Density Dependence in Demography and Dispersal Generates Fluctuating

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POPULATION BIOLOGY understanding of fluctuations in population size generated by erate f (nt (y)) offspring and then die. Second, a fraction p of density dependence in nonspatial models (4). We conjectured these offspring disperse. The probability that a dispersing indi- that density-dependent feedbacks might similarly generate fluc- vidual moves from location y to location x is given by the disper- tuating invasion speeds pursuing the suggestion first made in ref. sal kernel, k(x − y). The remaining fraction (1 − p) remains at 36. Because spread dynamics are jointly governed by demog- their natal location. Concatenating reproduction and dispersal, raphy (local births and deaths) and dispersal (spatial redistri- we have (40–43) bution), we considered several types of density feedbacks (37), ∞ Z including positive density dependence in population growth (i.e., Allee effects) at the low-density invasion front (38) and density- nt+1(x) = (1 − p)f nt (x) + p k(x − y)f nt (y) dy. [1] dependent movement (36, 39). −∞ Our analysis uncovered density-dependent mechanisms that can induce variability in invasion speed, with fluctuations rang- We will assume that f (1) = 1, so that the population has an equi- ing from stable two-point cycles to more complicated aperiodic librium at the carrying capacity nt (x) = 1, and that the tails of dynamics. By showing that simple invasion models can gener- the dispersal kernel k are thin (i.e., go to zero at least exponen- ate complex spread dynamics, our results reveal previously unde- tially fast), so that the probability that an individual disperses an scribed sources of variability in biological invasions and provide extremely large distance is exceedingly small. a roadmap for empirical studies to detect these processes in In general, both the dispersing fraction p and the dispersal ker- nature. nel k may depend on the population density at the natal location as does the reproduction function f . The way that the functions Models and Results f , p, and k depend on population density determine the dynam- We use integrodifference equations (11) to model population ics of Eq. 1. In the simplest case, the reproduction function f growth and spread. These models describe the change in pop- is strictly compensatory: that is, f is an increasing but deceler- 0 00 ulation density [nt (x)] from time t to time t + 1 as the result of ating function of density [f (n) > 0 and f (n) < 0]. For strictly demography and dispersal. First, individuals at location y gen- compensatory models, the population will spread at a constant

Fig. 1. Invasion dynamics under different types of density dependence and dispersal. (A) With compensatory growth at high densities, the wave shape and invasion speed are both constant, which is true with and without low-density Allee effects (AE) (overcompensatory model: σ2 = 0.25, a = 0, and r = 0.9) (SI Appendix, Fig. S1A). (B) With overcompensatory population growth and no Allee effect, population density exhibits ﬂuctuations behind the front, but the leading edge progresses at a constant speed (overcompensatory model: σ2 = 0.25, a = 0, and r = 2.7) (SI Appendix, Fig. S1A). (C) However, when overcompensation combines with low-density Allee effects, the invasion speed ﬂuctuates (overcompensatory model: σ2 = 0.25, a = 0.4, and r = 2.7) (SI Appendix, Fig. S1A). Variability in invasion speed can also occur when Allee effects combine with density dependence in (D) the proportion of dispersing offspring (SI Appendix, Fig. S1 B and C) (propensity model: a = 0.2, λ = 0, nˆ = 0.9, p0 = 0.05, pmax = 1, and α = 50) or (E and F) dispersal distance (SI Appendix, Fig. S1 B and D). In the latter model, dispersal distance (E) decreases with population density (distance model: a = 0.2, λ = 0, nˆ = 0.9, β = −50, 2 2 σ0 = 0.05, and σmax = 1) or (F) increases with density (distance model: parameters as in E except β = 50). Initial population densities are either (A–C) 2 or (D–F) 0.8 times the standard normal probability density truncated at |x| = 5.

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POPULATION BIOLOGY A old density a, individuals produce offspring at the constant per t = 103 capita rate λ. Alternatively, if the population size exceeds a, the population goes to carrying capacity. In the propensity model, population density influences the propensity to disperse (p). In particular, we assume that the proportion of offspring that disperse is given by a logistic function a of local population density [nt (x)] (Eq. 5) with four parameters: the minimum (p0) and maximum (pmax) dispersal proportions; a location parameter nˆ, which is the density at which the dis- 16 17 18 19 20 21 persal propensity is halfway between p0 and pmax; and a shape parameter α. The sign of α determines if the proportion dispersing increases (α > 0) or decreases (α < 0) with density (SI B Appendix, Fig. S1C). The larger the magnitude of α, the steeper t = 104 the density response, which is centered around nˆ. The propensity model can also generate invasions that spread at fluctuating speeds (Fig. 1D and SI Appendix, Fig. S7). We found that these fluctuations persist only when Allee effects are strong (0 ≤ λ < 1), that dispersal propensity increases with population density (α > 0), and that the dispersal response occurs a at a population density that is larger than the Allee threshold (nˆ > a). Fluctuations in speed are nearly always periodic (SI Appendix, Figs. S7C and S8 A–D) and of large amplitude, 16 17 18 19 20 21 altering the invasion speed by ∼100–750% relative to the mean speed (Fig. 2B). These large amplitude periodic fluctuations often include positive and negative speeds, meaning that inva- C sions alternate between steps forward and smaller steps back- t = 105 ward (Fig. 1D). As before, spreading speed fluctuations are created through variations in the dispersing population that pushes the invasion forward from behind the front (Fig. 1D). The magnitude of the push depends on the width of the region contributing dispersing individuals and the proximity of this region to the front (SI a Appendix, Fig. S2 G–L). When density dependence in dispersal is strong and positive (large α), the population directly adjacent to the front is below the Allee effect threshold (a) and therefore, 16 17 18 19 20 21 decays to zero (SI Appendix, Fig. S2 G and H). Farther behind Location (x) the front, density is above a but below the dispersal midpoint (nˆ); thus, this region of the population reproduces but does not D disperse (SI Appendix, Fig. S2 H and I). This action results in a 0.2 large push from behind the wave front that moves the invasion forward at the next time step when the nondispersing population eventually disperses (SI Appendix, Fig. S2 I–K). Subsequently, 0.15 the region of the nondispersing population is much smaller and farther from the invasion front at the next time step, resulting in a much smaller push (SI Appendix, Fig. S2K). With the distance model, we explore a second type of density- wave speed 0.1 dependent dispersal, where density alters the dispersal distance. 95 97 99 101 103 105 Here, all offspring disperse (P = 1), but density alters the variance (σ2) of the dispersal kernel (Eq. 6). Four parameters con- iteration 2 2 trol this dependence: σ0 and σmax, which are the lower and upper bounds of the variance; the location parameter nˆ, which is Fig. 3. Population density before [n(x); black curves] and after [f(n(x)); blue 2 the density at which dispersal variance is halfway between σ0 and curves] growth (overcompensatory model with Allee effects) (Eq. 3) at (A– 2 C) sequential time steps. Gray regions represent locations that go extinct σmax; and a shape parameter β. The dispersal variance increases because of Allee effects [light gray; n(x) < a)], and the solid points show with population density when β is positive and decreases with the edges of the wave. (D) The wave speed over time corresponds to A–C. density when β is negative. The larger the absolute value of β, Parameter values include r = 2.2,a = 0.4, and σ2 = 0.25. the sharper the response (SI Appendix, Fig. S1D). The distance model also produces the necessary spatiotemporal variability in population density behind the invasion front fluctuating invasion speeds when combined with Allee effects. to induce fluctuating invasion speeds (Fig. 1 E and F and SI Density-dependent dispersal, manifest either as density depen- Appendix, Fig. S7). As in the propensity model, the invasion dence in the propensity to disperse (p) or in the shape of the speed only fluctuates when Allee effects are strong (0 ≤ λ ≤ 1). dispersal kernel (k), can generate this high-density variability in However, unlike the propensity model, we find that persistent the pushing force as well. We show this result with two mod- fluctuations are possible when density-dependent dispersal is els (the “propensity model” and the “distance model,” respec- both positive (β > 0) and negative (β < 0) (Fig. 2C). The speed tively) (Materials and Methods), both built on a piecewise linear fluctuations are more frequently aperiodic (SI Appendix, Fig. growth function that is compensatory at high population den- S8 E–H) than the two-cycle fluctuations seen in the propensity (SI Appendix, Fig. S1B). We continue to include low-density sity model, with the largest amplitude when dispersal distance Allee effects. When the population size falls below the thresh- increases with density (β > 0) (Fig. 2C and SI Appendix, Fig.

4 of 6 | www.pnas.org/cgi/doi/10.1073/pnas.1618744114 Sullivan et al. 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(26–28) sion given stochasticity spread demographic (29–34) heterogeneity of and environmental dynamics of influences ecological pervasive often the the how affect and whether processes of these questions open (20–25). vari- remain studies they invasion highly cre- empirical However, the in can seen with speeds dependence consistent spreading able density dynamics, invasion intrinsic complex that ate forecasting. suggests invasion improve work may Our invasions biological of speed that invasions. conditions variable the for satisfy necessary nonetheless as identify may the we inconsis- study, way, superficially our this although with In (36), 57). tent Morris (16, and espe- biased Dwyer densities, sex by is study low movement at the abilities when Allee cially mate-finding Biolog- an which reducing (30). to effect, by models contribute Allee effect can predator–prey implicit movement other an density-dependent in to ically, occur rise to give known fact, is explicit in their in an may, (a dynamics predator–prey model without density that conjecture but prey We effect. movement) on Allee density-dependent depends of a distance with occur type dispersal can Working speeds predator (36). fluctuating Morris that when found and they Dwyer model, result of two-species this that Interestingly, with speeds. contrasts fluctuating of ingredient essary si h vropnaoymdl h itnemvdb iprigindi- dispersing [σ by density moved of distance independent the is model, viduals overcompensatory the in As i.S1 C Fig. , Appendix (SI dispersal density-dependent [n density population S1A): Fig. Appendix, (SI density high at overcompensation of possibility Model. Overcompensatory Cauchy). or normal (i.e., choice kernel to robust are results Qualitative hr 0 where Model. Propensity [σ (P model disperse this offspring in all density and of independent is Dispersal hruhyacutn o h ore fvraiiyi the in variability of sources the for accounting Thoroughly ≤ a < i.S1B Fig. Appendix, (SI 1 k (x p(n) f ee eue iercntn oe o growth: for model constant linear a used we Here, = (n) − y t (x = ; σ f ]vaalgsi omsmlrt te oeswith models other to similar form logistic a via )] ( p 2    = n) ) 0 ecmielwdniyAleefcswt the with effects Allee low-density combine We n 0 = + = exp(r √ 1).    1 2σ 1 λn for 1 + (1 2 2 (n) exp − exp .Dseslpoest eed nthe on depends propensity Dispersal ). σ p )for n)) 2 for = max : [   − σ n − n − 2 α(n ≥ s constant]. a , for < p a, a 0 2(x NSEryEdition Early PNAS n n − (58): ) > ≤ σ − n)] ˆ 2 2 a, (n) a. y ) 2 . = hyaluethe use all They 1.   . σ 2 constant], a , | f6 of 5 [5] [2] [3] [4]

POPULATION BIOLOGY Distance Model. For this model, we use the reproduction function (Eq. 4) but Table S1 has a list of parameters and definitions. Code to run these mod- assume that all offspring disperse (P = 1) after a dispersal distribution with els and recreate all figures is available from the Dryad Digital Repository: variance that is a logistic function of parental density [nt (x)] (SI Appendix, dx.doi.org/10.5061/dryad.69sq3. Fig. S1D); that is, ( ) σ2 − σ2 ACKNOWLEDGMENTS. We thank J. Pruszenski, E. Strombom, R. Williams, σ2(n) = σ2 + max 0 . [6] 0 1 + exp [ − β(n − nˆ)] and two anonymous reviewers for comments and support. The initial idea was developed during the 2014 ACKME Nantucket Mathematical We simulated each model for 200 iterations across a domain of length of Ecology retreat with input from participants and funding from the Woods 1,200 with 216 + 1 spatial nodes. Within each simulation, we defined the Hole Oceanographic Institute Sea Grant. The University of Minnesota (UMN) location of the invasion front at each time step as the location where the Minnesota Supercomputing Institute provided resources that contributed density of the invasion wave first exceeded a density threshold of 0.05. We to the research results reported within this paper (www.msi.umn.edu). The paper was funded in part by the Commonwealth Center for Humanities and then used this location to calculate (i) the instantaneous invasion speed Society, University of Louisville. L.L.S. and A.K.S. were supported by startup (i.e., the distance traveled by the front between consecutive time steps), funds from the UMN (to A.K.S.), B.L. was supported by National Science (ii) the mean invasion speed averaged over the last 50 time steps, and (iii) Foundation (NSF) Grant DMS-1515875, T.E.X.M. was supported by NSF Grant the amplitude of invasion speed fluctuations (the difference between the DEB-1501814, and M.G.N. was supported by NSF Grants DEB-1257545 and maximum and minimum speeds over the last 20 time steps). SI Appendix, DEB-1145017.

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