# The Azimuth Project Stochastic spatial host-consumer models

The idea is that we have two kinds of organism which move around in the world, and when they bump into each other, things happen (one of them can get eaten, for example). To focus on the non-spatial case first, let’s say that we have “herbivores” and “carnivores”. The number of herbivores changes because they can reproduce (rate $\sigma$) and get eaten (rate $\lambda$), while the number of carnivores changes because they reproduce and they die of natural causes (rate $\mu$). We have two species of organism, so we have two sets of creation and annihilation operators, which we could label with $c$ for carnivore and $h$ for herbivore. These satisfy

$[c, c^\dagger] = [h, h^\dagger] = 1$

and

$[c, h] = [c^\dagger, h^\dagger] = 0.$

We bundle what we know about the ecosystem at time $t$ into a probability vector $\left|\phi(t)\right\rangle$, which lives in the Fock space we build up by acting with the creation operators on the "vacuum" $\left|0\right\rangle$. The change in time of the probability vector is given by,

$\partial_t \left|\phi(t) \right\rangle = -H\left|\phi(t) \right\rangle,$

where the “stochastic Hamiltonian” $H$ is

$H = \lambda(1 - c^\dagger) c^\dagger c h^\dagger h + \mu(c^\dagger - 1) c + \sigma (1 - h^\dagger) h^\dagger h.$

Incorporating spatial extent into the model means promoting these creation and annihilation operators to sets thereof, indexed by spatial position, and adding appropriate movement terms to the stochastic Hamiltonian. Mobilia et al. look in particular at the sustainability transition, where the carnivores are consuming just enough to keep up their numbers and not go extinct. After a lot of redefinitions and rescalings, they get an action which is an integral of a Lagrangian density of the form

$\mathcal{L}[\tilde{\psi}, \psi] = \tilde{\psi}(\partial_t + D_c(r_c - \nabla^2))\psi - u\tilde{\psi}(\tilde{\psi} - \psi)\psi + \tau\tilde{\psi}^2 \psi^2.$

The field $\psi$ is a shifted and rescaled carnivore density, and $\tilde{\psi}$ is its conjugate “response field”. The various constants are mixed-together combinations of the parameters we started with. This is, incidentally, the same form of Lagrangian which defines the “Reggeon field theory” used back in the ’70s for high-energy scattering physics, with the last term providing what I think they’d call a quadruple-Pomeron vertex. Anyway, there are a fair many steps in the derivation of this Lagrangian which I’m not sure I could justify, and of course, once we’ve squeezed the dynamics we want to study into this formalism, we’d like to get answers out of it, and that appears to involve a whole lot of Feynmanesque diagrams and RG wizardry. Questions:

1. This Lagrangian $\mathcal{L}[\tilde{\psi}, \psi]$ was derived (breezily, with many steps left as exercises to the interested reader) for the situation near the carnivore sustainability transition. Can we use it to predict how the carnivore population will change over time near that phase transition? (This is probably a fairly minor modification of calculations which have already been done somewhere for the active-to-absorbing phase transition in 2D directed percolation.)

2. Can we estimate where that transition point will be, as a function of the growth, consumption and death rates which we plug into the model? (This, I think, is harder, but it might still be possible, if the work of Benitez and Wschebor is not completely wrong.)

### References

• H. Abarbanel and D. Bronzan (1974), “Pomeranchuk singularity in a Reggeon field theory with quartic couplings”, Physical Review D 9, 3304–12 (web)
• M. Mobilia, I. T. Georgiev, and U. C. Täuber (2006), “Phase transitions and spatio-temporal fluctuations in stochastic lattice Lotka–Volterra models”, Journal of Statistical Physics 128, 12: 447–83, arXiv:q-bio/0512039.
• Vollmayr-Lee’s talks on the “field theory approach to diffusion-limited reactions”, Boulder School for Condensed Matter and Materials Physics (2009)
• B. C. Stacey, A. Gros and Y. Bar-Yam (2011), “Beyond the mean field in host-pathogen spatial ecology”, arXiv:1110.3845 [nlin.AO].
• F. Benitez and N. Wschebor (2012), “Branching rate expansion around annihilating random walks” Physical Review E 86, 010104 arXiv:1203.0235 [cond-mat.stat-mech].
• U. C. Täuber (2012), “Population oscillations in spatial stochastic Lotka–Volterra models: A field-theoretic perturbational analysis”, Journal of Physics A 45 arXiv:1206.2303 [cond-mat.stat-mech].
• F. Benitez and N. Wschebor (2013), “Some exact results in branching and annihilating random walks” Physical Review E 87, 052132 arXiv:1207.6594 [cond-mat.stat-mech].
• F. Peruani and C. F. Lee (2013), “Fluctuations and the role of collision duration in reaction-diffusion systems” Europhysics Letters 102, 58001 arXiv:1305.4466 [cond-mat.soft].

category: blog