# The “Lévy or Diffusion” Controversy: How Important Is the Movement Pattern in the Context of Trapping?

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## Abstract

**:**

## 1. Introduction

## 2. BM vs. LW: Equivalence I

#### 2.1. Brownian Motion

#### 2.2. Condition of Equivalence

## 3. Time-Dependent Diffusion

#### Equivalence of Trap Counts: Brownian Motion vs. Diffusion in a Semi-Bounded Space

## 4. BM vs. LW: Equivalence II

#### 4.1. Stable Laws

#### 4.2. Equivalence of Trap Counts: Cauchy Walk vs. Diffusion

#### 4.3. Proposed Diffusion Coefficient

#### 4.4. Reproducing Lévy Trap Counts Using Diffusion

## 5. Discussion

## 6. Concluding Remarks

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Mean Field Numerical Solution

## Appendix B. Trap Count Recordings

**Table B1.**Trap count recordings for the cases (a) Holtsmark $S(\alpha =\frac{3}{2},{\gamma}_{h})$, (b) Cauchy $S(\alpha =1,{\gamma}_{c})$ and (c) symmetric-Lévy $S(\alpha =\frac{1}{2},{\gamma}_{l})$, with tail exponent $\alpha $ and scale parameter $\gamma $. Simulation details with all parameter values are given in the caption of Figure 5.

Time $\left(\mathit{t}\right)$ | ${\mathit{\gamma}}_{\mathit{h}}=0.01$ | $0.02$ | $0.04$ | ${\mathit{\gamma}}_{\mathit{c}}=0.0005$ | $0.002$ | $0.003$ | ${\mathit{\gamma}}_{\mathit{l}}=1\times {10}^{-6}$ | $4\times {10}^{-6}$ | $2\times {10}^{-5}$ |
---|---|---|---|---|---|---|---|---|---|

0.1 | 54 | 93.8 | 188 | 21.6 | 48.6 | 91 | 32.6 | 69.4 | 134.2 |

0.2 | 83.2 | 147.2 | 283.4 | 36.4 | 86.4 | 147.8 | 60.6 | 123.4 | 241.6 |

0.3 | 106.6 | 195.6 | 355.2 | 53.2 | 120.2 | 206.2 | 94 | 173 | 324.8 |

0.4 | 127.6 | 235.6 | 413.8 | 68.4 | 145 | 252.6 | 123.2 | 222.2 | 399.4 |

0.5 | 146.2 | 270.6 | 466 | 78.4 | 171 | 291.2 | 147.4 | 263.6 | 460.8 |

0.6 | 163.2 | 299.6 | 510.4 | 91.6 | 194.4 | 328.4 | 171.6 | 305 | 513 |

0.7 | 180.6 | 326.2 | 551.8 | 103.8 | 218.6 | 364.4 | 196 | 340.4 | 560. |

0.8 | 195.6 | 351.6 | 588.8 | 116.6 | 240.2 | 394.8 | 219.6 | 376.4 | 603. |

0.9 | 209.8 | 376.8 | 629.6 | 126 | 262.6 | 425.4 | 238.4 | 409.6 | 645.8 |

1.0 | 225.6 | 400.8 | 661 | 133.8 | 282.2 | 451.6 | 258 | 438.2 | 686.4 |

1.1 | 238.2 | 421.8 | 689.6 | 143.2 | 302.6 | 476 | 279 | 466.2 | 717.4 |

1.2 | 251.4 | 442.4 | 715.2 | 155.4 | 321.6 | 498.4 | 298.6 | 491.6 | 746.4 |

1.3 | 260.8 | 460.4 | 740.6 | 166.4 | 335.8 | 521.2 | 321.4 | 513 | 773.2 |

1.4 | 271 | 482 | 760.4 | 175.2 | 355 | 545 | 341.4 | 536.2 | 797 |

1.5 | 282.6 | 499 | 783 | 184.2 | 372.6 | 563.8 | 358.6 | 555.2 | 820.2 |

1.6 | 290.6 | 516.2 | 804 | 194.2 | 384.6 | 584 | 374.8 | 575.8 | 839 |

1.7 | 301 | 531.4 | 821.2 | 204.8 | 398 | 600.8 | 390.2 | 596 | 856.8 |

1.8 | 311 | 546.4 | 836.4 | 214.2 | 413 | 617.4 | 407.8 | 612.2 | 871.2 |

1.9 | 323 | 562.8 | 851.6 | 222.6 | 427 | 633.8 | 422.8 | 632 | 883 |

2.0 | 333.2 | 574 | 864 | 231.8 | 439.6 | 652.6 | 435 | 648.6 | 894.8 |

2.1 | 342.6 | 586.2 | 877.2 | 333.2 | 574 | 864 | 446.6 | 668.4 | 904.6 |

2.2 | 350.4 | 599 | 888.8 | 241.8 | 452.2 | 669 | 462 | 681 | 912.4 |

2.3 | 359.2 | 609.2 | 898.8 | 246.4 | 465.2 | 683.8 | 474.2 | 694.2 | 919.4 |

2.4 | 369.4 | 620.2 | 907.6 | 254.2 | 476.6 | 695.2 | 488.6 | 704 | 928.2 |

2.5 | 376.2 | 631 | 914.8 | 263 | 488.2 | 708.2 | 502.6 | 716.4 | 934.6 |

2.6 | 386.2 | 645.8 | 921.8 | 271.6 | 499.2 | 721.2 | 515 | 730.4 | 941.6 |

2.7 | 394.4 | 655.4 | 928.4 | 280 | 509 | 735.2 | 525.6 | 747.4 | 946.2 |

2.8 | 399.4 | 666.8 | 934.4 | 288 | 518.2 | 745.8 | 538 | 756.4 | 950.2 |

2.9 | 407.6 | 678.2 | 938.8 | 294.2 | 527.2 | 757 | 550 | 768.2 | 956.2 |

3.0 | 415 | 686.8 | 943.8 | 301.2 | 536.8 | 766.8 | 560.8 | 779.2 | 961.2 |

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1 | The randomness of animal movement is obviously an idealization, which, however, is well justified under certain conditions, e.g., see [18] for a detailed discussion of this issue. |

2 | This is the inverse of the error function defined by $\mathrm{erf}\left(z\right)=\frac{2}{\sqrt{\pi}}{\int}_{0}^{z}exp\left(\right)open="("\; close=")">-{z}^{\prime 2}$ |

3 | This specific type of random walk is of significant interest in foraging theory since an inverse square power-law distribution of flight lengths provides an optimal strategy to detect target sites provided that the sites are sparse and can be revisited [61]. Furthermore, see Section 2.2. |

4 | See Ahmed and Petrovskii [27] for a detailed description of the model previously proposed. |

**Figure 1.**(

**a**) Diffusive flux: Solid curves show the total flux $J\left(t\right)$ obtained from the diffusion model over time $0<t<5$ with the analytic solution given by (30), with fixed diffusion constants ${D}_{0}=0.05$, ${D}_{1}=0.15$ and varying Hurst exponents $H=\frac{1}{2}$ (standard diffusion), $H=\frac{3}{4}$ (super-diffusion) and $H=1$ (ballistic). The solution is defined over the semi-infinite domain $0<x<\infty $ with initial uniform distribution $u(x,t=0)={U}_{0}=200$ and point trap $u(x=0,t)=0$. Trap counts: Bold dots plot cumulative trap counts ${J}_{t}$ for Brownian motion with total population $N=1000$ recorded at times $t=0,\phantom{\rule{0.166667em}{0ex}}0.1,\phantom{\rule{0.166667em}{0ex}}0.2,\cdots ,\phantom{\rule{0.166667em}{0ex}}5$. Discrete time scale parameter is defined by combining (25) and (28), that is ${\sigma}^{2}\left(t\right)\approx 2({D}_{0}+{D}_{1}{t}^{2H-1})\Delta t$ with ${D}_{0},{D}_{1},H$ given above. Each individual executes a total of $S=5000$ steps with constant time step increment $\Delta t=0.001$ and total time $T=S\Delta t=5.$ Individuals initially uniformly distributed ${X}_{0}^{\left(n\right)}\sim U(0,L=5)$. The trap installed at position $x=0$ and simulations are conducted with the external boundary condition described in Section 2.1. Trap counts are replicated and averaged over ten realizations to reduce the effect of stochasticity. Numerical solution: The green dashed line represents the mean field numerical solution using the method of explicit finite differences. See Appendix A for further details on the numerical scheme. (

**b**) Absolute relative error: $A\left(t\right)$ plotted at times $t=hk$, $h=0.1$, $k=0,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}2,\cdots ,\phantom{\rule{0.166667em}{0ex}}50$, with average $\overline{A}=0.306$ (red), $0.157$ (blue) and $0.167$ (black), to illustrate the magnitude of the discrepancy between the analytic solution and trap counts (for the interpretation of the references to color in this figure legend, and all subsequent figures, the reader is referred to the web version of the article).

**Figure 2.**(

**a**) Probability density functions $\varphi \left(\xi \right)$ for stable laws; normal $\alpha =2$, Holtsmark $\alpha =\frac{3}{2}$, Cauchy $\alpha =1$ and symmetric-Lévy $\alpha =\frac{1}{2}$ with fixed scaling parameter $\gamma =1$ chosen for illustrative purposes (see (36)–(39)). (

**b**) Comparison of end tails for different $\alpha $, defined by (33).

**Figure 3.**(

**a**) Comparison of trap counts ${J}_{t}$ for the Cauchy walk $\mathbf{S}(\alpha =1,{\gamma}_{c}=0.002)$ (see (38)) against the diffusive flux $J\left(t\right)$ for anomalous diffusion for the three cases; standard diffusion $H=0.5,\{{D}_{0},{D}_{1}\}=\{0.1431,1.6730\}$, super-diffusion $H=0.75,\{{D}_{0},{D}_{1}\}=\{0.7263,1.1776\}$ and ballistic diffusion $H=1,\{{D}_{0},{D}_{1}\}=\{1.228,0.5844\}$ (see (30)). Trap counts were averaged over five realizations to reduce the effect of stochasticity. Total number of individuals $N=1000$ uniformly distributed over a finite domain $L=5$ with population density ${U}_{0}=200$. Each individual executes a total of $S=3000$ steps with constant time step increment $\Delta t=0.001$ and total time $T=S\Delta t=3.$ (

**b**) Relative error (%) measures the discrepancy between trap counts for the random walk and diffusion model defined by $\frac{J\left(t\right)-{J}_{t}}{N}$ plotted at times $t=0,\phantom{\rule{0.166667em}{0ex}}0.1,\phantom{\rule{0.166667em}{0ex}}0.2,\cdots ,\phantom{\rule{0.166667em}{0ex}}3$.

**Figure 4.**(

**a**) Growth function: Logistic $G\left(t\right)={D}_{1}t\left(\right)open="("\; close=")">1-\frac{t}{k}$, which reduces to linear growth $G\left(t\right)={D}_{1}t$ as $k\to \infty $. (

**b**) Diffusion coefficient: $D\left(t\right)={D}_{0}+{D}_{1}t\left(\right)open="("\; close=")">1-\frac{t}{k}$, which reduces to $D\left(t\right)={D}_{0}+{D}_{1}t{e}^{-\nu t}$ as $k\to \infty $. Parameter values (chosen for illustrative purposes): ${D}_{0}=2.5$, ${D}_{1}=10$, $\nu =0.8$ with different values $k=1.5,\phantom{\rule{0.166667em}{0ex}}5$, including the limiting case $k\to \infty $. (

**c**) Diffusive flux given by ((46)–(47)).

**Figure 5.**Simulation details: In accordance with the simulation setting in Section 2.1, $N=1000$ individuals are initially uniformly distributed along a 1D spatial domain $0<x<L=5$. After one time step $\Delta t=0.001$, each individual executes a single step, with the subsequent position defined by the recurrence relation (48). A total number of $S=3000$ steps is executed, with the total time of exposure $T=S\Delta t=3$. Prior to the simulation run, an impermeable external boundary is installed at $x=5$ ensuring that no individual can escape or enter the system at this end (no-migration/immigration), by forcing the condition: if ${X}_{i}^{\left(n\right)}>5$ at any instant in time, then ${X}_{i}^{\left(n\right)}=5$. The point trap at $x=0$ functions in the following way: if ${X}_{i}^{\left(n\right)}<0$ at any instant in time, then the individual is removed from the system, and the accumulated trap count increases by one. Consequently, the number of individuals in the population decrease as time flows, and an increasing stochastic trap count trajectory is formed. Trap counts: Bold dots depict cumulative trap counts ${J}_{t}$ recorded for the cases (a) Holtsmark, (b) Cauchy and (c) symmetric-Lévy at times $t=0,\phantom{\rule{0.166667em}{0ex}}0.1,\phantom{\rule{0.166667em}{0ex}}0.2,\phantom{\rule{0.166667em}{0ex}}\cdots ,\phantom{\rule{0.166667em}{0ex}}3$. Different scale parameters are considered for each respective case. Furthermore, trap counts are averaged over five realizations to reduce the effect of stochasticity (for the full list of recordings, see Table B1). Diffusive flux: Curves $J\left(t\right)$ shown for Model 2 in all three cases (red, blue and black curves). Model 1 shown only for the case corresponding to Holtsmark $\mathbf{S}(\alpha =1.5,{\gamma}_{h}=0.04)$ (magenta curve). All best-fit parameters are listed in Table 3.

**Figure 7.**Absolute relative error between trap counts and diffusive flux for the cases (

**a**) Holtsmark, (

**b**) Cauchy and (

**c**) symmetric-Lévy. Each color corresponds to those cases with scale parameters shown in Figure 5.

Standard Diffusion $\mathit{H}=\frac{1}{2}$ | Super Diffusion $\mathit{H}=\frac{3}{4}$ | Ballistic Diffusion $\mathit{H}=1$ | |
---|---|---|---|

Brownian trap counts | 0.306 | 0.157 | 0.167 |

Standard Diffusion $\mathit{H}=\frac{1}{2}$ | Super Diffusion $\mathit{H}=\frac{3}{4}$ | Ballistic Diffusion $\mathit{H}=1$ | |
---|---|---|---|

Brownian trap counts | 0.306 | 0.157 | 0.167 |

Cauchy trap counts | 2.035 | 0.849 | 1.120 |

**Table 3.**Best-fit parameters using a non-linear curve fitting tool (in the least squares sense) by fitting Model 1 (46) and Model 2 (47) against cumulative trap counts (see Table B1 for the complete list of recordings). The diffusion coefficients in Figure 6 are those plotted with highlighted parameters in the table below.

${\mathit{D}}_{0}$ | ${\mathit{D}}_{1}$ | $\mathit{\nu}$ | k | ${\mathit{D}}_{0}$ | ${\mathit{D}}_{1}$ | $\mathit{\nu}$ | |
---|---|---|---|---|---|---|---|

${\gamma}_{h}=$ 0.01 | 1.974 | 3.186 | 0.615 | 1286.032 | 1.9745 | 3.1834 | 0.6158 |

0.02 | 5.798 | 16.488 | 1.102 | 2846.675 | 5.7984 | 16.4881 | 1.1025 |

0.04 | 22.552 | 26.262 | 0.631 | 1.235 | 11.1403 | 133.0602 | 2.3539 |

${\gamma}_{c}=$ 0.0005 | 0.286 | 2.083 | 0.327 | 2598.906 | 0.2861 | 2.0828 | 0.3276 |

0.002 | 1.374 | 10.971 | 0.687 | 1918.029 | 1.3743 | 10.9709 | 0.6874 |

0.003 | 5.218 | 27.968 | 1.036 | 2589.831 | 5.2184 | 27.9676 | 1.0365 |

${\gamma}_{l}=$$1\times {10}^{-6}$ | 0.271 | 11.137 | 0.541 | 2608.857 | 0.2715 | 11.1369 | 0.5405 |

$4\times {10}^{-6}$ | 2.099 | 34.539 | 0.935 | 2613.735 | 2.0992 | 34.5388 | 0.9345 |

$2\times {10}^{-5}$ | 4.671 | 152.605 | 1.847 | 5202.465 | 4.6706 | 152.6045 | 1.8471 |

**Table 4.**Tabulated values of the average absolute relative error $\overline{A}$ as defined by (32), to compare the fit between Models 1 (46) and 2 (47) and trap counts. $\overline{A}$ is also included for the anomalous diffusion model; see Figure 3 and Table 2. Boxed values signify ‘equivalence’ between the diffusion and Lévy movement models.

Diffusion | ${\mathit{\gamma}}_{\mathit{h}}=0.01$ | $0.02$ | $0.04$ | ${\mathit{\gamma}}_{\mathit{c}}=0.0005$ | $0.002$ | $0.003$ | ${\mathit{\gamma}}_{\mathit{l}}={10}^{-6}$ | $4\times {10}^{-6}$ | $2\times {10}^{-5}$ |
---|---|---|---|---|---|---|---|---|---|

Standard $H=\frac{1}{2}$ | 2.035 | ||||||||

Super $H=\frac{3}{4}$ | 0.849 | ||||||||

Ballistic $H=1$ | 1.120 | ||||||||

Model 1 (46) | 0.139 | ||||||||

Model 2 (47) | 0.162 | 0.198 | 0.839 | 0.123 | 0.135 | 0.299 | 0.194 | 0.370 | 0.394 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Ahmed, D.A.; Petrovskii, S.V.; Tilles, P.F.C.
The “Lévy or Diffusion” Controversy: How Important Is the Movement Pattern in the Context of Trapping? *Mathematics* **2018**, *6*, 77.
https://doi.org/10.3390/math6050077

**AMA Style**

Ahmed DA, Petrovskii SV, Tilles PFC.
The “Lévy or Diffusion” Controversy: How Important Is the Movement Pattern in the Context of Trapping? *Mathematics*. 2018; 6(5):77.
https://doi.org/10.3390/math6050077

**Chicago/Turabian Style**

Ahmed, Danish A., Sergei V. Petrovskii, and Paulo F. C. Tilles.
2018. "The “Lévy or Diffusion” Controversy: How Important Is the Movement Pattern in the Context of Trapping?" *Mathematics* 6, no. 5: 77.
https://doi.org/10.3390/math6050077