STOCHASTIC SPATIAL LOTKA-VOLTERRA PREDATOR-PREY MODELS

Uwe C. Täuber (Personal webpage)

Department of Physics, Virginia Tech

Dynamical models of interacting populations have recently become of fundamental interest for the spontaneous formation of patterns and other intriguing features in non-equilibrium statistical physics. In turn, theoretical physics provides a toolbox for quantitative analysis for many paradigmatic models employed in biology and ecology. Stochastic, spatially extended models for predator-prey interaction display striking spatio-temporal structures [1,2] that are not captured by the Lotka-Volterra mean-field rate equations. These spreading activity fronts induce persistent correlations between predators and prey that can be studied through field-theoretic methods [3]. Introducing local restrictions on the prey population induces predator extinction. The critical dynamics at this continuous absorbing state transition is governed by the scaling exponents of critical directed percolation [4]. This lecture will also address the influence of spatially varying reaction rates: Fluctuations in rare favorable regions cause a remarkable increase in both predator and prey populations [5]. Intriguing novel features are found when variable interaction rates are affixed to individual particles rather than lattice sites. The ensuing stochastic dynamics combined with inheritance rules causes rapid time evolution for the rate distributions, with however overall neutral effect on stationary population densites [6]. When we subject the system to a periodically varying carrying capacity that describes seasonally oscillating availability of resources for the prey population, we observe intriguing stabilization of the two-species coexistence regime [7]. The lecture will finally briefly discuss noise-induced spontaneous pattern formation and the stabilization of vulnerable ecologies through immigration waves in systems with three cyclically competing species akin to spatial rock-paper-scissors games [8].
This research is supported by the U.S. National Science Foundation, Division of Mathematical Sciences under Award No. NSF DMS-2128587.

References:

[1] M. Mobilia, I.T. Georgiev, and U.C.T., J. Stat. Phys. 128, 447 (2007); arXiv:q-bio.PE/0512039.
[2] U. Dobramysl, M. Mobilia, M. Pleimling, and U.C.T., J. Phys. A: Math. Theor. 51, 063001 (2018); arXiv:1708.07055.
[3] U.C.T., J. Phys. A: Math. Theor. 45, 405002 (2012); arXiv:1206.2303.
[4] S. Chen and U.C.T., Phys. Biol. 13, 025005 (2016); arXiv:1511.05114.
[5] U. Dobramysl and U.C.T., Phys. Rev. Lett. 101, 258102 (2008); arXiv:0804.4127.
[6] U. Dobramysl and U.C.T., Phys. Rev. Lett. 110, 048105 (2013); arXiv:1206.0973.
[7] M. Swailem and U.C.T., preprint arXiv:2211.09276.
[8] S.R. Serrao and U.C.T., Eur. Phys. J. B 94, 175 (2021); arXiv:2105.08126.