Diffusion term

In the one-dimensional case, the deterministic diffusion equation is (1) where ρ(x, t) is the population density at time t and position x, and D is a constant diffusion coefficient (see e.g. [32]). Assuming the population starts in a localized region of space, the second moment of positions for large t is proportional to time: (2)

In our IBM, the diffusion process is modeled at the individual level via an isotropic random walk on a Cartesian grid [32]. Each grid cell has a fixed number of neighbors: 2 in the one-dimensional and 4 in the two-dimensional case. Each individual moves with probability P_move to a neighboring cell at a given time step. Movements are synchronous and independent within a time step τ, and thus the IBM is a discrete-time model [8].

This is in contrast with the so-called “continuous-time model”, where the timing of individual actions is based on a Poisson process so that only one action occurs at a time [8].

Our model is a binomial process, with success probability P_move, and sample size equal to the total number of individuals N₀. When N₀ P_move → 0 (or equivalently τ → 0), our model approaches the continuous-time process.

The large-scale diffusion coefficient is calculated using the microscopic parameters of a random walk (see also [32]): (3) P_direction is the probability to move to a specific neighbor of the current cell, Δx is equal to the distance between two neighboring cells (the length of the spatial step), and τ is the length of the time step; here Δx = 1 and τ = 1. For isotropic movement, P_direction = 1/n_neighbors.

S1 Appendix predicts that in the limit of decreasing spatial and temporal steps, the model approaches the diffusion Eq (1).

Growth term

A widely used deterministic model for density-dependent population dynamics is the logistic growth equation (4) ρ is the population density, r is the population growth rate, and K is the equilibrium density, the so-called carrying capacity [32] owing to limited resources in a spatial region.

There are many ways to define a stochastic model where the zero-noise limit approaches the logistic model (Eq (4)) [2, 21]. In our IBM, each individual can spawn (give birth to) a new one, or die, with probabilities β and δ, respectively. These reproduction and death behaviors depend on the local population density in the individual’s cell as follows: (5) (6) where b₀ and d₀ are the small-population limits of birth and death probabilities, respectively [16] (see Fig 1). N is the current number of individuals in the given cell. The deterministic population growth rate is r = (b₀ − d₀)/τ.

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Fig 1. The dependency of the birth β (Eq (5)) and the death probability δ (Eq (6)) on normalized local population density N/K.

Two different turnovers are considered: θ₁ and θ₂. When the number of individuals equals the carrying capacity N/K = 1, the equation β = δ = θ is fulfilled. Thus, as shown in the figure, the same carrying capacity can be reached with different turnovers [16].

https://doi.org/10.1371/journal.pone.0176101.g001

An additional parameter is the turnover θ, which is equal to the birth and death probabilities when the local population reaches its carrying capacity (β = δ = θ for N = K). Thus, θ defines the expected fraction of individuals that will be born and die at each time step when the cell’s population is K. In the model, θ is bounded in two directions: d₀ < θ < b₀, with d₀ < b₀. Note that θ only appears in the stochastic formulation and has no equivalent in the classical deterministic model. The same carrying capacity can be associated with different turnovers, and all models with same K, b₀, and d₀ converge to the same continuous model. To summarize, β, δ are interpreted as probabilities in the stochastic model, i.e., the parameters and the resulting growth rate r range from 0 to 1 by appropriate rescaling of the time unit τ.

The workflow during one time step t_i is given in Fig 2. All actions are simultaneous and independent. The synchronous update results in “overlapping generations”, i.e., individuals can survive several updates. The state of the system at time t_i depends only on the state at the previous time point t_i−1. Thus, the model is a Markov process.

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Fig 2. Growth model: Workflow during one time step t_i.

At each time step, the expected number of newborn and of dying individuals is βN and δN, respectively. The expected resulting difference in population size is 〈Δ〉 = (β − δ)N. We note that the linear dependency of β on N leads to β < 0 when N > b₀ K/(b₀ − θ). Values of β < 0 are interpreted here as death rates, such that δ* = |β| + δ for β < 0. In this way the convergence towards the logistic growth is fulfilled (see S2 Appendix).

https://doi.org/10.1371/journal.pone.0176101.g002

The time evolution of the probability P_N(t) to have N individuals at time t is the Master equation: (7)

The probability of transition from the state with N to the state with M individuals is denoted by T(M|N), and Δ ≡ |N − M|. For the classical one-step process only transitions including one individual Δ = 1 are allowed; instead, we allow all transitions 0 ≤ Δ ≤ N. (No more than N individuals can be born, as each individual can give birth at most to one newborn at each time step in our formulation.)

Multiple processes can lead to the same Δ for given N and M. For example, given two individuals N = 2, the transition from N = 2 to M = 3, Δ = 1, is possible by (i) birth of one new individual, or (ii) birth of two new individuals and death of one individual. The transition probability T(3|2) includes both (mutually exclusive) cases: (8) The first term on the right-hand side of Eq (8) describes the probability that case (i) happens. Here, β is the birth probability, (1 − β) is the probability that the second individual does not produce a new individual, (1 − δ)² is the probability that none of the two individuals dies. The first and the second binomial coefficients reflect different combinations due to the birth and death process, respectively. The second term on the right-hand side describes the probability of case (ii) (the general formulation of transition probabilities is given in S2 Appendix).

This Master equation of our IBM approaches the deterministic logistic equation in the continuum limit τ → 0 and K → ∞ (see S2 Appendix).

Growth-diffusion model

The growth-diffusion model tracks each individual as it decides on all three basic actions of birth, death, and movement. It follows that, in the appropriate limit τ → 0, Δx → 0, K → ∞, the IBM of growth-diffusion approaches the Fisher-Kolmogorov equation. In a homogeneous environment, changing K can be seen as a coordinate rescaling since for a fixed population density in individuals/km², a lower numbers of individuals in a cell corresponds to a higher spatial resolution (i.e. a smaller cell area). The frequency of updates, i.e., the inverse of the number of basic actions during one update, as well as the cell step size, i.e., Δx, affect both movement and growth. Our IBM is formally similar to the discrete-time/discrete-space IBM studied in [12]. As in [12], individuals’ movements are modeled as random walks and the zero-noise limit of the growth term approaches the logistic model. However, the detailed stochastic model of growth differs from the model presented here (Eqs (5) and (6)) [12].

The two-dimensional FK equation is (9) where ρ = ρ(x, y, t) is the density at position x and y and at time t [32]. Solutions give traveling waves of population expansion propagating asymptotically at constant speed. The minimal propagation speed is , where the abbreviation “det” stands for “deterministic” [32].

The update protocol of our IBM, i.e., the growth term (Fig 2) “plus” the movement term (section), is equivalent to the Forward Euler integration in that updates are performed according to the system’s state at the beginning of each time step [33]. The limit N → ∞ of 〈N(t + τ)〉 − 〈N(t)〉 calculated with Eq (7) (see also Eq (28) in S2 Appendix) and Eq (22) in S1 Appendix is a Forward Euler integrator of logistic growth and diffusion, respectively. By modeling a real system, the spatial resolution Δx and the temporal resolution τ should be chosen so that the resulting D, b₀, and d₀ can be correctly treated as probabilities. If this is true, the time step satisfies the Von Neumann criterion ([34]) which guarantees that the simulation is stable, i.e., numerical errors will not be amplified.

Diffusion term

We simulate the discrete-time one-dimensional random walk (see section, S1 Appendix), and compare the microscopic diffusion coefficients, i.e., the diffusion coefficient calculated using the microscopic parameters of a random walk such as P_move, P_direction, Δx, and τ (see Eq (3)), with the macroscopic diffusion coefficient, i.e., the diffusion coefficient calculated using large-scale properties of the system such as the variance of positions of all individuals at a given time (see Eq (2)). Simulation parameters and results are given in Table 1. Since individuals move independently, the slope of the variance, which is a linear function of time, is an unbiased estimator of D. The estimated macroscopic diffusivities are within the range of the microscopic coefficients (see Table 1). The macroscopic and microscopic views are in agreement as expected.

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Table 1. Simulation parameters used for the one-dimensional IBM of movement and resulting macroscopic diffusion coefficients.

N₀ = 3000 (i.e. simulations are far from continuous-time regime), Δx = 1, and τ = 1. The corresponding microscopic diffusion coefficients are calculated using Eq (3). To estimate the macroscopic diffusion coefficients we measure the variance in the position of all individuals at different time points. Since the variance of the position is a time series we calculate its time derivative and estimate the mean of the derivative. The macroscopic diffusion coefficients are then estimated using Eq (2).

https://doi.org/10.1371/journal.pone.0176101.t001

Growth term

We focus on parameters b₀, θ, and K. Hereafter, for ease of discussion, the death probability, d₀ = 0.

Asymptotic behavior for increasing K and decreasing θ.

First, we investigate the influence of fluctuations around equilibrium for various choices of K and θ. Simulation parameters are given in Table 2. To estimate the effective carrying capacity 〈N〉, i.e., the number of individuals in the steady state, N is averaged over time in 100 realizations for each parameter set, Fig 3.

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Table 2. Simulation parameters used for the IBM of growth.

d₀ = 0 for all parameter sets. The initial number of individuals for each simulation is N₀ = K. The system is simulated for 500 time steps.

https://doi.org/10.1371/journal.pone.0176101.t002

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Fig 3. IBM of growth: The normalized effective carrying capacity 〈N〉/K depends on K and the turnover θ.

Left panel (a) presents simulation results for b₀ = 0.8 and d₀ = 0. Symbols and colors indicate estimated 〈N〉/K for different turnover values shown in the legend. Lines represent the predicted effective carrying capacity 〈N〉 (Eq (11)) normalized by K. Right panel (b) presents simulation results for b₀ = 0.3 and d₀ = 0.

https://doi.org/10.1371/journal.pone.0176101.g003

The resulting effective carrying capacity 〈N〉 is lower than K.

The lower 〈N〉 owes to the non-linear properties of the stochastic logistic equation: (10) Z is the component of (relative) fluctuations and 〈Z〉 = 0 [21]. Following Renshaw [21], and noting that, for our binomial process, N ∼ K, probability of birth or death ∼θ, we can estimate the effective carrying capacity as (11) Fig 3 predicts that the relative deviation of 〈N〉 from K decreases with increasing K, as expected. Considering Eq (11): for increasing K, the deviation term becomes negligible, and 〈N〉/K → 1 as K → ∞.

If we decrease the turnover θ at constant b₀, the relative deviation of 〈N〉 from K changes. We consider two examples: b₀ = 0.3 and b₀ = 0.8 (Fig 3). The direction of this change depends on both θ and b₀. In the case b₀ = 0.3, a decrease in θ corresponds to a decrease in the fluctuations, and 〈N〉/K → 1 monotonically. On the other hand, when b₀ = 0.8, the behavior of the system is twofold: an initial decrease in θ from 0.7 to 0.4 leads to an increase of stochastic fluctuations, i.e., to a larger deviation of 〈N〉 from K. A further decrease in θ leads to a decrease fluctuations, and 〈N〉/K → 1.

To explain the dependence of the fluctuations on the turnover θ and on b₀, the second term of Eq (11) can be rewritten for the ratio of θ and b₀, α ≡ θ/b₀: (12) Fig 4 illustrates the dependency of f on the ratio α. Here, α varies from 0 to 1, since we are only interested in the biologically-relevant case of competition-driven lowering of the per-capita birth rate with increasing population density (θ < b₀).

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Fig 4. The term describing the deviation of 〈N〉 from K in dependency of the ratio α ≡ θ/b₀ (Eq (12)).

Different values of b₀ are used (see legend for corresponding colors and line styles). As an example, the red arrow shows how the deviation f changes when θ/b₀ = 0.875 and the τ is decreased. The black arrow shows how the deviation changes when θ/b₀ = 0.3 and the τ is decreased.

https://doi.org/10.1371/journal.pone.0176101.g004

We identify two regimes: b₀ ≤ 0.5 and b₀ > 0.5. When b₀ is held constant and b₀ ≤ 0.5, a decrease in the turnover θ, i.e., in the ratio α, always leads to a decrease in the deviation term f (Fig 4). When b₀ > 0.5, the behavior of the system is more complex. With an initial decrease in the turnover, f increases until it reaches its maximum value. With a further decrease of the turnover, f decreases. We may intuitively explain this by realizing that the variance in a binomial process is maximal when its probability of success (in our case θ) is 0.5. Different choices of parameters (b₀) position the system on either side of this maximum, and, as a consequence, can give rise to opposite behaviors of the fluctuations for decreasing θ.

Convergence towards the one-step process.

Here we indicate that by “slowing down” the IBM, i.e., decreasing τ and lowering the expected number of events within one update, the discrete-time model approaches the continuous-time birth-death process. To accomplish this, we change b₀ and θ while keeping their ratio constant. Note that this does not correspond to studying the same model at different time resolutions; rather, we are comparing IBMs with different update frequencies but same continuum limit (up to a rescaling of τ).

We explore the parameter space by spanning two orders of magnitude in θ and b₀. Three different ratios of θ/b₀ are considered; θ/b₀ = 0.875, 0.5, and 0.1. The total number of time steps is inversely proportional to b₀ and θ to fix the total simulation time. Further simulation details and simulation results are given in Fig 5, as well as the comparison with the approximation Eq (11) (three solid lines).

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Fig 5. IBM of growth: The effective carrying capacity 〈N〉 depends on the frequency of system updates.

The time step τ is changed by changing b₀ and θ keeping their ratio constant (d₀ = 0). A decreasing “physical” duration of τ corresponds to increasing values of log(1/b₀). The carrying capacity is K = 50. 50 realizations are performed for each set of parameters as indicated in the image. Shaded areas correspond to 〈N〉 ± SEM estimated in the simulations where SEM is the standard error of the mean. Solid lines correspond to the predicted 〈N〉 (Eq (11)). The red solid line corresponds to the deterministic case 〈N〉 = K.

https://doi.org/10.1371/journal.pone.0176101.g005

When θ/b₀ = 0.5, the effective carrying capacity 〈N〉 does not change as a function of the duration of the time step. Approximation Eq (11) predicts that, if θ = 0.5 ⋅ b₀, the deviation of 〈N〉 from K is constant: 〈N〉 = K − 1/2. In Fig 4, this case corresponds to the intersection of all curves for different values of b₀ at θ/b₀ = 0.5.

When the time step is small and θ → 0, the term describing the fluctuations Z is a Poisson distributed variable with Var(Z) = 2 Kθ that is monotonic in θ. Eq (11) then becomes 〈N〉∼K − 2θ/(2r − r²) which corresponds to the approximation of 〈N〉 in the continuous-time one-step process [21] and agrees with simulation results (Fig 5). In this regime, 〈N〉 becomes independent of τ.

Thus, the properties of the growth IBM around the equilibrium can be can be summarized using Eq (11). By normalizing Eq (11), we get (13) The normalized effective carrying capacity 〈N〉/K is shown as a two-dimensional contour plot in Fig 6. By considering the behavior of IBMs with more and more frequent updates, the trajectory of 〈N〉/K (Fig 6) can be represented by straight lines going towards the origin with slope determined by θ/b₀. For example when θ/b₀ = 0.6/0.8, a decrease of the time step leads to an increase of the noise, i.e., 〈N〉/K declines (trajectory shown by a dashed black arrow).

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Fig 6. Contour plot of 〈N〉/K in dependency on b₀ and the turnover θ.

The contours (colored solid lines) are calculated using Eq (13). The pattern of the dependency of 〈N〉/K on b₀ and θ remains the same for different K, but 〈N〉/K → 1 when K → ∞. Here, color legends are given for K = 10, K = 50, K = 100. The dashed black arrow shows how 〈N〉/K changes when the time step decreases and θ/b₀ = 0.6/0.8. The arrow head points towards the region of smaller time steps. The region θ > b₀ is not allowed.

https://doi.org/10.1371/journal.pone.0176101.g006

Growth-diffusion model

One-dimensional case.

The propagation speed v of the emerging traveling wave is a fundamental macroscopic property of the FK equation. We thus investigate the relation between v and the carrying capacity K, which is the parameter regulating fluctuations in continuous-time reaction-diffusion systems [22–24, 26]. In our simulations, K varies from 10 to 1000 individuals and b₀ = 0.8, 0.3, 0.1, and 0.01; θ = 10⁻¹ ⋅ b₀; d₀ = 0. The diffusion coefficient—as determined by microscopic parameters of the random walk (see section)—is set to D = 0.2. The initial number of individuals is N₀ = K, and their initial position is x₀ = 0. The simulation is stopped when the individuals reach x ≈ 300. Ten repetitions are performed for each parameter set.

These parameters are relevant for ecological dynamics, especially with respect to hunter-gatherer populations. For example, K in the range of 10 to 1000 individuals is consistent with hunter-gatherer group sizes [35]; the mean size of modern hunter-gatherer groups is ∼ 50 individuals, with a maximum of ∼ 270 and minimum of ∼ 13 individuals [36, 37]. The population density of modern hunter-gathers is ∼0.01 − 9.5 individuals/km² [37], so the spatial resolution of our simulations corresponds to 1 km²-100 km². Growth rates in the present simulations range between 0.01 and 0.8, whereas the intrinsic growth rate of foragers is ∼0.02/year [37].

We estimate the speed v by tracking the wave front position defined as x_K/2 where N = K/2. We measure v after the system stabilizes to linear speed (see section).

Fig 7 shows the relation between v_norm = v/v_det (see) and K. For all our parameter sets, the estimated speed v is lower than v_det and depends on K, contrary to the deterministic FK model, but seen previously in stochastic models [22–24, 26].

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Fig 7. IBM of growth-diffusion: The relation between the normalized speed v_norm and K in the one-dimensional case.

Left panel (a) presents v_norm when D = 0.2. Symbols show the measured normalized speed (see legend; θ = 10⁻¹ ⋅ b₀; d₀ = 0). Solid lines show the analytical approximation Eq (15) in respective colors. Right panel (b) presents the effect of D. Lines show the analytical approximation (Eq (15)) for D = 0.2 and D = 0.05 respectively. b₀ = 0.8 in both cases. Typical error bars are less than 1%.

https://doi.org/10.1371/journal.pone.0176101.g007

The lower propagation speed owes to the behavior at the tip of the traveling wave [22–24, 26]. A stochastic population with a lower number of individuals is more likely to go extinct due to fluctuations. At the tip of the traveling wave, N ≪ K, so extinctions of new “colonies” slow the wave front propagation [26].

To summarize, our parameter space exploration reveals tree types of asymptotic behavior toward higher normalized speed v_norm. Increasing K decreases relative fluctuations leading to fewer extinction events. At K ≥ 500, v reaches a plateau at values smaller than v_det: K → ∞ is not sufficient to reach the continuum limit.

Next, the normalized speed increases when growth and movement probabilities are decreased. The dependency of v_norm on r and D can be explained by the leading edge approximation of the traveling wave solution [28, 38]. Although the microscopic update rules of stochastic population-based models and our discrete-time IBM are different, they both have multiple births, deaths, and movements in a single time step.

The dispersion relation derived in [38] can be readily applied to our model, and it is quoted here for convenience: (14) D′ = P_move(1/n_neighbors) is the diffusion probability (Eq (3)). The front of the FK wave travels at the minimal speed where dv/dt = 0 (know as the “pulled” wave type) [28, 32]. This minimal speed approaches v_det when decreasing the growth probability(see Fig 7(a)) or decreasing D′ (see Fig 7(b)).

The approximation v ∼ min v(γ) (Eq (14)) is valid only in the limit K → ∞. However, [28] find an approximation that includes the impact of the carrying capacity K. We quote this result here as well: (15) [28]. γ₀ is the parameter that minimizes the dispersion relation for v. v′′(γ₀) is the second derivative of v with respect to γ at γ₀. The correction term L includes the dependency on ln K.

We compare approximation Eq (15) with v_norm estimated in our model. The approximation is given by solid lines in Fig 7. The simulated v_norm, especially in the limit of increasing K, is in rough agreement with Eq (15) as we expected, given the strong fluctuations regime we are considering, appropriate for ecological systems. Furthermore, approaching the limit of continuous time of the growth term (b₀ < 0.1) and with an increased carrying capacity (K ≥ 500), the estimated speed approaches asymptotically the deterministic speed. This asymptotic convergence to v_det is consistent with previous results on continuous-time FK equation [22–24, 26].

Note that the K → ∞ limit of the reaction-diffusion IBM agrees with a slow-down of the wave due to the Forward Euler time step. In particular, the slow-down is pronounced at high r and D (see also Fig 7 and the approximation Eq (14)).

Two-dimensional case.

We perform simulations on a 1000x100 rectangular grid. The values of K, b₀, d₀, θ are equal to the parameters used for the one-dimensional case, and D = 0.2. The individuals are confined to a 1000 by 100 cells domain. At the beginning of the simulation (t = 0), individuals are arranged in a line parallel to the y-axis, at x₀ = 0. We use this “line setup” due to its comparability with the one-dimensional case with a traveling wave moving in one direction. The initial number of individuals is N₀ = K in each of these grid cells. In accordance to the one-dimensional case, the simulation is stopped when the individuals reach x ≈ 300.

We calculate the y-averaged wave profile along the x-axis, i.e., the mean number of individuals at a given position x, and, in analogy to the one-dimensional case, the speed is determined by estimating x_K/2 as a function of time.

Results are presented in Fig 8.

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Fig 8. IBM of growth-diffusion: The relation between the normalized speed v_norm and the carrying capacity in the two-dimensional case.

Colors and symbols indicate b₀ values shown in the legend. D = 0.2, θ = 10⁻¹ ⋅ b₀, and d₀ = 0.

https://doi.org/10.1371/journal.pone.0176101.g008

As in the one-dimensional case, the speed v is consistently lower than the deterministic speed v_det. Furthermore, v increases with (i) an increase of the carrying capacity K and (ii) a decrease of r: the two asymptotic behaviors discussed for the one-dimensional case can still be identified. Additionally, the speed is higher in the two-dimensional case compared with the one-dimensional case. The difference is especially pronounced for K ≤ 50 and b₀ ≥ 0.3 corresponding to stronger fluctuations in the number of individuals. In the one-dimensional case, the wave is momentarily halted when the population in the farthest nonempty cell at the (zero-dimensional) tip of the wave goes extinct. In the two-dimensional case, the propagation of the wave is slowed down, but is not stopped completely since the probability that extinction happens in all cells at the (one-dimensional) front of the wave along the y-axis is very low. Furthermore, in the two-dimensional case, “extinct” cells at the wave front can be populated by non-extinct neighbors on each side. This effect is enhanced by the higher total number of individuals in the system. Thus, the extinction at the tip of the wave has a more dramatic effect on the one-dimensional domain.

Dispersal into the Americas

Understanding the timing and the dynamics of the colonization of the Americas by modern humans during the late Pleistocene is a challenging problem. Recent genomic evidence shows that all Native Americans diverged from their Siberian ancestor 20 ka (1 ka—“kilo-annum”– represents 1000 years before present) and no earlier than 23 ka [39]. The time of population differentiation supports the hypothesis that people could reach southern South America by 14.6 ka (site Monte Verde), before the Clovis sites in North America were established (around 13 ka) [39–41] (see also Table 3). Archeological sites like Schaefer, Hebior, and Page-Ladson also suggest a human presence in North America by 14.6 ka [42, 43]. Other important archeological sites with radiocarbon dates are given in Table 3. In the context of the timing of the colonization, the opening of ice-free corridors in the Laurentide and Cordilleran ice sheets that covered North America played a crucial role. Namely, the coastal corridor opened around 16 ka and the interior corridor around 14 ka [39].

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Table 3. Archaeological dates and simulated arrival times during the late Pleistocene dispersal in the Americas.

Dates are given in thousand years before present (ka). The second column shows archaeological dates and the corresponding references. The subsequent columns show the difference between the simulated arrival times and the archaeological dates for the used sets of parameters. Here, the earliest archaeological date is used to calculate the difference in case that a range of dates is presented for a site. Positive values of the difference indicate that the individuals arrive at the site earlier than shown by the archaeological data. Negative values indicate that the individuals arrive latter.

https://doi.org/10.1371/journal.pone.0176101.t003

In the following, we explore the properties of the IBM model with a suite of simulations of the colonization of the Americas. The model corresponds to a basic scenario involving as few parameters as possible. First, we test the match between the simulated arrival times and empirical data on the age of archaeological sites. Further, we check if the stochastic fluctuations contribute significantly to the simulation results for the relevant range of parameters. In other words, we investigate if and how the resolution of the stochastic IBM simulation changes the propagation speed and thus the arrival times.

Simulations are performed on a global spherical grid based on a tessellation of a regular icosahedron. Most of the grid’s nodes have 6 neighbors, i.e., spatial elementary units are hexagons. Exceptions on the entire spherical grid are 12 nodes that have only 5 neighbors. (The altitude maps are produced using a global relief model [45].)

As previously mentioned, individuals live on the grid’s cells. Furthermore, on cells, environmental variables such as altitude and ice cover are defined. We use paleo-topography data [46] and update ice cover and see level every 500 years. The movement of individuals is restricted now to the “habitable” cells: individuals do not move into the sea, to cells whose altitude is higher than 2500 m, and to cells covered by ice. In this implementation of the model, the movement probability is a function of both accessible and inaccessible cells, according to the following equation where is the total number of the neighbors and is the number of inaccessible neighbors. This ensures that the local diffusion coefficient, in the direction where movement is possible, is not affected by the anisotropy of the surrounding environment.

We use four different sets of parameters (see Table 4). The last column of Table 4 shows the total number of individuals at the end of each simulation.

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Table 4. Parameter sets used for the simulations of the late Pleistocene dispersal in the Americas.

K is the carrying capacity of every cell. N_tot is the total number of individuals in the Americas at the end of the corresponding simulation. For all parameter sets, the resulting growth rate is 0.06/yr, the diffusion coefficient is D ∼ 180 km²/yr, and the deterministic speed is v_det ∼ 6.57km/yr. Δx is the distance between two neighboring nodes, i.e., the spatial resolution.

https://doi.org/10.1371/journal.pone.0176101.t004

In the reference simulation (set 1), the spatial resolution is Δx ∼ 60 km, leading to a cell area of ∼3700 km². The temporal resolution is τ = 1yr.

The diffusion coefficient, the growth rate, and the carrying capacity (and the resulting speed) are chosen to be consistent with the estimated large-scale parameters of hunter-gatherer groups (see also section) [35–37]. For instance, the growth rate at the half of the carrying capacity is set to 3% per year, (which we assume is also equal to the turnover, giving an average individual lifespan of 33 yr when N = K), what corresponds to the growth-rate of a non-expanding population [35]. The diffusion coefficient is D ∼ 180 km²/yr. The resulting deterministic speed is v_det ∼ 6.57 km/yr. This value is in the middle of the estimated colonization speeds in the Americas during the late Pleistocene, namely 5-8 km/yr [47]. This propagation speed in the Americas is very high compared to other regions, e.g., 0.5 km/yr is the estimated speed of the colonization of Europe by anatomically modern humans [47]. Finally, the carrying capacity is chosen so that the resulting density is around 0.1 individuals per km².

We run a second IBM with different spatial resolution: Δx ∼ 30 km (parameter set 2, Table 4). In a third simulation, we change the time resolution, whereby one simulation step corresponds to 5 years (parameter set 3). Note, that the large-scale parameters r, θ/τ, D, v, and ρ are the same for all three IBMs.

Finally, we consider a setting with a low carrying capacity K = 10, leading to a population density of ρ = 0.003 individuals / km² (parameter set 4). Compared to sets 1-3, the population density changes. Other large-scale parameters stay the same.

The dispersal into the Americas starts 18 ka from one occupied grid cell located in Beringia (latitude: 64° N, longitude: 159° W). The initial number of individuals in this cell is N₀ = K for each set of parameters (Table 4). Thus, the colonization starts from only one group of individuals and there is no prolonged invasion from Beringia that would replace prior populations.

Model simulations are performed with the QHG code [31]. Due to the discrete-time algorithm, simulations can be parallelized, e.g., the QHG code uses shared-memory parallelization [31]. By using 32 threads, the reference simulation runs ∼12 times faster (∼45 min), than on a single thread.

During the simulations, the arrival time is recorded at each cell, i.e., the time at which the cell is colonized for the first time. Archeological dates and simulated arrival times at selected sites are given in Table 3. Fig 9 shows maps of overall arrival times at the time when individuals reach the southern tip of South America (t = 13 ka). Fig 10 predicts how far individuals disperse at t = 13.5 ka, i.e., 2.5 ka after the ice-free corridor opens. (In all our simulations, the corridor opens at 16 ka.)

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Fig 9. Maps of the arrival times.

The snapshots are taken at 13 ka (1 ka—“kilo-annum” represents 1000 years before present). Thus, the latest arrival times correspond to 13 ka (shown in magenta). The earliest arrival times correspond to 16 ka (in light blue). Panels (a), (b), (c), and (d) show simulations with the parameter sets 1, 2, 3, and 4, respectively. Color codings of the arrival times are given in the legend. The white area corresponds to the land area without individuals: ice covered territory, mountains, and the territory that is not yet occupied, e.g., the white area in Patagonia (especially pronounced on panel (d) due to a slower wave).

https://doi.org/10.1371/journal.pone.0176101.g009

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Fig 10. Snapshots of the simulations at t = 13.5 ka (2.5 ka after the ice-free corridor opens).

The left panel shows the simulation with the parameter set 1, the right panel shows the simulation with the parameter set 4. The number of individuals N per cell is shown. Color coding for N is given in the legend. Furthermore, the sea is colored dark blue and the unoccupied territory is colored white. Panel (b) indicates that the wave does not have enough time to reach Patagonia, i.e., a large territory in South America remains unoccupied (white coloring of the territory).

https://doi.org/10.1371/journal.pone.0176101.g010

The simulation with parameter set 1 predicts that individuals need ∼3 ka after the opening of the ice-free corridor to reach Patagonia (Figs 9 and 10, Table 3). In general, the arrival times at particular sites are in the range of archaeological estimates, or individuals arrive earlier (e.g. Paisley Cave, see Table 3). An exception is the Monte Verde site at which individuals arrive ∼1000 years later than predicted. As discussed in the literature, it is difficult to construct a FK-inspired model so that the population reaches Monte Verde on time or earlier [47]. This even holds true when the deterministic propagation speed is very high, e.g., 8 km/yr [47]. To fulfill the archaeological arrival times at Monte Verde with the IBM, one might need to increase the diffusion coefficient or to change the assumptions of the movement term, e.g., by considering coastal migration with Lévy-flight properties [48, 49].

Similar results and timings are obtained when the spatial resolution increases (parameter set 2).

When the time resolution is decreased, e.g., 1 simulation step corresponds to 5 years, the propagation speed slows down due to increased stochastic fluctuations (parameter set 3, see also section). Southern south America is reached ∼400 years later than in the case of a higher time resolution (parameter set 1). Thus, this low time resolution introduces an additional effect to take into consideration when running an IBM. Even a stronger slowdown of the wave can be observed when the carrying capacity is reduced to K = 10 (parameter set 4). In this case, Patagonia is reached ∼700 years later than in the simulation with parameter set 1. Thus, the relative amount of stochastic fluctuations plays a significant role in the simulations’ outcome. Assuming a starting date of 18 ka for the dispersal into the Americas, this corresponds to ∼4% of the total time required to colonize these continents.