WKB theory of large deviations in stochastic populations

Abstract

Stochasticity can play an important role in the dynamics of biologically relevant populations. These span a broad range of scales: from intra-cellular populations of molecules to population of cells and then to groups of plants, animals and people. Large deviations in stochastic population dynamics -such as those determining population extinction, fixation or switching between different statesare presently in a focus of attention of statistical physicists. We review recent progress in applying different variants of dissipative WKB approximation (after Wentzel, Kramers and Brillouin) to this class of problems. The WKB approximation allows one to evaluate the mean time and/or probability of population extinction, fixation and switches resulting from either intrinsic (demographic) noise, or a combination of the demographic noise and environmental variations, deterministic or random. We mostly cover well-mixed populations, single and multiple, but also briefly consider populations on heterogeneous networks and spatial populations. The spatial setting also allows one to study large fluctuations of the speed of biological invasions. Finally, we briefly discuss possible directions of future work.

Master equations and the theory of stochastic path integrals

“…Information on the real-space approach can be found in [63,64,146,171,172,177,[354][355][356][357][358][359][360][361][362][363]. The recent review of Assaf and Meerson provides an in-depth discussion of the applicability of the real-and momentum-space approaches [364].…”

“…We outlined some of these methods in section 2.3. "Real-space" WKB approximations, which employ an exponential ansatz for the probability distribution rather than for the generating function, were not discussed in this review (information on these approximations can be found in [63,64,146,171,172,177,[354][355][356][357][358][359][360][362][363][364]). WKB approximations often prove helpful in computing a mean extinction time or a (quasi)stationary probability distribution.…”

Large fluctuations in anti-coordination games on scale-free graphs

“…Hence, with the expression of φ C , when ρ * < 1/2 and ρ > ρ * this gives This shows that when initially the fraction of cooperators is not too low, metastability occurs prior to fixation and the leading contribution to the MFT (27) is independent of the initial condition [55][56][57][58][59][60][61]. It stems from this result that fixation occurs much more rapidly for the VM on scale-free networks with 2 < ν < 3 than with the LD.…”

“…While these models shed light on intriguing properties of evolution on complex graphs, such as the fact that it depends on the microscopic details of the update rules, they cannot describe evolutionary processes characterized by a metastable species coexistence prior to fixation like in the paradigmatic anti-coordination games (ACGs) [1][2][3]54], see e.g. [55][56][57][58][59][60][61]. In spite of the importance of systems like ACGs, their properties have been investigated mostly on regular lattices [19,62,63] and on small-world networks [64,65].…”

Reconstructing an epigenetic landscape using a genetic ‘pulling’ approach

“…(20), and employ the WKB approximation, π m,n = π(x, y) = e −N S(x,y) , where S(x, y) is the action, N 1 is the typical protein population size at the high state, and x = m/N and y = n/N are the mRNA and protein concentrations, respectively. This yields a stationary Hamilton-Jacobi equation H(x, y, ∂ x S, ∂ y S) = 0 with Hamiltonian [69,79] While a numerical solution can be found for any set of parameters, in order to make analytical progress we consider the limit where the mRNA lifetime is short compared to that of the protein, γ 1, which holds in bacteria. In this limit, the mRNA concentration and momentum, x(t) and p x (t), instantaneously equilibrate to some (slowly varying) functions of y and p y [80].…”