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.