Perturbative field-theoretical analysis of three-species cyclic predator-prey models

The RPS model consists of three particle species, subject to the cyclically coupled stochastic competition reactions

$$\begin{align} A_1+A_2 &\xrightarrow{\lambda_1^{^{\prime}}} 2A_1, \nonumber\\ A_2+A_3 &\xrightarrow{\lambda_2^{^{\prime}}} 2A_2, \nonumber\\ A_3+A_1 &\xrightarrow{\lambda_3^{^{\prime}}} 2A_3. \end{align} \tag{ 1 }$$

In this paper, we consider the cyclic-symmetric case, such that $\lambda_1^{^{\prime}} = \lambda_2^{^{\prime}} = \lambda_3^{^{\prime}} = \lambda^{^{\prime}}$. In this limit, the system displays a discrete S₃ symmetry among the three species. A brief analysis of the general asymmetric case is presented in appendix A. We note that every species interacts via a standard non-linear LV predation reaction with the subsequent species in the cycle, consuming a 'prey' particle and reproducing at the same instant. The total number of individuals is unchanged by all reactions, hence particle number conservation holds globally and locally (except for hops to neighboring lattice sites, see below).

We consider a model wherein particles from all three species perform random walks on a d-dimensional hyper-cubic lattice with L^d sites and lattice constant c. We do not restrict the number of particles per lattice site, hence we do not consider finite local carrying capacities here (the total number of particles is fixed). The rate at which particles hop between sites is given by $D/c^2$, where D denotes a macroscopic diffusion constant. The reactions (1) occur on-site, and only if two particles of differing species are present. Reaction products are put on the same lattice point as the reactants.

In the limit of large diffusivities (relative to the reaction rates $\lambda^{^{\prime}}$) the system can be considered well-mixed. Hence, the RPS rules can be approximated by the three coupled mean-field rate equations for the homogenized species concentrations and with the volume reactivities $\lambda = c^{-d} \lambda^{^{\prime}}$:

$$\begin{align} \frac{\mathrm{d}a_1(t)}{\mathrm{d}t} &=\lambda a_1(t) \big[ a_2(t)-a_3(t) \big],\nonumber \\[3pt] \frac{\mathrm{d}a_2(t)}{\mathrm{d}t} &=\lambda a_2(t) \big[ a_3(t)-a_1(t) \big], \nonumber\\[3pt] \frac{\mathrm{d}a_3(t)}{\mathrm{d}t} &=\lambda a_3(t) \big[ a_1(t)-a_2(t) \big]. \end{align} \tag{ 2 }$$

This system of ordinary differential equations yields non-linear oscillations around a neutral fixed-line which is determined by the initial conditions. The fixed-line steady-state concentrations can be obtained by setting the time derivatives to zero, resulting in

$$\begin{align} a_i^{\infty}=\frac{\rho}{3} , \quad \forall i \in \{1, 2, 3\} , \end{align} \tag{ 3 }$$

with the conserved total population density $\rho = a_1 + a_2 + a_3 = \textrm{const}$, parameterizing the fixed-line. Linearization about this three-species coexistence fixed-line yields the stability matrix

$$\begin{align} S_{\mathrm{RPS}} = \frac{\lambda\rho}{3} \begin{pmatrix} 0 & +1 & -1 \\ -1 & 0 & +1 \\ +1 & -1 & 0 \end{pmatrix} , \end{align} \tag{ 4 }$$

with eigenvalues $\{0, -i\omega_0 , i\omega_0 \}$. Since the non-zero eigenvalues are purely imaginary, the mean-field RPS system performs perpetual non-linear oscillations around the coexistence fixed point with frequency (in the linearized approximation) $\omega_0 = \rho\lambda/\sqrt{3}$.

The bulk part of the Doi–Peliti action for the stochastic spatially extended RPS model follows directly from the reactions (1) and reads ⁴

$$\begin{align} \mathcal{A}^{\mathrm{RPS}} = \sum_{i=1,2,3} \int \mathrm{d}t \, \mathrm{d}^dx \, \hat{a}_i \left( \partial_t-D\nabla^2 \right) a_i + \lambda \sum_{i=1,2,3} \int \mathrm{d}t \, \mathrm{d}^dx \, \hat{a}_i \left( \hat{a}_{i+1} -\hat{a}_i \right) a_i a_{i+1} . \end{align} \tag{ 5 }$$

For convenience, here we drop all position and time indices $(\vec{x},t)$ on the fields and identify $a_4 = a_1$. The first term describes the random nearest-neighbor hopping of the particles in the system, while the second contribution originates from the nonlinear reactions (1). As the auxiliary field $\hat{a}_i(\vec{x}, t)$ corresponds to a projection dual state, with average $\langle \hat{a}_i (\vec{x},t) \rangle = 1$, a Doi shift $\tilde{a}_i(\vec{x}, t) = \hat{a}_i(\vec{x}, t)-1$ is conveniently applied to have the new field averaged to $\langle \tilde{a}_i(\vec{x}, t) \rangle = 0$. After the Doi shift and ignoring boundary terms, the action becomes

$$\begin{align} \mathcal{A}^{\mathrm{RPS}} = \sum_{i=1,2,3} \int \! \mathrm{d}t \, \mathrm{d}^dx \, \tilde{a}_i \left( \partial_t-D\nabla^2 \right) a_i + \lambda\! \sum_{i=1,2,3} \int \! \mathrm{d}t \mathrm{d}^dx \left( \tilde{a}_i+1 \right) \left( \tilde{a}_{i+1}-\tilde{a}_i \right) a_i a_{i+1} . \end{align} \tag{ 6 }$$

This shifted action may now be viewed as a Janssen–De Dominicis response functional [42, 43] that represents the stochastic dynamics in terms of generalized Langevin equations. The $\tilde{a}_i$ fields play the role of response fields and their coupling to the particle densities, shown in the terms that are second order in these fields, entails the presence of multiplicative noise terms. This comparison leads to the formulation of equivalent Langevin stochastic differential equations encoded in the action (6),

$$\begin{align} \partial_t a_i =D \nabla^2 a_i + \lambda a_i \left( a_{i+1}-a_{i-1} \right) + \zeta_i , \end{align} \tag{ 7 }$$

where $\zeta_i(\vec{x},t)$ are the components of multiplicative noise in the system with vanishing means and correlations

$$\begin{align} \langle \zeta_i(\vec{x}_1,t_1) \, \zeta_j(\vec{x}_2, t_2) \rangle = 2 Z_{ij} \, \delta^{(d)}(\vec{x}_1-\vec{x}_2) \delta(t_1-t_2) , \end{align} \tag{ 8 }$$

with the noise correlation matrix

$$\begin{align} Z=\lambda \begin{pmatrix} \ a_1a_2 & -\frac12 a_1a_2 & -\frac12 a_1a_3 \\[6pt] -\frac12 a_1a_2 & \ a_2a_3 & -\frac12 a_2a_3 \\[6pt] -\frac12 a_1a_3 & -\frac12 a_2a_3 & \ a_1a_3 \end{pmatrix} . \end{align} \tag{ 9 }$$

Note that the noise auto-correlations Z_ii are always determined by the concentration of the predator species A_i and its respective prey $A_{i+1}$, and the scale is set by the predation rate λ. Hence the noise directly associated with a given species is solely determined by its role as predator.

Before we proceed with the perturbation theory analysis, we quickly comment on the conserved Noether current associated with the total particle number preservation in the stochastic reaction processes (1). This conservation law corresponds to a global U(1) symmetry in the Doi–Peliti action (5) for the RPS model, namely it remains invariant under the U(1) gauge transformation

$$\begin{align} \hat{a}_i^{^{\prime}} = e^{-i\theta} \hat{a}_i , \quad a_i^{^{\prime}} = e^{i\theta} a_i , \end{align} \tag{ 10 }$$

where θ is an arbitrary phase angle. The conservation law follows from the action (5) and the symmetry transformation (10) and assumes the usual form of a continuity equation

$$\begin{align} \partial_t j_0 + \nabla \cdot \vec{j} = 0 , \end{align} \tag{ 11 }$$

with

$$\begin{align} j_0 = \sum_i \hat{a}_ia_i , \quad \vec{j} = -D \sum_i \left( \hat{a}_i \nabla a_i - a_i \nabla \hat{a}_i \right) . \end{align} \tag{ 12 }$$

When choosing the Doi field $\hat{a}_i = 1$, a_i represents the density of particle species A_i and equation (12) turns into the diffusion equation for the conserved total particle number density,

$$\begin{align} \partial_t \sum_{i=1,2,3} a_i = D \nabla^2 \sum_{i=1,2,3} a_i . \end{align} \tag{ 13 }$$

We note that the symmetry (10) corresponds to the freedom of choosing the phases of the probability state a_i and its dual projected state $\hat{a}_i$.

To start, we transform the fields to describe the fluctuations around the stationary fixed-point species concentrations. To this end we employ the linear transformation

$$\begin{align} c_i(\vec{x},t) = a_i(\vec{x},t) - \frac{\rho}{3} , \quad \tilde{c}_i(\vec{x},t) = \tilde{a}_i(\vec{x},t) , \end{align} \tag{ 14 }$$

which implies $\langle c_i \rangle = 0$. In the symmetric RPS model, there is both total particle number conservation and cyclic permutation symmetry among the three distinct species. These two symmetries combined imply vanishing additive counterterms to the stationary concentrations due to fluctuations. The action for these new fluctuating fields now reads

$$\begin{align} \mathcal{A}^{\mathrm{RPS}} = \int \! \mathrm{d}t \, \mathrm{d}^dx \sum_i \bigg[& \tilde{c}_i \left( \partial_t - D\nabla^2 \right) c_i - \frac{\lambda \rho^2}{3} \tilde{c}_i \left( \tilde{c}_i - \tilde{c}_{i+1} \right) - \frac{\lambda\rho}{3} \tilde{c}_i \left( \tilde{c}_{i+1} - \tilde{c}_{i+2} \right) - \frac{\lambda\rho}{3} \tilde{c}_i^2 (c_i+c_{i+1}) \nonumber\\ & + \frac{\lambda\rho}{3} \tilde{c}_i \tilde{c}_{i+1} (c_i+c_{i+1}) - \lambda \tilde{c}_i c_i (c_{i+1}-c_{i+1}) - \lambda \tilde{c}_i^2 c_ic_{i+1} + \lambda \tilde{c}_i \tilde{c}_{i+1} c_i c_{i+1} \bigg] ,\nonumber\\ \end{align} \tag{ 15 }$$

where we again identify $c_4 = c_1$ and $c_5 = c_2$ for convenience. The quadratic part in the above action can be diagonalized by means of the following linear transformation

$$\begin{align} \begin{pmatrix} c_1 \\ c_2 \\ c_3 \end{pmatrix} = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & -\frac{1+i\sqrt{3}}{2} & -\frac{1-i\sqrt{3}}{2} \\[6pt] 1 & -\frac{1-i\sqrt{3}}{2} & -\frac{1+i\sqrt{3}}{2} \\[6pt] 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} \phi_o \\ \phi_+ \\ \phi_- \end{pmatrix} , \end{align} \tag{ 16 }$$

and

$$\begin{align} \begin{pmatrix} \tilde{c}_1 \\ \tilde{c}_2 \\ \tilde{c}_3 \end{pmatrix} = \frac{1}{\sqrt{3}} \begin{pmatrix} 1 & -\frac{1-i\sqrt{3}}{2} & -\frac{1+i\sqrt{3}}{2} \\[6pt] 1 & -\frac{1+i\sqrt{3}}{2} & -\frac{1-i\sqrt{3}}{2} \\[6pt] 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} \tilde{\phi}_o \\ \tilde{\phi}_+ \\ \tilde{\phi}_- \end{pmatrix} . \end{align} \tag{ 17 }$$

The resulting action becomes $\mathcal{A}^{\mathrm{RPS}} = \mathcal{A}^{\mathrm{RPS}}_0 + \mathcal{A}^{\mathrm{RPS}}_{\mathrm{int}}$, with the Gaussian part

$$\begin{align} \mathcal{A}^{\mathrm{RPS}}_0 = \int\!\mathrm{d}t\,\mathrm{d}^dx \Big[ \tilde{\phi}_+ \left( \partial_t-D\nabla^2+i\omega_0 \right)\! \phi_+ + \tilde{\phi}_o \left( \partial_t-D\nabla^2 \right)\! \phi_o + \tilde{\phi}_- \left( \partial_t-D\nabla^2-i\omega_0 \right)\! \phi_- \Big] ,\nonumber\\ \end{align} \tag{ 18 }$$

and the nonlinear contributions (vertices)

$$\begin{align} \mathcal{A}^{\mathrm{RPS}}_{\mathrm{int}} &= \int\!\mathrm{d}t \, \mathrm{d}^dx \, \bigg[ - \lambda\rho^2 \tilde{\phi}_+ \tilde{\phi}_- + i\frac{\lambda\rho}{3} \tilde{\phi}_o \left( \tilde{\phi}_+\phi_+ - \tilde{\phi}_-\phi_- \right) - \frac{2\lambda\rho}{\sqrt{3}} \, \tilde{\phi}_+ \tilde{\phi}_- \phi_o \nonumber\\[6pt] &\qquad\qquad\quad\ \, - i\lambda \left (\tilde{\phi}_+ \phi_-^2 - \tilde{\phi}_- \phi_+^2 - \tilde{\phi}_+ \phi_+ \phi_o + \tilde{\phi}_- \phi_- \phi_o \right) \nonumber\\[6pt] &\qquad\qquad\quad\ \, - i\frac{\lambda\rho}{6} \left[ (1-i\sqrt{3}) \tilde{\phi}_+^2 \phi_- - (1+i\sqrt{3}) \tilde{\phi}_-^2 \phi_+ \right] + \textrm{four-point vertices} \bigg] . \end{align} \tag{ 19 }$$

Here, $\omega_0 = \lambda\rho/\sqrt{3}$ denotes the zeroth-order oscillation frequency of the $\phi_{+/-}$ modes. $\mathcal{A}^{\mathrm{RPS}}_0$ is the diagonalized harmonic part of the action, while $\mathcal{A}^{\mathrm{RPS}}_{\mathrm{int}}$ represents the 'interaction' contributions for the perturbative expansion. We omit the explicit expressions for the four-point vertices as they will not contribute to the dispersion relations at one-loop order, which shall be clear in the calculation below. It is manifest that $\phi_o = (a_1+a_2+a_3-\rho)/\sqrt{3}$ represents the fluctuation of the total particle number density. Due to the conservation law (13), the φ_o mode is purely diffusive, and its exact dispersion relation in the harmonic part of the action acquires no fluctuation corrections. The $\phi_+$ and $\phi_-$ modes may be viewed as the left- and right-rotating waves in the system. At tree level, they are purely oscillating modes without dissipation, i.e. the real part of the mass term vanishes.

We have applied a field-theoretical perturbation theory to one-loop order and calculated the renormalized diffusion constant D_r , damping constant γ_r , and oscillation frequency ω_r . ⁵ To all orders in the fluctuation expansion extending beyond the mean-field approximation, there should be no correction to the two-point vertex function $\Gamma_{\tilde{\phi}_{o}\phi_{o}}^{(1,1)}$ or propagator self-energy for the φ_o mode, whence it retains its tree-level purely diffusive dispersion relation as dictated by the conservation law. For the $\phi_\pm$ modes, the one-loop Feynman diagrams for the two-point vertex functions $\Gamma_{\tilde{\phi}_{\pm}\phi_{\pm}}^{(1,1)}$ are displayed in figure 2. The solid lines represent the $\phi_+$ and $\phi_-$ propagators, while the dashed lines indicate the diffusive mode φ_o . In our convention, time and hence momentum always flow from right to left in the Feynman diagrams. The analytic expression for the two-point vertex function $\Gamma_{\tilde{\phi}_{\pm}\phi_{\pm}}^{(1,1)}$ reads

$$\begin{align} \Gamma_{\tilde{\phi}_{\pm}\phi_{\pm}}^{(1,1)}(p,\omega) & = i\omega+Dp^2+u_0 \pm i\omega_0 + \frac{\sqrt{3}\lambda\omega_0}{6D}\int_k I\!\left(\frac{u_0\pm i\omega_0}{2D}\right) \nonumber\\[6pt] & \quad -\frac{\sqrt{3}\lambda\omega_0}{6D}(1\pm i\sqrt{3})\int_k I\!\left(\frac{u_0\mp i\omega_0}{D}\right) - \frac{\lambda\omega_0^2}{D^2}\int_k\frac{1}{k^2+\frac{u_0}{D}} \, I\!\left(\frac{u_0 \mp i\omega_0}{D} \right) , \end{align} \tag{ 20 }$$

where $\int_k$ is short-hand for the d-dimensional wavevector integral $\int \! \mathrm{d}^dk / (2\pi)^d$, and the function I is defined as

$$\begin{align} I(m^2)=\frac{1}{k^2+m^2+\frac{i\omega+Dp^2}{2D}+p\cdot k} . \end{align} \tag{ 21 }$$

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The damping constant u₀ in equation (20) is introduced to regularize the IR singularities that emerge in later calculations. A nonzero renormalized u_r will be generated by the fluctuations, but it is of higher order in the perturbation expansion; thus, we need to take $u_0\rightarrow 0$ at the end. This two-point function can also be expressed with the renormalized parameters as

$$\begin{align} \Gamma_{\tilde{\phi}_{\pm}\phi_{\pm}}^{(1,1)}(p,\omega)=Z_{\phi_\pm} \left( i\omega+D_rp^2+u_r\pm i\omega_r \right) , \end{align} \tag{ 22 }$$

where $Z_{\phi_\pm}$ absorbs all related wave function renormalizations (ultraviolet/UV divergences) in (20). The renormalized diffusivity D_r , damping u_r , and oscillation frequency ω_r can be inferred accordingly from the explicit one-loop result (20) and (22). We obtain the following formal expressions for D_r , u_r , and ω_r ,

$$\begin{align} D_r&=D + \frac{\sqrt{3}\lambda\omega_0}{6dD}\int_k\frac{k^2}{(k^2 \pm \frac{i\omega_0}{2D})^3} - \frac{\sqrt{3}\lambda\omega_0}{6dD}(1\pm i\sqrt{3})\int_k\frac{k^2}{(k^2 \mp \frac{i\omega_0}{D})^3} \nonumber\\[6pt] &\quad - \frac{\lambda\omega_0^2}{dD^2}\int_k \frac{1}{(k^2\mp \frac{i\omega_0}{D})^3} , \nonumber\\[6pt] u_r\pm i\omega_r &= \pm i\omega_0 \Bigg[1-\frac{\lambda}{2D}\int_k \frac{1}{k^2 + \frac{u_0}{D}} \pm \frac{i\sqrt{3}\lambda}{6D}\int_k\left(\frac{1}{k^2 \mp \frac{i\omega_0}{D}} - \frac{1}{k^2 \pm \frac{i\omega_0}{2D}}\right) \nonumber\\[6pt] & \qquad\qquad +\frac{\sqrt{3}\lambda\omega_0}{12D^2}\int_k\frac{1}{(k^2 \pm \frac{i\omega_0}{2D})^2} - \frac{\sqrt{3}\lambda\omega_0}{12D^2}(1\mp i\sqrt{3})\int_k\frac{1}{(k^2 \pm \frac{i\omega_0}{2D})^2} \Bigg] . \end{align} \tag{ 23 }$$

Hence we indeed see that a non-zero damping u_r is generated at one-loop order from the fluctuations, in agreement with Monte Carlo simulation data [14]. The IR divergence at one-loop order that appears when in the renormalized oscillation frequency ω_r in low dimensions $d \leqslant 2$ is caused by the contribution of the massless φ_o mode as u₀ is set to zero. The IR divergence at one-loop order that appears in the renormalized oscillation frequency ω_r in low dimensions $d \leqslant 2$ is caused by the superposition of $\phi_+$ and $\phi_-$ modes as u₀ is set to zero. Our analysis of the one-loop results shows that the $\phi_\pm$ modes are inherently massive, as they acquire non-zero damping u_r . Thus, the IR divergence can be resolved simply by maintaining a finite value for u₀. For dimensions $d \geqslant 2$, there are also UV divergences present. Nevertheless, all systems of interest have a natural cutoff in the UV limit, which is defined by the lattice constant c. The renormalized variables in different physically accessible dimensions are presented below.

2.5.1.  d = 1.

In one dimension, the renormalized parameters are

$$\begin{align} \mathrm{Re} D_r &= D+\lambda\sqrt{\frac{D}{\omega_0}}\left(\frac{7\sqrt{2}}{64}+\frac{\sqrt{6}}{192} -\frac{\sqrt{3}}{48}\right) , \nonumber\\[8pt] \omega_r &= \omega_0\Bigg[1-\frac{\lambda}{4D}\sqrt{\frac{D}{u_0}}-\frac{\lambda}{D}\sqrt{\frac{D}{\omega_0}} \left(\frac{\sqrt{3}}{8} + \frac{\sqrt{6}}{32} + \frac{\sqrt{2}}{32}\right)\Bigg] , \nonumber\\[8pt] u_r &= \frac{\lambda\omega_0}{D}\sqrt{\frac{D}{\omega_0}}\left(\frac{\sqrt{3}}{8} - \frac{\sqrt{6}}{32} + \frac{\sqrt{2}}{32}\right) ; \end{align} \tag{ 24 }$$

some numerical results are depicted in figure 3. We observe that the renormalized frequency ω_r diverges when $u_0 \rightarrow 0$. This IR divergence indicates strong fluctuation corrections to the oscillation frequency in the perturbative regime where reaction rates are small, and $u_0 \to u_r \sim \lambda$ is of first order in the reactivity. However, numerical simulations are invariably performed outside this regime for the sake of computational efficiency, as large reaction rates are used to avoid long relaxation times. Thus, no strong fluctuation-induced renormalization have been encountered in the simulations. To one-loop order, $u_r \gt 0$, which indicates the stability of the system's spatially homogeneous ground state with respect to fluctuations.

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2.5.2.  d = 2.

In two dimensions, the renormalized variables read

$$\begin{align} \mathrm{Re} D_r &= D+ \frac{3\lambda}{32\pi} , \nonumber\\[8pt] \omega_r &= \omega_0\Bigg[1-\frac{\lambda}{8D\pi}\ln \bigg(\frac{D\Lambda^2}{u_0}\bigg)-\frac{\lambda}{D\pi}\left(\frac{1}{16}+\frac{\sqrt{3}\pi}{24}\right)\Bigg] , \nonumber\\[8pt] u_r &= \frac{\sqrt{3}\lambda\omega_0}{48D\pi}\left(3+2\ln 2\right) . \end{align} \tag{ 25 }$$

Note that we have explicitly introduced the cutoff $\Lambda \sim \pi / c$ to regularize the UV divergence for the renormalized oscillation frequency. We plot ω_r and the damping parameter u_r in figure 4. The renormalized diffusion constant D_r only linearly depends on the parameter λ. As in the one-dimensional case, the oscillation frequency ω_r diverges as $u_0\rightarrow 0$. We note that the damping constant u_r is also positive at d = 2.

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2.5.3.  d = 3.

In three dimensions, we may safely set $u_0 = 0$ and the renormalized system parameters follow as

$$\begin{align} \mathrm{Re} D_r &= D+ \frac{\lambda}{\pi}\sqrt{\frac{\omega_0}{D}}\left(\frac{\sqrt{3}}{192} -\frac{\sqrt{6}-5\sqrt{2}}{384}\right) , \nonumber\\ \omega_r &= \omega_0\Bigg[1-\frac{\lambda}{D\pi} \sqrt{\frac{\omega_0}{D}}\left(\frac{\sqrt{3}}{96} + \frac{5\sqrt{6}}{192} + \frac{\sqrt{2}}{64}\right)\Bigg] , \nonumber\\ u_r &= \frac{\lambda\omega_0}{D\pi}\sqrt{\frac{\omega_0}{D}}\left(-\frac{\sqrt{3}}{96} + \frac{5\sqrt{6}}{192} - \frac{\sqrt{2}}{64}\right) . \end{align} \tag{ 26 }$$

We notice that for d > 2 the IR divergences disappear and fluctuation effects become generally weak. u_r is also positive for d = 3, as displayed in figure 5.

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We have found that in dimensions d = 1, 2, and 3, the diffusivity D experiences an upward shift, indicating that fluctuations enhance the diffusion. Our results, depicted in figures 3–5, show that as the reaction rate increases, the characteristic frequency ω_r shifts to smaller values, as the reactions drive the system toward a spatially more homogeneous distribution, leading to slower oscillations. The decline in the characteristic frequencies is in accordance with the numerical simulation data in [14]. In contrast with the LV model, Monte Carlo simulations of the RPS system do not appear to show strong renormalization effects [14], although both models feature a logarithmic divergence in two dimensions. The IR divergence in the RPS model appears as a consequence of the fact that the corrections are built using the Gaussian theory which has zero damping, precisely as in the LV model. The positive fluctuation-induced damping µ_r , in contrast to the possibly negative one in the LV model, indicates that the system remains stable against the spontaneous emergence of spatio-temporal structures.

The following discussion of the mean-field theory, Doi–Peliti action, and Langevin representation for the spatially extended stochastic ML model was laid out in detail in [20]. Here we summarize the pertinent points needed for our comparison with the RPS model and the computation of the fluctuation corrections to one-loop order. Following the conventions in [20], the reactions in the ML model read

$$\begin{align} B_i+B_{i+1} &\xrightarrow{\sigma^{^{\prime}}} B_i , \nonumber\\ B_ i&\xrightarrow{\mu} 2B_i , \nonumber\\ 2B_i &\xrightarrow{\kappa^{^{\prime}}} B_i , \end{align} \tag{ 27 }$$

where $i = 1,2,3$ denotes the three competing species, and we identify $B_4 = B_1$ as before. In contrast to the RPS system, the reactions in the ML model do not conserve the total particle number. The first two reactions implement predation and reproduction independently, while the third reaction implements 'soft' site occupation constraint to effectively represent a finite carrying capacity. As in the RPS model, we consider a model wherein particles from all three species perform random walks on a d-dimensional hyper-cubic lattice with L^d sites and lattice constant c. In the large diffusivity limit, the dynamics is governed by the mean-field rate equations

$$\begin{align} \frac{\mathrm{d}b_1(t)}{\mathrm{d}t}&=b_1(t)\Big(-\sigma b_3(t)+\mu-\kappa b_1(t) \Big) , \nonumber\\[8pt] \frac{\mathrm{d}b_2(t)}{\mathrm{d}t}&=b_2(t)\Big(-\sigma b_1(t)+\mu-\kappa b_2(t) \Big) , \nonumber\\[8pt] \frac{\mathrm{d}b_3(t)}{\mathrm{d}t}&=b_3(t)\Big(-\sigma b_2(t)+\mu-\kappa b_3(t) \Big) , \end{align} \tag{ 28 }$$

where $\sigma = c^d\sigma^{^{\prime}}$ and $\kappa = c^d\kappa^{^{\prime}}$ are the volume reaction rates. Instead of a fixed line defined by the initial condition, the ML system displays a unique fixed point at mean-field level. By setting the time derivatives to be 0, the steady-state concentrations are found to be

$$\begin{align} b_i^{\infty}=\frac{\mu}{\sigma+\kappa} ,\quad \forall i \in \{1, 2, 3\} , \end{align} \tag{ 29 }$$

and the associated stability matrix reads

$$\begin{align} S_{\mathrm{ML}}=-\frac{\mu}{\sigma+\kappa} \begin{pmatrix} \kappa & 0 & \sigma\\ \sigma & \kappa & 0\\ 0 & \sigma & \kappa \end{pmatrix} . \end{align} \tag{ 30 }$$

Its eigenvalues at the coexistence fixed point are $\{-\mu, -\mu (2\kappa-\sigma\pm i\sqrt{3}\sigma) / 2(\sigma+\kappa)\}$. The first eigenvalue $-\mu$ is always negative which implies the stability of the corresponding eigenmode, namely the exponential decay of the total particle number, see below. The imaginary part of the two complex conjugate eigenvalues, $\pm\sqrt{3}\mu\sigma/2(\sigma+\kappa)$, represents the frequency of temporal oscillations for the associate modes, whose amplitudes are either exponentially damped or growing. When $2\kappa\gt \sigma$, the real part of the complex eigenvalues is negative and the limit circles are stable. Otherwise, for $2\kappa\lt \sigma$, the limit circles are unstable and one observes the spontaneous formation of spiral structures in the system. In the vicinity of the Hopf bifurcation at $2\kappa = \sigma$, the time evolution of the two modes corresponding to the complex conjugate eigenvalues becomes much slower than the fast relaxing mode, which introduces a natural time scale separation. As a consequence of the critical slowing down near the Hopf bifurcation, the fast relaxing mode can be integrated out and the system is effectively governed by the complex time-dependent Ginzburg–Landau equation [20].

The Doi–Peliti action follows from the reactions of the ML model and reads

$$\begin{align} \mathcal{A}^{\mathrm{ML}} & =\sum_{i=1,2,3}\int\!\mathrm{d}t\,\mathrm{d}^dx \bigg[\hat{b}_i \left( \partial_t -D\nabla^2\right)\! b_i+\mu\,\hat{b}_ib_i \Bigl( 1-\hat{b}_i \Bigr)+\kappa\,\hat{b}_ib_i^2 \Bigl( \hat{b}_i-1 \Bigr) \nonumber\\ & \qquad \quad \qquad \qquad\ +\sigma\,\hat{b}_ib_ib_{i+1}\Bigl( \hat{b}_{i+1}-1 \Bigr) \bigg] . \end{align} \tag{ 31 }$$

This action does not obey the U(1) global symmetry present in the RPS model; indeed, the total particle number is not conserved under the ML reactions (27). Following the Doi shift to the fluctuating auxiliary fields $\tilde{b}_i(\vec{x},t) = \hat{b}_i(\vec{x},t)-1$, the action becomes

$$\begin{align} \mathcal{A}^{\mathrm{ML}}& =\sum_{i=1,2,3}\int\!\mathrm{d}t\,\mathrm{d}^dx \bigg[ \tilde{b}_i \left( \partial_t-D\nabla^2-\mu \right)\! b_i-\mu\,\tilde{b}_i^2b_i+\kappa\,\tilde{b}_ib_i^2 \Bigl( \tilde{b}_i+1 \Bigr) \nonumber\\ & \qquad \quad \qquad \qquad\ +\sigma\,\tilde{b}_{i+1}b_ib_{i+1} \Bigl( \tilde{b}_ i+1 \Bigr) \bigg] . \end{align} \tag{ 32 }$$

As in the RPS model above, we may now view this shifted action as a Janssen–De Dominicis functional which is equivalent to the corresponding generalized Langevin equations

$$\begin{align} \partial_tb_i=D\nabla^2b_i+\mu b_i-\kappa b_i^2-\sigma b_{i}b_{i+2}+\xi_i , \end{align} \tag{ 33 }$$

where $\xi_i(\vec{x},t)$ represent the multiplicative noise components with correlators

$$\begin{align} \langle\xi_i(\vec{x}_1,t_1)\,\xi_j(\vec{x}_2,t_2)\rangle=2\Xi_{ij}\,\delta^{(d)}(\vec{x}_1-\vec{x}_2)\delta(t_1-t_2) , \end{align} \tag{ 34 }$$

with

$$\begin{align} {\Xi}=\begin{pmatrix} \mu b_1-\kappa b_1^2 & -\frac{\sigma}{2}b_1b_2 & -\frac{\sigma}{2}b_1b_3 \\[6pt] -\frac{\sigma}{2}b_1b_2 & \mu b_2-\lambda b_2^2 & -\frac{\sigma}{2}b_2b_3 \\[6pt] -\frac{\sigma}{2}b_1b_3 & -\frac{\sigma}{2}b_2b_3 & \mu b_3-\lambda b_3^2 \end{pmatrix} . \end{align} \tag{ 35 }$$

Before diagonalizing the quadratic action, we first shift to fluctuating fields, $\tilde{d}_i(\vec{x},t) = \tilde{b}_i(\vec{x},t)$ and $d_i(\vec{x},t) = b_i(\vec{x},t)-\frac{\mu}{\sigma+\kappa}-C$. Here C is a counterterm which encodes the fluctuation corrections to the average concentrations. Owing to the cyclic symmetry among the three different species, we only need to introduce a single counterterm. The harmonic part of the action in terms of the new fields $\tilde{d}_i$ and d_i reads

$$\begin{align} \begin{aligned} \mathcal{A}^{\mathrm{ML}}_0=\sum_i\int\!\mathrm{d}t\,\mathrm{d}^dx \bigg[ \tilde{d}_i \left( \partial_t-D\nabla^2+\frac{\kappa\mu}{\kappa+\sigma}+(2\kappa+\sigma)C \right)\!d_i +\left( \frac{\sigma\mu}{\kappa+\sigma}+\sigma C \right)\!\tilde{d}_id_{i+1} \bigg] . \end{aligned} \end{align} \tag{ 36 }$$

Since the counterterm C is of first order in the perturbative expansion parameters, up to zeroth order the harmonic part of the action can be diagonalized by the following linear transformation

$$\begin{align} \begin{pmatrix} \tilde{d}_1 \\ \tilde{d}_2 \\ \tilde{d}_3 \\ \end{pmatrix}=\frac{1}{\sqrt{3}} \begin{pmatrix} 1 & -\frac{1-i\sqrt{3}}{2} & -\frac{1+i\sqrt{3}}{2} \\[6pt] 1 & -\frac{1+i\sqrt{3}}{2} & -\frac{1-i\sqrt{3}}{2} \\[6pt] 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} \tilde{\psi}_o\\ \tilde{\psi}_+\\ \tilde{\psi}_- \end{pmatrix} , \end{align} \tag{ 37 }$$

and

$$\begin{align} \begin{pmatrix} d_1 \\ d_2 \\ d_3 \\ \end{pmatrix}=\frac{1}{\sqrt{3}} \begin{pmatrix} 1 & -\frac{1+i\sqrt{3}}{2} & -\frac{1-i\sqrt{3}}{2} \\[6pt] 1 & -\frac{1-i\sqrt{3}}{2} & -\frac{1+i\sqrt{3}}{2} \\[6pt] 1 & 1 & 1 \end{pmatrix} \begin{pmatrix} \psi_o\\ \psi_+\\ \psi_- \end{pmatrix} . \end{align} \tag{ 38 }$$

After applying this linear transformation, the action can be expressed as $\mathcal{A}^{\mathrm{ML}} = \mathcal{A}^{\mathrm{ML}}_0+\mathcal{A}^{\mathrm{ML}}_s+\mathcal{A}^{\mathrm{ML}}_{\mathrm{int}}$, representing, respectively, the harmonic, source, and non-linear interaction terms. Again, we omit the four-point vertices, since they will not contribute to the dispersion relation renormalizations at one-loop order. The other terms are

$$\begin{align} \mathcal{A}^{\mathrm{ML}}_0=\int\!\mathrm{d}t\,\mathrm{d}^dx \bigg[ &\tilde{\psi}_o \left( \partial_t-D\nabla^2+\mu+2(\sigma+\kappa)C \right)\!\psi_o +\tilde{\psi}_+ \left(\partial_t-D\nabla^2+\gamma_0+i\nu_0\right)\!\psi_+ \nonumber\\[8pt] &+\tilde{\psi}_-\left(\partial_t-D\nabla^2+\gamma_0-i\nu_0\right)\psi_- +\frac12 \left( \sigma+4\kappa+i\sqrt{3} \sigma \right)\!C \tilde{\psi}_+\psi_+ \nonumber\\[8pt] & +\frac12 \left( \sigma+4\kappa-i\sqrt{3}\sigma \right)\!C \tilde{\psi}_-\psi_- \bigg] , \end{align} \tag{ 39 }$$

$$\begin{align} \mathcal{A}^{\mathrm{ML}}_s=\int\!\mathrm{d}t\,\mathrm{d}^dx \bigg[ &\left( \mu C+(\kappa+\sigma)C^2 \right) \left( \sqrt{3}\tilde{\psi}_o+\tilde{\psi}_o^2 \right) \nonumber\\[8pt] &+ \left( -\frac{3\mu^2\sigma}{(\kappa+\sigma)^2} + 2\mu\frac{\kappa-2\sigma}{\kappa+\sigma} C +(2\kappa-\sigma)C^2 \right)\! \tilde{\psi}_+\tilde{\psi}_-\bigg] , \end{align} \tag{ 40 }$$

$$\begin{align} \mathcal{A}^{\mathrm{ML}}_\mathrm{int} & = \int\!\mathrm{d}t\,\mathrm{d}^dx \bigg[ \left( \frac{\mu(\kappa-2\sigma)}{\sqrt{3}(\kappa+\sigma)}+\frac{2\kappa-\sigma}{\sqrt{3}}\,C \right) \left( \tilde{\psi}_+^2\psi_-+\tilde{\psi}_-^2\psi_+ \right) + \frac{\kappa+\sigma}{\sqrt{3}}\tilde{\psi}_o\psi_o^2 \nonumber\\[8pt] &\quad +\left( \frac{\mu}{\sqrt{3}}+\frac{2(\kappa+\sigma)}{\sqrt{3}}\,C \right)\!\tilde{\psi}_o^2\psi_o +\left( \frac{\mu(2\kappa-\sigma)}{\sqrt{3}(\kappa+\sigma)}+\frac{4\kappa+\sigma}{\sqrt{3}}\,C \right)\! \left( \tilde{\psi}_o\tilde{\psi}_+\psi_+ + \tilde{\psi}_o\tilde{\psi}_-\psi_- \right) \nonumber\\[8pt] &\quad +\frac{\sqrt{3}}{6} \left( 2\kappa-\sigma+\sqrt{3}i\sigma \right)\!\tilde{\psi}_-\psi_+^2 + \frac{\sqrt{3}}{6} \left( 2\kappa-\sigma-\sqrt{3}i\sigma \right)\!\tilde{\psi}_+\psi_-^2 +\frac{2\kappa-\sigma}{\sqrt{3}} \,\tilde{\psi}_o\psi_+\psi_- \nonumber\\[8pt] &\quad + \frac{\sqrt{3}}{6}\!\left( 4\kappa+\sigma-\sqrt{3}i\sigma \right)\!\tilde{\psi}_-\psi_o\psi_- + \frac{\sqrt{3}}{6}\!\left( 4\kappa+\sigma+\sqrt{3}i\sigma \right)\!\tilde{\psi}_+\psi_o\psi_+ + \textrm{four-point vertices} \bigg] ,\nonumber\\ \end{align} \tag{ 41 }$$

where $\gamma_0 = \mu(2\kappa-\sigma)/2(\sigma+\kappa)$ and $\nu_0 = \sqrt{3}\mu\sigma/2(\sigma+\kappa)$. The ψ_o mode corresponds to the fluctuation of the total particle density and decays exponentially at tree level. In contrast to the RPS model, the ML $\psi_\pm$ modes display non-vanishing dissipation γ₀ already on the mean-field level. As mentioned above, a Hopf bifurcation occurs at $2\kappa = \sigma$; when $2\kappa \lt \sigma$, the system is unstable and spiral structures are spontaneously generated. In the perturbative regime, the assumed tiny fluctuation corrections should not change the overall stability features, but will only shift the Hopf bifurcation point by a small amount.

Prior to calculating the renormalized quantities, we need to determine the counterterm C by requiring the average fluctuation of the total density to be zero, $\langle\psi_o\rangle = 0$. Up to one-loop order, the contributions to $\langle\psi_o\rangle$ are shown in figure 6. We note that the second diagram in figure 6 is of second order and thus can be dropped. The corresponding analytic expression results in

$$\begin{align} C=-\frac{\mu\sigma(2\kappa-\sigma)}{2D(\kappa+\sigma)^2}\int_k\frac{1}{k^2+\frac{\gamma_0}{D}} . \end{align} \tag{ 42 }$$

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We may now proceed to the fluctuation renormalization of the two-point vertex functions $\Gamma^{(1,1)}_{\tilde{\psi}_o\psi_o}$ and $\Gamma^{(1,1)}_{\tilde{\psi}_{\pm}\psi_{\pm}}$, which encode the self-energies entering the dispersion relation of the ψ_o and $\psi_\pm$ modes. As the total particle number is not conserved in the ML model, the dispersion relation acquires non-trivial corrections from the perturbation expansion. The one-loop diagrams that contribute to the vertex function $\Gamma^{(1,1)}_{\tilde{\psi}_o\psi_o}$ are pictured in figure 7. The last diagram is of higher order and thus can be omitted; this results in

$$\begin{align} \Gamma^{(1,1)}_{\tilde{\psi}_o\psi_o}(p,\omega) & = i\omega+Dp^2+\mu+2(\kappa+\sigma)C -\frac{\mu(\kappa+\sigma)}{3D}\int_k I\!\left(\frac{\mu}{D}\right) \nonumber\\[8pt] &\quad -\frac{2(\sigma+\kappa)\gamma_0}{3D\mu} \left(\gamma_0-\sqrt{3}\nu_0\right) \int_k I\!\left(\frac{\gamma_0}{D} \right) - \frac{2\sqrt{3}(\sigma+\kappa)\gamma_0\nu_0}{3D^2\mu}\,(\gamma_0+\mu) \int_k\frac{1}{k^2+\frac{\gamma_0}{D}} \, I\!\left(\frac{\gamma_0}{D}\right) .\nonumber\\ \end{align} \tag{ 43 }$$

Provided $\gamma_0 \gt 0$, all corrections at one-loop level are real, and no imaginary part appears in the mass term of the ψ_o mode, hence there are no total particle number oscillations. However, for $\gamma_0 \lt 0$ the system exhibits emergent oscillations of the total particle number, indicating the spontaneous formation of spatio-temporal structures. As the renormalized two-point vertex function can also be written as

$$\begin{align} \Gamma^{(1,1)}_{\tilde{\psi}_o\psi_o}(p,\omega)=Z_{\phi_o} \left( i\omega + D_{r}^op^2 +\mu_r \right) , \end{align} \tag{ 44 }$$

the renormalized diffusivity $D_{r}^o$ and mass parameter µ_r can be calculated accordingly. Here, $Z_{\phi_o}$ absorbs all wave function renormalizations. Since the rotating wave modes $\psi_\pm$ acquires a different diffusivity renormalization from the o mode, we carefully distinguish the renormalized quantities $D_r^{\pm}$ and $D_r^{o}$.

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The explicit expressions for $D_{r}^o$ and µ_r read

$$\begin{align} \mu_r &= \mu \bigg[ 1-\frac{2(\sigma+\kappa)}{3D}\frac{\gamma_0}{\mu} \int_k \frac{1}{k^2+\frac{\gamma_0}{D}}-\frac{\sigma+\kappa}{3D} \int_k \frac{1}{k^2+\frac{\mu}{D}} - \frac{\mu(\sigma+\kappa)}{6D^2} \int_k \frac{1}{(k^2+\frac{\mu}{D})^2} \nonumber\\[8pt] &\quad\quad\ -\frac{(\sigma+\kappa)\gamma_0}{3D^2} \left( 1+\frac{2\sqrt{3}\nu_0\gamma_0}{\mu^2} \right) \int_k\frac{1}{(k^2+\frac{\gamma_0}{D})^2} - \frac{\sqrt{3}(\sigma+\kappa)}{3D^3} \frac{\nu_0\gamma_0(\gamma_0+\mu)}{\mu}\int_k \frac{1}{(k^2+\frac{\gamma_0}{D})^3} \bigg] , \nonumber\\[8pt] D_{r}^o &= D - \frac{\mu(\sigma+\kappa)}{3dD} \int_k \frac{k^2}{(k^2+\frac{\mu}{D})^3} -\frac{2(\sigma+\kappa)}{3dD}\frac{\gamma_0}{\mu}\left(\gamma_0-\sqrt{3}\nu_0\right) \int_k \frac{k^2}{(k^2+\frac{\gamma_0}{D})^3} \nonumber\\[8pt] &\quad - \frac{2\sqrt{3}(\sigma+\kappa)}{3dD^2}\frac{\nu_0\gamma_0}{\mu} \left( \gamma_0+\mu \right) \int_k \frac{k^2}{(k^2+\frac{\gamma_0}{D})^4}. \end{align} \tag{ 45 }$$

After evaluating the integrals one arrives at

$$\begin{align} \mu_r & = 1 - \frac{\Gamma(1-d/2)}{2^d\pi^{d/2}} \left( \frac{\sigma+\kappa}{3D}\Big(\frac{\mu}{D}\Big)^{d/2-1} +\frac{2(\sigma+\kappa)\gamma_0}{3D\mu} \Big(\frac{\gamma_0}{D}\Big)^{d/2-1} \right) \nonumber\\[8pt] & \quad - \frac{\Gamma(2-d/2)}{2^d\pi^{d/2}} \biggl[\frac{\mu(\sigma+\kappa)}{6D^2}\Big(\frac{\mu}{D}\Big)^{d/2-2} +\frac{(\sigma+\kappa)\gamma_0}{3D^2} \biggl( 1+\frac{2\sqrt{3}\nu_0\gamma_0}{\mu^2} \biggr) \Big(\frac{\gamma_0}{D}\Big)^{d/2-2}\biggr] \nonumber\\[8pt] & \quad - \frac{\Gamma(3-d/2)}{2^d\pi^{d/2}}\frac{\sqrt{3}(\sigma+\kappa)\nu_0\gamma_0}{3D^3\mu}\left(\gamma_0+\mu\right)\Big(\frac{\gamma_0}{D}\Big)^{d/2-3} ,\nonumber \\[8pt] D_r^o & = D - \frac{\Gamma(2-d/2)}{2^{d+2}\pi^{d/2}} \left[ \frac{\mu(\sigma+\kappa)}{3D} \Big(\frac{\mu}{D}\Big)^{d/2-2} + \frac{2(\sigma+\kappa)\gamma_0}{3D\mu}\left(\gamma_0-\sqrt{3}\nu_0\right) \Big(\frac{\gamma_0}{D}\Big)^{d/2-2} \right] \nonumber\\[8pt] & \quad - \frac{\Gamma(3-d/2)}{3 \cdot 2^{d+2}\pi^{d/2}}\frac{2\sqrt{3}(\sigma+\kappa)}{3D^2\mu}\nu_0\gamma_0 (\gamma_0+\mu) \Big(\frac{\gamma_0}{D}\Big)^{d/2-3} . \end{align} \tag{ 46 }$$

For $d\geqslant 2$, UV divergences in µ_r are manifest; but since the lattice constant c serves as a natural UV cutoff in lattice models, we will not discuss these UV divergences further.

The renormalized vertex function $\Gamma^{(1,1)}_{\tilde{\psi}_{\pm}\psi_{\pm}}$ can also be calculated according to the one-loop diagrams in figure 8, resulting in

$$\begin{align} \Gamma^{(1,1)}_{\tilde{\psi}_{\pm}\psi_{\pm}}(p,\omega) & = i\omega+Dp^2+\gamma_0 \pm i \nu_0 +\frac{\sigma+\kappa}{\mu} \left(\gamma_0+\mu\pm i\nu_0\right) C \nonumber\\[8pt] & \quad - \frac{(\kappa+\sigma)(\gamma_0-\sqrt{3}\nu_0)}{3D\mu}\,(\gamma_0\mp i\nu_0) \int_k I\!\left(\frac{\gamma_0\mp i\nu_0}{D}\right) \nonumber\\[8pt] & \quad -\frac{(\sigma+\kappa)\gamma_0}{3D\mu}\,(\gamma_0+\mu\pm i\nu_0) \int_k I\!\left(\frac{\gamma_0+\mu\pm i\nu_0}{2D}\right) \nonumber\\[8pt] & \quad -\frac{2\sqrt{3}(\sigma+\kappa)\nu_0}{3D^2\mu}\,(\gamma_0^2+\nu_0^2) \int_k\frac{1}{k^2+\frac{\gamma_0}{D}}\, I\!\left(\frac{\gamma_0\mp i\nu_0}{D}\right) \nonumber\\[8pt] & \quad -\frac{\sqrt{3}(\sigma+\kappa)\gamma_0\nu_0}{3D^2\mu}\,(\gamma_0+\mu\pm i\nu_0) \int_k\frac{1}{k^2+\frac{\gamma_0}{D}}\,I\!\left(\frac{\gamma_0+\mu\pm i\nu_0}{2D}\right) . \end{align} \tag{ 47 }$$

Upon invoking the relation with renormalized quantities

$$\begin{align} \Gamma^{(1,1)}_{\tilde{\psi}_{\pm}\psi_{\pm}}(p,\omega) = Z_{\phi_\pm} \left( i\omega + D_{r}^\pm p^2 +\gamma_r \pm i\nu_r \right) , \end{align} \tag{ 48 }$$

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the expressions for the renormalized parameters γ_r and ν_r are readily inferred,

$$\begin{align} \gamma_r \pm i\nu_r & = \gamma_0 \pm i\nu_0 +(\sigma+\kappa) \bigg[ M_1^{(\pm)}\frac{\Gamma(1-d/2)}{2^d\pi^{d/2}}\left(\frac{\gamma_0}{D}\right)^{d/2-1}\nonumber\\[8pt] & \quad +M_2^{(\pm)}\frac{\Gamma(1-d/2)}{2^d\pi^{d/2}}\left(\frac{\gamma_0^2+\nu_0^2}{D^2}\right)^{\!(d-2)/4} \exp\!\left(\mp i \frac{d-2}{2} \theta\right) \nonumber\\[8pt] & \quad +M_3^{(\pm)}\frac{\Gamma(1-d/2)}{2^d\pi^{d/2}}\left(\frac{(\mu+\gamma_0)^2+\nu_0^2}{4D^2}\right)^{\!(d-2)/4} \exp\!\left(\pm i \frac{d-2}{2} \eta\right) \nonumber\\[8pt] & \quad +M_4^{(\pm)}\frac{\Gamma(2-d/2)}{2^d\pi^{d/2}}\left(\frac{\gamma_0^2+\nu_0^2}{D^2}\right)^{\!(d-4)/4} \exp\!\left(\mp i \frac{d-4}{2} \theta\right) \nonumber\\[8pt] & \quad +M_5^{(\pm)}\frac{\Gamma(2-d/2)}{2^d\pi^{d/2}}\left(\frac{(\mu+\gamma_0)^2+\nu_0^2}{4D^2}\right)^{\!(d-4)/4} \exp\!\left(\pm i \frac{d-4}{2} \eta\right) \bigg] , \end{align} \tag{ 49 }$$

where the coefficients $M_i^{(\pm)}$ are defined in equation (B.2) in the appendix, and the angles θ and η are given by $\tan \theta = \nu_0 / \gamma_0$ and $\tan \eta = \nu_0 / (\gamma_0+\mu)$. We note that at odd dimensions d, the first term in the bracket in equation (49) switches from real to imaginary as γ₀ changes its sign; however, at even dimensions, this term is always real. Finally, the renormalized diffusivity reads

$$\begin{align} D_r^{\pm} & = D -\frac{\kappa+\sigma}{d\,2^d\pi^{d/2}} \bigg[ \Gamma(1-d/2)P_1^{(\pm)} \left(\frac{\gamma_0^2+\nu_0^2}{D^2}\right)^{\!(d-2)/4} \exp\!\left(\mp i\frac{d-2}{2}\theta\right) \nonumber\\[8pt] & \quad +\Gamma(2-d/2)P_2^{(\pm)}\left(\frac{\gamma_0^2+\nu_0^2}{D^2}\right)^{(d-4)/4} \exp\!\left(\mp i\frac{d-4}{2}\theta\right) \nonumber\\[8pt] & \quad +\frac{\Gamma(3-d/2)}{2}P_3^{(\pm)}\left(\frac{\gamma_0^2+\nu_0^2}{D^2}\right)^{\!(d-6)/4} \exp\!\left(\mp i\frac{d-6}{2}\theta\right) \nonumber\\[8pt] & \quad + \Gamma(1-d/2)Q_1^{(\pm)}\left(\frac{(\mu+\gamma_0)^2+\nu_0^2}{4D^2}\right)^{\!(d-2)/4} \exp\!\left(\pm i\frac{d-2}{2}\eta\right) \nonumber\\[8pt] & \quad + \Gamma(2-d/2)Q_2^{(\pm)}\left(\frac{(\mu+\gamma_0)^2+\nu_0^2}{4D^2}\right)^{\!(d-4)/4} \exp\!\left(\pm i\frac{d-4}{2}\eta\right) \nonumber\\[8pt] & \quad + \frac{\Gamma(3-d/2)}{2}Q_3^{(\pm)}\left(\frac{(\mu+\gamma_0)^2+\nu_0^2}{4D^2}\right)^{\!(d-6)/4} \exp\!\left(\pm i\frac{d-6}{2}\eta\right)\nonumber\\[8pt] & \quad -\Gamma(1-d/2)\left(P_1^{(\pm)}+Q_1^{(\pm)}\right)\left(\frac{\gamma_0}{D}\right)^{\!(d-2)/2} \bigg] . \end{align} \tag{ 50 }$$

In the appendices, we provide additional details and the definitions of the various coefficients, as well as explicit evaluations for $d = 1,2,3$. Here, we focus on the behavior of the damping parameter µ_r across different dimensions. It is important to note that µ_r is generally a complex number: Its imaginary part conveys information about spatial oscillations, while its real part represents either exponential decay or growth of the average particle density.

3.4.1.  d = 1.

In one dimension, the renormalized damping parameter µ_r reads

$$\begin{align} \mu_r = \mu \left[ 1-\frac{5(\sigma+\kappa)}{24D}\sqrt{\frac{D}{\mu}}-\frac{\sigma+\kappa}{D} \sqrt{\frac{D}{\gamma_0}} \left( \frac{1}{12}+\frac{\gamma_0}{3\mu}+\frac{\sqrt{3}\nu_0}{16\mu} +\frac{\sqrt{3}\nu_0}{16\gamma_0}+\frac{\sqrt{3}\nu_0\gamma_0}{6\mu^2} \right) \right] . \end{align} \tag{ 51 }$$

For $\gamma_0 \lt 0$, µ_r acquires an imaginary part, indicating oscillatory behavior. However, if $\gamma_0 \gt 0$, there is only damping. We display different scenarios in figure 9.

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3.4.2.  d = 2.

At two dimensions, we need to introduce the UV cutoff $\Lambda \sim \pi/c$; the damping parameter becomes

$$\begin{align} \mu_r = \mu \Bigg[ & 1-\frac{\sigma+\kappa}{6D\pi}\frac{\gamma_0}{\mu}\ln\frac{D\Lambda^2}{\gamma_0} -\frac{\sigma+\kappa}{12D\pi}\ln\frac{D\Lambda^2}{\mu}-\frac{\sigma+\kappa}{24D\pi} \nonumber\\[8pt] & - \frac{\sigma+\kappa}{12D\pi} \left( 1+\frac{\sqrt{3}\nu_0}{2\mu}+\frac{\sqrt{3}\nu_0}{2\gamma_0} +\frac{2\sqrt{3}\nu_0\gamma_0}{\mu^2} \right) \Bigg] . \end{align} \tag{ 52 }$$

As in the one-dimensional case, when $\gamma_0 \gt 0$, we have pure damping, whereas the system displays population oscillations if $\gamma_0 \lt 0$. The damping coefficients µ_r at d = 2 for different bare parameters are plotted in figure 10.

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3.4.3.  d = 3.

In three dimensions, the damping parameter reads

$$\begin{align} \mu_r = \mu \left[1 + \frac{\sigma+\kappa}{16D\pi}\sqrt{\frac{\mu}{D}}+\frac{\sigma+\kappa}{6D\pi} \sqrt{\frac{\gamma_0}{D}} \left( -\frac{1}{4}+\frac{\gamma_0}{\mu}-\frac{\sqrt{3}\nu_0}{16\mu} -\frac{\sqrt{3}\nu_0}{16\gamma_0}-\frac{\sqrt{3}\nu_0\gamma_0}{2\mu^2} \right) \right] . \end{align} \tag{ 53 }$$

Again, oscillations emerge in the region where $\gamma_0 \lt 0$. The real and imaginary parts of the damping parameter µ are depicted in figure 11.

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For $\nu_0 = 1$, the damping µ_r decreases in one and two dimensions, but increases in three dimensions in the stable region where $\gamma_0 \gt 0$, as shown in figures 9–11. This indicates that the reactions effectively slow down the relaxation processes in one and two dimensions, but speed them up in three dimensions. However, due to the highly nonlinear dependence on the parameters µ₀, γ₀, and ν₀, a more general conclusion cannot be made. It is worth noting that in the unstable region, an imaginary part of µ_r is generated, formally resulting from the subtraction of an inadequate homogeneous steady state. Yet these emergent oscillations manifestly indicate the instability with respect to spontaneous formation of spatio-temporal patterns in this regime. We emphasize again that the one-loop fluctuation corrections should merely induce small quantitative corrections, and cannot induce qualitative changes in the region where perturbation theory is applicable. The ML system thus maintains a bifurcation point, below which the homogeneous ground state is rendered unstable and spiral structures emerge.

It is well-established that in sufficiently large spatial systems, the ML model displays spontaneously emerging dynamic spiral structures in individual simulation runs (these are of course averaged out in ensemble averages). Here, we propose that one crucial necessary condition for the formation of such persistent spatio-temporal patterns is the existence of a stable and uniform oscillation frequency at the local lattice site level, which then allows spatially extended coherent oscillatory features.

Each site in a lattice subject to stochastic reactions and spreading processes can be regarded as a separate system that is coupled to a particle reservoir (in the thermodynamic limit). In the ML model, at the (linearized) mean-field level, the local oscillation frequency is uniquely determined by the reaction rates, as is also apparent in the diagonalized ML Doi–Peliti action (39) at tree level. A similar definition of the oscillation frequency at linearized mean field level can also be obtained in the LV model [39]. However, in the RPS model at tree level, the oscillation frequency is set by the global conserved particle number ρ. As the particle number at each site is changing all the time owing to its coupling to the environment that serves as a nonlocal reservoir, there does not exist a unique characteristic oscillation frequency for each site during any single run stochastic realization. Ultimately, these nonlocal effects originate from the long-range correlations introduced by the global particle number conservation law, whose relevance in the context of pattern formation was demonstrated in [11]. The average of the oscillation frequency with a fixed total particle number is given by the expression in the diagonalized RPS Doi–Peliti action (18).

These straightforward tree-level arguments are readily generalized to all (loop) orders in the perturbation expansion. In the ML model, no nonlinear terms that depend on the total particle density (or any other global quantity) are present in the action, and thus the renormalized frequency will also be independent of the particle number density. This is not the case for the RPS model, where ρ manifestly enters the vertices. A perhaps more straightforward way to understand this distinction invokes the fixed points (stationary species densities) of the systems. In the RPS system, there exists a fixed line that is parameterized by the conserved total particle number. For each lattice site in the RPS model, the population oscillations wander along this fixed line with a mean that corresponds to the averaged local total particle number at this specific site. In contrast, the ML and LV models have unique fixed points, independent of any global constraints that originate from conservation laws, and consequently all lattice sites are governed by identical oscillation frequencies determined by these fixed points.

To illustrate these points, we plot the time evolution of the population density of a single species at two randomly chosen lattice sites in a single simulation run for the RPS and ML models in figures 12 and 13, respectively. For the RPS species density, the time intervals separating different peaks are not regularly spaced for the two lattice sites. Correspondingly, its Fourier transform displays multiple peaks with almost equal intensities, and no well-defined characteristic frequency is discernible. In contrast, the peaks in the ML species density time evolution are evenly separated, and hence a dominant Fourier peak emerges. These numerical observations support our analysis that the frequency in a single run in the ML model is well-defined, while that is clearly not the case for the RPS system. Yet a well-defined oscillation frequency clearly constitutes a necessary condition to form spatio-temporal patterns.

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Although the arguments in this section are formulated in the framework of individual sites in a regular lattice, they are readily extended to an effective unit cell that is similar in size to the characteristic diffusion length scale, or to models defined on a continuum which require a finite reaction range. As long as the system size is much larger than any of these small length scales (that provide suitable ultraviolet cutoffs for the continuum field theory), the specific microscopic details implemented in numerical simulations are not expected to have a significant impact on our conclusions.

While we posit above that a stable oscillation frequency at each lattice site is a necessary requirement to form oscillatory spatio-temporal patterns such as spirals, this is not a sufficient condition. Another crucial ingredient is of course that the spatially homogeneous stationary state is rendered unstable; fluctuations of a certain wavevector range will then spontaneously generate spatial patterns [45, 46].

In the stochastic spatially extended LV model [39], in the absence of site restrictions, the population oscillation damping vanishes in the Gaussian approximation. To one-loop order, a negative damping frequency is generated which indicates an instability of the spatially uniform system, inducing nontrivial particle transport that in turn drives the formation of expanding evasion-pursuit activity waves. In the ML model, the damping coefficient is already nonzero in the Gaussian approximation. As stated above, when $2\kappa \lt \sigma$, the limit cycles of the ML model become unstable and spiral structures emerge. Yet within the realm of a perturbative approach, any fluctuation corrections will only shift the Hopf bifurcation point, but cannot qualitatively change the system's overall features. In contrast, the RPS system behaves similarly to the LV system, as we encounter a vanishing damping coefficient in the Gaussian approximation. However, at one-loop level, the generated mass term is positive indicating merely a generated finite damping constant, as opposed to its negative counterpart in the LV model. Hence the RPS model remains inert against spontaneous pattern formation. Higher-order terms in the fluctuation expansion should not overturn the sign of the one-loop results within the perturbative regime, which renders this argument perturbatively robust.

In this part, we briefly investigate the effect of spatially disordered, uniformly distributed quenched reaction probabilities on the fluctuation corrections. We start with the Doi–Peliti action

$$\begin{align} \mathcal{A} = \int\!\mathrm{d}t\, \mathrm{d}^dx\left[\mathcal{L}_0({\hat{a}_i,a_i})+\mathcal{L}^{^{\prime}}({\hat{a}_i,a_i})\right] , \end{align} \tag{ 54 }$$

where $\mathcal{L}_0$ and $\mathcal{L}^{^{\prime}}$ are (usually polynomial) functions of the fields $\{\hat{a}_i\}$ and $\{a_i\}$. We introduce spatial disorder to $\mathcal{L}^{^{\prime}}$, which we take to represent stochastic reactions while $\mathcal{L}_0$ encodes particle diffusions. The action with quenched spatial disorder is then given by

$$\begin{align} \mathcal{A}_D = \int\!\mathrm{d}t\, \mathrm{d}^dx \, \mathcal{L}_0 + \int \mathrm{d}^dx \, \eta(\vec{x}) \int\!\mathrm{d}\mathcal{L}^{^{\prime}} ; \end{align} \tag{ 55 }$$

here, we assume $\eta(\vec{x})$ to be uniformly distributed in the interval $[0,2]$ with mean $\overline{\eta} = 1$. Note that the reaction rates are only spatially disordered and remain fixed in time. The average of any observables follows from equation (A.9) through

$$\begin{align} \overline{\langle O(t) \rangle}\propto \prod_{\vec{x}}\int_0^2 \frac{\mathrm{d}\eta(\vec{x})}{2}\ \int \prod_{i} \mathcal{D}[\hat{a}_i]\mathcal{D}[a_i]\,O(\{\hat{a}_i,a_i\}) \, e^{-\mathcal{A}_D} , \end{align} \tag{ 56 }$$

where the overline denotes the quenched disorder average. We can readily integrate out the disorder $\eta(\vec{x})$ first, since the observable O is independent of the random variables η, and arrive at

$$\begin{align} \overline{\langle O(t)\rangle}\propto \int\prod_{i}\mathcal{D}[\hat{a}_i]\mathcal{D}[a_i]\,O(\{\hat{a}_i,a_i\})\, e^{-\mathcal{A}_D^{^{\prime}}} , \end{align} \tag{ 57 }$$

with

$$\begin{align} \mathcal{A}_D^{^{\prime}} = \mathcal{A} + \int\mathrm{d}^dx \ln\left[ \frac{\sinh(\int\!\mathrm{d}t\ \mathcal{L}^{^{\prime}})}{\int\!\mathrm{d}t\ \mathcal{L}^{^{\prime}}}\right] . \end{align} \tag{ 58 }$$

Incorporating disorder in the original Doi–Peliti action (54) thus effectively leads to an additional term that is nonlocal in the time domain, owing to the temporally fixed reaction rates. However, in the perturbative regime where $\mathcal{L}^{^{\prime}}$ is small, this extra term can be expanded near $\mathcal{L}^{^{\prime}} = 0$:

$$\begin{align} \ln\left[\frac{\sinh(\int\mathrm{d}t\ \mathcal{L}^{^{\prime}})}{\int\mathrm{d}t\ \mathcal{L}^{^{\prime}}}\right] = \frac{1}{3}\int\!\mathrm{d}t \,\mathrm{d}t^{^{\prime}} \,\mathcal{L}^{^{\prime}}_t\,\mathcal{L}_{t^{^{\prime}}} + \mathcal{O}(\mathcal{L}^{^{\prime} 3}) , \end{align} \tag{ 59 }$$

where the labels t and t^ʹ distinguish the different time dependences. It is evident from this expansion that the extra temporally nonlocal term 'entangles' different replicas of the system. However, as the first term is already second-order in $\mathcal{L}^{^{\prime}}$, it is of higher order than the original action $\mathcal{A}$, and should not markedly affect the system's fluctuation corrections in the naive perturbation regime, at least to low loop orders. This observation may account for the insensitivity of the RPS and ML models to quenched randomness in the reaction rates, as reported in [14, 17]. We remark that the stronger effects of varying predation rates λ in the LV system can be largely traced to the sensitivity ${\sim} 1/\lambda$ of the stationary species densities already on the mean-field approximation level [35].

The RPS rules (1) mandate that particle numbers are discrete, hence the occupation numbers of lattice sites can be written as positive integers $n_{i \alpha}$, where the index i accounts for different particle species, while α enumerates the sites on a d-dimensional hyper-cubic lattice. The master equation for the local, on-site RPS reactions then reads:

$$\begin{align} \begin{split} \frac{\partial P(n_{i \alpha};t)}{\partial t} & =\sum_{i=1,2,3} \lambda^{^{\prime}}_i \bigl[ (n_{i \alpha}-1)(n_{i+1 \alpha}+1)P(n_{i \alpha}-1,n_{i+1 \alpha}+1;t) \\ & \qquad \qquad \quad\ -n_{i \alpha}n_{i+1 \alpha} P(n_{i \alpha},n_{i+1 \alpha};t) \bigr] , \end{split} \end{align} \tag{ A.1 }$$

where the index i wraps around (i.e. i = 4 is to be identified with i = 1). Note that for brevity we have not included hopping to adjacent lattice sites here. As an initial state, we assume a uniform distribution of particles with an average initial number of particles N_i per lattice site of species i. This corresponds to a Poisson distribution for the occupation number of each species i at all lattice sites α, $P(n_{i \alpha};t) = \prod_{i = 1,2,3} N_i^{n_{i \alpha}} e^{-N_i}/(n_{i \alpha}!)$. The discrete nature of the possible states of the RPS systems suggests the introduction of a product Fock space state vector

$$\begin{align} |\Phi(t)\rangle = \sum_{\{n_{i \alpha}\}}P(n_{i \alpha};t) \prod_{i=1,2,3} \prod_{\alpha=1}^{L^d}|n_{i \alpha}\rangle , \end{align} \tag{ A.2 }$$

where the $|n_{i \alpha}\rangle$ represent the occupation states of species i on lattice site i. In analogy with the quantum-mechanical harmonic oscillator, the single-site states (and thereby the full state vector) can be acted upon by bosonic ladder operators obeying the commutation relations $[a_{i \alpha},a_{j \beta}] = 0$ and $[a_{i \alpha},a_{j \beta}^\dagger] = \delta_{\alpha\beta}\delta_{ij}$. The occupation number eigenstates are constructed via $a_{i \alpha}|n_{i \alpha}\rangle = n_{i \alpha}|n_{i \alpha}-1\rangle$, $a_{i \alpha}^\dagger|n_{i \alpha}\rangle = |n_{i \alpha}+1\rangle$, and the empty state $|0\rangle$ is defined by $a_{i \alpha}|0\rangle = 0$.

The time evolution of the state vector (A.2) follows directly from the master equation (A.1) and can be written in the form

$$\begin{align} \frac{\partial}{\partial t}|\Phi(t)\rangle = -H|\Phi(t)\rangle \Longleftrightarrow |\Phi(t)\rangle=e^{-Ht}|\Phi(0)\rangle , \end{align} \tag{ A.3 }$$

where H denotes the (time-independent) Liouville operator which can be split into a diffusion and a reaction term, $H = H_\textrm{diff}+H_\textrm{reac}$, where the on-site reaction contribution is a sum of local terms $H_\textrm{reac} = \sum_{i = 1}^{L^d}H_\alpha$, and specifically for the RPS model

$$\begin{align} H_\alpha=\sum_{i=1,2,3}\lambda_i^{^{\prime}}\left(a_{i+1 \alpha}^\dagger - a_{i \alpha}^\dagger\right) a_{i \alpha}^\dagger a_{i \alpha}a_{i+1 \alpha} . \end{align} \tag{ A.4 }$$

Similarly, since on-lattice diffusion is implemented by particles performing simple jumps between nearest-neighbor lattice sites, the diffusion part of H reads

$$\begin{align} H_\mathrm{diff}=\sum_{i=1,2,3}\frac{D_i}{c^2}\sum_{\langle \alpha\beta \rangle} \left(a_{i \alpha}^\dagger-a_{i \beta}^\dagger\right)(a_{i \alpha}-a_{i \beta}) , \end{align} \tag{ A.5 }$$

where $\langle \alpha\beta \rangle$ indicates a sum over all possible nearest-neighbor lattice site pairs in the system.

Following the steps of [39–41], we write averages for observables $O = O(\{n_{i \alpha}\})$ as a multi-dimensional integral over coherent states

$$\begin{align} \langle O(t)\rangle\propto\int\prod_{i=1,2,3}\prod_{\alpha=1}^{L^d} d\psi_{i \alpha}^*d\psi_{i \alpha} \, O(\{\psi_{i \alpha}\}) \, e^{-\mathcal{A}(\psi_{i \alpha}^*,\psi_{i \alpha};t)} , \end{align} \tag{ A.6 }$$

where the $\psi_{i \alpha}$ and $\psi_{i \alpha}^*$ are complex eigenvalues describing the coherent right and left eigenstates of the ladder operators $a_{i \alpha}$ and $a_{i \alpha}^\dagger$, respectively. The coherent-state 'action' is given by

$$\begin{align} \mathcal{A}(\psi_{i \alpha}^*,\psi_{i \alpha};t^{^{\prime}}) & = \sum_{i=1,2,3} \sum_{\alpha=1}^{L^d} \biggl[ \int_0^{t^{^{\prime}}}\mathrm{d}t\ \psi_{i \alpha}^*(t) \frac{\partial\psi_{i \alpha}(t)}{\partial t}dt-\psi_{i \alpha}(t^{^{\prime}}) - N_i \psi_{i \alpha}^*(0) \biggr] \nonumber\\[8pt] & \quad + \int_0^{t^{^{\prime}}} \mathrm{d}t\ H\bigl( a_{i \alpha}^\dagger\to\psi_{i \alpha}^*(t),a_{i \alpha} \to \psi_{i \alpha}(t)\bigr) , \end{align} \tag{ A.7 }$$

where we have to replace the ladder operators by their eigenvalues in the Liouville operator H.

In the spatial continuum limit (lattice constant c → 0) we may replace the sum over lattice sites with a d-dimensional volume integral $\sum_{\alpha = 1}^{L^d} \to c^{-d}\int d^dx$, and the discretely spaced coherent-state values with continuous fields $\psi_{i \alpha}(t)\to c^d a_i(\vec{x},t)$ and $\psi_{i \alpha}^*(t)\to 1+\tilde{a}_i(\vec{x},t)$. Hence, the 'bulk' part of action (not considering the terms from the initial conditions at t = 0 and the projection states at $t = t^{^{\prime}}$) of the RPS system is given by

$$\begin{align} \mathcal{A} = \int \mathrm{d}t\ \mathrm{d}^dx \Biggl [ & \sum_{i=1,2,3}\tilde{a}_i \biggl( \partial_t-D_\alpha\nabla^2\biggr) a_i + \lambda_1a_1a_2(\tilde{a}_1+1)(\tilde{a}_2-\tilde{a}_1) \nonumber\\[8pt] & + \lambda_2a_2a_3(\tilde{a}_2+1)(\tilde{a}_3-\tilde{a}_2) + \lambda_3a_3a_1(\tilde{a}_3+1)(\tilde{a}_1-\tilde{a}_3) \Biggr] . \end{align} \tag{ A.8 }$$

In the continuum limit we can thus write averages in the following coherent-state path integral form

$$\begin{align} \langle O(t)\rangle\propto\int\prod_{i=1,2,3} \mathcal{D}[\tilde{a}_i]\mathcal{D}[a_i]\,O(\{a_i\})\, e^{-\mathcal{A}(\tilde{a}_i,a_i;t)} . \end{align} \tag{ A.9 }$$

Our aim here is to derive stochastic partial differential (Langevin) equations for the species concentrations that accurately capture the intrinsic reaction noise. To this end, we note that the Janssen–de Dominicis response functional [40, 42, 43, 55]

$$\begin{align} \mathcal{A} = \int \mathrm{d}t\ \mathrm{d}^dx \sum_i\tilde{a}_i\biggl(\partial_t a_i-D_i\nabla^2a_i-F_i[\{a_i\}] -\sum_j L_{ij}[\{a_i\}]\tilde{a}_j \biggr) \end{align} \tag{ A.10 }$$

is equivalent to the set of SPDEs

$$\begin{align} \partial_t a_i = D_i\nabla^2 a_i(\vec{x},t)+F_i[\{a_i(\vec{x},t)\}]+\zeta_i(\vec{x},t) , \end{align} \tag{ A.11 }$$

with the associated noise (cross-)correlations

$$\begin{align} \langle\zeta_i(\vec{x},t)\zeta_j(\vec{x}^{^{\prime}},t^{^{\prime}})\rangle=2L_{ij}[\{a_i(\vec{x},t)\}] \delta(t-t^{^{\prime}})\delta(\vec{x}-\vec{x}^{^{\prime}}) . \end{align} \tag{ A.12 }$$

This correspondence allows the immediate derivation of a coupled Langevin equation formulation of any system that exhibits an action functional of the form (A.10). Hence, via a direct comparison with the action of the RPS system (A.7), we can extract the deterministic part of the SPDEs describing the RPS system

$$\begin{align} F_1=(\lambda_1a_2-\lambda_3a_3)a_1 , \quad F_2=(\lambda_2a_3-\lambda_1a_1)a_2 , \quad F_3=(\lambda_3a_1-\lambda_2a_2)a_3 , \end{align} \tag{ A.13 }$$

which equal the right-hand side of the mean-field equations, as they should. Furthermore, the effective noise correlations are given by the matrix L_ij :

$$\begin{align} &L_{11}=\lambda_1a_1a_2 , \quad L_{12}=-\frac{\lambda_1}{2}a_1a_2 , \quad L_{13}=-\frac{\lambda_3}{2}a_1a_3 , \nonumber\\[8pt] &L_{22}=\lambda_2a_2a_3 , \quad L_{23}=-\frac{\lambda_2}{2}a_2a_3 , \quad L_{33}=\lambda_3a_1a_3 . \end{align} \tag{ A.14 }$$

Hence, the SPDEs (A.11) can be constructed from the mean-field equations by including a term that accounts for diffusion and multiplicative noise terms obeying the given (cross-)correlations. Note that the noise auto-correlations L_ii are always determined by the concentration of the predator species A_i and its respective prey $A_{i+1}$, and the scale is set by the associated predation rate λ_i . Thus, the noise directly associated with a given species is solely determined by its role as predator.

In order to investigate the asymmetric 'corner' limit of the RPS system, we re-define the interaction rates as $\lambda_1 = \lambda/x$, $\lambda_2 = \lambda$ and $\lambda_3 = \kappa\lambda$. The dimensionless variable x varies in the interval $(0,1]$ and describes the asymmetry of the rates, while the equally dimensionless parameter κ is of order unity and describes the difference between the predation rates of species A₂ and A₃. We are interested in the limit x → 0 in which the predation reactions between species A₁ and A₂ dominate. The concentrations at the coexistence fixed point become

$$\begin{align} (\Omega_1,\Omega_2,\Omega_3)=\rho(x,\kappa x,1-[1+\kappa]x)+\mathcal{O}(x^2) . \end{align} \tag{ A.15 }$$

Hence, the densities of species A₁ and A₂ become small as x → 0, while species A₃ makes up most of the overall species abundance. This is the 'corner' limit in which RPS can be approximated by a two-species LV system, with the third, most abundant species serving as a mean-field like reservoir to feed the first species, and to provide the effective spontaneous death reaction for the second species, as explained above [56]. Species A₁ thus effectively turns into prey, while A₂ becomes the sole predator species. The noise correlation matrix L in this corner case reads

$$\begin{align*} & L_{11}=\frac{\lambda}{x}a_1a_2 , \quad L_{12}=-\frac{\lambda}{2x}a_1a_2 , \quad L_{13}\approx-\frac{\kappa\lambda}{2}a_1\rho , \\[8pt] & L_{22}\approx \lambda a_2\rho , \quad\,\ L_{23}\approx-\frac{\lambda}{2}a_2\rho , \qquad\! L_{33}=\kappa\lambda a_1\rho . \end{align*}$$

The noise strength of species A₁, as well as the noise cross-correlations between species A₁ and A₂, are inversely proportional to the large rate scaling factor x, indicating that fluctuations of species A₁ and A₂ (the LV predator and prey, respectively) become strong in the limit x → 0. Indeed, writing the resulting effective SPDEs in the limit of large λ₁ and assuming a homogeneous and stationary distribution of species A₃ yields

$$\begin{align} \partial_t a_1 &\approx D_1\nabla^2a_1 + \lambda_1a_1a_2 - \rho \left( 1 - \frac{\lambda_2+\lambda_3}{\lambda_1} \right) \lambda_3a_1 + \zeta_1 , \end{align} \tag{ A.16 }$$

$$\begin{align} \partial_t a_2 &\approx D_2\nabla^2a_2 + \rho \left( 1 - \frac{\lambda_2+\lambda_3}{\lambda_1} \right) \lambda_2a_2 - \lambda_1a_1a_2+ \zeta_2 , \end{align} \tag{ A.17 }$$

with the noise correlations

$$\begin{align} \langle\zeta_1(\vec{x},t)\zeta_1(\vec{x}^{^{\prime}},t^{^{\prime}})\rangle &= 2\lambda_1a_1a_2 \delta(\vec{x}-\vec{x}^{^{\prime}})\delta(t-t^{^{\prime}})\ , \end{align} \tag{ A.18 }$$

$$\begin{align} \langle\zeta_1(\vec{x},t)\zeta_2(\vec{x}^{^{\prime}},t^{^{\prime}})\rangle&=-\lambda_1a_1a_2 \delta(\vec{x}-\vec{x}^{^{\prime}})\delta(t-t^{^{\prime}})\ , \end{align} \tag{ A.19 }$$

$$\begin{align} \langle\zeta_2(\vec{x},t)\zeta_2(\vec{x}^{^{\prime}},t^{^{\prime}})\rangle&=2\lambda_2\rho \left( 1 - \frac{\lambda_2+\lambda_3}{\lambda_1} \right) a_2 \delta(\vec{x}-\vec{x}^{^{\prime}})\delta(t-t^{^{\prime}}) . \end{align} \tag{ A.20 }$$

This set of Langevin equations precisely matches those derived directly for the LV model [39].

In order to gain more insight into the role of fluctuations in the RPS system, we study the non-linear vertices arising from the Doi–Peliti action (A.7). To this end, we first need to diagonalize the action by transforming to appropriate field combinations. We then list the resulting vertices that capture fluctuation corrections beyond the Gaussian mean-field approximation.

To start, we transform the fields to describe the fluctuations around the fixed-point species concentrations. To this end we employ the linear transformation

$$\begin{align} a_i(\vec{x},t)=\Omega_i+c_i(\vec{x},t) , \quad \tilde{a}_i(\vec{x},t)=\tilde{c}_i(\vec{x},t) , \end{align} \tag{ A.21 }$$

here ignoring higher-order shifts of the steady-state coexistence concentrations induced by stochastic fluctuations (i.e. the counter-terms or additive renormalizations in [39]). The action for these new fluctuating fields becomes

$$\begin{align} \mathcal{A}=\int \mathrm{d}t\ \mathrm{d}^dx \left[ \sum_{i=1,2,3} \tilde{c}_i \left( \partial_t-D_i\nabla^2 \right) c_i+\bar{\mathcal{A}} \right] , \end{align} \tag{ A.22 }$$

with the reduced part

$$\begin{align} \bar{\mathcal{A}}&=\frac{-1}{\bar{\lambda}^2}\Bigl[ \lambda_1(\tilde{c}_1+1)(\tilde{c}_1-\tilde{c}_2) (\bar{\lambda}c_1+\lambda_2^{^{\prime}}\rho)(\bar{\lambda}c_2+\lambda_3\rho)\nonumber\\[6pt] & \quad + \lambda_2^{^{\prime}}(\tilde{c}_2+1)(\tilde{c}_2-\tilde{c}_3) (\bar{\lambda}c_2+\lambda_3\rho)(\bar{\lambda}c_3+\lambda_1\rho)\nonumber\\[6pt] & \quad + \lambda_3(\tilde{c}_3+1)(\tilde{c}_3-\tilde{c}_1) (\bar{\lambda}c_3+\lambda_1\rho)(\bar{\lambda}c_1+\lambda_2\rho)\Bigr] , \end{align} \tag{ A.23 }$$

and $\bar{\lambda} = \lambda_1+\lambda_2+\lambda_3$. The harmonic part of this action can be cast in a bilinear matrix form $\bar{s}_h = \sum_{ij} \tilde{c}_j A_{ij}c_i$ with the mass matrix

$$\begin{align} A=\frac{\rho}{\bar{\lambda}} \begin{pmatrix} 0 & -\lambda_1\lambda_2 & \lambda_2\lambda_3\\ \lambda_1\lambda_3 & 0 & -\lambda_2\lambda_3\\ -\lambda_1\lambda_3 & \lambda_1\lambda_2 & 0 \end{pmatrix}=-A_s , \end{align} \tag{ A.24 }$$

where A_s is the stability matrix of the system at mean-field level. We note that it reduces to the stability matrix (4) in the symmetric limit $\lambda_1 = \lambda_2 = \lambda_3 = \lambda$.

Our goal is to find a transformation that diagonalizes the mass matrix A, and thus the harmonic part of the action (A.7), if we set all diffusivities equal, $D_i = D$. The matrix A is asymmetric, hence we make use of its orthogonal left and right eigenvectors $\vec{u}_i A = e_i\vec{u}_i$ and $A\vec{v}_i = e_i\vec{v}_i$, respectively. The resulting eigenvector matrices $Q = (\vec{u}_1,\vec{u}_2,\vec{u}_3)$ and $P = (\vec{v}_1,\vec{v}_2,\vec{v}_3)$ read

$$\begin{align} Q= \begin{pmatrix} 1 & 1 & 1 \\[6pt] 1 & -\frac{\lambda_2(\lambda_1+\lambda_2)}{\lambda_2\lambda_3+i\Lambda} & -\frac{\lambda_2(\lambda_1+\lambda_2)}{\lambda_2\lambda_3-i\Lambda} \\[10pt] 1 & -\frac{\lambda_2(\lambda_2+\lambda_3)}{\lambda_1\lambda_2-i\Lambda} & -\frac{\lambda_2(\lambda_2+\lambda_3)}{\lambda_1\lambda_2+i\Lambda} \end{pmatrix} , \end{align} \tag{ A.25 }$$

where $\Lambda = \sqrt{\lambda_1\lambda_2\lambda_3(\lambda_1+\lambda_2+\lambda_3)}$, and

$$\begin{align} P= \begin{pmatrix} 1 & 1 & 1 \\[6pt] \frac{\lambda_3}{\lambda_2} & -\frac{\lambda_3(\lambda_1+\lambda_2)}{\lambda_2\lambda_3-i\Lambda} & -\frac{\lambda_3(\lambda_1+\lambda_2)}{\lambda_2\lambda_3+i\Lambda} \\[10pt] \frac{\lambda_1}{\lambda_2} & -\frac{\lambda_1(\lambda_2+\lambda_3)}{\lambda_1\lambda_2+i\Lambda} & -\frac{\lambda_1(\lambda_2+\lambda_3)}{\lambda_1\lambda_2-i\Lambda} \end{pmatrix} . \end{align} \tag{ A.26 }$$

The right and left eigenvector matrices then transform the mass matrix to the diagonal form $Q^TAP(Q^TP)^{-1} = \textrm{diag}(e_i)$. Defining new fields $\tilde{\phi}_i$ and φ_i according to the transformation $\tilde{c}_i = \sum_j Q^T_{ji}\tilde{\phi}_j$ and $c_i = \sum_j P_{ij}\phi_j$, we arrive at ⁶

$$\begin{align} \tilde{c}_1& = \tilde{\psi}+\tilde{\phi}_++\tilde{\phi}_- , \nonumber \\[8pt] \tilde{c}_{2/3}& = \tilde{\psi}-\frac{\lambda_2}{\lambda_1+\lambda_3}(\tilde{\phi}_++\tilde{\phi}_-)\pm \frac{i\Lambda/\lambda_{^3/_1}}{\lambda_1+\lambda_3}(\tilde{\phi}_+-\tilde{\phi}_-) ,\\[8pt] c_1& = \psi+\phi_++\phi_- , \nonumber \end{align} \tag{ A.27 }$$

$$\begin{align} c_{2/3}& = \frac{\lambda_{^3/_1}}{\lambda_2}\psi-\frac{\lambda_{^3/_1}}{\lambda_1+\lambda_3} (\phi_++\phi_-)\mp\frac{i\Lambda/\lambda_{2}}{\lambda_1+\lambda_3}(\phi_+-\phi_-) . \end{align} \tag{ A.28 }$$

It is already obvious from this structure that the fields $\tilde{\psi}$ and ψ describe the fluctuation of the total population, while the other fields are oscillatory in nature. Employing these transformations, one arrives at the diagonalized harmonic action

$$\begin{align} \mathcal{A}_0& = \int dt\int d^dx(\lambda_1+\lambda_2+\lambda_3) \Bigl[\frac{1}{\lambda_2}\tilde{\psi} \left( \partial_t-D\nabla^2 \right) \psi \nonumber\\[8pt] &\quad +\frac{1}{\lambda_1+\lambda_3}\tilde{\phi}_+ \left( \partial_t-D\nabla^2+i\omega_0 \right) \phi_+ +\frac{1}{\lambda_1+\lambda_3}\tilde{\phi}_- \left( \partial_t -D\nabla^2-i\omega_0 \right) \phi_- \Bigr] . \end{align} \tag{ A.29 }$$

The field $\psi = \lambda_2(c_1+c_2+c_3)/(\lambda_1+\lambda_2+\lambda_3)$ is massless, encodes no reactions, and is purely diffusive, as it represents the total local concentration of all three species. The corresponding harmonic propagator is given in Fourier space by

$$\begin{align} \langle\tilde{\psi}(\vec{q},\omega)\,\psi(\vec{q}^{^{\prime}},\omega^{^{\prime}})=\frac{\lambda_2} {\lambda_1+\lambda_2+\lambda_3}\frac{(2\pi)^{d+1}\delta(\vec{q}+\vec{q}^{^{\prime}})\delta(\omega+\omega^{^{\prime}})} {-i\omega+Dq^2} , \end{align} \tag{ A.30 }$$

while the oscillating field propagators display poles at finite eigenfrequencies $\mp\omega_0$, similar to the LV case [39]. The action transformed to the new fields becomes quite cumbersome to write out in full, hence we merely provide the coefficients of the possible field combinations in the vertices in table A1. Note that we omit the coefficients of any four-point vertices as these do not contribute to one-loop order corrections.

Table A1. Coefficients of the vertices in the action after the transformation to fluctuating fields.

Two-point (noise) sources
$\tilde{\phi}_\pm^2$	$+\frac{\rho\lambda_2}{(\lambda_1+\lambda_3)^2} \bigl(\rho\frac{\lambda_1^2(\lambda_2-\lambda_3)-\lambda_3^2(\lambda_1-\lambda_2)} {\lambda_1+\lambda_2+\lambda_3}\pm i\omega_0[\lambda_1-\lambda_3]\bigr)$
$\tilde{\phi}_+\tilde{\phi}_-$	$-\frac{2\rho^2\lambda_2}{\lambda_1+\lambda_3} \frac{\lambda_1\lambda_2+\lambda_1\lambda_3+\lambda_2\lambda_3} {\lambda_1+\lambda_2+\lambda_3}$
Merging three-point vertices
$\tilde{\phi}_\pm\phi_\pm\psi$	$\pm 2i\frac{\omega_0}{\rho\lambda_2} \frac{(\lambda_1+\lambda_2+\lambda_3)^2}{\lambda_1+\lambda_3}$
$\tilde{\phi}_\pm\phi_\pm^2$	$+\bigl(\frac{\lambda_1+\lambda_2+\lambda_3} {\lambda_1+\lambda_3}\bigr)^2\bigl([\lambda_1-\lambda_3]\pm i\frac{\omega_0}{\rho\lambda_2} [\lambda_1-2\lambda_2+\lambda_3]\bigr)$
$\tilde{\phi}_\pm\phi_\mp^2$	$-\bigl(\frac{\lambda_1+\lambda_2+\lambda_3} {\lambda_1+\lambda_3}\bigr)^2\bigl([\lambda_1-\lambda_3]\pm i\frac{\omega_0}{\rho\lambda_2} [\lambda_1+2\lambda_2+\lambda_3]\bigr)$
Splitting three-point vertices
$\psi\tilde{\phi}_\pm^2$	$\frac{2}{(\lambda_1+\lambda_3)^2}\bigl(\rho[\lambda_1^2 (\lambda_2-\lambda_3)+\lambda_3^2(\lambda_2-\lambda_1)]\pm i\omega_0[\lambda_1-\lambda_3] [\lambda_1+\lambda_2+\lambda_3]\bigr)$
$\psi\tilde{\phi}_+\tilde{\phi}_-$	$-\frac{4\rho}{\lambda_1+\lambda_3} \bigl(\lambda_1\lambda_2+\lambda_1\lambda_3+\lambda_2\lambda_3\bigr)$
$\phi_\pm\tilde{\psi}\tilde{\phi}_\pm$	$\pm2i\omega_0\frac{\lambda_1+\lambda_2+\lambda_3} {\lambda_1+\lambda_3}$
$\phi_\pm\tilde{\phi}_\pm^2$	$\frac{\rho}{(\lambda_1+\lambda_3)^2}\bigl(\lambda_1\lambda_2 [\lambda_1-\lambda_2]-\lambda_1\lambda_3[\lambda_1+\lambda_3]+2\lambda_2\lambda_3^2\pm \frac{\lambda_2\rho}{i\omega_0}[\lambda_2(\lambda_1^2-\lambda_3^2)+\lambda_1\lambda_3 (\lambda_2+\lambda_3-2\lambda_1)]\bigr)$
$\phi_\pm\tilde{\phi}_\mp^2$	$-\frac{\rho}{(\lambda_1+\lambda_3)^3} \bigl(\lambda_1^2\lambda_2[\lambda_1+3\lambda_2-\lambda_3]+\lambda_1\lambda_3 [\lambda_1 \lambda_2]^2+2(\lambda_1^2+\lambda_2^2)\lambda_3^2+\lambda_1\lambda_3^3\pm \frac{\lambda_2\rho}{i\omega_0}[-\lambda_1^3\lambda_2+\lambda_1\lambda_3^3 +\lambda_2\lambda_3^3+\lambda_1^2\lambda_3(2\lambda_2+\lambda_3)]\bigr)$
$\phi_\pm\tilde{\phi}_+\tilde{\phi}_-$	$\frac{2\rho}{(\lambda_1+\lambda_3)^2} \bigl(2\lambda_1\lambda_2^2+[\lambda_2^2-\lambda_1^2]\lambda_3-[\lambda_1+\lambda_2] \lambda_3^2\mp\frac{\lambda_2\rho}{i\omega_0}[\lambda_1^2(\lambda_2+\lambda_3) -\lambda_2\lambda_3^2-\lambda_1\lambda_2\lambda_3]\bigr)$

For the ψ_o mode, the renormalized parameters read

$$\begin{align} \mu_r & = \mu \left[ 1-\frac{5(\sigma+\kappa)}{24D}\sqrt{\frac{D}{\mu}}-\frac{\sigma+\kappa}{D} \sqrt{\frac{D}{\gamma_0}} \left( \frac{1}{12}+\frac{\gamma_0}{3\mu}+\frac{\sqrt{3}\nu_0}{16\mu} +\frac{\sqrt{3}\nu_0}{16\gamma_0}+\frac{\sqrt{3}\nu_0\gamma_0}{6\mu^2} \right) \right] , \nonumber\\[8pt] D^{0}_r & = D-\frac{\sigma+\kappa}{48}\sqrt{\frac{D}{\mu}}-\frac{\sigma+\kappa}{24}\sqrt{\frac{D}{\gamma_0}} \left( \frac{\gamma_0}{\mu}-\frac{\sqrt{3}\nu_0}{2\mu}+\frac{\sqrt{3}\nu_0}{2\gamma_0} \right) . \end{align} \tag{ C.1 }$$

For the $\psi_\pm$ modes, when $\gamma_0 \geqslant 0$, the renormalized parameters are

$$\begin{align} \gamma_r & = \gamma_0 + (\sigma+\kappa)\sqrt{\frac{D}{\nu_0}} \ \bigg[ \frac12 \mathrm{Re}M_1 \sqrt{\frac{\nu_0}{\gamma_0}}+\frac{1}{2} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-1/4} \left( \mathrm{Re}M_2\cos\frac{\theta}{2}-\mathrm{Im}M_2\sin\frac{\theta}{2} \right) \nonumber\\[8pt] & \quad +\frac{\sqrt{2}}{2} \left( 1+\frac{(\gamma_0+\mu)^2}{\nu_0^2} \right)^{\!-1/4} \left( \mathrm{Re}M_3\cos\frac{\eta}{2}+\mathrm{Im}M_3\sin\frac{\eta}{2} \right) \nonumber\\[8pt] & \quad + \frac{D}{4\nu_0}\left(1+\frac{\gamma_0^2}{\nu_0^2}\right)^{\!-3/4} \left( \mathrm{Re}M_4\cos\frac{3\theta}{2}-\mathrm{Im}M_4\sin\frac{3\theta}{2} \right) \nonumber\\[8pt] & \quad + \frac{\sqrt{2}D}{2\nu_0} \left (1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-3/4} \left( \mathrm{Re}M_5\cos\frac{3\eta}{2}+\mathrm{Im}M_5\sin\frac{3\eta}{2} \right) \bigg] , \nonumber\\[8pt] \nu_r & = \nu_0 +(\sigma+\kappa)\sqrt{\frac{D}{\nu_0}} \ \bigg[ \frac12 \mathrm{Im}M_1 \sqrt{\frac{\nu_0}{\gamma_0}}+\frac{1}{2} \left( 1+\frac{\gamma_0^2}{\nu_0^2}\right)^{\!-1/4} \left( \mathrm{Re}M_2\sin\frac{\theta}{2}+\mathrm{Im}M_2\cos\frac{\theta}{2} \right) \nonumber\\[8pt] & \quad +\frac{\sqrt{2}}{2} \left( 1+\frac{(\gamma_0+\mu)^2}{\nu_0^2} \right)^{\!-1/4} \left( -\mathrm{Re}M_3\sin\frac{\eta}{2}+\mathrm{Im}M_3\cos\frac{\eta}{2} \right) \nonumber\\[8pt] & \quad + \frac{D}{4\nu_0} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-3/4} \left( \mathrm{Re}M_4\sin\frac{3\theta}{2}+\mathrm{Im}M_4\cos\frac{3\theta}{2} \right) \nonumber\\[8pt] & \quad + \frac{\sqrt{2}D}{2\nu_0} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2}\right)^{\!-3/4} \left( -\mathrm{Re}M_5\sin\frac{3\eta}{2}+\mathrm{Im}M_5\cos\frac{3\eta}{2} \right) \bigg] ; \end{align} \tag{ C.2 }$$

yet if $\gamma_0 \lt 0$, the first term in both parameters changes:

$$\begin{align} \gamma_r & = \gamma_0 + (\sigma+\kappa)\sqrt{\frac{D}{\nu_0}} \, \bigg[ \!-\frac12 \mathrm{Im}M_1 \sqrt{\frac{\nu_0}{|\gamma_0|}}+\frac12 \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-1/4}\! \left( \mathrm{Re}M_2\cos\frac{\theta}{2}-\mathrm{Im}M_2\sin\frac{\theta}{2} \right) \nonumber\\[8pt] & \quad +\frac{\sqrt{2}}{2} \left( 1+\frac{(\gamma_0+\mu)^2}{\nu_0^2} \right)^{\!-1/4} \left( \mathrm{Re}M_3\cos\frac{\eta}{2}+\mathrm{Im}M_3\sin\frac{\eta}{2} \right) \nonumber\\[8pt] & \quad + \frac{D}{4\nu_0} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-3/4} \left( \mathrm{Re}M_4\cos\frac{3\theta}{2}-\mathrm{Im}M_4\sin\frac{3\theta}{2} \right) \nonumber\\[8pt] & \quad + \frac{\sqrt{2}D}{2\nu_0} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-3/4} \left( \mathrm{Re}M_5\cos\frac{3\eta}{2}+\mathrm{Im}M_5\sin\frac{3\eta}{2} \right) \bigg] , \nonumber\\[8pt] \nu_r & = \nu_0 +(\sigma+\kappa)\sqrt{\frac{D}{\nu_0}} \ \bigg[ \frac12 \mathrm{Re}M_1 \sqrt{\frac{\nu_0}{|\gamma_0|}} +\frac12 \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-1/4} \left( \mathrm{Re}M_2\sin\frac{\theta}{2}+\mathrm{Im}M_2\cos\frac{\theta}{2} \right) \nonumber\\[8pt] & \quad +\frac{\sqrt{2}}{2} \left(1 +\frac{(\gamma_0+\mu)^2}{\nu_0^2} \right)^{\!-1/4} \left( -\mathrm{Re}M_3\sin\frac{\eta}{2}+\mathrm{Im}M_3\cos\frac{\eta}{2}\right) \nonumber\\[8pt] & \quad + \frac{D}{4\nu_0} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-3/4} \left( \mathrm{Re}M_4\sin\frac{3\theta}{2}+\mathrm{Im}M_4\cos\frac{3\theta}{2} \right) \nonumber\\[8pt] & \quad + \frac{\sqrt{2}D}{2\nu_0} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-3/4} \left( -\mathrm{Re}M_5\sin\frac{3\eta}{2}+\mathrm{Im}M_5\cos\frac{3\eta}{2} \right) \bigg] . \end{align} \tag{ C.3 }$$

The renormalized diffusivity is not affected by the sign of γ₀,

$$\begin{align} \mathrm{Re}D_r & = D +(\kappa+\sigma)\sqrt{\frac{D}{\nu_0}} \ \bigg[ -\frac12 \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-1/4} \mathrm{Im}P_1\sin\frac{\theta}{2} - \frac12 \mathrm{Re}Q_1 \sqrt{\frac{\nu_0}{\gamma_0}} \nonumber\\[8pt] & \quad +\frac{D}{4\nu_0} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-3/4} \left(\mathrm{Re}P_2\cos\frac{3\theta}{2}-\mathrm{Im}P_2\sin\frac{3\theta}{2} \right) \nonumber\\[8pt] & \quad +\frac{3D^2}{16\nu_0^2} \left( 1+\frac{\gamma_0^2}{\nu_0^2}\right)^{\!-5/4} \left( \mathrm{Re}P_3\cos\frac{5\theta}{2}-\mathrm{Im}P_3\sin\frac{5\theta}{2} \right) \nonumber\\[8pt] & \quad +\frac{\sqrt{2}}{2} \left(1 +\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-1/4} \left( \mathrm{Re}Q_1\cos\frac{\eta}{2}+\mathrm{Im}Q_1\sin\frac{\eta}{2} \right) \nonumber\\[8pt] & \quad +\frac{\sqrt{2}D}{2\nu_0} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-3/4} \left( \mathrm{Re}Q_2\cos\frac{3\eta}{2}+\mathrm{Im}Q_2\sin\frac{3\eta}{2} \right) \nonumber\\[8pt] & \quad +\frac{3\sqrt{2}D^2}{4\nu_0^2} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-5/4} \left( \mathrm{Re}Q_3\cos\frac{5\eta}{2}+\mathrm{Im}Q_3\sin\frac{5\eta}{2} \right) \bigg] . \end{align} \tag{ C.4 }$$

For the ψ_o mode, the renormalized parameters read

$$\begin{align} \mu_r & = \mu \Bigg[ 1-\frac{\sigma+\kappa}{6D\pi}\frac{\gamma_0}{\mu}\ln\frac{D\Lambda^2}{\gamma_0} -\frac{\sigma+\kappa}{12D\pi}\ln\frac{D\Lambda^2}{\mu}-\frac{\sigma+\kappa}{24D\pi} \nonumber\\[10pt] & \quad - \frac{\sigma+\kappa}{12D\pi} \left( 1+\frac{\sqrt{3}\nu_0}{2\mu}+\frac{\sqrt{3}\nu_0}{2\gamma_0} +\frac{2\sqrt{3}\nu_0\gamma_0}{\mu^2} \right) \Bigg] , \nonumber\\[10pt] D^o_r & = D-\frac{\sigma+\kappa}{48\pi}-\frac{\sigma+\kappa}{24\pi} \left( \frac{\gamma_0}{\mu} -\frac{2\sqrt{3}\nu_0}{3\mu}+\frac{\sqrt{3}\nu_0}{3\gamma_0} \right) . \end{align} \tag{ C.5 }$$

For the $\psi_\pm$ modes, the renormalized parameters are

$$\begin{align} \gamma_r & = \gamma_0 +\frac{\sigma+\kappa}{2\pi} \bigg[ \frac{\mathrm{Re}M_1}{2} \ln\frac{D\Lambda^2}{\gamma_0} + \left( \mathrm{Re}M_2+\mathrm{Re}M_3\right)\ln\Lambda -\theta\mathrm{Im}M_2+\eta\mathrm{Im}M_3 \nonumber\\[10pt] & \quad -\frac14 \left[ \mathrm{Re}M_2\ln \frac{\gamma_0^2+\nu_0^2}{D^2}+\mathrm{Re}M_3 \ln \frac{(\mu+\gamma_0)^2+\nu_0^2}{4D^2} \right] \nonumber\\[10pt] & \quad +\frac{D}{2\nu_0} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-1/2} \left(\mathrm{Re}M_4\cos\theta-\mathrm{Im}M_4\sin\theta \right) \nonumber\\[10pt] & \quad +\frac{D}{\nu_0} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-1/2} \left(\mathrm{Re}M_5\cos\eta + \mathrm{Im}M_5\sin\eta\right) \bigg] , \nonumber\\[10pt] \nu_r & = \nu_0 +\frac{\sigma+\kappa}{2\pi} \bigg[ \frac{\mathrm{Im}M_1}{2} \ln \frac{D\Lambda^2}{\gamma_0} +\left(\mathrm{Im}M_2+\mathrm{Im}M_3\right) \ln\Lambda+\theta\mathrm{Re}M_2 - \eta\mathrm{Re}M_3 \nonumber\\[10pt] & \quad -\frac14 \left[ \mathrm{Im}M_2\ln \frac{\gamma_0^2+\nu_0^2}{D^2} \mathrm{Im}M_3 \ln \frac{(\mu+\gamma_0)^2+\nu_0^2}{4D^2} \right] \nonumber\\[10pt] & \quad +\frac{D}{2\nu_0} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-1/2} \left(\mathrm{Re}M_4\sin\theta+\mathrm{Im}M_4\cos\theta\right) \nonumber\\[10pt] & \quad +\frac{D}{\nu_0} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-1/2} \left(-\mathrm{Re}M_5\sin\eta + \mathrm{Im}M_5\cos\eta\right) \bigg] , \end{align} \tag{ C.6 }$$

and the renormalized diffusivity is

$$\begin{align} \mathrm{Re}D_r = D + \frac{\kappa+\sigma}{2\pi} \bigg[ &-\frac{\mathrm{Re}Q_1}{8} \ln \frac{\nu_0^2+(\mu+\gamma_0)^2}{4\gamma_0^2} +\frac14 \left(\eta\mathrm{Im}Q_1 - \theta\mathrm{Im}P_1\right) \nonumber\\[8pt] &+\frac{D}{4\nu_0}\left(1+\frac{\gamma_0^2}{\nu_0^2}\right)^{\!-1/2}\left(\mathrm{Re}P_2\cos\theta-\mathrm{Im}P_2\sin\theta\right)\nonumber\\[8pt] &+\frac{D^2}{8\nu_0^2} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-1} \left(\mathrm{Re}P_3\cos2\theta-\mathrm{Im}P_3\sin2\theta \right) \nonumber\\[8pt] &+\frac{D}{2\nu_0} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2}\right)^{\!-1/2} \left(\mathrm{Re}Q_2\cos\eta+\mathrm{Im}Q_2\sin\eta\right) \nonumber\\[8pt] &+\frac{D^2}{2\nu_0^2} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-1} \left(\mathrm{Re}Q_3\cos2\eta+\mathrm{Im}Q_3\sin2\eta\right) \bigg] . \end{align} \tag{ C.7 }$$

For the ψ_o mode, the renormalized parameters read

$$\begin{align} \mu_r & = \mu \left[ 1 + \frac{\sigma+\kappa}{16D\pi}\sqrt{\frac{\mu}{D}}+\frac{\sigma+\kappa}{6D\pi}\sqrt{\frac{\gamma_0}{D}} \left( \!-\frac14+\frac{\gamma_0}{\mu}-\frac{\sqrt{3}\nu_0}{16\mu}-\frac{\sqrt{3}\nu_0}{16\gamma_0}-\frac{\sqrt{3}\nu_0\gamma_0}{2\mu^2} \right)\!\right] , \nonumber\\[8pt] D^o_r & = D-\frac{\sigma+\kappa}{96\pi}\sqrt{\frac{\mu}{D}}-\frac{\sigma+\kappa}{48\pi}\sqrt{\frac{\gamma_0}{D}}\left( \frac{\sqrt{3}\nu_0}{6\gamma_0}+\frac{\gamma_0}{\mu}-\frac{5\sqrt{3}\nu_0}{6\mu} \right) . \end{align} \tag{ C.8 }$$

For the $\psi_\pm$ modes, if $\gamma_0 \geqslant 0$, the renormalized parameters are

$$\begin{align} \gamma_r & = \gamma_0 +\frac{\sigma+\kappa}{4\pi}\sqrt{\frac{\nu_0}{D}} \ \bigg[ -\mathrm{Re}M_1\sqrt{\frac{\gamma_0}{\nu_0}}-\left(1+\frac{\gamma_0^2}{\nu_0^2}\right)^{1/4}\left(\mathrm{Re}M_2\cos\frac{\theta}{2}+\mathrm{Im}M_2\sin\frac{\theta}{2}\right) \nonumber\\[8pt] &\quad - \frac{\sqrt{2}}{2} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!1/4} \left(\mathrm{Re}M_3\cos\frac{\eta}{2}-\mathrm{Im}M_3\sin\frac{\eta}{2}\right) \nonumber\\[8pt] &\quad + \frac{D}{2\nu_0} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-1/4} \left(\mathrm{Re}M_4\cos\frac{\theta}{2}-\mathrm{Im}M_4\sin\frac{\theta}{2}\right) \nonumber\\[8pt] &\quad + \frac{\sqrt{2}D}{2\nu_0 }\left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-1/4} \left(\mathrm{Re}M_5\cos\frac{\eta}{2}+\mathrm{Im}M_5\sin\frac{\eta}{2}\right) \bigg] , \nonumber\\[8pt] \nu_r & = \nu_0 +\frac{\sigma+\kappa}{4\pi}\sqrt{\frac{\nu_0}{D}} \ \bigg[ -\mathrm{Im}M_1 \sqrt{\frac{\gamma_0}{\nu_0}} - \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!1/4} \left(-\mathrm{Re}M_2\sin\frac{\theta}{2}+\mathrm{Im}M_2\cos\frac{\theta}{2}\right) \nonumber\\[8pt] &\quad - \frac{\sqrt{2}}{2} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!1/4} \left(\mathrm{Re}M_3\sin\frac{\eta}{2}+\mathrm{Im}M_3\cos\frac{\eta}{2}\right) \nonumber\\[8pt] &\quad + \frac{D}{2\nu_0} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-1/4} \left(\mathrm{Re}M_4\sin\frac{\theta}{2}+\mathrm{Im}M_4\cos\frac{\theta}{2}\right) \nonumber\\[8pt] &\quad + \frac{\sqrt{2}D}{2\nu_0} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-1/4} \left(-\mathrm{Re}M_5\sin\frac{\eta}{2}+\mathrm{Im}M_5\cos\frac{\eta}{2}\right) \bigg] . \end{align} \tag{ C.9 }$$

When $\gamma_0 \lt 0$ and the system is rendered unstable, the renormalized parameters become

$$\begin{align} \mu_r & = \left[ 1 + \frac{\sigma+\kappa}{16D\pi}\sqrt{\frac{\mu}{D}}+\frac{\sigma+\kappa}{6D\pi}\sqrt{\frac{\gamma_0}{D}} \left( -\frac14+\frac{\gamma_0}{\mu}-\frac{\sqrt{3}\nu_0}{16\mu} -\frac{\sqrt{3}\nu_0}{16\gamma_0}-\frac{\sqrt{3}\nu_0\gamma_0}{2\mu^2} \right) \right] , \nonumber\\[8pt] D^o_r & = D-\frac{\sigma+\kappa}{96\pi}\sqrt{\frac{\mu}{D}}-\frac{\sigma+\kappa}{48\pi}\sqrt{\frac{\gamma_0}{D}} \left (\frac{\sqrt{3}\nu_0}{6\gamma_0}+\frac{\gamma_0}{\mu}-\frac{5\sqrt{3}\nu_0}{6\mu} \right) . \end{align} \tag{ C.10 }$$

For the $\psi_\pm$ modes, for $\gamma_0 \geqslant 0$, the renormalized parameters are

$$\begin{align} \gamma_r & = \gamma_0 +\frac{\sigma+\kappa}{4\pi}\sqrt{\frac{\nu_0}{D}} \ \bigg[ -\mathrm{Im}M_1 \sqrt{\frac{|\gamma_0|}{\nu_0}}-\left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!1/4} \left(\mathrm{Re}M_2\cos\frac{\theta}{2}+\mathrm{Im}M_2\sin\frac{\theta}{2}\right) \nonumber\\[8pt] &\quad - \frac{\sqrt{2}}{2} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!1/4} \left(\mathrm{Re}M_3\cos\frac{\eta}{2}-\mathrm{Im}M_3\sin\frac{\eta}{2}\right) \nonumber\\[8pt] &\quad + \frac{D}{2\nu_0} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-1/4} \left(\mathrm{Re}M_4\cos\frac{\theta}{2}-\mathrm{Im}M_4\sin\frac{\theta}{2}\right) \nonumber\\[8pt] &\quad + \frac{\sqrt{2}D}{2\nu_0} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-1/4} \left(\mathrm{Re}M_5\cos\frac{\eta}{2}+\mathrm{Im}M_5\sin\frac{\eta}{2}\right) \bigg] , \nonumber\\[8pt] \nu_r & = \nu_0 +\frac{\sigma+\kappa}{4\pi}\sqrt{\frac{\nu_0}{D}} \ \bigg[ \mathrm{Re}M_1 \sqrt{\frac{|\gamma_0|}{\nu_0}} - \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!1/4} \left(-\mathrm{Re}M_2\sin\frac{\theta}{2}+\mathrm{Im}M_2\cos\frac{\theta}{2}\right) \nonumber\\[8pt] &\quad - \frac{\sqrt{2}}{2} \left (1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!1/4} \left(\mathrm{Re}M_3\sin\frac{\eta}{2}+\mathrm{Im}M_3\cos\frac{\eta}{2}\right) \nonumber\\[8pt] &\quad + \frac{D}{2\nu_0} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-1/4} \left(\mathrm{Re}M_4\sin\frac{\theta}{2}+\mathrm{Im}M_4\cos\frac{\theta}{2}\right) \nonumber\\[8pt] &\quad + \frac{\sqrt{2}D}{2\nu_0} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-1/4} \left(-\mathrm{Re}M_5\sin\frac{\eta}{2}+\mathrm{Im}M_5\cos\frac{\eta}{2}\right) \bigg] . \end{align} \tag{ C.11 }$$

The renormalized diffusivity is not affected by the sign of γ₀,

$$\begin{align} \mathrm{Re}D_r & = D +\frac{\kappa+\sigma}{3\pi}\sqrt{\frac{\nu_0}{D}} \ \bigg[ -\frac14 \left( 1+ \frac{\gamma_0^2}{\nu_0^2} \right)^{\!1/4} \mathrm{Im}P_1\sin\frac{\theta}{2} + \frac14 \mathrm{ReQ_1} \sqrt{\frac{\gamma_0}{\nu_0}} \nonumber\\[8pt] &\quad +\frac{D}{8\nu_0} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-1/4} \left(\mathrm{Re}P_2\cos\frac{\theta}{2}-\mathrm{Im}P_2\sin\frac{\theta}{2}\right) \nonumber\\[8pt] &\quad +\frac{D^2}{32\nu_0^2} \left( 1+\frac{\gamma_0^2}{\nu_0^2} \right)^{\!-3/4} \left(\mathrm{Re}P_3\cos\frac{3\theta}{2}-\mathrm{Im}P_3\sin\frac{3\theta}{2}\right) \nonumber\\[8pt] &\quad -\frac{\sqrt{2}}{8} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!1/4} \left(\mathrm{Re}Q_1\cos\frac{\eta}{2}-\mathrm{Im}Q_1\sin\frac{\eta}{2}\right) \nonumber\\[8pt] &\quad +\frac{\sqrt{2}D}{8\nu_0} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-1/4} \left(\mathrm{Re}Q_2\cos\frac{\eta}{2}+\mathrm{Im}Q_2\sin\frac{\eta}{2}\right) \nonumber\\[8pt] &\quad +\frac{\sqrt{2}D^2}{16\nu_0^2} \left( 1+\frac{(\mu+\gamma_0)^2}{\nu_0^2} \right)^{\!-3/4} \left(\mathrm{Re}Q_3\cos\frac{3\eta}{2}+\mathrm{Im}Q_3\sin\frac{3\eta}{2}\right) \bigg] . \end{align} \tag{ C.12 }$$