Population dynamics with spatial structure and an Allee effect

Abstract

Population dynamics including a strong Allee effect describe the situation where long-term population survival or extinction depends on the initial population density. A simple mathematical model of an Allee effect is one where initial densities below the threshold lead to extinction, whereas initial densities above the threshold lead to survival. Mean-field models of population dynamics neglect spatial structure that can arise through short-range interactions, such as competition and dispersal. The influence of non-mean-field effects has not been studied in the presence of an Allee effect. To address this, we develop an individual-based model that incorporates both short-range interactions and an Allee effect. To explore the role of spatial structure we derive a mathematically tractable continuum approximation of the IBM in terms of the dynamics of spatial moments. In the limit of long-range interactions where the mean-field approximation holds, our modelling framework recovers the mean-field Allee threshold. We show that the Allee threshold is sensitive to spatial structure neglected by mean-field models. For example, there are cases where the mean-field model predicts extinction but the population actually survives. Through simulations we show that our new spatial moment dynamics model accurately captures the modified Allee threshold in the presence of spatial structure.

1. Introduction

Mathematical models of biological population dynamics are routinely built upon the classical logistic growth model where a population tends to a finite carrying capacity density for all positive initial densities [1–3]. While the logistic growth model is reasonable in some situations, there are other situations where the long-term survival of a population depends on the initial density, often called an Allee effect [4,5]. A strong Allee effect is associated with a net negative growth rate at low densities leading to population extinction below the Allee threshold density. By contrast, a net positive growth rate at higher densities leads to the survival of the population when the initial density is greater than the Allee threshold [6–8]. Another type of Allee effect, known as the weak Allee effect, describes population growth with a reduced but positive growth rate at low densities [5]. Unlike the strong Allee effect, a weak Allee effect does not exhibit any threshold density due to the net positive growth rate. In this study, we focus on the strong Allee effect since we are interested in exploring threshold effects and factors that influence the Allee threshold. Initial evidence for the Allee effect came from ecological systems for various plant and animal populations [6,9–15], whereas more recent studies suggest a role for the Allee effect in populations of biological cells [16–21].

Most mathematical models of Allee population dynamics invoke a mean-field assumption [2,6,22,23] where, either implicitly or explicitly, interactions between individuals are assumed to occur in proportion to the average density. Such models neglect spatial correlations between the locations of individuals [24]. When short-range interactions are present, such as short-range competition, the mean-field approximation can become inaccurate [25–28]. Short-range interactions can lead to the development of spatial structure that can affect the overall population dynamics [29–31]. Spatial structure in biological populations includes both clustering and segregation [32–37]. Stochastic individual-based models (IBMs) offer a straightforward means of exploring population dynamics without invoking a mean-field approximation [38,39]. However, IBM approaches are computationally prohibitive for large populations and provide limited mathematical insight into the population dynamics, for example, how particular biological mechanisms affect the carrying capacity or the Allee threshold [40].

A continuum approximation of the IBM in terms of the dynamics of spatial moments that tracks the evolution of the density of individuals, pairs, triplets and so on, is a useful way to study how short-range interactions and spatial structure can influence population dynamics [41,42]. Law et al. developed a spatial moment model, called the spatial logistic model, which quantifies the impact of spatial structure on classical logistic growth dynamics [29]. This work shows that spatial structure has a strong impact on the carrying capacity density. More recently, the spatial logistic model has been extended to consider other relevant mechanisms including interspecies and intraspecies interactions, neighbour-dependent motility bias, predator–prey dynamics and chase–escape interactions [43–46].

In this work, we present an IBM and a novel spatial moment dynamics approximation that incorporates a strong Allee effect. This model is a generalization of the spatial logistic model [29]. Localized density-dependent interactions, such as short-range competition, short-range cooperation and short-range offspring dispersal, are incorporated. In the limit of large-scale interactions where the mean-field approximation is valid, both the IBM and the spatial moment model are consistent with the classical mean-field Allee growth model. By contrast, when spatial structure is present, we find that the Allee threshold density can be very sensitive to the spatial structure. For example, under a combination of short-range competition and short-range dispersal, all initial densities lead to population extinction, whereas the classical mean-field model predicts that the population will survive. By contrast, the new spatial moment approximation gives an accurate prediction of the long-time outcome even when strong spatial structure is present.

In this work we broadly consider two different types of results. In the first set we focus on population-level outcomes with regard to whether a population eventually becomes extinct or whether it survives. We study these problems using a classical mean-field model, a new spatial moment dynamics model as well as using repeated, identically prepared stochastic simulations. For these results we are very careful to select initial conditions in the IBM simulations so that the vast majority of the repeated simulations lead to the same long-time outcome. For example, we consider parameter choices and initial conditions where more than 99% of identically prepared IBM simulations all lead to either long-term extinction or long-term survival. The extremely small proportion of outliers are then excluded from the calculation of ensemble data. In the second set of results we focus on repeated IBM simulations in situations where stochastic effects can lead to either long-term survival or long-term extinction. We characterize this transition in terms of a survival probability, defined as the fraction of IBM realizations in which the population does not become extinct. Both the classical mean-field model and the new spatial moment dynamics model are deterministic approximations of the IBM and do not describe any stochastic effects.

2. IBM

The IBM describes the dynamics of N(t) individuals, initially distributed randomly on a continuous two-dimensional domain of size L × L. The location of the nth individual is ${\mathbf{\text{x}}}_n\in {\mathbb{R}}^2$, and periodic boundary conditions are imposed. Individuals undergo birth, death and movement events, with event rates influenced by interactions between individuals. The IBM is developed for a spatially homogeneous, translationally invariant environment, where the probability of finding an individual in a small region, averaged over multiple realizations of the IBM, is independent of the location of that region [43,46]. Hence the model is relevant to populations that do not involve macroscopic gradients in the density of individuals [47].

Competition between individuals influences the death rate, modelling increased mortality as a result of competition for limited resources. We use an interaction kernel, ${\displaystyle {\omega }_c(|\mathit{\xi }|)}$, to describe the competition a particular reference individual experiences from another individual at a displacement, ξ. We specify the competition kernel to be a function of separation distance, |ξ|,

$${\omega }_c(|\mathit{\xi }|)={\gamma }_c\exp (-\frac{|\mathit{\xi }{|}^2}{2{\sigma }_c^2}),$$ 2.1

where γ_c > 0 and σ_c > 0 are the competition strength and range, respectively. Specifying the competition kernel to be Gaussian means that the impact of competition is a decreasing function of separation distance, |ξ|. We define a random variable, X_n, that measures the neighbourhood density of the nth individual weighted over by the competition kernel as,

$$X_n=\sum _{\begin{array}{c}k=1\\ k\ne n\end{array}}^{N(t)}{\omega }_c(|{\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n|).$$ 2.2

We consider the death rate of the nth individual to be some function of X_n,

$${\mathcal{D}}_n=F(X_n).$$ 2.3

For a specific choice of F(X_n), the key parameters controlling how competition influences the death rate are σ_c and γ_c. Figure 1 illustrates two scenarios for the simplest choice of F(X_n) = X_n. The arrangement of agents in figure 1a,c are identical but we consider a long-range competition kernel (large σ_c) in figure 1a,b and a short-range competition kernel (small σ_c) in figure 1c,d. In figure 1 we compute the death rate of a test individual located at any location x. This allows us to treat ${\mathcal{D}}_n$ as a continuous function of position. Level curves of ${\mathcal{D}}_n$ are superimposed in figure 1a,c and we see that the differences in the length-scale of interaction leads to very different local death rates. For example, when competition is long-range in figure 1a,b the death rate for the relatively isolated green agent is ${\mathcal{D}}_n=0.275$ whereas when the competition is short-range in figure 1c,d the death rate of the same agent is very different, ${\mathcal{D}}_n=0$. A similar set of results with a different F(X_n) in section 1 of the electronic supplementary material shows similar results. (In particular, ${\mathcal{D}}_n=0.076$ and ${\mathcal{D}}_n=0$ for long-range and short-range interactions, respectively).

Figure 1.

Visualizing long- and short-range competition interactions. (a,c) Locations of individuals (dots) superimposed with level curves of ${\mathcal{D}}_n$ for long- and short-range competition, respectively. (b,d) shows the long- (σ_c = 4.0) and short-range (σ_c = 0.5) competition kernels. Here, F(X_n) = X_n and γ_c = 0.1. (Online version in colour.)

We also consider a cooperative interaction between individuals that enhances the proliferation rate of individuals [18]. This is a model for sexual reproduction or some other mutualistic interaction in which reproductive fitness increases in the presence of near neighbours. Similar to the competition kernel, we define a cooperation kernel,

$${\omega }_p(|\mathit{\xi }|)={\gamma }_p\exp (-\frac{|\mathit{\xi }{|}^2}{2{\sigma }_p^2}),$$ 2.4

to account for the contribution of a neighbour at a displacement ξ to the reference individual’s proliferation rate. Here, γ_p > 0 and σ_p > 0 represent the strength and range of the interaction, respectively. As with competition, we define a random variable, Y_n, that measures the neighbourhood density weighted by the cooperation kernel,

$$Y_n=\sum _{\begin{array}{c}k=1\\ k\ne n\end{array}}^{N(t)}{\omega }_p(|{\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n|).$$ 2.5

The proliferation rate of the nth individual is taken to be some function of Y_n,

$${\mathcal{P}}_n=G(Y_n).$$ 2.6

When an individual undergoes proliferation, a daughter agent is placed at a displacement sampled from a dispersal kernel, μ_p(ξ), which we choose to be a bivariate normal with mean zero and standard deviation σ_d.

For simplicity, we assume the movement rate is density-independent with a constant rate, m. An individual undergoing a motility event traverses a displacement, (lcos (θ), lsin (θ)) sampled from a movement kernel, μ_m(ξ) . The direction of movement, θ ∈ [0, 2π] is uniformly distributed. The distance moved, l, is sampled from a relatively narrow, truncated Gaussian distribution with mean, μ_s, and standard deviation, σ_s, where σ_s < μ_s/4. To ensure l is positive, the Gaussian is truncated so that μ_s − 4σ_s < l < μ_s + 4σ_s.

This IBM is an extension of the spatial stochastic logistic model [29], often simply called the spatial logistic model, which focuses on understanding the impact of short-range interactions and spatial structure on the classical logistic growth model [1]. In the spatial logistic model, the death rate is taken to be the sum of the competition from the neighbours and the proliferation rate is constant, so there is no cooperation. Furthermore, the spatial logistic growth model does not involve any agent motility. We recover the spatial logistic model as a particular case of our model when we set F(X_n) = X_n, G(Y_n) = p and m = 0. In figure 2a–c, we summarize the dynamics of the spatial logistic model when the interactions are long-range and the mean-field approximation is valid. Under these conditions the death rate is a linearly increasing function of density and the proliferation rate is constant, as shown in figure 2a. This choice leads to an unstable steady state ${\overline{Z}}_1^{(1)}=0$ and a stable steady-state ${\overline{Z}}_1^{(2)}>0$, as shown in figure 2b. Hence the mean-field implies that all initial population densities will eventually tend to ${\overline{Z}}_1^{(2)}$ as t → ∞, as in figure 2c.

Figure 2.

Comparison of the spatial logistic and Allee effect models under mean-field conditions. (a,d) Proliferation (red) and death rates (black) as functions of density. (b,e) Density growth rate as a function of density. Stable and unstable equilibrium points are highlighted with filled and empty cyan dots, respectively. (c,f ) Dynamics for both models with thecyan lines indicating the equilibrium densities. (Online version in colour.)

Our generalized IBM framework accommodates various nonlinear functional forms for ${\mathcal{D}}_n$ and ${\mathcal{P}}_n$. In figure 2d we choose ${\mathcal{D}}_n$ to be a concave up quadratic function and ${\mathcal{P}}_n$ to be a linearly increasing function of density. This leads to three equilibrium densities: ${\overline{Z}}_1^{(1)}$, ${\overline{Z}}_1^{(2)}$ and ${\overline{Z}}_1^{(3)}$, as in figure 2e. Here, ${\overline{Z}}_1^{(1)}$ and ${\overline{Z}}_1^{(3)}$ are stable equilibria, and ${\overline{Z}}_1^{(2)}$ is an unstable equilibrium. The population dynamics here with the long-range interactions where the mean-field approximation is valid is shown in figure 2f . Here we see that populations with an initial density less than the Allee threshold, ${\overline{Z}}_1^{(2)}$, eventually go extinct. By contrast, any initial density greater than the Allee threshold, ${\overline{Z}}_1^{(2)}$, eventually tends to ${\overline{Z}}_1^{(3)}$. This is the simplest mean-field model of an Allee effect in which the net population growth rate is a cubic function of population density [3]. For the spatial logistic model, the stability of the mean-field equilibrium depends on the model parameters. For example, the unstable equilibrium point at ${\overline{Z}}_1^1=0$, becomes a stable equilibrium point (population extinction) when competition and dispersal are short-range. Results in §5 reveal that similar observations hold for the IBM considered here.

We simulate the IBM using the Gillespie algorithm [48] that is described in section 2 of the electronic supplementary material. The population dynamics arising from the IBM is analysed by considering the average density of individuals, Z₁(t) = N(t)/L². Information about the spatial configuration of the population can be studied in terms of the average density of pairs of individuals expressed as a pair correlation function, C(|ξ|, t) [31,35,49,50]. The pair-correlation function denotes the average density of pairs of individuals with separation distance |ξ|, at a time, t, normalized by the density of pairs in a population with the complete absence of spatial structure. Therefore, for a population without any spatial structure, C(|ξ|, t) = 1. When C(|ξ|, t) < 1, there are fewer pairs of individuals with a separation distance, |ξ|, than in a population without any spatial structure. We refer to this spatial configuration as being segregated. When C(|ξ|, t) > 1, we have more pairs of individuals with a separation distance, |ξ|, than we would have in a population without any spatial structure and this spatial configuration is referred to as being clustered.

3. Spatial moment dynamics

In this section, we construct a continuum approximation of the IBM in terms of the dynamics of spatial moments. The first spatial moment, Z₁(t), for a point process of the kind considered in the IBM is defined as the average density of individuals [29,43]. The second spatial moment, Z₂(ξ, t), is the average density of pairs of individuals separated by a displacement of ξ. The third spatial moment, Z₃(ξ, ξ′, t), is the average density of a triplet of individuals separated by displacements ξ and ξ′, respectively. A formal definition of spatial moments is provided in section 3 of the electronic supplementary material. It is possible to define higher-order moments similarly, but for the present study, we restrict our attention to the first three spatial moments [43,51].

To derive dynamical equations for the evolution of the spatial moments, we need to find expressions for the continuum analogue of the discrete event rates given in equation (2.3) and equation (2.6). This can be achieved by finding the expected death and proliferation rates, ${\displaystyle \mathbb{E}[D_n]=\mathbb{E}[F(X_n)]}$ and ${\displaystyle \mathbb{E}[P_n]=\mathbb{E}[G(Y_n)]}$, respectively. To calculate these expected rates, expand F(X_n) and G(Y_n) in a Taylor series about $\overline{X}=\mathbb{E}[X_n]$ and $\overline{Y}=\mathbb{E}[Y_n]$. For the death rate we have

$$\begin{array}{rl}\mathbb{E}[F(X_n)]& =\mathbb{E}[F(\overline{X})+F^{\prime }(\overline{X})(X_n-\overline{X})+\frac{F^{″}(\overline{X})}{2!}{(X_n-\overline{X})}^2+\frac{F^{‴}(\overline{X})}{3!}{(X_n-\overline{X})}^3+\dots ],\\ & =F(\overline{X})+\frac{F^{″}(\overline{X})}{2!}\hspace{0.17em}\text{Var}[X_n]+\frac{F^{‴}(\overline{X})}{3!}\mathbb{E}[{(X_n-\overline{X})}^3]+\dots .\end{array}$$ 3.1

While our IBM can incorporate any choice of F(X_n), the higher-order terms in the truncated Taylor series in equation (3.1) are, in general, non-zero. The most straightforward choice of F(X_n) to generate the Allee effect is a quadratic, and this choice has the additional benefit that third and higher derivatives vanish, so we have

$$\mathbb{E}[F(X_n)]=F(\overline{X})+\frac{F^{″}(\overline{X})}{2!}\hspace{0.17em}\text{Var}[X_n].$$ 3.2

The computation of expected death rates reduces to calculating ${\displaystyle \mathbb{E}[X_n]}$ and ${\displaystyle \text{Var}[X_n]}$, and substituting these into equation (3.2). If we suppose the L × L domain is divided into ${\displaystyle M=L^2/\delta A}$ subregions, each of area δA, where these subregions are sufficiently small such that each subregion contains at most one individual, we have

$$\mathbb{E}[X_n]=\mathbb{E}[\sum _{\begin{array}{c}k=1\end{array}}^M{\omega }_c(|{\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n|)\hspace{0.17em}{\mathbb{I}}_{\delta A}({\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n)],$$ 3.3

where, the indicator function, ${\displaystyle {\mathbb{I}}_{\delta A}({\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n)=1}$, if an individual is present in a region of area δA at a displacement ${\displaystyle {\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n}$, and ${\displaystyle {\mathbb{I}}_{\delta A}({\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n)=0}$, otherwise. Using the property of the indicator function that $\mathbb{E}[{\mathbb{I}}_{\delta A}({\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n)]=\mathbb{P}[{\mathbb{I}}_{\delta A}({\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n)=1]$ [52], we have,

$$\mathbb{E}[X_n]=\sum _{\begin{array}{c}k=1\end{array}}^M{\omega }_c(|{\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n|)\hspace{0.17em}\mathbb{P}[{\mathbb{I}}_{\delta A}({\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n)=1].$$ 3.4

In the continuum limit, the right-hand side of equation (3.4) is equivalent to multiplying the conditional probability of having an individual in a small window of size δA at a displacement ξ from the reference individual, with the corresponding interaction kernel and integrating over all possible displacements as δA → 0 [43]. The conditional probability for the presence of a neighbour individual is Z₂(ξ, t) δA/Z₁(t). A derivation of the conditional probability is provided in section 4 of the electronic supplementary material. Hence we have,

$$\mathbb{E}[X_n]=\int {\omega }_c(|\mathit{\xi }|)\hspace{0.17em}\frac{Z_2(\mathit{\xi },t)}{Z_1(t)}\hspace{0.17em}\text{d}\mathit{\xi }.$$ 3.5

Note that the previous spatial moment models, including the spatial logistic model, assume that F(X_n) is a linear function [29,30,40,43]. In that case, the death rate in equation (3.2) depends only on $\mathbb{E}[X_n]$. But in the more general case where F(X_n) is nonlinear, such as an Allee effect, we also require information about the variance. Therefore, we compute Var[X_n] in a similar fashion,

$$\begin{array}{rl}\text{Var}[X_n]& =\text{Var}[\sum _{\begin{array}{c}k=1\end{array}}^M{\omega }_c(|{\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n|)\hspace{0.17em}{\mathbb{I}}_{\delta A}({\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n)],\\ & =\sum _{\begin{array}{c}k=1\end{array}}^M{\omega }_c^2(|{\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n|)\hspace{0.17em}\text{Var}[{\mathbb{I}}_{\delta A}({\mathbf{\text{x}}}_k-{\mathbf{\text{x}}}_n)]\\ & +\sum _{\begin{array}{c}i=1,j=1\\ i\ne j\end{array}}^M{\omega }_c(|{\mathbf{\text{x}}}_i-{\mathbf{\text{x}}}_n|)\hspace{0.17em}{\omega }_c(|{\mathbf{\text{x}}}_j-{\mathbf{\text{x}}}_n|)\hspace{0.17em}\text{Cov}[{\mathbb{I}}_{\delta A}({\mathbf{\text{x}}}_i-{\mathbf{\text{x}}}_n),\hspace{0.17em}{\mathbb{I}}_{\delta A}({\mathbf{\text{x}}}_j-{\mathbf{\text{x}}}_n)].\end{array}$$ 3.6

Following a similar procedure used in the computation of the continuum analogue of $\mathbb{E}[X_n]$ in equation (3.5), we derive the expression for Var[X_n] as,

$$\begin{array}{rl}\text{Var}[X_n]& =\int {\omega }_c^2(|\mathit{\xi }|)\left(\frac{Z_2(\mathit{\xi },t)}{Z_1(t)}\right)\text{d}\mathit{\xi }\\ & +\iint {\omega }_c(|{\mathit{\xi }}^{\prime }|)\hspace{0.17em}{\omega }_c(|{\mathit{\xi }}^{″}|)(\frac{Z_3({\mathit{\xi }}^{\prime },{\mathit{\xi }}^{″},t)}{Z_1(t)}-\frac{Z_2({\mathit{\xi }}^{\prime },t)\hspace{0.17em}{Z}_2({\mathit{\xi }}^{″},t)}{Z_1^2(t)})\text{d}{\mathit{\xi }}^{\prime }\text{d}{\mathit{\xi }}^{″}.\end{array}$$ 3.7

For brevity, we omit the intermediate steps involved in the derivation of Var[X_n] here. These details are provided in section 5 of the electronic supplementary material.

We consider specific functional forms as $F(X_n)=d+X_n^2$ and G(Y_n) = p + Y_n. We make these choices because they are the simplest scenario that result in a strong Allee effect. To proceed, we compute the expected death rate of an individual, D₁(t), by substituting the expressions for $\mathbb{E}[X_n]$ and Var[X_n] in equation (3.2) to give,

$$\begin{array}{rl}{D}_1(t)& =d+{(\int {\omega }_c(|\mathit{\xi }|)\hspace{0.17em}\frac{Z_2(\mathit{\xi },t)}{Z_1(t)}\hspace{0.17em}\text{d}\mathit{\xi })}^2+\int {\omega }_c^2(|\mathit{\xi }|)\frac{Z_2(\mathit{\xi },t)}{Z_1(t)}\hspace{0.17em}\text{d}\mathit{\xi }\\ & +\iint {\omega }_c(|\mathit{\xi }|)\hspace{0.17em}{\omega }_c(|{\mathit{\xi }}^{\prime }|)(\frac{Z_3(\mathit{\xi },{\mathit{\xi }}^{\prime },t)}{Z_1(t)}-\frac{Z_2(\mathit{\xi },t)\hspace{0.17em}{Z}_2({\mathit{\xi }}^{\prime },t)}{Z_1^2(t)})\text{d}\mathit{\xi }\hspace{0.17em}\text{d}{\mathit{\xi }}^{\prime }.\end{array}$$ 3.8

Similarly the expected proliferation rate for an individual is,

$$P_1(t)=p+\int {\omega }_p(|\mathit{\xi }|)\hspace{0.17em}\frac{Z_2(\mathit{\xi },t)}{Z_1(t)}\hspace{0.17em}\text{d}\mathit{\xi }.$$ 3.9

The dynamics of the first moment depend solely on the balance between proliferation and death. The movement of individuals does not result in a change in the population size. Hence the time evolution of the first spatial moment is given by,

$$\frac{\text{d}}{\text{d}t}{Z}_1(t)=\underset{\text{Increase in density due to proliferation}}{\underbrace{P_1(t)Z_1(t)}}-\stackrel{\text{Decrease in density due to death}}{\overbrace{D_1(t)Z_1(t).}}$$ 3.10

Note that the dynamics of the first moment depends on the second and third moments through equations (3.8)–(3.9), and to solve the dynamics of the first moment, we need to specify the values of these higher-order moments.

Now we derive the dynamical equation for the density of pairs of individuals. For the derivation, we need to calculate the event rates of individuals while they are in a pair with another individual at a displacement, ξ. The conditional probability of finding an individual at displacement ξ′, given that a pair of individuals exist with separation displacement ξ, is ${\displaystyle Z_3(\mathit{\xi },{\mathit{\xi }}^{\prime },t)\hspace{0.17em}\delta A/Z_2(\mathit{\xi },t)}$. The derivation of the expression for the conditional probability is given in section 4 of the electronic supplementary material. Using this expression for the conditional probability, and following the same procedures used to arrive at equation (3.5) and equation (3.7), we compute $\mathbb{E}[X_n]$ and Var[X_n] for an individual that forms a pair with another individual at a displacement ξ. Hence, the expected death rate of an individual, conditional on the presence of a neighbour at a displacement ξ, is given by,

$$D_2(\mathit{\xi },t)=\hspace{0.17em}d+{(\int {\omega }_c(|{\mathit{\xi }}^{\mathbf{\prime }}|)\hspace{0.17em}\frac{Z_3(\mathit{\xi },{\mathit{\xi }}^{\mathbf{\prime }},t)}{Z_2(\mathit{\xi },t)}\hspace{0.17em}\text{d}{\mathit{\xi }}^{\mathbf{\prime }}+{\omega }_c(|\mathit{\xi }|))}^2+\int {\omega }_c^2(|{\mathit{\xi }}^{\mathbf{\prime }}|)\frac{Z_3(\mathit{\xi },{\mathit{\xi }}^{\mathbf{\prime }},t)}{Z_2(\mathit{\xi },t)}\hspace{0.17em}\text{d}{\mathit{\xi }}^{\mathbf{\prime }}.$$ 3.11

Note that the subscript in D₂(ξ, t) indicates the fact that we are computing the expected rate for an individual that forms a pair with another individual at displacement ξ. The additional factor of ω_c(|ξ|) in the second term of equation (3.11) accounts for the direct influence of the individual at displacement ξ. Similarly, the expected proliferation rate of an individual, conditional on the presence of a neighbour at a displacement ξ, is,

$$P_2(\mathit{\xi },t)=p+\int {\omega }_p(|{\mathit{\xi }}^{\mathbf{\prime }}|)\hspace{0.17em}\frac{Z_3(\mathit{\xi },{\mathit{\xi }}^{\mathbf{\prime }},t)}{Z_2(\mathit{\xi },t)}\hspace{0.17em}\text{d}{\mathit{\xi }}^{\mathbf{\prime }}+{\omega }_p(|\mathit{\xi }|).$$ 3.12

The time evolution of Z₂(ξ, t) depends on the creation of new pairs and the loss of existing pairs. The schematic in figure 3 illustrates possible ways in which movement, proliferation or death event leads to the creation or destruction of pairs of individuals separated by a displacement of ξ. Figure 3a represents a pair of individuals at a separation displacement of ξ. A movement of either individual destroys this pair, as does the death of either individual. Figure 3b,c demonstrates two different ways to generate a new pair separated by a displacement ξ. When an individual among the pair separated by a displacement ξ^′ + ξ in figure 3b moves or places a daughter agent a displacement ξ′, a new pair is formed at a displacement ξ. In this case, the movement and proliferation occur with rates μ_m(ξ^′) m and μ_p(ξ^′)P₂(ξ^′ + ξ, t), respectively. Another possibility for the creation of a pair with separation displacement ξ is when a single individual, as shown in figure 3c, places a daughter agent over a displacement −ξ. The rate for this event is μ_p( − ξ) P₁(t). The dynamics of the second moment is obtained by combining these possibilities as,

$$\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial }t}{Z}_2(\mathit{\xi },t)=\underset{\text{Loss of pairs at displacement}\mathit{\xi }\text{due to the death of either individual}}{\underbrace{-\hspace{0.17em}2D_2(\mathit{\xi },t)Z_2(\mathit{\xi },t)}}\\ & \underset{\text{Loss of pairs at displacement}\mathit{\xi }\text{due to the movement of either individual}}{\underbrace{-\hspace{0.17em}2m\hspace{0.17em}{Z}_2(\mathit{\xi },t)}}\\ & \underset{\text{Formation of pairs at displacement}\hspace{0.17em}\mathit{\xi }\hspace{0.17em}\text{due to the proliferation of individuals that form a pair at displacement}\$\mathit{\xi }+{\mathit{\xi }}^{\prime }}{\underbrace{+\hspace{0.17em}2\int {\mu }_p({\mathit{\xi }}^{\mathbf{\prime }})P_2(\mathit{\xi }+{\mathit{\xi }}^{\mathbf{\prime }},t)Z_2(\mathit{\xi }+{\mathit{\xi }}^{\mathbf{\prime }},t)\hspace{0.17em}\text{d}{\mathit{\xi }}^{\mathbf{\prime }}}}\\ & \underset{\text{Formation of pairs at displacement}\hspace{0.17em}\mathit{\xi }\text{due to the movement of individuals that form a pair at displacement}\hspace{0.17em}\mathit{\xi }+{\mathit{\xi }}^{\prime }}{\underbrace{+\hspace{0.17em}2m\int {\mu }_m({\mathit{\xi }}^{\mathbf{\prime }})\hspace{0.17em}{Z}_2(\mathit{\xi }+{\mathit{\xi }}^{\mathbf{\prime }},t)\hspace{0.17em}\text{d}{\mathit{\xi }}^{\mathbf{\prime }}}}\\ & \underset{\text{Formation of pairs at displacement}\mathit{\xi }\text{due to the placement of a daughter individual at a displacement}-{\mathit{\xi }}^{\prime }}{\underbrace{+\hspace{0.17em}2{\mu }_p(-\mathit{\xi })P_1(t)Z_1(t).}}\end{array}$$ 3.13

Figure 3.

Possible events leading to a change in pair density. Red dots represent existing individuals and black open circles indicate potential locations of an individual after a movement or proliferation event. (a) A pair separated by a displacement ξ. Movement or death of either individual destroys the pair. (b) A pair separated by a displacement ξ + ξ^′. A movement or placement of a daughter over a displacement of ξ^′ creates a new pair separated by displacement ξ. (c) A single individual where the placement of a daughter at displacement  − ξ creates a new pair with a displacementξ. (Online version in colour.)

Since the event rates in equations (3.11)–(3.12) depend on the third-order moment, Z₃(ξ, ξ′, t), we need some expression for the third moment to solve the system, and we anticipate that the dynamics of the third moment will depend upon higher moments. To deal with this hierarchy of equations we use the Power-2 asymmetric moment closure approximation [29,30]

$$Z_3(\mathit{\xi },{\mathit{\xi }}^{\prime },t)=\frac{4Z_2(\mathit{\xi },t)Z_2({\mathit{\xi }}^{\prime },t)+Z_2(\mathit{\xi },t)Z_2({\mathit{\xi }}^{\prime }-\mathit{\xi },t)-Z_2({\mathit{\xi }}^{\prime },t)Z_2({\mathit{\xi }}^{\prime }-\mathit{\xi },t)-Z_1^4(t)}{5\hspace{0.17em}{Z}_1(t)},$$ 3.14

to approximately close the system in terms of the first and second moments only. Other closure approximations, such as the power-1 closure, the symmetric power-2 closure and the Kirkwood superposition approximation [29,53], are possible, and we compare the accuracy of these four different closure approximations in section 6 of the electronic supplementary material. Details about the numerical methods involved in solving the dynamical equation for the second moment, equation (3.13), is provided in section 7 of the electronic supplementary material and MATLAB code to implement the algorithm is available on Github.

4. Mean-field dynamics

Under the classical mean-field approximation, interactions between individuals occur in proportion to the average density, and there is no spatial structure. These conditions correspond to having long-range interactions between individuals. Comparing the solutions of the classical mean-field model, IBM simulations and the solution of the new spatial moment dynamics model will provide insight into how spatial structure influences the population dynamics.

In terms of spatial moments, the mean-field implies that we have $Z_2(\mathit{\xi },t)=Z_1^2(t)$ [30,31], which means that the expected death rate from equation (3.8) simplifies to,

$$D_1(t)=d+Z_1^2(t){(\int {\omega }_c(|\mathit{\xi }|)\hspace{0.17em}\text{d}\mathit{\xi })}^2+Z_1(t)\int {\omega }_c^2(|\mathit{\xi }|)\hspace{0.17em}\text{d}\mathit{\xi }.$$ 4.1

Similarly, the expected proliferation rate in equation (3.9) simplifies to

$$P_1(t)=p+Z_1(t)\int {\omega }_p(|\mathit{\xi }|)\hspace{0.17em}\text{d}\mathit{\xi }.$$ 4.2

Since the interaction kernels have the property that ${\displaystyle \int {\omega }_c(|\mathit{\xi }|)\hspace{0.17em}\text{d}\mathit{\xi }=2\pi {\gamma }_c{\sigma }_c^2}$ and ${\displaystyle \int \hspace{0.17em}{\omega }_c^2(|\mathit{\xi }|)\hspace{0.17em}\text{d}\mathit{\xi }=\pi {\gamma }_c^2{\sigma }_c^2}$, we substitute the mean-field death and proliferation rates in equations (4.1)–(4.2) into equation (3.10) to give

$$\frac{\text{d}}{\text{d}t}{Z}_1(t)=(p-d)Z_1(t)+(2\pi {\gamma }_p\hspace{0.17em}{\sigma }_p^2-\pi {\gamma }_c^2{\sigma }_c^2)Z_1^2(t)-4{\pi }^2{\gamma }_c^2\hspace{0.17em}{\sigma }_c^4\hspace{0.17em}{Z}_1^3(t),$$ 4.3

which leads to three equilibrium densities:

$$\begin{array}{rl}{\overline{Z}}_1^{(1)}& =0,\\ {\overline{Z}}_1^{(2)}& =\frac{2{\gamma }_p{\sigma }_p^2-{\gamma }_c^2{\sigma }_c^2}{8\pi {\gamma }_c^2{\sigma }_c^4}-\frac{\sqrt{{(2{\gamma }_p{\sigma }_p^2-{\gamma }_c^2{\sigma }_c^2)}^2+16{\gamma }_c^2{\sigma }_c^4(p-d)}}{8\pi {\gamma }_c^2{\sigma }_c^4},\\ {\overline{Z}}_1^{(3)}& =\frac{2{\gamma }_p{\sigma }_p^2-{\gamma }_c^2{\sigma }_c^2}{8\pi {\gamma }_c^2{\sigma }_c^4}+\frac{\sqrt{{(2{\gamma }_p{\sigma }_p^2-{\gamma }_c^2{\sigma }_c^2)}^2+16{\gamma }_c^2{\sigma }_c^4(p-d)}}{8\pi {\gamma }_c^2{\sigma }_c^4},\end{array}\}$$ 4.4

where ${\overline{Z}}_1^{(3)}>{\overline{Z}}_1^{(2)}>{\overline{Z}}_1^{(1)}$ for the parameters we consider in this study. The equilibrium points ${\overline{Z}}_1^{(1)}$ and ${\overline{Z}}_1^{(3)}$ are stable, whereas ${\overline{Z}}_1^{(2)}$ is unstable for the parameter regimes considered in this study. In this classical mean-field context, ${\overline{Z}}_1^{(2)}$ is the Allee threshold and ${\overline{Z}}_1^{(3)}$ is the carrying capacity density. The equilibrium densities and dynamics associated with equation (4.3) are depicted in figure 2e,f . The derivation for the classical cubic mean-field description of the Allee effect, equation (4.3), shows how individual-level competitive and cooperative interactions result in a population-level Allee effect.

5. Results and discussion

We now present IBM simulation results together with numerical solutions of both the spatial moment and the mean-field models to explore the influence of spatial structure on the population dynamics. In each case that we consider (figures 4–7) we plot the time evolution of the average density of individuals from repeated, identically prepared IBM simulations. Information about the spatial structure of the population is given in terms of the pair correlation function computed at the end of the simulation. Since the IBM is stochastic, there is a non-zero probability that any individual simulation will lead to extinction, regardless of whether the average outcome is that the population would survive. In the first set of results we present we take care to choose parameters and initial conditions such that at least 99% of the 1000 identically prepared simulations lead to the same long-term population-level outcome (i.e. extinction or survival) and any outliers, if present, are excluded from the calculation of ensemble averages. For each parameter combination, we consider three different initial conditions: (i) ${\overline{Z}}_1^{(1)}<Z_1(0)<{\overline{Z}}_1^{(2)}$; (ii) ${\overline{Z}}_1^{(2)}<Z_1(0)<{\overline{Z}}_1^{(3)}$; and (iii) $Z_1(0)>{\overline{Z}}_1^{(3)}$. In the IBM simulations we control the initial density by choosing a different value of N(0).

Figure 4.

Long-range interactions and dispersal kernels (σ_c = σ_p = σ_d = 4.0) with weak interaction strengths (γ_c = 0.009 and γ_p = 0.009) lead to mean-field conditions. (a–c) Initial locations of individuals (dots) for three different initial population sizes, N(0) = 80, 240 and 400, respectively. (d–f ) Locations of individuals at t = 30. (g–i) Density of individuals as a function of time. Black solid lines correspond to averaged results from 1000 identically prepared IBM realizations, red dashed lines correspond to the solutions of the spatial moment dynamics model and green solid lines are the solution of the mean-field model. The cyan lines show the equilibrium densities. (j–l) Pair-correlation function, C(|ξ|, 30). The parameter values are d = 0.4, p = 0.2, m = 0.1, μ_s = 0.4and σ_s = 0.1. (Online version in colour.)

Figure 7.

Short-range cooperation and dispersal promotes population growth. In this case we consider short-range cooperation σ_p = 0.5 and an intermediate-range dispersal σ_d = 2.0 with γ_p = 0.576. (a–c) The initial locations of individuals (red dots) for three different population sizes, N(0) = 105, 240 and 400. (d–f ) show the locations of individuals at t = 30. (g–i) show the density of individuals as a function of time. Black solid lines correspond to the averaged results from 1000 realizations of the IBM, red dashed lines correspond to the solutions of spatial moment dynamics and green solid lines correspond to the solution of the mean-field model.The cyan lines show the critical densities. (j–l) show the C(|ξ|, t) computed at t = 30 as a function of separation distance. The parameter values are d = 0.4, p = 0.2, m = 0.1, μ_s = 0.4 and σ_s = 0.1. (Online version in colour.)

As a starting point we consider a simple case with relatively weak long-range interactions so that the mean-field approximation is valid in figure 4. Choosing long-range dispersion and competition kernels ensures that there is minimal correlation between the locations of agents in the simulations. As expected, results with ${\overline{Z}}_1^{(1)}<Z_1(0)<{\overline{Z}}_1^{(2)}$ lead to extinction, and results with ${\overline{Z}}_1^{(2)}<Z_1(0)<{\overline{Z}}_1^{(3)}$ and $Z_1(0)>{\overline{Z}}_1^{(3)}$ lead to the population eventually settling to the carrying capacity density by t = 30. The comparison of the averaged IBM results, the mean-field and spatial moment models in figure 4g–i, shows that all three approaches are consistent and the eventual long-time population contains no spatial structure since the pair correlation function is unity in figure 4k,l. Note that we do not show the pair correlation function in figure 4j since, in this case, the long-time result is that the population goes extinct.

Having verified that both the IBM and spatial moment dynamics model replicate solutions of the mean-field model for relatively weak long-range interaction and dispersal kernels, we now focus on short-range interactions that can lead to spatial structure.

(a). Short-range competition reduces the Allee threshold

Results in figure 5 are presented in the same format as in figure 4 except that we consider short-range competition by setting σ_c = 0.5 without changing the cooperation or dispersal kernels.

Figure 5.

Short-range competition (σ_c = 0.5 and γ_c = 0.448) reduces the Allee threshold. (a–c) Initial locations of individuals (red) for three different population sizes, N(0) = 80, 160 and 240, respectively. (d–f ) Locations of individuals at t = 30. (g–i) Density of individuals as a function of time. Black solid lines correspond to the averaged results from 1000 identically prepared IBM realizations, red dashed lines correspond to the solutions of the spatial moment dynamics model and green solid lines are the solution of the mean-field model.The cyan lines show the equilibrium densities. (j–l) Pair-correlation function, C(|ξ|, 30). The parameter values are d = 0.4, p = 0.2, m = 0.1, μ_s = 0.4 and σ_s = 0.1. (Online version in colour.)

Figure 5a–c shows the initial randomly placed populations, N(0) = 80, 160 and 240, respectively. Results in the left-most column with $Z_1(0)=80/400<{\overline{Z}}_1^{(2)}$ lead to extinction. Results in the central column with $Z_1(0)=160/400<{\overline{Z}}_1^{(2)}$ are very interesting because the initial density is below the classical mean-field Allee threshold and so standard models would predict extinction, yet we see that the population grows to reach a positive carrying capacity density. This difference is caused by the spatial structure, which we can see in figure 5k is segregated at short distances. When competition is short-range and the population is segregated, individuals in the IBM experience less competition than would be expected in a population without spatial structure under mean-field conditions. This decrease in competition means that the population increases despite the initial density being less than the mean-field Allee threshold. These observations highlight the interplay between the competition and dispersal ranges. In figure 6, we present results when both competition and dispersal are short-range, and we demonstrate that those results are entirely different from those in figure 5. We observe a discrepancy between the IBM and the spatial moment model results for the transient dynamics, approximately between t = 10 and t = 20, in figure 5h. We attribute this discrepancy to the accuracy of the moment closure approximation since similar results are observed in lattice-based models [54].

Figure 6.

Short-range competition and short-range dispersal drive the population to extinction. In this case we consider short-range competition and dispersal (σ_c = σ_d = 0.5) with γ_c = 0.488. (a–c) Initial locations of individuals (red) for three different population sizes, N(0) = 80, 240 and 400, respectively. (d–f ) Locations of individuals at t = 30.(g–i) Density of individuals as a function of time. Black solid lines correspond to the averaged results from 1000 identically prepared IBM realizations, red dashed lines correspond to the solutions of spatial moment dynamics model and green solid lines are the solution of the mean-field model. The cyan lines show the equilibrium densities. (j–l) Pair-correlation function, C(|ξ|, 10). The parameter values are d = 0.4, p = 0.2, m = 0.1, μ_s = 0.4 and σ_s = 0.1. (Online version in colour.)

Results in the right-most column in figure 5 show that when the initial density is above the mean-field Allee threshold, $Z_1(0)>{\overline{Z}}_1^{(2)}$, we see that the population increases to reach the same carrying capacity density is in figure 5h. Here we find that the carrying capacity density reached by the IBM is much higher than the mean-field carrying capacity density. This means that the classical mean-field model underpredicts the long-time density. By contrast, the spatial moment dynamics model leads to an accurate prediction of the averaged density from the IBM. This result, that the spatial structure can impact the carrying capacity density, is consistent with previous observations for the spatial logistic model [29], but our observation that spatial structure changes the Allee threshold has not been reported previously.

(b). Short-range competition and dispersal encourage population extinction

Results in figure 6 consider both short-range competition and short-range dispersal by setting σ_c = σ_d = 0.5. The format of the results in figure 6 is the same as in figures 4 and 5. An additional set of results in section 8 of the electronic supplementary material present some cases where we consider just short-range dispersal.

IBM simulations in figure 6 show that short-range dispersal and competition leads to clustering and extinction. When the dispersal is short-range, daughter individuals are placed in close proximity to the parent individual, which leads to the formation of clusters. Short-range competition means that the competition within those clusters is strong, and significantly increases the local death rate of individuals. In the classical mean-field model we expect extinction to occur only when $Z_1(0)<{\overline{Z}}_1^{(2)}$ but our IBM results show that the population goes extinct when we set $Z_1(0)>{\overline{Z}}_1^{(2)}$ in figure 6h, and even more surprisingly, we see that the IBM simulations lead to extinction even when $Z_1(0)>{\overline{Z}}_1^{(3)}$ in figure 6i. This means that the spatial structure in this case leads the population to extinction. While the mean-field model completely fails to predict the observed extinction in the IBM, we find that the spatial moment model accurately reproduces the population dynamics of the IBM.

(c). Short-range cooperation and intermediate-range dispersal promotes population growth

We now consider a further case where IBM simulations are qualitatively different from the classical mean-field model. Results in figure 7 correspond to short-range cooperation (σ_p = 0.5) and intermediate-range dispersal (σ_d = 2.0). Some clustering in figure 7d–f is evident, and this clustering is induced by the intermediate-range dispersal and leads to enhanced proliferation because of strong short-range cooperation. This case is very interesting because we observe population growth even when the initial density is below the classical mean-field Allee threshold in figure 7g. For all three choices of initial density in figure 7, the population survives and eventually reaches a carrying capacity density that is greater than the carrying capacity density predicted by the classical mean-field model.

(d). Survival probability

We conclude our results by using the IBM to estimate the survival probability, $\mathbb{P}(\text{survival})$, as a function of initial density in figure 8. To calculate $\mathbb{P}(\text{survival})$, we perform a large number of identically prepared realizations over a sufficiently long period of time, t = 100. From these simulations we record the fraction of realizations in which the population does not become extinct in this time interval. Results in figure 8a show $\mathbb{P}(\text{survival})$ for a population with short-range competition. The mean-field Allee threshold for this choice of parameters is ${\overline{Z}}_1^{(2)}=0.43$. By contrast, we find that a certain proportion of populations with $Z_1(0)<{\overline{Z}}_1^{(2)}$ can survive. This difference between the classical mean-field result is due to the presence of spatial structure, and for this choice of parameters we have a segregated population, as illustrated in figure 5. Another factor that influences the survival probability is the stochastic nature of the IBM. Even in the absence of spatial structure, the stochasticity can lead to a continuous transition of $\mathbb{P}(\text{survival})$ from 0 to 1 in the IBM, but possibly quite narrow and centred around the mean-field model Allee threshold. The spatial structure can shift the transition left or right (depending on whether it makes survival more or less likely) and potentially broaden the transition curve. Results in figure 8b show $\mathbb{P}(\text{survival})$ for a population with short-range cooperation and intermediate-range dispersal where the classical mean-field Allee threshold is ${\overline{Z}}_1^{(2)}=0.3$. Again we see that populations with an initial density below the Allee threshold can survive. This difference between the classical mean-field result is due to the presence of spatial structure, which in this case leads to clustering, as shown in figure 7, and the stochasticity of the IBM.

Figure 8.

Survival probability as a function of Z₁(0) for: (a) short-range competition (σ_c = 0.5, γ_c = 0.488), and (b) short-range cooperation and intermediate-range dispersal (σ_p = 0.5, σ_d = 2.0, γ_p = 0.576). Cyan lines show the classical mean-field Allee threshold and black dots show $\mathbb{P}(\text{survival})$ estimated from 100 identically prepared IBM realizations.The other parameter values are d = 0.4, p = 0.2, m = 0.1, μ_s = 0.4 and σ_s = 0.1. (Online version in colour.)

6. Conclusion and outlook

In this study we consider an IBM that describes population dynamics that incorporates short-range interactions and spatial structure. The model construction allows us to incorporate a strong Allee effect, and in particular to explore the impact of spatial structure on the Allee threshold. Classical mathematical models of population dynamics that incorporate an Allee effect are based on making a mean-field approximation. This approximation implies that the population is well mixed, and the probability of finding an individual at a particular location is independent of the locations of other individuals. As a result, interactions between individuals are proportional to N(t)(N(t) − 1) ≈ N²(t). This amounts to the neglect of spatial structure, such as clustering and segregation.

We explore how short-range competition, short-range cooperation and short-range dispersal lead to spatial structure in a dynamic population and we focus on examining how this spatial structure influences the Allee threshold density. Overall, we find that the Allee threshold can be very sensitive to the presence of spatial structure to the point that classical mean-field predictions are invalid. For example, when we consider short-range dispersal and short-range competition, we find that the population becomes extinct, despite the fact that the classical mean-field model predicts that the population will always survive when $Z_1(0)>{\overline{Z}}_1^{(2)}$. While our IBM results disagree with the classical mean-field prediction when the spatial structure is present, we also derive and solve a novel spatial moment dynamics model that is able to accurately capture how the Allee threshold depends upon spatial structure and we find that the spatial moment model reliably predicts population dynamics when spatial structure is present. Our results on the estimation of $\mathbb{P}(\text{survival})$ show that a certain proportion of populations seemed to have non-zero survival probability despite the fact that the initial population density is below the Allee threshold due to the presence of spatial structure. Note that the spatial moment model developed in this study is deterministic and hence cannot be used to estimate $\mathbb{P}(\text{survival})$. Instead, the spatial moment model predicts a threshold initial density, above which the population survives. Overall our results show that the spatial moment model predicts this threshold more accurately than the mean-field model.

There are many potential avenues to extend the features in this study. For example, in this work we make the simplest possible assumption that agent movement is random and density-independent. This feature could be refined. For example, if the model was applied to study a population of biological cells, it might be more appropriate to consider a density-dependent movement rate and some directional bias where individuals are either attracted to or repelled from other agents in their neighbourhood [55,56]. In this study, we restrict our exploration to a simple population where all individuals are of the same type. Another interesting extension of the model would be to consider a multi-species where the total population consists of individuals from various distinct species [14,57]. Our IBM simulation results suggest that the spatial structure and the demographic noise emerging from the stochastic nature of individual birth, mortality and interaction events may reshape the fixed points and the bistability shown by mean-field models of the Allee effect. A formal analysis of the impact of these factors on the stability of the fixed points will be an interesting extension of the present work.

Data accessibility

MATLAB implementations for the IBM and the numerical solution of the spatial moment dynamics model are available on Github.

Authors' contributions

All authors conceived the study. A.S. carried out the stochastic simulations, performed the mathematical derivations and wrote code to solve the governing equations. All authors interpreted the results. A.S. drafted the manuscript. All authors edited the manuscript and approved the final version.

Competing interests

We declare we have no competing interests.

Funding

This study is supported by the Australian Research Council (grant no. DP200100177). M.J.P. is partly supported by Te Pūnaha Matatini, a New Zealand Centre of Research Excellence.