Many models for the movement of particles and individuals are based on the diffusion equation, which, in turn, can be derived from an uncorrelated random walk or a position-jump process. In those models, individuals have a location but no well-defined velocity. An alternative, and sometimes more accurate, model is based on a correlated random walk or a velocity-jump process, where individuals have a well defined location and velocity. The latter approach leads to hyperbolic equations for the density of individuals, rather than parabolic equations that result from the diffusion process. Almost all previous work on correlated random walks considers a homogeneous landscape, whereas diffusion models for uncorrelated walks have been extended to spatially varying environments. In this work, we first derive the equations for a correlated random walk in a one-dimensional spatially varying environment with either smooth variation or piecewise constant variation. Then we show how to derive the so-called parabolic limit from the resulting hyperbolic equations. We develop homogenization theory for the hyperbolic equations, and show that taking the parabolic limit and homogenization are commuting actions. We illustrate our results with two examples from ecology: the persistence and spread of a population in a patchy heterogeneous landscape.
Citation: Frithjof Lutscher, Thomas Hillen. Correlated random walks in heterogeneous landscapes: Derivation, homogenization, and invasion fronts[J]. AIMS Mathematics, 2021, 6(8): 8920-8948. doi: 10.3934/math.2021518
Many models for the movement of particles and individuals are based on the diffusion equation, which, in turn, can be derived from an uncorrelated random walk or a position-jump process. In those models, individuals have a location but no well-defined velocity. An alternative, and sometimes more accurate, model is based on a correlated random walk or a velocity-jump process, where individuals have a well defined location and velocity. The latter approach leads to hyperbolic equations for the density of individuals, rather than parabolic equations that result from the diffusion process. Almost all previous work on correlated random walks considers a homogeneous landscape, whereas diffusion models for uncorrelated walks have been extended to spatially varying environments. In this work, we first derive the equations for a correlated random walk in a one-dimensional spatially varying environment with either smooth variation or piecewise constant variation. Then we show how to derive the so-called parabolic limit from the resulting hyperbolic equations. We develop homogenization theory for the hyperbolic equations, and show that taking the parabolic limit and homogenization are commuting actions. We illustrate our results with two examples from ecology: the persistence and spread of a population in a patchy heterogeneous landscape.
[1] | Y. Alqawasmeh, F. Lutscher, Movement behaviour of fish, harvesting-induced habitat degradation and the optimal size of marine reserves, Theor. Ecol., 12 (2019), 453–466. doi: 10.1007/s12080-019-0411-x |
[2] | Y. Alqawasmeh, F. Lutscher, Persistence and spread of stage-structured populations in heterogeneous landscapes, J. Math. Biol., 78 (2019), 1485–1527. doi: 10.1007/s00285-018-1317-8 |
[3] | A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Providence: AMS Chelsea Publishing, 2010. |
[4] | R. S. Cantrell, C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Hoboken: Wiley, 2003. |
[5] | B. Choi, Y.-J. Kim, Diffusion of biological organisms: Fickian and Fokker–Planck type diffusions, SIAM J. Appl. Math., 79 (2019), 1501–1527. doi: 10.1137/18M1163944 |
[6] | J. Chung, Y. J. Kim, O. Kwong, C. W. Yoon, Biological advection and cross diffusion with parameter regimes, AIMS Mathematics, 4 (2020), 1721–1744. |
[7] | Y. Dolak, T. Hillen, , Cattaneo models for chemotaxis, numerical solution and pattern formation, J. Math. Biol., 46 (2003), 153–170. doi: 10.1007/s00285-002-0173-7 |
[8] | J. P. Duncan, R. N. Rozum, J. A. Powell, K. M. Kettenring, Multi-scale methods predict invasion speeds in variable landscapes, Theor. Ecol., 10 (2017), 287–303. doi: 10.1007/s12080-017-0329-0 |
[9] | A. Einstein, Zur Theorie der Brownschen Bewegung, Ann. Phy., 19 (1906), 371–381. |
[10] | L. Fahrig, Effect of habitat fragmentation on the extinction threshold: a synthesis, Ecol. Appl., 12 (2002), 346–353. |
[11] | R. Filliger, M.-O. Hongler, Supersymmetry in random two-velocity processes, Physica A, 332 (2004), 141–150. doi: 10.1016/j.physa.2003.09.048 |
[12] | R. A. Fisher, The advance of advantageous genes, Ann. Eugenics, 7 (1937), 355–369. doi: 10.1111/j.1469-1809.1937.tb02153.x |
[13] | J. F. Fryxell, Predictive modelling of patch use by terrestrial herbivores. In H.H.T. Prins and F. van Langevelde, editors, Dynamics of Foraging Resource Ecology: Spatial and Temporal, chapter 6A, pages 105–123. Springer, 2008. |
[14] | M. J. Garlick, J. A. Powell, M. B. Hooten, L. R. McFarlane, Homogenization of large-scale movement models in ecology, Bull. Math. Biol., 73 (2011), 2088–2108. doi: 10.1007/s11538-010-9612-6 |
[15] | S. Goldstein, On diffusion by discontinuous movements and the telegraph equation, Quart. J. Mech. Appl. Math., 4 (1951), 129–156. doi: 10.1093/qjmam/4.2.129 |
[16] | K. P. Hadeler, Nonlinear propagation in reaction transport systems. In S. Ruan and G. Wolkowicz, editors, Differential Equations with Applications to Biology. The Fields Institute Lecture Series, AMS, 1998. |
[17] | K. P. Hadeler, Reaction transport systems in biological modelling. In V. Capasso and O. Diekmann, editors, Mathematics Inspired by Biology, Lect. Notes Math. 1714, pages 95–150, Heidelberg, 1999. Springer Verlag. |
[18] | K. P. Hadeler, Topics in Mathematical Biology, Heidelberg, Springer, 2018. |
[19] | K. P. Hadeler, R. Illner, P. van den Driesche, A disease transport model. In G. Lumer and L. Weiss, editors, Evolution equations and their applications in physical and life sciences, pages 369–385, New York, 2000. Marcel Dekker. |
[20] | T. Hillen, A Turing model with correlated random walk, J. Math. Biol., 35 (1996), 49–72. doi: 10.1007/s002850050042 |
[21] | T. Hillen, Invariance principles for hyperbolic random walk systems, J. Math. Ana. Appl., 210 (1997), 360–374. doi: 10.1006/jmaa.1997.5411 |
[22] | T. Hillen, Hyperbolic models for chemosensitive movement, Math. Mod. Meth. Appl. Sci., 12 (2002), 1007–1034. doi: 10.1142/S0218202502002008 |
[23] | T. Hillen, On the ${L}^2$-moment closure of transport equations: The general case, Disc. Cont. Dyn. Syst. B, 5 (2005), 299–318. doi: 10.3934/dcdsb.2005.5.299 |
[24] | T. Hillen, Existence theory for correlated random walks on bounded domains, Canad. Appl. Math. Quart., 18 (2010), 1–40. |
[25] | E. E. Holmes, Are diffusion models too simple? A comparison with telegraph models of invasion, Am. Nat., 142 (1993), 779–795. doi: 10.1086/285572 |
[26] | D. D. Joseph, L. Preziosi, Heat waves, Rev. Mod. Phys., 61 (1998), 41–73. |
[27] | M. Kac, A stochastic model related to the telegrapher's equation, Rocky MT J. Math., 4 (1956), 497–509. |
[28] | F. Lutscher, Modeling alignment and movement of animals and cells, J. Math. Biol., 45 (2002), 234–260. doi: 10.1007/s002850200146 |
[29] | F. Lutscher, A. Stevens, Emerging patterns in a hyperbolic model for locally interacting cell systems, J. Nonlin. Sci., 12 (2002), 619–640. |
[30] | G. A. Maciel, F. Lutscher, How individual response to habitat edges affects population persistence and spatial spread, Am. Nat., 182 (2013), 42–52. doi: 10.1086/670661 |
[31] | G. A. Maciel, F. Lutscher, Allee effects and population spread in patchy landscapes, J. Biol. Dyn., 9 (2015), 109–123. doi: 10.1080/17513758.2015.1027309 |
[32] | G. A. Maciel, F. Lutscher, Movement behavior determines competitive outcome and spread rates in strongly heterogeneous landscapes, Theor. Ecol., 11 (2018), 351–365. doi: 10.1007/s12080-018-0371-6 |
[33] | J. D. Murray, Mathematical Biology, New York: Springer, 1989. |
[34] | H. G. Othmer, A continuum model for coupled cells, J. Math. Biol., 17 (1983), 351–369. |
[35] | H. G. Othmer, S. R. Dunbar, W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263–298. doi: 10.1007/BF00277392 |
[36] | O. Ovaskainen, S. J. Cornell, , Biased movement at a boundary and conditional occupancy times for diffusion processes, J. Appl. Prob., 40 (2003), 557–580. doi: 10.1239/jap/1059060888 |
[37] | J. R. Potts, T. Hillen, M. A. Lewis, , Edge effects and the spatio-temporal scale of animal movement decisions, Theor. Ecol., 9 (2015), 233–247. |
[38] | J. Powell, N. E. Zimmermann, Multiscale analysis of active seed dispersal contributed to resolving Reid's paradox, Ecology, 85 (2004), 490–506. doi: 10.1890/02-0535 |
[39] | H. Schwetlick, Travelling fronts for multidimensional nonlinear transport equations, Ann. I H Poincarè (C) Ana. Nonlin., 17 (2000), 523–550. |
[40] | N. Shigesada, K. Kawasaki, E. Teramoto, Traveling periodic waves in heterogeneous environments, Theor. Popul. Biol., 30 (1986), 143–160. doi: 10.1016/0040-5809(86)90029-8 |
[41] | P. Turchin, Quantitative Analysis of Movement: measuring and modeling population redistribution of plants and animals, Sunderland: Sinauer Assoc., 1998. |
[42] | B. Yurk, C. Cobbold, Homogenization techniques for population dynamics in strongly heterogeneous landscapes, J. Biol. Dyn., 12 (2018), 171–193. doi: 10.1080/17513758.2017.1410238 |
[43] | E. Zauderer, Correlated random walks, hyperbolic systems and Fokker-Planck equations, Math. Comupt. Model., 17 (1993), 43–47. |