Interspecific competition slows range expansion and shapes range boundaries

Significance

Range expansion is how invasive species spread and how species track habitats shifting from climate change, so understanding this process is a key applied and basic challenge. Ecologists have long theorized that a competitor can slow or even halt range expansion. However, there has never been definitive empirical evidence of interspecific competition affecting expansion dynamics, and forecasts of spread generally do not consider competitive interactions. We demonstrate experimentally that competition between species can slow range expansion across multiple generations. We also show that competition and density-dependent dispersal continuously alter the shape of the expanding range boundary such that the shape changes over time. Therefore, accurate forecasts of range expansion must account for interactions between competing species.

Abstract

Species expanding into new habitats as a result of climate change or human introductions will frequently encounter resident competitors. Theoretical models suggest that such interspecific competition can alter the speed of expansion and the shape of expanding range boundaries. However, competitive interactions are rarely considered when forecasting the success or speed of expansion, in part because there has been no direct experimental evidence that competition affects either expansion speed or boundary shape. Here we demonstrate that interspecific competition alters both expansion speed and range boundary shape. Using a two-species experimental system of the flour beetles Tribolium castaneum and Tribolium confusum, we show that interspecific competition dramatically slows expansion across a landscape over multiple generations. Using a parameterized stochastic model of expansion, we find that this slowdown can persist over the long term. We also find that the shape of the moving range boundary changes continuously over many generations of expansion, first steepening and then becoming shallower, due to the competitive effect of the resident and density-dependent dispersal of the invader. This dynamic boundary shape suggests that current forecasting approaches assuming a constant shape could be misleading. More broadly, our results demonstrate that interactions between competing species can play a large role during range expansions and thus should be included in models and studies that monitor, forecast, or manage expansions in natural systems.

Biological invasions and climate-induced range shifts are drastically altering the structure and functioning of ecological communities worldwide (1–5). Both processes are examples of range expansion into new areas, a fundamental ecological phenomenon driven by population growth and dispersal (6). In nature, range expansion almost invariably involves species moving into areas occupied by competing species (7, 8). Interspecific competition affects both population growth (9, 10) and dispersal (11), so logically should influence range expansion and range boundary dynamics. However, most studies do not consider competitive interactions when forecasting the speed or dynamics of range expansions (12–14). The lack of attention given to competitive interactions may explain recent failures to predict the success and expansion speed of invasive species (15, 16). Similarly, competitive interactions could determine whether species can keep pace with a shifting environment induced by climate change (17). Whether the accuracy of such forecasts could be improved with data and models of competitive interactions remains an open question in ecology (6, 15–19). Theoretical models of range expansion show that interspecific competition can slow expansion (20) and change the shape of range boundaries (21), but empirical examples of expansion being affected by competition are largely limited to correlative studies (22) and short-term transplant (23) or competitor-removal experiments (10). Direct experimental evidence that competition can significantly affect the spatial dynamics of range expansion in natural systems over both short or long time scales is essentially absent. Strong evidence would provide compelling justification to include competitive interactions in predictive models.

One of the best known and most comprehensive studies of range expansion with a competitor examined the invasion of the gray squirrel into the habitat of the red squirrel in Britain (24). The expansion speed of the gray squirrel from 1960 to 1981 could be predicted by a modified version of the Fisher wave equation (25). Interestingly, while the two-species population model used in the study included a term for the competitive effect of the resident red squirrel on the invading gray squirrel, the best match between the data and the model occurred when the gray squirrel was assumed to be entirely unaffected by interspecific competition. Subsequent research in this system also suggests that the range contraction of the resident red squirrel may have been driven not by direct competition with the gray squirrel but by a pathogen (26), adding further ambiguity to the role of interspecific competition on range dynamics. Finally, this observational study represents only a single replicate of range expansion with a competitor, making it problematic to infer mechanism from an inherently variable process (27). Resolving uncertainty about the impacts of interspecific competition on range expansion requires an approach capable of: 1) examining range expansion in systems where competition is sufficiently strong and 2) replicating the expansion process to account for stochastic variation. Both can be achieved with a manipulative laboratory experiment.

We examined the expansion of the red flour beetle, Tribolium castaneum, across replicated, artificial landscapes, consisting of discrete habitat patches coupled by dispersal, in which an interspecific competitor (Tribolium confusum) was either absent (“no competition”) or present (“competition”). Park’s classic nonspatial experiments with the same species demonstrated that interspecific competition negatively impacted population growth of both species and eventually led to extinction by competitive exclusion within single patches (28–30). Accurate models of Tribolium population growth with and without competitors have since been developed (31, 32), and it is therefore an ideal system with which to combine experiments and modeling to more clearly understand how range expansion is affected by moderate to strong interspecific competition.

Experimental landscapes were one-dimensional arrays, consisting of 16 discrete patches (plastic boxes) connected by small holes drilled into the sides, and filled with standard flour medium (SI Appendix, Fig. S1). All landscapes were started by allowing 50 adult T. castaneum to mate and lay eggs in patch 1 of each landscape. For the competition treatments only, 50 adults of the competitor beetle, T. confusum, laid eggs in each of patches 9–16. We then documented abundance across space of both species over eight nonoverlapping generations; each 6-wk generation consisting of discrete growth and then dispersal (24 h) phases. We established 15 replicates for each of the two landscape treatments and censused all beetles in every patch once per generation. (One replicate [competition] was removed mid-way through the study due to a laboratory mishap and is not included in the analysis.) In addition to the landscapes, to observe nonspatial dynamics and to obtain data for parameterizing a model of population growth, we established single-patch treatments initiated with the eggs of 50 T. castaneum, 50 T. confusum, or 50 each of both species, measuring abundances over the same schedule.

This system has been used previously to study range expansion in the absence of interspecific competitors (27) and is well-suited to examining expansion involving multiple species. First, there is a long history of studying competition between T. castaneum and T. confusum, so much is already known about how they interact, which is predominantly through egg cannibalism (28–30, 33). Second, experiments are conducted under tightly controlled conditions to minimize the influence of confounding extrinsic environmental factors on competitive interactions, growth, and dispersal. Third, by enforcing nonoverlapping generations with discrete growth and dispersal phases, we are able to mimic simple but common seasonal lifecycles. Finally, laboratory experiments provide data with high spatiotemporal resolution for multiple replicates, enabling a comprehensive view of the possible effects of interspecific competition on expansion speed and range boundaries.

Results and Discussion

Over eight generations, we observed clear and consistent differences in the mean abundances of the focal beetle (T. castaneum) between landscapes with and without an interspecific competitor (Fig. 1). During generations 1 through 3, mean abundances across space were similar between treatments. This was expected since expanding populations in the landscapes with a competitor typically did not encounter heterospecifics before generation 3, and so initial expansion occurred in the absence of interspecific competitors regardless of treatment. Following contact with competitor populations, mean abundances of T. castaneum in its expanding range boundary were lower than in the same patches of competitor-free landscapes. By generation 8, mean abundances in patches 5–16 of landscapes with a competitor were significantly lower than in landscapes without a competitor (bootstrap 95% confidence intervals did not overlap; compare Fig. 1 A and B).

Fig. 1.

Mean abundances during range expansion of T. castaneum in (A) no competition and (B) competition landscapes over eight generations, showing that competition slowed expansion and steepened the expanding range boundary. All landscapes were started with 50 T. castaneum in patch 1 (generation 0; not shown). In addition, for the competition landscapes, 50 T. confusum were initially added to each of patches 9–16. Solid squares connected by lines indicate the nonzero mean abundances of T. castaneum (y axis) in patches (x axis) for successive generations (red to violet, generation 1 through 8; see legend). Shaded areas are bootstrapped 95% confidence intervals for generation 8. The mean abundances of the competitor, T. confusum, are also shown in B with open triangles connected by dotted lines. T. castaneum spread from left to right, while T. confusum spread from right to left. Means are of 15 replicates. Data from individual replicates are shown in SI Appendix, Figs. S5 and S6.

The lower mean abundances of T. castaneum across the more distant patches in the competition treatments clearly demonstrates that range expansion can be slowed by interspecific competition. The slowdown is directly attributable to competition between T. castaneum and T. confusum for two reasons. First, the only difference between experimental treatments was the presence of the competitor. Second, consistent with prior studies of Tribolium, we found that in single-patch replicates, T. castaneum had lower mean abundances when paired with T. confusum compared to monocultures (SI Appendix, Fig. S2). Based on prior studies (28–30, 33) and our fitted model (Materials and Methods), the reduction in T. castaneum abundance arises from interspecific cannibalism by the high numbers of resident T. confusum. We note that while such interspecific competition was strong enough to reduce abundances in single-patch replicates, we found competitive exclusion occurred in only a small number of single-patch replicates after eight generations (T. castaneum went extinct in 7% of two-species patches), indicating that interspecific competition was moderate but nevertheless sufficient to slow expansion into occupied habitat.

Having demonstrated that interspecific competition can slow range expansion over the short term, we used simulations of a model parameterized from experimental data to examine the effects of interspecific competition over longer time periods. Specifically, we examined the impact of competition on expansion speed and the shape of the expanding range boundary over 100 generations. To parameterize the spatially explicit, two-species model, we first separately fitted a range of submodels for population growth and dispersal (Materials and Methods) using training data on population growth from only the single-patch replicates and data on dispersal from only the first two generations of the experimental landscapes. These models account for multiple forms of stochasticity known to affect both growth and dispersal processes in the Tribolium system (27, 32). We selected the submodels with the lowest Akaike information criterion (AIC) score for each of growth and dispersal. The best fitting population growth submodels included interspecific cannibalism for both species and an Allee effect for T. confusum. The best fitting dispersal submodels included a form of density-dependent dispersal. These submodels described the training data well (SI Appendix, Figs. S3 and S4).

To validate the fitted model, we compared simulations of the model to the full eight generations of range expansion in the experimental landscapes. The parameterized model predicted the expansion dynamics of both species in the experimental landscapes well over the eight generations of the experiment given the strong overlap between the 95% prediction intervals and the data (Fig. 2 A and B; generations 1–7 shown in SI Appendix, Figs. S5 and S6). Since the fitted model described well the observed dynamics over multiple generations, we simulated this model for 100 generations in larger 200-patch landscapes (n = 10,000) to generate longer-term realizations of range dynamics.

Fig. 2.

Predictions of our spatiotemporal model (stochastic growth + dispersal) compared to data from experimental landscapes after eight generations. Predictions are forecasts from the initial abundance at generation 0 for (A) no competition and (B) competition landscapes. Replicates are shown as connected lines. The spatiotemporal model—shown as a polygon representing the 95% prediction intervals—was parameterized independently with data from single-patch replicates and, for the dispersal component of the model, only the first and second generation of expansion. Data and predictions for both T. castaneum (black) and T. confusum (brown) are shown.

According to the parameterized model, the expansion of T. castaneum would continue to be slowed by the presence of T. confusum (Fig. 3A). For the no competition simulations, by generation 100, mean abundance of T. castaneum was high (>100) in patches 1–174. In contrast, for the competition simulations, mean abundance of T. castaneum was >100 in patches 1–100 only. Combined with our experimental data, these results show that the presence of the competitor T. confusum greatly decreases the expansion speed over most abundance thresholds of the invasion wave of T. castaneum. Interestingly, in both the experiment and the long-term simulations, expansion speed was little affected at the lowest abundance thresholds, the very tip of the range boundary that included the furthest forward individuals (SI Appendix, Fig. S12). We show below that this more sustained expansion speed at the tip of the range boundary was due to density-dependent dispersal.

Fig. 3.

Predicted mean abundances (>0) in landscapes over 100 generations for (A) the fitted model and (B) the fitted model but with the density-dependent dispersal parameter, $G_n$, of the invader, T. castaneum set to zero. Dashed green lines show the expansion of T. castaneum in the absence of a competitor. Solid gray lines show the expansion of T. castaneum in the presence of competitor T. confusum, and dotted brown lines show the contraction of T. confusum. Lines are drawn every 10 generations beginning at generation 10 (leftmost) and ending in a bold line at generation 100. For each treatment, expansion was simulated using the stochastic spatiotemporal model parameterized with experimental data. Lines are means of 10,000 simulations.

This variation in expansion speeds at different abundance thresholds means that competition altered the shape of the T. castaneum range boundary. Furthermore, we found range boundary shape to be dynamic with a long transient period. Whereas the shape of the moving range boundary was largely constant when T. castaneum expanded into unoccupied territory (Figs. 1A and 3A, dashed green lines), the boundary clearly changed shape over time when T. castaneum expanded into occupied territory (Figs. 1B and 3A, solid gray lines). At first, the range boundary steepened and shortened as the two species met in both the experiment and simulations (Figs. 1B and 3A), but in the simulations it then steadily became shallower and elongated over time (Fig. 3A and SI Appendix, Fig. S11). The simulations also predicted a gradual range contraction for the competitor T. confusum in the long term (Fig. 3A, dotted brown lines), with 66% of replicates containing no T. confusum in patches 1–50 at generation 100. Crucially, both the contraction in the range of T. confusum and the shape change from steep to shallow in the expanding range boundary of T. castaneum were not observable in the short-term experimental data alone (Fig. 1). Thus, initial dynamics of range boundaries between interacting species may not predict their long-term dynamics.

By investigating the effect of different model parameters (Materials and Methods), we found that the shape of the range boundary is determined largely by an interplay between interspecific competition, especially the competitive effect of the resident, and the dependence of invader dispersal on the combined density of both species (Figs. 3 and 4 and SI Appendix, Fig. S9). Without density-dependent dispersal, competition slows expansion and elongates the range boundary, making it shallower (Fig. 3B), but competition combined with density-dependent dispersal in the invader dramatically elongates the range boundary (Fig. 3A and SI Appendix, Fig. S9). In effect, competition from the resident resists the moving abundance mass of the invader, but density-dependent dispersal in the invader has a spearheading effect that drives individuals forward to form a long, low-abundance tip to the range boundary (Fig. 3A and SI Appendix, Fig. S9, column 3). Sensitivity analysis shows that the resident competitive effect, ${\alpha }_{nm}$, and invader density-dependent dispersal, $G_n$, are the major influences on the shape and also shows that the interplay of interspecific interactions changes over time (Fig. 4): early on, range boundary shape is most sensitive to the shallowing influence of invader dispersal, but over time the shape becomes more sensitive to the competitive effect of the resident, which first steepens the range boundary but then elongates it. The competitive effect of the invader on the resident, ${\alpha }_{mn}$, while less potent, counteracts the competitive resistance of the resident and makes the range boundary steeper (Fig. 4 and SI Appendix, Fig. S9, column 1). Density-dependent dispersal in the resident, $G_m$, slows the invader but has little influence on its range boundary shape, slightly steepening it (Fig. 4 and SI Appendix, Fig. S9, column 4). These changing influences of interspecific interactions cause the shape of a species’ range boundary to change steadily over many generations—at least the first 50 generations in the case of our Tribolium model (SI Appendix, Fig. S11).

Fig. 4.

Sensitivity of the wave shape in competitive landscapes to different model parameters over time. Sensitivity is the percent change in the exponent, $b$, describing the shape of the front of the expanding wave per percent change in competition parameters $a_{mn}$ (effect of invader on resident), $a_{nm}$ (effect of resident on invader), and density-dependent dispersal parameters $G_n$ (invader) and $G_m$ (resident). Wave shape was estimated from the expected values of simulations of the fitted models at 95–105% of the fitted parameter values. Negative values on the y axis indicate the wave becomes shallower for an increase in the parameter, while positive values indicate the wave becomes steeper for an increase in the parameter.

The shape of a moving range boundary is important for several reasons. First, boundary shape determines how a species expands into new areas and can therefore inform management decisions. For instance, when moving range boundaries have long tails (e.g., locations with many individuals are far from locations with few individuals), managers could expect longer lags between when a species first appears and when it becomes abundant in an area. Second, traveling wave solutions such as the Fisher wave speed are derived from asymptotic behavior when expanding ranges have constant shape or profile; that is, when all moving parts of the range travel at the same speed. Third, when all parts of a moving range travel at the same speed, the speed of the furthest occupied location (i.e., abundance ≥ 1) can be used as a proxy for the speed of other abundance thresholds. The latter two points are especially relevant because the Fisher wave speed is frequently used to predict expansion speeds (16), and high-resolution abundance data for expanding species are often limited. Our findings from simulations of the parameterized model, showing that long-term expansion into occupied habitat can change the shape of a moving range boundary over many generations, suggest that relying on the asymptotic behavior of models or on occupancy data alone could be misleading when expansion occurs in the presence of competing species.

We examined expansion in one dimension, which approximates some natural situations reasonably well, such as spread along a river or coastline. However, expansion in nature will usually be in two or even three dimensions. Mathematically, the analysis of multidimensional expansion is much more difficult than the one-dimensional case (34), a difficulty that is compounded by features common to natural systems such as complicated initial range geometry, heterogeneous environments, and heterogeneous distributions of competitors. Beyond the case of approximately one-dimensional natural systems, we could expect analogous behavior to our one-dimensional system if expansion is along a planar front. If an invasion starts from a small propagule and expands radially, a heuristic analysis (34) shows an initial increase in the rate of expansion for a single species. The situation for two species would be more complex, but since initially the invading species would be rare, it should still show an initial increase in the expansion rate. More complex behavior is certainly possible due to instabilities along fronts, heterogeneities, and other complications, but a study of these cases is beyond the scope of our current work even in the purely theoretical realm.

The results of our microcosm experiment should be taken as a demonstration of what is possible when species expand into landscapes occupied by competitors. Importantly, we chose species and conditions where the effects of competition were moderately strong. Our species, T. castaneum and T. confusum, are closely related and exhibit similar resource requirements and general life history. In more natural systems, species invading new habitats can be more distantly related to resident species, with less niche overlap and weaker competition, as was apparently the case for gray and red squirrels in Britain (24). Nevertheless, niche overlap is generally high among plants (35), and many plant species are experiencing range shifts (14). Our findings demonstrate that if there is at least moderate competition between species, range expansion will be slowed. We also show that it is possible to accurately predict range expansion using a stochastic spatiotemporal model and that such models can reveal long-term effects not easily observed over the shorter time scales of most observational and experimental studies of range dynamics.

Conclusion

We demonstrated that interspecific competition can affect range expansion in two important ways. First, we found that the expansion of a focal species through a landscape can be slowed by the presence of an interspecific competitor across multiple generations. This finding was consistent across many replicates. In line with theory, this suggests range shifts caused by climate change or human introductions can be slowed by interspecific competition, increasing the chance of extinction or control. Therefore, effective monitoring and management of expanding species could be improved by considering competitive interactions, either as a factor in predicting future expansion, or as a form of biocontrol, as is already done in some plant systems (36). Our findings concerning the shape of the range boundary are also important. In the absence of competition and density-dependent dispersal, the shape of the expanding range boundary is approximately constant across short and long time scales. However, when an expanding species meets a competitor, interspecific effects on population growth and density-dependent dispersal cause the shape of the range boundary to change both qualitatively and quantitatively over many generations. Thus, approaches that assume a constant shape for the moving range boundary, such as the Fisher wave speed, will not be accurate for many ecological systems. Rather, as we describe here, models parameterized from data on the key processes of population growth and dispersal may be important for accurately forecasting range expansions over time scales relevant for ecological decision making.

Materials and Methods

Experimental Landscapes.

We examined the spread of the flour beetle Tribolium castaneum (Herbst, 1797) across artificial “landscapes” with and without the competitor Tribolium confusum (du Val, 1863). Landscapes were composed of 16 plastic boxes (dimensions: 4 × 4 x 6 cm; herein “patches”), each containing 30-mL mixtures of 95% organic wheat flour and 5% brewer’s yeast, and arranged into a single row of patches (i.e., a one-dimensional linear array). Patches were held together by elastic bands, and each patch had one 2-mm hole drilled into two opposing sides to allow for beetles to move between patches at specified dispersal times (patches 1 and 16 did not have holes on the outside side). During the nondispersal period of the beetle life cycle (see below), these holes were blocked by pieces of acetate. A partial view of an experimental landscape is shown in SI Appendix, Fig. S1.

We established 15 replicate landscapes for each of two biotic treatments (“no competition” and “competition”). No competition landscapes were founded by adding to patch 1 exactly 50 T. castaneum beetles from large, long-running stock cultures (5,000–10,000 individuals) raised on flour-yeast mixtures in temperature- and humidity-controlled incubators. All other patches in these landscapes did not initially contain beetles. The competition landscapes were founded by adding 50 T. castaneum to patch 1 and 50 T. confusum to each of patches 9 through 16. All adults added of both species were 41 d old, and, except during censuses, all landscapes were kept in temperature- and humidity-controlled incubators (29.6 °C and 60–70% relative humidity).

We also established 20 single-patch replicates for each combination of the two beetle species: T. castaneum alone (50 adults), T. confusum alone (50 adults), and T. castaneum + T. confusum (50 adults of each species). Landscapes and single-patch replicates were kept in the same incubator for the duration of the experiment.

Once beetles were added to patches, we enforced a 6-wk life cycle, which mimics seasonal or semelparous organisms. The life cycle was as follows: 1) Reproduction: Adult beetles were given 24 h (± 10 min) to copulate and lay eggs in their current patch; 2) Removal of adults: Adult beetles were removed from their patch using a sieve. The flour mixture and any beetle eggs were returned to the patch; 3) Development: Patches were left undisturbed for 40 d for beetles to hatch into larvae and for the larvae to grow into adults; 4) Dispersal: Holes connecting adjacent patches were unblocked (i.e., the strip of acetate was removed), and beetles were allowed to disperse freely for 24 h (± 10 min); 5) Sieving and flour replacement: Holes connecting patches were blocked, and beetles were sieved from each patch. Adult beetles were separated by species, censused, and then returned to the same patch number of a newly made landscape containing fresh flour mixture.

Censuses during step 5 were performed as follows: When there were 75 or fewer adults of the same species in a patch, all individuals of that species were counted and, as an error check, weighed (as a group) using a microbalance. This procedure has an error less than 1% (32). When there were more than 75 adults in a patch, we weighed 50 adults, then all adults, and used the two measurements to estimate the total number of beetles in the patch. Previous work has shown that this technique has an error of 1–2%, which is less than the error associated with counting large numbers of beetles (32).

Due to the significant time and labor required to fully process landscapes/patches (starting/stopping dispersal, sieving and counting adults, replacing flour), replicates were separated in time across the 6-wk life cycle. Three landscapes and four single-patch replicates of each treatment were processed every week for 5 wk every cycle. Stock populations were similarly kept on this staggered schedule so that the initial beetles were always exactly 41 d old. While all our stock populations were highly inbred (originating from a small number of founders and maintained in B.A.M.'s laboratory for several years prior), a staggered schedule could lead to small genetic differences between sets of replicates. To ameliorate this possibility, during the 6-mo preparation for the experiment, each generation we held over 5% of beetles from previous weeks and mixed them with stock populations in other weeks. We assumed that such gene flow minimized genetic differences between stock populations prior to the initiation of the experiment.

To control for temperature and humidity variation within incubators, every week we removed all landscapes and single-patch controls from the incubators (including those not being processed that week) and then added them back to the incubators in random locations (obtained using a random number generator). This ensured that any additional environmental differences between landscapes (due to their position within incubators) were completely uncorrelated with treatment, replicate, or week number.

The experiment began on December 20, 2016, and ended on December 26, 2017, lasting for eight full generations of the life cycle described above. Due to a laboratory mishap, we did not collect a generation of data for one competition landscape. Data from all other generations of this replicate are included in our analyses. Additionally, laboratory mishaps led to the destruction of one competition landscape prior to the end of the experiment and to the removal of two single-patch replicates for T. castaneum (alone) and three single-patch replicates containing both species.

Estimating Parameters for Population Growth.

To investigate the processes determining population growth within a patch we contrasted the fit of a range of growth models to data from the single-patch replicates. The models were a family of stochastic Ricker models first derived for T. castaneum in monoculture (32) and later extended to the two-species case (37). We further extend these models by adding multiple forms of Allee effects (i.e., positive density-dependence). We fully derive these extensions in SI Appendix and briefly describe the basic structure of the two-species models below.

The deterministic analog to our stochastic models of the growth of population $N$ (T. castaneum) is given by:

$$N_{t+1}=N_t{R}_n{e}^{-\left({\alpha }_{nn}{N}_t+{\alpha }_{nm}{M}_t\right)}$$

where $M_t$ is the population size of the interspecific competitor (T. confusum) at time $t$, $a_{nn}$ is the search rate of T. castaneum for T. castaneum eggs, and ${\alpha }_{nm}$ is the interspecific search rate of T. confusum for T. castaneum eggs. The equation for population $M$ (T. confusum) has the same form:

$$M_{t+1}=M_t{R}_m{e}^{-\left({\alpha }_{mm}{M}_t+{\alpha }_{mn}{N}_t\right)}$$

where $R_m$ is the density-independent growth rate of species $M$, ${\alpha }_{mm}$ is the intraspecific search rate, and ${\alpha }_{mn}$ is the interspecific search rate.

We incorporate stochasticity by treating the underlying individual-based, demographic processes as random variables. The basic stochastic model (“Poisson Ricker model”) assumes that both per capita offspring production and density-dependent survival are stochastic, describing them with Poisson and binomial random variables, respectively. This is a model of demographic stochasticity. The other models include this demographic stochasticity and incorporate different combinations of three additional sources of random variation: stochastic sex determination (i.e., some beetles are female, and only females give birth), demographic heterogeneity (i.e., random differences in $R$ between individuals in a population), and environmental heterogeneity (i.e., random differences in $R$ across space or time). Each model has a different probability mass function given in SI Appendix, Table S1.

Estimating the Dispersal Parameters.

For each species, we fit two models of dispersal to data from the first generation of the landscape replicates. In addition, since such data were limited, we considered second generation data from landscape replicates that either did not disperse or that experienced dispersal <3 individuals, representing a very low probability of establishing outside the initial patch. We did not consider data beyond generation 2 to ensure that data from subsequent generations could be used to validate the fitted model.

The first dispersal model was a stochastic diffusion model, previously derived in (27) to describe the dispersal of T. castaneum in one-dimensional landscapes over the dispersal period. The model is a continuous-time stochastic model (“Diffusion”) with deterministic analog:

$$\frac{d{N}_x}{dt}=D\left(N_{x-1}-H{N}_x+N_{x+1}\right)$$

where $N_x$ is the number of individuals of a species in patch $x$, and where $x-1$ is to the left of patch $x$, $x+1$ is to the right of patch $x$, $D$ is the diffusion coefficient, and $H$ is the number of holes in a patch (in our 16 patch landscapes, patches 2–15 have two holes, while patches 1 and 16 have only 1). The dispersal kernel is obtained by solving the above equation for initial conditions representing the number of beetles of the same species in different patches before dispersal. This produces a vector of the expected number of individuals in each patch after dispersal across a landscape. The vector of expected values (one value for each patch) is then converted to success probabilities, $p_x=\frac{N_x}{{\displaystyle \sum N_i}}$, of a multinomial distribution. An alternative approach that ultimately leads to the same probabilities is to consider the dispersal of individuals from one patch to another as a homogeneous Poisson process (i.e., exponential waiting times between dispersal events) and to solve the associated Kolmorogov forward equation to obtain the occupancy probabilities of the different patches.

While this model has been shown to provide a reasonable description of T. castaneum dispersal in experimental landscapes (27), it is largely incapable of producing a pattern commonly observed in our landscapes; that is, higher abundances after dispersal outside of the starting patch (starting patch = patch 1; SI Appendix, Fig. S4). In a simple deterministic diffusion model, this pattern cannot occur (i.e., starting patches will always contain more individuals), and even when dispersal is stochastic, such a pattern is highly improbable (SI Appendix, Fig. S4).

Dispersal in our system could be density-dependent; however, a dependence of dispersal rate on densities that change dynamically during dispersal is not capable of producing the “hump” seen in patch 2 of the experimental data (SI Appendix, Fig. S4). In contrast, the “conditioning” of flour medium (i.e., reduction of patch and food quality) by high densities of beetles during the population growth phase is known to affect beetle behavior (38) and has the potential to lead to larger numbers of beetles in nonstarting patches. Thus, we tested an additional continuous-time dispersal model (“Fouling”) with the following deterministic analog:

$$\frac{d{N}_x}{dt}=D_{x-1}{N}_{x-1}-H{D}_x{N}_x+D_{x+1}{N}_{x+1}$$

where $D_x$ is the diffusion coefficient associated with patch $x$. To allow prior densities (i.e., densities before dispersal) to affect diffusion, we set $D_x=D{e}^{G{N}_x\left(0\right)}$ where $D$ is the intrinsic diffusion coefficient and $G$ is a “fouling” coefficient representing the per capita effect of density, $N_x\left(0\right)$, in patch $x$ at the start of the dispersal period. In this model, when $G$ is positive, beetles are more likely to disperse away from patches that contained high densities. Since $D_x$ depends on the number of beetles immediately prior to dispersal, the density-dependent factor does not change dynamically during dispersal, and thus we can incorporate stochasticity in dispersal in the same way as for model 1. For simulating the two-species model (see below), we assumed that the dispersal response to fouling depended on the combined density of both species.

Optimization and Model Selection.

Details on optimization and model selection (growth and dispersal) are provided in SI Appendix.

Validating the Fitted Two-Species Model of Range Expansion.

To validate the fitted model, we compared simulations of the model with spatiotemporal data from the experimental landscapes. As described above, parameters of the top scoring stochastic growth submodel (negative binomial-binomial-demographic [NBBd] for T. castaneum, negative binomial-binomial-environment-Allee [NBBe Allee] for T. confusum) were estimated using only data from the single-patch controls, and thus the estimates are entirely independent of our landscape data. Similarly, the parameters of the dispersal submodel were estimated using data from only the first and part of the second generation of spread, and thus the estimates are entirely independent of the landscape data from generation 3 on.

We simulated 10,000 realizations of the model over eight generations using the parameter estimates in SI Appendix, Tables S2 and S3. Initial conditions were the same as the initial setup of the laboratory experiment (50 T. castaneum adults in patch 1 and, where relevant, 50 T. confusum in patches 9–16). Simulations represent forecasts of the model out to generation 8 starting from initial conditions.

Quantifying Uncertainty in the Estimates of the Growth and Dispersal Parameters.

We used nonparametric bootstrapping to quantify uncertainty in the parameter estimates (39). To do this, we optimized the best fit growth model (NBBd for T. castaneum and NBBe Allee for T. confusum) to 5,000 samples (with replacement) of the relevant growth data (single-patch replicates). Optimization converged for 99% of samples. The marginal 95% confidence intervals were obtained from percentiles of the bootstrap samples. SI Appendix, Fig. S7 shows these intervals as well as the point estimates reported earlier and used in the long-term simulations.

Similarly, we optimized the best fit dispersal model to 5,000 samples of the dispersal data (where each landscape was a replicate) and obtained 95% confidence intervals from the bootstrap samples (SI Appendix, Fig. S8). Comparing SI Appendix, Figs. S7 and S8, it is clear that uncertainty in the dispersal parameters was generally larger than for the growth parameters.

Long-Term Simulations.

To examine how competition affected range expansion over the long term, we simulated the top-ranked spatiotemporal model using the parameter estimates in SI Appendix, Tables S2 and S3. We simulated no competition and competition treatments in larger 200-patch landscapes with the same starting conditions as in our experiment (i.e., 50 T. castaneum in patch 1 and, for competition landscapes only, 50 T. confusum in all patches >8). For each treatment, we simulated the model 10,000 times for 100 generations.

Effect of Interspecific Parameters on Range Boundary Shape.

To examine how interactions between competitors affected the shape of the range boundary, we simulated the spatiotemporal model, varying one interspecific parameter at a time while holding the other parameters constant at their fitted values. For each parameter combination, we simulated the model 10,000 times and calculated the expected values of the stochastic model as the mean abundances in each patch across simulations. To plot the dynamics of the model we varied the interspecific competition parameters ($a_{nm}$, $a_{mn}$) and density-dependent dispersal parameters (here separate for each species: $G_n$, $G_m$) over a range from 0 to 125% of their estimated values.

To conduct a sensitivity analysis, we varied these parameters from 95 to 105% of their fitted values in 1% increments. As a univariate summary of the range boundary shape, we used the exponent, $b$, of the exponential model

$$N_x=a{e}^{-bx}$$

where $N_x$ is the abundance in patch $x$, $a$ is an estimated nuisance parameter that scales the leftmost abundance, and large values of $b$ indicate a steep range boundary. Using nonlinear least squares, we fitted this model to the calculated expected abundances in the tail of the range boundary, where the tail was taken to be the spatial region at the wave front defined by abundances below a threshold of 20. This model provided an excellent approximating summary of the range boundary shape allowing us to track the change in the shape over time (SI Appendix, Fig. S10). We further reduced Monte Carlo simulation noise in the estimates of $b$ by smoothing the resulting curve of shape versus generation (SI Appendix, Fig. S11). We calculated the sensitivity of the shape as the slope of the linear regression of the estimated exponents (standardized by the exponent when the model parameter was at 100%) against the parameter percentages. Thus, sensitivity is the percent change in the shape per percent change in the parameter.

Code Availability.

All data and code necessary for reproducing these analyses and figures have been deposited in Dryad (https://doi.org/10.5061/dryad.f4qrfj6sp) (40) and are available at GitHub, https://github.com/legault/Range_expansion_competition.