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II.3 Pulled waves propagate without locking in DSDT models Not all invasions in DSDT models are locked; see Figs. and . In fact, it can be rigorously demonstrated that [MATH] is a smooth function for pulled waves that are driven by the growth and dispersal at the very tip of the invasion front. Invasions are guaranteed... |
The main result of this calculation is that pulled waves in DSDT models behave as in CSCT models. Specifically, the population density at long times depends only on [MATH] and is described by a continuous function [MATH] with [MATH] given by |
[EQUATION] where [MATH] In the absence of an Allee effect, numerical simulations show excellent agreement with these predictions. The invasion velocities predicted by Eq. ( 12 ) perfectly matched the simulation results for the piecewise-linear, Beverton-Holt, and Hill model (Fig. A). In addition, the observed populatio... |
II.4 Pushed waves: A second type of unlocked expansions in DSDT models To understand the transition from locked to unlocked fronts, we examined how the invasion dynamics change with the strength of an Allee effect. For reaction-diffusion waves in CSCT models, a critical strength of the Allee effect is necessary to tran... |
We then asked whether locked and pulled waves are the only two classes of expansions in DSDT models or whether DSDT models also support unlocked waves that are pushed rather than pulled. To answer this question, we obtained the phase diagram of invasion velocities in the piecewise-linear growth model. Specifically, we ... |
One might wonder whether the existence of pushed waves in Fig. A could be attributed to finite [MATH] and [MATH] in our simulations. Indeed, velocities plateaus smaller than [MATH] and [MATH] are missed by the procedure described above. To exclude the artifacts due to finite resolution, we considered a slice of the pha... |
The transitions between different modes of propagation in DSDT models is more complex than in other models. For CSCT model, the transition between pulled and pushed waves occurs at a specific strength of the Allee effect and is independent of [MATH] . In contrast, the phase boundary depends on both [MATH] and [MATH] in... |
II.5 Response to perturbations To better understand the differences between pulled, pushed, and locked invasions, we examined how they respond to perturbations. In particular, we were interested in how the effect of a small perturbation decays in time and whether this effect vanishes at long times or not. To this end, ... |
We expected that a perturbation of pulled and pushed fronts should lead to a nonzero shift of the front profile. This expectation follows from the following argument. The movement of unlocked fronts is described by a continuous function [MATH] , which is a solution of Eq. ( ). Because [MATH] is continuous and Eq. ( ) i... |
There are also differences in how fronts approach the long-time steady state following a perturbation. Exponential relaxation is expected for locked waves, since each of the discrete profiles is a stable fixed point of the dynamics. In contrast, pulled waves are expected to show a much slower relaxation with perturbati... |
To determine the relaxation to perturbations, we compared the profile of the wave just after perturbation to the asymptotic profile. The asymptotic profile was determined by waiting [MATH] time steps after perturbation and fitting the following [MATH] profiles using a cubic spline. The profiles at each time point after... |
II.6 Locked invasions are robust to fluctuations Since locked waves return to exactly the same steady state following perturbations, they should be robust to demographic and environmental fluctuations. To establish the limits of this robustness, we modified the piecewise-linear growth model and Eq. ( ) to include demog... |
Demographic fluctuations were described by [EQUATION] where [MATH] refers to binomial sampling from [MATH] with probability [MATH] . Temporal fluctuations were modeled by drawing [MATH] from a uniform distribution from [MATH] to [MATH] at each time step in Eq. ( ). Spatial heterogeneity was included by using a differen... |
For all three scenarios, we observed that velocity locking is extremely robust to fluctuations (Fig. 11 ). Although small plateaus are progressively washed out by stronger fluctuations, large plateaus remain clearly visible even when fluctuations are about 25% of the mean. Thus, velocity locking should be readily obser... |
II.7 Velocity locking suppresses front diffusion Given that velocity plateaus remain even in the presence of strong fluctuations, we decided to examine how velocity locking affects stochastic properties of invasion fronts. The consequences of demographic and environmental noise are understood relatively well in CSCT mo... |
We applied this procedure of estimating [MATH] to DSDT models where we defined the position of the front as the sum of the population density normalized by the carrying capacity. Figure 12 A shows that noisy invasions in DSDT models also exhibit diffusive front wandering and, therefore, can be assigned an effective dif... |
The vanishing of [MATH] for locked invasions provides a convenient way to detect velocity locking in situations where one cannot modify the model parameters and confirm the existence of a velocity plateau. Zero diffusion could also be beneficial in technological applications that require coherence or reproducibility. I... |
II.8 Locked waves due to positive density-dependent dispersal Mode locking requires nonlinear dynamics. So far, we focused on the Allee effect as the source of this nonlinearity. In the context of range expansions, density-dependent dispersal could provide an alternative mechanism. To test this hypothesis, we modified ... |
[EQUATION] and used a simple linear dependence of [MATH] on [MATH] [EQUATION] For [MATH] , we used the piecewise-linear model without an Allee effect ( [MATH] ). |
The linear dependence of [MATH] on [MATH] (Eq. 15 ) was observed in some species morisita:linear_diffusion, and has been frequently used in mathematical modeling murray:mathematical_biology, kawasaki:density_dependent_diffusion, . For [MATH] , the dispersal is highest at the edge of the front, and the invasions are pul... |
Indeed, the transition from pulled to pushed expansions has been recently demonstrated in a CSCT model with density-dependent dispersal kawasaki:density_dependent_diffusion, , which can be specified by the following partial differential equation |
[EQUATION] For logistic growth and linear [MATH] kawasaki:density_dependent_diffusion, obtained exact expressions for the velocity and front shape of pushed waves: |
[EQUATION] This formula is applicable for [MATH] . For lower values of [MATH] , the expansion is pulled, and its velocity is given by [MATH] . We note that the results in kawasaki:density_dependent_diffusion, are based on a general mapping between equations that determine expansion velocity in models with density-depen... |
In DSDT model, increasing the ratio of [MATH] also shows a clear transition form pulled to pushed and locked expansions. For pulled waves, [MATH] is smooth while expansions at higher [MATH] exhibit velocity plateaus (Fig. 13 A). The phase diagram in the [MATH] space is qualitative similar to that in [MATH] space, but p... |
Density-dependent dispersal has been described in many species, and it can also be easily engineered in microbes dispersal:review, hwa:engineered_dispersal, dispersal:sheep, . Therefore, nonlinearities in dispersal could provide an alternative route to velocity locking in populations of living organisms. |
II.9 Evolution in locked invasions Given that small changes in model parameters do not affect the velocities of locked fronts, we were interested to determine whether velocity locking affects species evolution. Specifically, we asked whether velocity locking eliminates the selection for higher dispersal or growth, whic... |
[EQUATION] where [MATH] denotes the intermediate step between [MATH] and [MATH] , and [MATH] is the piecewise-linear growth map. The second equation implements growth in density while preserving the the ratio of resident and mutant from the migration step. |
We found that mutants with different dispersal rate can take over the population even though the mutant and the resident have identical invasion velocities (Fig. 14 ). Similar to previous findings for CSCT models korolev:arrest, , selection for both faster and slower dispersal was possible depending on the strength of ... |
II.10 Velocity locking in two spatial dimensions Since many range expansions occur in two rather than in one spatial dimensions, we sought to determine how spatial dimensionality affects velocity locking. To answer this question, we generalized Eq. ( ) to two spatial dimensions: |
[EQUATION] The results of two-dimensional simulations are shown in Fig. 15 . We found that velocity plateaus are still present in two spatial dimensions; in addition, the fronts of the expansions assume characteristic geometric shapes that depend on the magnitude of the expansion velocity. |
II.11 Velocity locking in CSCT models with spatial and temporal periodicity Although velocity locking arises most naturally in DSDT models, it is the spatio-temporal periodicity rather than discreteness that is required for velocity locking. To demonstrate this explicitly, we simulated the following CSCT model with the... |
[EQUATION] To specify the periodic variation, we used a simple cosine with an offset: [EQUATION] where [MATH] and [MATH] are temporal and spatial periods, and [MATH] and [MATH] set the magnitude of the spatial variation. Note that the growth rate is identical to that in Eq. ( ) with [MATH] except for the temporal varia... |
Equation ( 19 ) was solved numerically as described in Methods. Velocity was computed by fitting the front position to a linear function of time. The front position was defined as the furthest site with population density greater than half the average bulk density at the time. The velocity of invasion is shown in Fig. ... |
III Discussion In this paper, we examined the properties of range expansions in fragmented habitats with seasonal growth. Both ecological conditions are quite common in natural and laboratory populations keitt:pinning, zhang:death_galaxy, fisman:seasonality, fahrig:fragmentation, wilson:coexistence, kefi:desert, abraha... |
When both space and time are discrete, range expansions can fundamentally differ from the predictions of continuous models. In particular, invasions can proceed in a step-like or pulsed fashion with population densities at the front assuming a discrete set of values that repeat periodically. Such invasions have been ob... |
Why does velocity locking occur only in DSDT models and not for pulled waves? The answer lies in the translational symmetry of the traveling wave solutions. Consider an expansion that has reached a steady state and expands at constant velocity with a time-invariant density profile in the reference frame co-moving with ... |
Invasions in DSCT models also possess translational symmetry because two consecutive density profiles separated by any time are both solutions of the underlying mathematical model. Since time is continuous, there is again a continuous family of solutions, which are also related by spatial translations since density pro... |
For pulled waves, velocity locking does not occur because of another continuous symmetry unrelated to spatial and temporal continuity. The velocity of pulled invasions is determined by the dynamics at the expansion edge, where the population density is small and all nonlinear terms can be neglected. Since linear equati... |
This symmetry based analysis predicts that models that do not exhibit velocity locking should have continuous profiles in the comoving reference frame. This prediction agrees with the known results and our simulations. When space is continuous, the density profile is clearly a continuous function, which translates in s... |
The remaining question is why velocity locking occurs in DSDT models with significant nonlinearities in dispersal or growth. An intuitive argument was proposed in Ref. carretero:locking_pre, that noticed that the infinite dimensional system of coupled difference equations in DSDT models can be reduced to a single diffe... |
[EQUATION] The function [MATH] is necessarily periodic, so the dynamics can be restricted to [MATH] for DSDT models or to [MATH] for periodic CSCT models (in this case one considers [MATH] at times [MATH] and [MATH] ). Because of this periodicity [MATH] is often referred to as a circle map. |
Mode locking in circle maps is well understood. For [MATH] that are monotonic, Eq. ( 21 ) predicts an aperiodic increase of [MATH] similar to the dynamics in pulled waves bak:devils_staircase, . However, once nonlinearities are strong enough to make [MATH] non-monotonic, the circle maps can lock into periodic oscillati... |
bak:devils_staircase, fat_fractal:prl, arnold:scaling, . In the context of range expansions, this means that velocity plateaus account for only part of the [MATH] plot—the Devil’s staircase is not complete. Therefore, at some values of [MATH] front propagation occurs without strict periodicity and velocity locking even... |
Velocity locking could have important implications for the management of invasive species because the invariance of invasion rate under parameter variation could dramatically change how invaders respond to interventions. Indeed, a management strategy could be deemed ineffective when it does not result in any change of ... |
Velocity locking also affects the dynamics and evolution of the invading population. We found that locked fronts propagate quasi-deterministically without exhibiting any diffusive wandering due to demographic or environmental fluctuations. In contrast, unlocked fronts have a nonzero effective diffusion constant that be... |
Our results show that that velocity locking occurs in both one and two spatial dimensions and is extremely robust. It persists despite external perturbations, demographic noise, environmental fluctuations, and habitat heterogeneity. Velocity locking also does not require perfect discreteness of space and time; approxim... |
While the main focus of our work has been on biological invasions, velocity locking could also have implications beyond ecology. Indeed, front propagation arises in quantum chromodynamics marquet:qcd, , entanglement spreading schachenmayer:entanglement, jurcevic:entanglement, , chemical kinetics douglas:assembly_wave, ... |
IV Methods IV.1 Simulations of CSCT, CSDT, and DSCT models For CSCT models, simulations in Fig. A were performed using the pdepe function in MatLab with error tolerance of [MATH] . Simulations in Fig. B were performed using finite-difference approach with discretization [MATH] and [MATH] |
For DSCT models, we used the cubic [MATH] defined in Eq. ( ) and solved Eq. ( ) for [MATH] patches using the standard Runge-Kutta method in MatLab |
In all models, the habitat was empty initially, but the boundary conditions were chosen to mimic an expansion from a region where the population is well established. Specifically, population density was set to the carrying capacity on the left edge of the habitat. The boundary condition on the right edge was absorbing,... |
IV.2 Simulations of DSDT models Simulations were performed as follows. Each generation, population densities were updated according to Eq. ( ) with reflecting boundary conditions on both ends of the simulation box. The size of the simulation box contained at least [MATH] patches, and its position was periodically adjus... |
Because our main goal was to demonstrate that periodic front propagation arises due to velocity locking, we were careful to exclude other sources of oscillations. In particular, we considered [MATH] without over-compensatory dynamics and limited the range of dispersal rate to [MATH] . For higher values of [MATH] , the ... |
We list parameters used in Figure here. In panel A for CSCT, [MATH] and cubic growth with [MATH] [MATH] , and [MATH] . For DSCT, [MATH] and cubic growth with [MATH] [MATH] , and [MATH] . For CSDT, [MATH] and piecewise-linear growth with [MATH] [MATH] , and [MATH] . For DSDT, [MATH] and piecewise-linear growth with [MAT... |
IV.3 Simulations of CSCT models with spatial and temporal periodicity in Fig. 16 To handle the time and space dependent coefficients in Eq. ( 19 ), simulations were performed with the following discretization scheme |
[EQUATION] where [MATH] and [MATH] were the time and space discretizations used for the numerical solution. Runs were performed at two separate discretization values to ensure that results were independent of the choice of discretization. |
Appendix A Velocities of pulled expansions A.1 Velocity in pulled DSDT models Here, we derive the velocity of pulled expansions for DSDT models shown in Fig. . Our calculation closely follows that of van Saarloos saarloos:review, and is given here for completeness. |
When expansions are pulled, their properties are determined by the dynamics at the expansion front, where [MATH] is small and [MATH] can be approximated as [MATH] where [MATH] . In the absence of an Allee effect, this approximation yields the maximal possible growth rate; therefore, the bulk dynamics cannot push the po... |
The linearization of Eq. ( ) for DSDT models yields [EQUATION] with [MATH] This equation is solved through Fourier transforms defined as |
[EQUATION] with the following result [EQUATION] where [MATH] is the Fourier transform of the initial population density. We then shift to the comoving reference frame with a spatial coordinate [MATH] and perform the inverse Fourier transform: |
[EQUATION] This integral can be evaluated using the saddle point approximation bender:orszag, , which is appropriate for large [MATH] when the transient dynamics are over. The saddle point approximation states that the integral is dominated by a region near [MATH] , a particular value of [MATH] in the complex plane for... |
[EQUATION] Since this is a complex equation, both real and imaginary parts need to be zero, which yields two real equations for the three unknowns: the real and imaginary parts of [MATH] and [MATH] . The third required equation comes from the requirement that [MATH] does not increase to infinity or diminish to zero at ... |
[EQUATION] It is easy to see that Eqs. ( 27 ) and ( 28 ) are satisfied only by purely imaginary [MATH] , which yields the expected exponential decay of [MATH] with [MATH] as [MATH] from the first exponent in Eq. ( 26 ). The velocity [MATH] and asymptotic decay rate [MATH] are then given by the following system of equat... |
[EQUATION] [EQUATION] The second equation is equivalent to the requirement that [MATH] minimizes [MATH] in the first equation. Thus, we can alternatively express [MATH] as |
[EQUATION] A.2 Velocity in pulled DSCT models The derivation for the pulled velocity of DSCT models mirrors the derivation above. The linearization of Eq. ( ) for DSCT models yields |
[EQUATION] where [MATH] As above, we solve this equation via Fourier transform, inverse Fourier transform and moving to the comoving reference frame with [MATH] . This gives the analog of Eq. ( 26 |
[EQUATION] Like above, we evaluate the integral via the saddle point approximation and use the requirement that [MATH] does not increase to infinity or diminish to zero at long times. Thus, we arrive at the pulled velocity for DSCT models |
[EQUATION] A.3 Velocity in pulled CSDT models The pulled velocity for CSDT models has been derived kot:csdt, . For completeness, we reproduce their results using the same method as in the previous subsections. The linearization of Eq. ( 35 ) for CSDT models yields: |
[EQUATION] where [MATH] We solve this equation via Fourier transform in space and then perform the inverse Fourier transform and a shift into the comoving reference frame with [MATH] |
[EQUATION] where [MATH] is the Fourier transform of the dispersal kernel. We then evaluate the integral via the saddle point approximation, using the requirement that [MATH] does not increase to infinity or diminish to zero at long times. The result reads |
[EQUATION] Note that the Fourier transform of the dispersal kernel evaluated at [MATH] is equivalent to calculating the moment generating function of [MATH] , which was used by Ref. kot:csdt, |
For the piecewise-linear model with Laplace kernel used in simulations, the above result reduces to [EQUATION] The pulled velocities for all four classes of models are compared to simulations in Fig. B. |
Appendix B Expansions locked at [MATH] in the piecewise-linear model We now show that [MATH] indeed exhibits exact plateaus by explicitly finding traveling-wave solutions of Eq. ( ) with [MATH] for the piecewise-linear model. Since velocity locking occurs only in the presence of an Allee effect, we will assume that [MA... |
[EQUATION] where [MATH] is the amplitude of the solution that should be determined from matching to the solution behind the front, and [MATH] is the spatial decay rate chosen to satisfy Eq. ( 23 ) with [MATH] , i.e. we require that |
[EQUATION] To analyze the nature of solutions to this equation, it is convenient to define [MATH] and rewrite the equation as follows: |
[EQUATION] For [MATH] , i.e. for the strong Allee effect, the right hand side is monotonically decaying from [MATH] to [MATH] as [MATH] increases from its minimal value of [MATH] to [MATH] . As a result, there is a unique solution [MATH] of Eq. ( 40 ). When the Allee effect is weak ( [MATH] ), the right hand side of eq... |
[EQUATION] Clearly, for [MATH] , Eq. ( 42 ) sets an upper bound on the migration rates consistent with [MATH] . We also note that, when Eq. ( 40 ) has two solutions, only the larger one corresponds to a pushed expansion. This situation is completely analogous to that for reaction-diffusion equations discussed in Ref. s... |
With [MATH] defined by the largest solution of Eq. ( 40 ), the solution for [MATH] is then given by the following ansatz: [EQUATION] |
where [MATH] determines the initial position of the front. In the following, we assume that [MATH] for simplicity. Note that we set [MATH] due to the matching requirement. Indeed, for the exponential profile to be preserved in the nonlinear model, the right most point of the bulk solution must also satisfy Eq. ( 40 ) i... |
Three other conditions are necessary for the ansatz to hold. (i) The right most point of the bulk region should not fall below [MATH] following migration; otherwise, it will fall below the carrying capacity and would not be able to send enough migrants to its neighbor that needs to reach the carrying capacity in the ne... |
First condition: [EQUATION] Second condition: [EQUATION] Third condition: [EQUATION] In our simulations, the first condition was never violated, so it might be unnecessary at least for [MATH] . The last two conditions can be used to obtain analytical expressions for [MATH] and [MATH] by changing inequalities to equalit... |
For completeness, we also provide the solutions for the critical values of [MATH] discussed above. From Eqs. ( 40 ) and ( 45 ), we conclude that |
[EQUATION] provided [MATH] is the largest root of Eq. ( 40 ). From Eqs. ( 40 ) and ( 46 ), we conclude that [EQUATION] provided [MATH] is the largest root of Eq. ( 40 ), and [MATH] is the positive root of the following equation: |
[EQUATION] When [MATH] is the only or the smallest root of Eq. ( 40 ), there is no velocity locking. Further, note that in addition to Eq. ( 47 ), [MATH] is bounded above by [MATH] when [MATH] and the requirement that [MATH] in our simulations. |
Acknowledgements Preliminary work was carried out by Vipul Vachharajani, Eugene Yurtsev, and Jeff Gore, who observed velocity locking in simulations. This work was partially supported by a grant from the Simons Foundation (#409704, Kirill S. Korolev), by the Cottrell Scholar Award (#24010, Kirill S. Korolev), and by a ... |
# Source: arxiv 1806.02485 # Title: Stochastic Block Models are a Discrete Surface Tension # Sections: all # Downloaded: 2026-03-02T08:57:34.636638+00:00 |
\headers SBM is discrete surface tensionZ. M. Boyd, M. A. Porter, and A. L. Bertozzi Stochastic Block Models are a Discrete Surface Tension |
Abstract Networks, which represent agents and interactions between them, arise in myriad applications throughout the sciences, engineering, and even the humanities. To understand large-scale structure in a network, a common task is to cluster a network’s nodes into sets called “communities”, such that there are dense c... |
keywords: networks, community structure, data clustering, stochastic block models (SBMs), Merriman–Bence–Osher (MBO) scheme, geometric partial differential equations |
{AMS} 65K10, 49M20, 35Q56, 62H30, 91C20, 91D30, 94C15 Introduction The study of networks, in which nodes represent entities and edges encode interactions between entities |
, can provide useful insights into a wide variety of complex systems in myriad fields, such as granular materials , disease spreading |
, criminology , and more. In the study of such applications, the analysis of large data sets — from diverse sources and applications — continues to grow ever more important. |
The simplest type of network is a graph, and empirical networks often appear to exhibit a complicated mixture of regular and seemingly random features |
. Additionally, it is increasingly important to study networks with more complicated features, such as time-dependence , multiplexity |
, annotations , and connections that go beyond a pairwise paradigm . One also has to worry about “features” such as missing information and false positives |
Nevertheless, it is convenient in the present paper to restrict our attention to undirected, unweighted graphs for simplicity. To try to understand the large-scale structure of a network, it can be very insightful to coarse-grain it in various ways |
. The most popular type of clustering is the detection of assortative “communities,” in which dense sets of nodes are connected sparsely to other dense sets of nodes |
. A statistically principled approach is to treat community detection as a statistical inference problem using a model such as a stochastic block model (SBM) |
. The detection of communities has given fascinating insights into a variety of applications, including brain networks , social networks |
, granular networks , protein interaction networks , political networks , and many others. One of the most popular frameworks for detecting communities is to use an SBM, a generative model that can produce networks with community structure |
One uses an SBM for community detection by fitting an observed graph to a statistical model to attempt to infer the most probable community assignment for each node. SBMs can incorporate a variety of features, including degree heterogeneity |
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