Abstract
Density dependence plays an important role in population regulation, and has a long history in ecology as a mechanism that can induce local density fluctuations. Yet much less is known about how these endogenous processes affect spatial population dynamics. Biological invasions occur through the combined action of population growth (demography), and movement (dispersal), making them relevant for understanding how density dependence regulates spatial spread. While classical ecological theory suggests that many invasions move at a constant speed, empirical work is illuminating the highly variable nature of biological invasions, which can lead to non-constant spreading speeds. Here, we explore endogenous density dependence as a mechanism for inducing variability in biological invasions. We constructed a set of integrodifference population models that incorporate classic population fluctuation mechanisms to determine how density dependence in demography, including Allee effects, and in dispersal affects the speed of biological invasions. We show that density dependence is a key factor in producing fluctuations in spreading speed when Allee effects are acting on population densities that fluctuate locally. We show that the necessary density fluctuations can arise from either a nonmonotone population growth function where densities fluctuate locally (e.g., overcompensatory population growth), or from density-dependent dispersal when the population growth function results in constant local densities. As density dependence in both demography and dispersal are common, this mechanism of variability may influence many invading organisms.
Significance Statement Controlling the spread of biological invasions reduces the cost of mitigating invasive species. However, predicting empirical invasive population spread is difficult as evidence shows the speed of this movement can be highly variable. Here, we provide a novel mechanism for this variability, showing that internal population dynamics can lead to fluctuations in the speed of biological invasions through the combined action of density dependence in demography and dispersal. Speed fluctuations occur through the creation of a variable pushed invasion wave, that moves forward not from small populations at the invasion front, but instead from larger, more established populations that “jump” forward past the previous invasion edge. Variability in the strength of the push generates fluctuating invasion speeds.
Introduction
Fluctuations in population size have fueled a now-classic debate over whether populations are governed by extrinsic environmental factors or by intrinsic self-limitation (reviewed in Kingsland 1995). One of the most important advances of twentieth-century ecology was the discovery that intrinsic density feedbacks can cause population densities to fluctuate, even in constant environments (May, 1974; Turchin, 2003; Costantino et al., 1997). This discovery helped resolve the important role of density dependence in population regulation, revealing that strong regulating forces can generate dynamics that are superficially consistent with no regulation at all. The long history and textbook status of fluctuations in local population size contrast strongly with relatively poor understanding of fluctuations in the spatial dimension of population growth: spread across landscapes.
Understanding and predicting the dynamics of population spread take on urgency in the era of human-mediated biological invasions and range shifts in response to climate change. The velocity of spread, or “invasion speed”, is a key summary statistic of an expanding population and an important tool for ecological forecasting (Fagan et al., 2002). Estimates of invasion speed are often derived from regression methods that describe change in spatial extent with respect to time (Miller and Tenhumberg, 2010; Lubina and Levin, 1988; Andow et al., 1990; van den Bosch et al., 1992). Implicit in this approach is the assumption that the true spreading speed is constant and deviations from it represent “error” in the underlying process, or in human observation of the process. This assumption is reinforced by long-standing theoretical predictions that, under a wide range of conditions, a spreading population will asymptotically reach a constant velocity that is determined by the “pulling” force of rare long-distance movement and rapid population growth at the low-density leading edge (Weinberger, 1982; Skellam, 1951; Kot et al., 1996; Neubert and Caswell, 2000). This conventional wisdom of long-term constant invasion speeds is still widely applied, and support for this result is found in lab experiments under controlled conditions (Gandhi et al., 2016).
In contrast to classic theoretical and empirical approaches that emphasize a longterm constant speed, there is growing recognition that spread dynamics can be highly variable and idiosyncratic (Melbourne and Hastings, 2009; Miller and Inouye, 2013; Johnson et al., 2006; Peltonen et al., 2002; Walter et al., 2015; Robertson et al., 2009; Michaels, 1984; Chen, 2014). Some models predict fluctuating spreading speeds due to extrinsic factors such as environmental heterogeneity (Neubert and Caswell, 2000) or interactions with other species (Dwyer and Morris, 2006). Indeed, empirical studies of invasive organisms often attribute temporal variation in speed to differences in the environments encountered by the invading population (e.g., Andow et al. 1990; Peltonen et al. 2002). An alternative hypothesis is that endogenous mechanisms generate fluctuations in spreading speed, even in a homogeneous landscape, mirroring the potential for endogenous fluctuations in local population size in temporally constant environments. Endogenous fluctuations in spreading speed (which we define here as any variability in spreading speed through time, ranging from two-cycle oscillations to chaos), have been surprisingly neglected by the large theoretical literature on biological invasion (but see Johnson et al. 2006, Shaw et al. unpublished) and would be easily missed by empirical studies that were not looking for them. Understanding whether such fluctuations are possible and the conditions under which they occur would help resolve sources of variability in invasion speed, and facilitate management objectives for range expansion by native and exotic species.
Here, we develop mathematical models of spatial spread to ask whether the velocity of an expanding population can fluctuate, even in a spatially and temporally uniform environment, and to identify conditions under which such endogenous fluctuations may occur. As a starting point, we take inspiration from the relatively complete understanding of endogenous fluctuations in local population density, which arise from time-lagged density feedbacks (i.e., populations persistently overshoot and undershoot their carrying capacity). We conjectured that density feedbacks should be similarly important for fluctuating invasion speeds. Because spread dynamics are jointly governed by demography (local births and deaths) and dispersal (spatial redistribution), we considered several types of density feedbacks (Sakai et al., 2001), including density-dependent movement (Matthysen, 2005) and positive density dependence in population growth (i.e., Allee effects) at the low-density invasion front (Taylor and Hastings, 2005). Allee effects cause invasion waves to be “pushed” from behind their leading edge (Kot et al., 1996; Wang et al., 2002) and we show them to be an important ingredient of fluctuations in the speed of spatial spread.
We began by asking whether conditions that promote fluctuations in local density also promote fluctuations in spatial spread. We then asked whether fluctuations in invasion speed are possible even when population growth produces constant local population densities. We discovered several mechanisms, all arising from density dependence in demography and/or dispersal, that can induce endogenous fluctuations in invasion speed, ranging from stable two-point cycles to apparent chaos. By demonstrating that simple invasion models can generate complex spread dynamics, our results reveal previously undescribed sources of variability in biological invasions and provide a roadmap for empirical studies to detect these processes in nature.
The Models
We start with an integrodifference population model for population spread through a spatially uniform environment (Kot et al., 1996):
Here nt(x) is the population density at time t and location x, and is a function of two sequential processes: local demography and dispersal. We assume non-overlapping generations where adults nt(x) generate f(nt(x)) offspring, that then disperse. The distribution of dispersal distances (the dispersal kernel) is given by k(x – y, σ2) and is the probability that an individual disperses from location y to location x (where the probability depends only on the distance x – y), with σ2 as the variance of the kernel. For all models, we describe dispersal using a Laplace probability density function (Wang et al., 2002).
Fluctuating Non-spatial Density
We first ask if a growth function f(n) that promotes fluctuations in local density also promotes fluctuations in spreading speed. We therefore consider the case of overcompensatory population growth, where density can overshoot the carrying capacity. Long-standing theory suggests that compensatory population growth, with or without Allee effects, leads to constant invasion speeds (Weinberger, 1982; Lui, 1985; Wang et al., 2002). Additionally, when Allee effects are not present, overcompensatory growth does not give rise to fluctuations in invasion speed (Li et al., 2009). Here, we investigate whether adding an Allee effect to an overcompensatory growth function can induce fluctuating invasion speeds. We define the growth term of equation 1 as the Ricker function which is modified to include the possibility of a strong Allee effects (Fig S1a). Here, r is the intrinsic growth rate, and nthresh is the Allee effect threshold, which is the critical density below which the population goes extinct. We refer to this model as the “overcompensatory model” throughout.
We simulated the model across a range of r and nthresh parameter values (Fig. 1a), each for 200 iterations using MATLAB (MATLAB, 2014). Here, we fixed the variance of the dispersal kernel to σ2 = 0.25. Within each simulation, we defined the invasion front at each time step as the location where the density of the invasion wave was first above the detection threshold of 0.05 (Fig. 2a-e). We then used this location to calculate the instantaneous invasion speed as the distance travelled by the front between consecutive time steps (Fig. 2f). To determine if the invasion speed fluctuated, we quantified how this speed changed through time.
Overcompensatory Model Results
We found that the overcompensatory model can generate fluctuating invasion speeds, but only if Allee effects and fluctuations in local population density are present (Fig. 1a, S2). In populations experiencing Allee effects, the fluctuations in local population density are propagated in space, which promotes fluctuations in invasion speed. This can occur even when the local population would crash in a nonspatial model due to high population density fluctuations. Fluctuations in speed occur within limited parameter space; when the Allee effect threshold (nthresh) is too large the spreading population crashes, and if the growth rate (r) is too small the population invades at a constant speed (Fig. 1a). The observed fluctuations have a small amplitude, and range from 2-cycle oscillations to apparently chaotic (Fig. S3).
In this model, fluctuations in speed are induced via a pushed invasion wave, and the Allee threshold determines the magnitude of fluctuations by altering the size of the push. The invasion front moves forward, not from the low density leading edge, but instead from populations farther back in the wave that jump forward past the invasion front and push the wave ahead. When the population density at any location is smaller than the Allee threshold, which occurs at the edge of the wave, the population decays to zero before the next time step. Populations just above the Allee threshold become large after reproduction, but as the adult population size of n(x) increases beyond the Allee threshold, the offspring population size f(n(x)) declines, as defined by the Ricker growth function (Fig. S1a). Therefore, when reproduction occurs (transition between n(x) and f(n(x)), Fig. 2a-b), the populations with highest density become populations of low density, and populations with density just above the Allee threshold become high density. Through time, this creates variability in the size of the push, or the region contributing to the wave front (Fig. 2b vs d), which leads to fluctuations in the invasion speed.
Constant Non-spatial Density
We next examine a model where the population growth function (f(n)) results in constant local population densities to determine if this case can also produce fluctuating invasion speeds. Here, we consider a version of the basic model (eqn. 1) where the growth function is piecewise linear, with the form where nk is the carrying capacity density, and nthresh is the critical density above which the population reaches its max density (Fig. S1b). As before, nthresh is the Allee effect threshold; below this point population growth is less than one when strong Allee effects are present. However, we now include a parameter λ that describes the strength of the Allee effect. For λ = nk/nthresh there is no Allee effect. For 0 ≤ λ ≤ 1 there is a strong Allee effect where population size decreases at low density (Fig. S1b). For 1 < λ < nk/nthresh there are weak Allee effects, but we only briefly touch on these results, as they do not produce fluctuations in speed.
Here we explore density dependence in two aspects of dispersal: the propensity (the fraction of individuals that disperse), and the distances that dispersing individuals travel. We again use the general integrodifference form for our invasion model (eqn. 1), but incorporate a piecewise linear growth function (eqn. 3).
When dispersal propensity is density-dependent we let the probability of dispersal be given by which is a logistic form similar to other models with density-dependent dispersal (Smith et al., 2008). Here, ξt(x) = ϵnt(x) + (1 – ϵ)f(nt(x)) is a weighted combination of the local density of adults nt(x), and offspring f(nt(x), where ϵ is the relative weighting of these two densities. The dispersal propensity can depend on only the adult population density (ϵ = 1), on only the offspring density (ϵ = 0), or on some combination of both (0 < ϵ < 1). In this formulation, is the dispersal threshold, α is a shape parameter that controls the steepness at the threshold, and p0 and pmax are lower and upper bounds on the propensity, respectively (Fig. S1c). The sign of a determines if dispersal increases or decreases with density, indicating positive or negative density dependence, respectively. From this, we get the integrodifference model which we refer to as the “propensity model” throughout.
Alternatively, when the dispersal distance is density-dependent, we let the dispersal kernel variance be modified by the weighted population density (ϵ) and have the form where β is a shape parameter that controls the steepness at the threshold, and and are lower and upper bounds on the variance, respectively (Fig. S1d). Here, the sign of β determines if density dependence is positive or negative. This results in our final integrodiffernce model which we refer to as the “distance model” throughout.
We simulated both of these models (eqns. 5 and 7) for 200 iterations, across parameter values ϵ = [0,0.5,1], 0 ≤ λ ≤ 2 (strong to weak Allee effects), and λ = 5 (no Allee effect), and nthresh = 0.2 for both models, with −100 ≤ α ≤ 100, po = 0.05, and pmax = 1 for the propensity model, and −100 ≤ β ≤ 100, , and for the distance model. As before, we calculated the instantaneous invasion speed to determine if it fluctuated through time.
Propensity Model Results
With the propensity model (eqn. 5), speed fluctuations occur only when the propensity to disperse depends at least in part on the adult population density (ϵ > 0), Allee effects are present (0 ≤ λ < 1), and dispersal propensity increases with increasing population density (α > 0). We did not find evidence of fluctuations with weak Allee effects (1 < λ < nk/nthresh) (Fig. S4a). The amplitude of these fluctuations increase as the strength of the Allee effect, and density dependence increases (Fig. 1b). Fluctuations in this model always occur as 2-cycle oscillations, and are stronger when it takes a larger density to trigger movement (increasing ) (Fig. S4a). Alternatively, when dispersal propensity decreases with increasing denisty (α < 0), the invasion moves at a constant speed.
Here, density-dependent dispersal induces the density fluctuations needed to create speed fluctuations, but only when combined with Allee effects, which generates a pushed wave (SI Appendix). As before, spreading speed fluctuations are created through variations in the magnitude of the push that reaches the edge of the invasion. The magnitude of a push depends on the width of the region contributing to the push, and the proximity of this region in relation to the wave front (Fig. 2g-k). Directly adjacent to the wave edge the population is below the Allee effect threshold (nthresh) and therefore decays to zero (Fig. 2g). Farther from the edge, the population density is above the Allee effect threshold, but below the dispersal threshold . Thus this region of the population behind the wave front reproduces, but does not disperse (Fig. 2h,i). This results in a large push from behind the wave front that moves the invasion front forward at the next time step (Fig. 2i-k). Subsequently, the region of the non-dispersing population is much smaller and farther from the invasion front at the next time step, resulting in a much smaller push (Fig. 2k).
Distance Model Results
For the distance model (eqn. 7), we again find that invasion speed only fluctuates via pushed waves when density-dependent dispersal depends at least partially on the adult population density (ϵ > 0), and strong Allee effects are present (0 ≤ λ ≤ 1). However, unlike the propensity model, we find fluctuations when density-dependent dispersal is both positive (β > 0) and negative (β < 0) (Fig. 1c, S4b, SI Appendix). The speed fluctuations exhibit more chaotic dynamics (Fig. S5) than the two-cycle fluctuations seen in the propensity model, with larger amplitude fluctuations when the dispersal distance increases with density (β > 0), than when it decreases with density (β < 0) (Fig. 1c, S5). In general, fluctuations are larger as both Allee effects and density dependence are stronger. Additionally, when increasing densities increase the dispersal distance (β > 0), the fluctuation amplitude increases as it takes a larger density to trigger long distance movement, and this trend is opposite when increasing densities decrease dispersal distance (β < 0) (Fig. S4b).
When the dispersal distance shows positive density dependence (Fig. 2m-r), populations at densities above the dispersal threshold will disperse long distances, and those below will disperse short distances. While the short distance dispersers are always directly adjacent to the wave edge after reproduction, each push forward is made up of a combination of both short and long distance dispersers. The size of this push changes depending on the proportion of the push made up of each type of disperser, and this proportion changes with time, creating fluctuating invasion speeds. For example, when a small peak in population density is above the dispersal threshold, a small population mass disperses long distances and the front advances a short distance (Fig. 2m-o). However, when a larger peak of the population density is above the dispersal threshold, a larger population disperses long distances, and the wave advances a longer distance (Fig. 2o-q).
We also find spreading speed fluctuations when the dispersal distance has negative density dependence, that also result from variation in the proportion of the population that disperse short and long distances. Here, however, when populations are above the dispersal threshold, they disperse a short distance, and when they are below the dispersal threshold, they disperse long distances. In the negative density-dependent dispersal case, long distance disperses are always adjacent to the wave edge, and pushes with a small proportion of long distance dispersers move less far (Fig. 2s-u) than those made up of more long distance dispersers (Fig. 2u-w).
Discussion
Here we demonstrate that fluctuations in invasion speed can be induced solely through endogenous population dynamics, a previously undescribed mechanism behind invasion variability. Specifically, we show that Allee effects acting on fluctuating local population densities are necessary to create these variable invasion speeds. This occurs when the population growth function produces both fluctuating and constant local population densities. In the former case, overcompensatory growth produces the necessary fluctuating local population densities, in the latter case these density fluctuations are created through density-dependent dispersal. In our models, fluctuations in spreading speed occur through a form of variable pushed wave. While pushed waves are common (Gandhi et al., 2016; Mendez et al., 2011; van Saarloos, 2003; Garnier et al., 2012), especially given the known influence of Allee effects at the low density wave edge (Taylor and Hastings, 2005; Shigesada and Kawasaki, 1997), our models show that this pushing force generated by the Allee effect can lead to endogenous variability in spreading speed when accompanied by mechanisms that create density-dependence, as this combined action creates variability in the pushing force through time. This result has potential to be more consistent with the highly variable data seen from empirical invasion studies (Johnson et al., 2006; Peltonen et al., 2002; Walter et al., 2015; Robertson et al., 2009; Michaels, 1984; Chen, 2014) Allee effects are often considered in relation to invasions (Taylor and Hastings, 2005), as the leading edge of an invasion tends to experience low population densities (Shigesada and Kawasaki, 1997), and Allee effects are seen in many organisms (Kramer et al., 2009; Morris, 2002). In isolation Allee effects have been shown to influence small populations at the invasion edge by decreasing low-density vital rates (e.g., reproduction Veit and Lewis 1996), which can lead to decreased invasion speeds (Wang and Kot, 2001). We show here that Allee effects can also generate fluctuations in spreading speed, but they must be acting on populations that also have some form of local density fluctuations. This combination of Allee effects and density fluctuations due to dispersal have been shown to induce oscillations in a serious North American forest invader, the Gypsy Moth (Johnson et al., 2006). In our models, these fluctuations are driven by variable pushed waves. One major new insight our results provide, is that vital rates must be not only examined in populations at low density, but also in those at high-density, as the invasion front is being driven by high-density populations.
In our models, density-dependent dispersal, which is displayed in many organisms (Bonenfant et al., 2009; Matthysen, 2005; Fronhofer et al., 2015; Denno and Peterson, 1995), was a main source of local population density fluctuations. Its effect ranged depending on whether responses to density were positive or negative, and if it altered either the propensity to disperse or the dispersal distance. In the propensity model, fluctuations were seen when the propensity to disperse increased with increasing density. Positive density-dependent dispersal propensity is most notable in insects, as wingless aphids can produce winged morphs when densities become high (Harrison, 1980; Johnson, 1965), and some butterflies increase movement in response to mate density avoidance (Baguette et al., 1998). This movement can create an Allee effect if it reduces mate finding abilities at low densities, especially when the movement is sex biased (Shaw et al. unpublished). In the distance model, fluctuations were seen under both positive and negative density dependence. Mobile organisms can increase their dispersal distance with increasing density by altering behavioral responses (Matthysen, 2005). Alternatively, dispersal distances can decrease with density when crowding decreases reproductive output and dispersal ability (Marchetto et al., 2010; Donohue, 1998; Matthysen, 2005), or in animals (notably small mammals) with strong group behavior (Ims and Andreassen, 2005; Andreassen and Ims, 2001; Matthysen, 2005). The empirical studies on density-dependent dispersal tend to match where we find fluctuating invasion speeds in our models, indicating we have explored relevant parameter space.
Coupling models and empirical data has proven to be a fruitful approach to understanding the mechanisms behind fluctuations in non-spatial population density (e.g., Turchin 2003; Costantino et al. 1997), yet we have much less coupled data in spatial systems (Bolker et al., 2003). We propose that examining highly variable empirical invasion data (Melbourne and Hastings, 2009) in light of our theoretical results could provide a novel mechanism by which variable invasions occur. To identify the density-dependent mechanisms acting on invaders, empirical data on the combination of fluctuation periodicity, amplitude, and the long-term shape of the wave would be necessary. Given the difficulty of collecting long-term data, some patterns might be easier to identify than others. The strong 2-cycle speed fluctuations generated when invaders experience both Allee effects and density-dependent dispersal propensity would likely be the most evident in data. We recognize that while many invaders may experience Allee effects, or density-dependent dispersal, the likelihood of both endogenous processes acting on an invading population simultaneously (which is required to generate speed fluctuations) is unknown. Teasing out the signature of these endogenous mechanisms from data may prove difficult, given an inherently het-erogeneous and stochastic world, yet we encourage empiricists to re-examine variable invasion data in the context of these density-dependent mechanisms.
Understanding the basic mechanisms behind invasion variability would allow for better forecasting, and ultimately improved control, of biological invasions. While fluctuations in invasion speed have been found due to exogenous factors including habitat patchiness, predator-prey dynamics, and climatic variability (Dwyer and Morris, 2006; Neubert et al., 2000; Peltonen et al., 2002), we show here, that internal density-dependent population dynamics can also induce fluctuating invasion speeds. These results provide a new focus for understanding variable invasions.
Supporting Information
Acknowledgements
LLS and AKS were supported by startup funds from the University of Minnesota to AKS, BTL was supported by NSF DMS-1515875. The initial idea was developed during the 2014 ACKME Nantucket Mathematical Ecology retreat with input from participants and funding from Sea Grant. The manuscript was greatly improved by comments from E. Strombom and R. Williams. The authors acknowledge the Minnesota Supercomputing Institute (MSI) at the University of Minnesota for providing resources that contributed to the research results reported within this paper. URL: http://www.msi.umn.edu.
Appendix: Fluctuations in population density
A. The propensity model
We demonstrate that the term outside the integral in model (6), which determines the dynamics of the individuals remaining sedentary, can generate fluctuations in population density for positive α and ϵ. This term may be described by G(n) = [1 – p(ξ(n))]f(n) where p(ξ) is given by (4) with ξt(x) replaced by ξ, and ξ is given by (5) with nt(x) replaced by n. ξ is a function of n. Clearly G(0) = 0, and
It is easily seen that p′(ξ) > 0 for α > 0, and ξ′(n) > 0 for 0 < ϵ ≤ 1 and n ≠ nthresh.
If λ = 0, G(n) equals zero for 0 ≤ n < nthresh, has a jump at nthresh, and then becomes decreasing since G′(n) = — p′(ξ)ξ′(n)nmax < 0 for n > nthresh.
If λ > 0, and G′(n) = − p′(ξ(n))ξ′(n)nmax < 0 for n > nthresh. This shows that G(n) increases initially, and decreases for n > nthresh.
We have shown that for positive α and ϵ and for λ ≥ 0, G(n) is a nonmonotone function and it generates fluctuations in population density.
B. The distance model
We show that the integrand of model (8) produces fluctuations in density for β ≠ 0 and ϵ > 0. We use ξ(x) to denote ξt(x) given by (5) with nt(x) replaced by n(x), and σ2(ξ(x)) to denote σ2(ξt(x)) given by (7) with ξt(x) replaced by ξ(x).ξ(x) can be viewed as a function of n(x). The Laplace kernel k(x – y,σ2(ξ) takes the form of
Let
We use to denote the derivative of with respect to n, and H′(x – y, n) to denote the partial derivatives of H with respect to n. Then
For n(y) > nthresh, f′(n(y)) = 0. We therefore have that for n(y) > nthresh,
Since ϵ > 0, for n ≠ nthresh, . It follows that .
If λ = 0, then H(x – y,n(y)) equals zero for 0 ≤ n(y) < nthresh, has a jump at nthresh, and then decreases in n(y) in the interval when β > 0 or in the interval when β < 0.
If λ > 0, . We have that H(x – y, n(y)) increases in n(y) when n(y) is small, and decreases in n(y) in the interval when β > 0 or in the interval when β < 0.
We have proven that for positive ξ and for λ ≥ 0, H(x — y, n(y)) produces fluctuations in density in the interval when β > 0 or in the interval when β < 0.