the intrinsic growth rate. Because of the choice of sparsity function, a species could have an Allee effect or not. Therefore, the maximum population density is *K*(*x*) and is known as the environment carrying capacity. Throughout the paper, we have assume that the growth rate per capita (7) [as well as (5)–(10)] satisfies:

For any (x in <

For any (x in <

From the assumptions, the function (u_2(x)) indicates the crowding effects of the population at *x*, which may vary by location but it has a uniform upper bound, *K*(*x*). The function (u_1(x)) is where *f*(*t*, *x*, *u*) attains the maximum value. Here we still allow logistic growth in cases where (u_1(x) = 0) . The constant *N* is the uniform upper bound of the growth rate per capita.

The graphs on top row are growth rate *uf*(*t*, *x*, *u*), and the ones on lower row are growth rate per capita *f*(*t*, *x*, *u*), where **a** logistic for (M
ightarrow 0) **b** weak Allee effect for (M=1.1+ cos (pi x)) **c** strong Allee effect for (M=-(1.1+ cos (pi x))) with (K(x)=2.0+ cos (pi x)) , (d=1.0) , (r(x)=1.2) over (Omega =[0,1]) . In all figure *f*(*t*, *x*, *u*) follows (7)

The function *f*(*t*, *x*, *u*) could be logistic or having Allee effect based on the following assumptions (see also Fig. 2):

Logistic type when (f(t,x,0)>0) , (u_1(x)=0) and (f(t,x,cdot )) is decreasing in ([0,u_2(x)]) (Fig. 2a).

Weak Allee effect type when there exists (x_s>0) such that (f(t,x,0)ge 0) for (x>x_s) and (f(t,x,0)le 0) for (x<x_s) in a non-empty open domain (Omega _ssubset Omega) . Also (u_1(x)>0) , (f(t,x,cdot )) is increasing in ([0,u_1(x)]) , and (f(t,x,cdot )) is decreasing in ([u_1(x),u_2(x)]) (Fig. 2b).

Strong Allee effect type when (f(t,x,0)< 0) , (u_1(x)>0) , (f(t,x,u_1)>0) , (f(t,x,cdot )) is increasing in ([0, u_1(x)]) , (f(t,x,cdot )) is decreasing in ([u_1(x),u_2(x)]) (Fig. 2c).

Throughout the paper, we will consider *f*(*t*, *x*, *u*) as defined in (7), which satisfies (h1)–(h3). To estimate the long-time dynamics of the species whether they will persist or become extinct steady state solution and eigenvalue-eigenfunction method are used to determine the dynamical behavior.

## Allee Effects and the Risk of Biological Invasion

The Allee effect is a nonlinear phenomenon exhibited in the population dynamics of sparse populations in which the per capita population growth rate increases with increasing population density. In sufficiently sparse populations, the Allee effect may lead to extinction and is known to generate a threshold in the probability of establishment when presented as a function of introduced population size or density. As introduced populations are generally small, Allee effects are probably common in biological invasions and their consideration is necessary for accurately assessing the risk of invasion by many species, including all sexually reproducing species. *Bythotrephes longimanus*, an invasive, freshwater, cladoceran zooplankter from Europe, is one such species. Here, I review a previously published model of the Allee effect for continuously sexually reproducing species. Then, I develop a new model for seasonally parthenogenetic species such as *Bythotrephes*, and thereby demonstrate the potential consequences of Allee effects. This result underscores the importance of considering nonlinear phenomena, including thresholds, when conducting risk analysis for biological introductions.

## Conclusions

In this work, we examine the link between threshold effects in population growth mechanisms and Allee effects. An abrupt change in growth mechanisms, which we refer to as a *binary switch*, is thought to be a common feature of biological population dynamics. Despite the ubiquitous nature of local binary switches in population dynamics, an explicit connection to Allee effects has not been considered. To explore this connection in greater detail, we examine a population density growth model, in which the proliferation and death rates vary with the local density of the population. By incorporating a local binary switch in these proliferation and death rates, we greatly reduce the size of the parameter space while explicitly incorporating a biologically realistic threshold effect in the proliferation and death rates.

To provide insight into the qualitative features of population dynamics arising in the Binary Switch Model, we examine the presence and stability of the resulting equilibria. We show that when the binary switch occurs at some intermediate population density and the high-density death rate is not too large, a diverse family of Allee effects is supported by the model. Among these Allee effects are: (i) logistic growth, when no binary switch is present (ii) the Weak Allee effect, which modifies the simpler logistic growth model without changing its equilibria or their stability (iii) an Extinction regime, where all population densities will eventually go extinct (iv) the Strong Allee effect, where population below a critical density will go extinct rather than grow, and (v) the Hyper-Allee effect, which has two distinct positive stable population densities. Furthermore, we show that there are additional forms of Allee effects at the boundaries in the parameter space that separate these five main classes of Allee effects.

Along with exhibiting a wide range of Allee effects, the Binary Switch Model has a restricted parameter regime, making the interpretation of the local binary switch clearer while requiring fewer parameters to identify when calibrating to experimental data. To demonstrate these advantages, we calibrate the Binary Switch Model to experimental data sets arising in cell biology. Not only can the Binary Switch Model provide a good match to all experimental data across three different initial densities, we also find that the parameters used to match the data provide a more explicit interpretation of the underlying local growth mechanisms arising in the population. Specifically, we confirm that the experimental data are inconsistent with the standard logistic model and that the phenomena is best explained by a binary switch at low density. We conclude that the Binary Switch Model is useful to theorists and experimentalists alike in providing insight into binary switches at the individual scale that produce Allee effects at the population scale.

While one of the merits of the Binary Switch Model is to show how a single local binary switch gives rise to a variety of Allee effects, further extensions of the modelling framework can be made. For instance, additional switches can be incorporated into the modelling framework, representing populations whose proliferation and death rates change at more than one density. We anticipate that this kind of extension would lead to additional forms of Allee effects in the resulting population dynamics. Another potential modification would be to generalise the notion how we measure local density. In this work, we take the simplest possible approach use the number of nearest neighbours on a hexagonal lattice to represent the local density. Several generalisations, such as working with next nearest neighbours or working with a weighted average of nearest neighbours, could be incorporated into our modelling framework (Fadai et al. 2020 Jin et al. 2016a). Again, we expect that such extensions would lead to an even richer family of population dynamics models. We leave these extensions for future considerations.

Department of Mathematics, Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, Tucson, Arizona, 85721, United States

**Received** May 2014 **Revised** September 2014 **Published** April 2015

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P. A. Abrams, Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods, *Ecology Letters*, **4** (2001), 166-175. doi: 10.1046/j.1461-0248.2001.00199.x. Google Scholar

W. C. Allee, *Animal Aggregations, a Study in General Sociology*, University of Chicago Press, Chicago, 1931. Google Scholar

W. C. Allee, *The Social Life of Animals*, 3rd edition, William Heineman Ltd, London and Toronto, 1941. Google Scholar

W. C. Allee, O. Park, T. Park and K. Schmidt, *Principles of Animal Ecology*, W. B. Saunders Company, Philadelphia, 1949. Google Scholar

D. S. Boukal and L. Berec, Single-species Models of the Allee effect: Extinction boundaries, sex Ratios and mate Encounters, *Journal of Theoretical Biology*, **218** (2002), 375-394. doi: 10.1006/jtbi.2002.3084. Google Scholar

F. Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, *TREE*, **14** (1999), 405-410. doi: 10.1016/S0169-5347(99)01683-3. Google Scholar

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J. M. Cushing, Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations, *Journal of Biological Dynamics*, **8** (2014), 57-73. doi: 10.1080/17513758.2014.899638. Google Scholar

J. M. Cushing and J. Hudson, Evolutionary dynamics and strong Allee effects, *Journal of Biological Dynamics*, **6** (2012), 941-958. doi: 10.1080/17513758.2012.697196. Google Scholar

B. Dennis, Allee effects: Population growth, critical density, and the chance of extinction, *Natural Resource Modeling*, **3** (1989), 481-538. Google Scholar

F. Dercole and S. Rinaldi, *Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications*, Princeton University Press, Princeton, New Jersey, 2008. Google Scholar

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R. Lande, Natural selection and random genetic drift in phenotypic evolution, *Evolution*, **30** (1976), 314-334. Google Scholar

R. Lande, A quantitative genetic theory of life history evolution, *Ecology*, **63** (1982), 607-615. Google Scholar

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, *Theoretical Population Biology*, **43** (1993), 141-158. doi: 10.1006/tpbi.1993.1007. Google Scholar

J. Lush, *Animal Breeding Plans*, Iowa State College Press, Ames, Iowa, USA, 1937. Google Scholar

S. P. Otto and T. Day, *A Biologist's Guide to Mathematical Modeling in Ecology and Evolution*, Princeton University Press, Princeton, New Jersey, USA, 2007. Google Scholar

I. Scheuring, Allee effect increases dynamical stability in populations, *Journal of Theoretical Biology*, **199** (1999), 407-414. doi: 10.1006/jtbi.1999.0966. Google Scholar

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, *Theoretical Population Biology*, **64** (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8. Google Scholar

T. L. Vincent and J. S. Brown, *Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics*, Cambridge University Press, New York, 2005. doi: 10.1017/CBO9780511542633. Google Scholar

G. Wang, X.-G. Liang and F.-Z. Wang, The competitive dynamics of populations subject to an Allee effect, *Ecological Modelling*, **124** (1999), 183-192. doi: 10.1016/S0304-3800(99)00160-X. Google Scholar

##### References:

P. A. Abrams, Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods, *Ecology Letters*, **4** (2001), 166-175. doi: 10.1046/j.1461-0248.2001.00199.x. Google Scholar

W. C. Allee, *Animal Aggregations, a Study in General Sociology*, University of Chicago Press, Chicago, 1931. Google Scholar

W. C. Allee, *The Social Life of Animals*, 3rd edition, William Heineman Ltd, London and Toronto, 1941. Google Scholar

W. C. Allee, O. Park, T. Park and K. Schmidt, *Principles of Animal Ecology*, W. B. Saunders Company, Philadelphia, 1949. Google Scholar

D. S. Boukal and L. Berec, Single-species Models of the Allee effect: Extinction boundaries, sex Ratios and mate Encounters, *Journal of Theoretical Biology*, **218** (2002), 375-394. doi: 10.1006/jtbi.2002.3084. Google Scholar

F. Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, *TREE*, **14** (1999), 405-410. doi: 10.1016/S0169-5347(99)01683-3. Google Scholar

F. Courchamp, L. Berec and J. Gascoigne, *Allee Effects in Ecology and Conservation*, Oxford University Press, Oxford, Great Britain, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001. Google Scholar

J. M. Cushing, Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations, *Journal of Biological Dynamics*, **8** (2014), 57-73. doi: 10.1080/17513758.2014.899638. Google Scholar

J. M. Cushing and J. Hudson, Evolutionary dynamics and strong Allee effects, *Journal of Biological Dynamics*, **6** (2012), 941-958. doi: 10.1080/17513758.2012.697196. Google Scholar

B. Dennis, Allee effects: Population growth, critical density, and the chance of extinction, *Natural Resource Modeling*, **3** (1989), 481-538. Google Scholar

F. Dercole and S. Rinaldi, *Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications*, Princeton University Press, Princeton, New Jersey, 2008. Google Scholar

L. Edelstein-Keshet, *Mathematical Models in Biology*, Classics in Applied Mathematics 46, SIAM, Philadelphia, USA, 2005. doi: 10.1137/1.9780898719147. Google Scholar

S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, *Journal of Biological Dynamics* , **4** (2010), 397-408. doi: 10.1080/17513750903377434. Google Scholar

D. S. Falconer and T. F. C. Mackay, *Introduction to Quantitative Genetics*, Pearson Education Limited, Prentice Hall, Essex, England, 1996. Google Scholar

F. A. Hopf and F. W. Hopf, The role of the Allee effect in species packing, *Theoretical Population Biology*, **27** (1985), 27-50. doi: 10.1016/0040-5809(85)90014-0. Google Scholar

M. R. S. Kulenovic and A.-A. Yakubu, Compensatory versus overcompensatory dynamics in density-dependent leslie models, *Journal of Difference Equations and Applications*, **10** (2004), 1251-1265. doi: 10.1080/10236190410001652711. Google Scholar

R. Lande, Natural selection and random genetic drift in phenotypic evolution, *Evolution*, **30** (1976), 314-334. Google Scholar

R. Lande, A quantitative genetic theory of life history evolution, *Ecology*, **63** (1982), 607-615. Google Scholar

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, *Theoretical Population Biology*, **43** (1993), 141-158. doi: 10.1006/tpbi.1993.1007. Google Scholar

J. Lush, *Animal Breeding Plans*, Iowa State College Press, Ames, Iowa, USA, 1937. Google Scholar

S. P. Otto and T. Day, *A Biologist's Guide to Mathematical Modeling in Ecology and Evolution*, Princeton University Press, Princeton, New Jersey, USA, 2007. Google Scholar

I. Scheuring, Allee effect increases dynamical stability in populations, *Journal of Theoretical Biology*, **199** (1999), 407-414. doi: 10.1006/jtbi.1999.0966. Google Scholar

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, *Theoretical Population Biology*, **64** (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8. Google Scholar

T. L. Vincent and J. S. Brown, *Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics*, Cambridge University Press, New York, 2005. doi: 10.1017/CBO9780511542633. Google Scholar

G. Wang, X.-G. Liang and F.-Z. Wang, The competitive dynamics of populations subject to an Allee effect, *Ecological Modelling*, **124** (1999), 183-192. doi: 10.1016/S0304-3800(99)00160-X. Google Scholar

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Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. *Kinetic & Related Models*, 2011, 4 (1) : 187-213. doi: 10.3934/krm.2011.4.187

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## Component, group and demographic Allee effects in a cooperatively breeding bird species, the Arabian babbler (Turdoides squamiceps)

In population dynamics, inverse density dependence can be manifested by individual fitness traits (component Allee effects), and population-level traits (demographic Allee effects). Cooperatively breeding species are an excellent model for investigating the relative importance of Allee effects, because there is a disproportionately larger benefit to an individual of being part of a large group. As a consequence, larger groups have greater performance than small groups, known as the group Allee effect. Although small populations of cooperative breeders may be prone to all levels of Allee effects, empirical evidence for the existence of a demographic Allee effects is scarce. To determine the extent to which Allee effects are present in a cooperatively breeding species, we used a comprehensive 35-year life history database for cooperatively breeding Arabian babblers (Turdoides squamiceps). Firstly, we confirmed the existence of a component Allee effect by showing that breeding individuals in large groups receive greater benefits than those in small groups second, we confirmed the existence of group Allee effect by showing that larger groups survive longer. And thirdly, we identified a demographic Allee effect by showing that per capita population growth rate is positively affected by population density. Finally, we found that emigration and immigration rates, although dependent on group size, do not buffer against component and group-level Allee effects becoming a demographic Allee effect. Our finding of the existence of all three levels of Allee effects in a cooperative breeder may have important implications for future research and conservation decisions.

**Keywords:** Allee effect Arabian babblers Cooperation Density dependence Group size population dynamics.

## Mathematical modeling of population dynamics with Allee effect

Allee effect that refers to a positive relationship between individual fitness and population density provides an important conceptual framework in conservation biology. While declining Allee effect causes reduction in extinction risk in low-density population, it provides a benefit in limiting establishment success or spread of invading species. Population models that incorporated Allee effect confer the fundamental role which plays for shaping the population dynamics. In particular, non-spatial predator–prey and invasion models have shown the influence of Allee effects on population dynamics, and spatial models have illustrated its critical roles for pattern formation and the manifestation of traveling wave fronts. We highlight all such no-spatial and spatial population models and their contributions in deeper understanding of population dynamics. In addition, we briefly outline the trends for future research on Allee effect which we think are interesting and widely open.

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## Population genetic consequences of the Allee effect and the role of offspring-number variation

A strong demographic Allee effect in which the expected population growth rate is negative below a certain critical population size can cause high extinction probabilities in small introduced populations. But many species are repeatedly introduced to the same location and eventually one population may overcome the Allee effect by chance. With the help of stochastic models, we investigate how much genetic diversity such successful populations harbor on average and how this depends on offspring-number variation, an important source of stochastic variability in population size. We find that with increasing variability, the Allee effect increasingly promotes genetic diversity in successful populations. Successful Allee-effect populations with highly variable population dynamics escape rapidly from the region of small population sizes and do not linger around the critical population size. Therefore, they are exposed to relatively little genetic drift. It is also conceivable, however, that an Allee effect itself leads to an increase in offspring-number variation. In this case, successful populations with an Allee effect can exhibit less genetic diversity despite growing faster at small population sizes. Unlike in many classical population genetics models, the role of offspring-number variation for the population genetic consequences of the Allee effect cannot be accounted for by an effective-population-size correction. Thus, our results highlight the importance of detailed biological knowledge, in this case on the probability distribution of family sizes, when predicting the evolutionary potential of newly founded populations or when using genetic data to reconstruct their demographic history.

**Keywords:** family size founder effect genetic diversity introduced species stochastic modeling.

Copyright © 2014 by the Genetics Society of America.

### Figures

The expected number of surviving…

The expected number of surviving offspring per individual (see Equation 1) as a…

Consequences of the Allee effect…

Consequences of the Allee effect for the population genetics and dynamics of successful…

Population-genetic consequences of the Allee…

Population-genetic consequences of the Allee effect under the mate-finding model where offspring-number variation…

Overview over the various mechanisms…

Overview over the various mechanisms by which the Allee effect influences the amount…

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