Critical behavior of a phase transition in the dynamics of interacting populations

Dear editor,

We are pleased to resubmit our manuscript entitled “Critical behavior of a phase transition in the dynamics of interacting populations” for publication as an article in SciPost. We thank the referees for their careful reading and expert and useful feedback on the manuscript. Both referees judge that the work merits publication in Scipost, pointing its significance, interest and novelty. The referees comment on the presentation, and provide feedback on how to improve it. In response to the referees’ feedback, we have made considerable changes to the manuscript, which significantly improved the presentation. We hope that with these changes, the manuscript may now be accepted for publication in SciPost. Below we give a point-by-point response to the referees’ comments.

Sincerely,

Thibaut Arnoulx de Pirey and Guy Bunin

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Response to the First Referee
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General response: We thank the referee for their critical reading and helpful feedback. We were glad to read the referee’s positive assessments of the manuscript’s content. To address the referee’s criticisms on the presentation, we have made considerable changes to the manuscript, with the aim of improving readability, and making it more accessible to both experts and readers from a wider background, as detailed below. Regarding the above mentioned reference [21], the present scope of the work is to study in detail the dynamics in the vicinity of the critical point, which was not the focus of [21].

Point-by-point responses:

1) We thank the referee for these valuable comments. Following the referee's suggestion, we have significantly changed the presentation of Sec. II of the manuscript. On top of giving expressions for the values of the relevant critical exponents, we now give a physical picture of what the dynamics are doing close to the transition. This proceeds by distinguishing the different species by their time-averaged growth rate, and highlighting the key role of the small fraction of those for which it is small.

2) We thank the referee for their insightful comments about the presentation of our work. Following the referee's comment, we have extensively rewritten Sec. 5. This rewriting includes putting some of the algebra to the appendix and rephrasing what stayed in the main text to make the flavor of the calculation much more transparent. We believe technical aspects nonetheless deserve to be part of the main text since this is one of the first times DMFT equations with temporal fluctuations are explicitly solved in a model where the effective dynamics of the individual degrees of freedom is non-Gaussian (unlike spherical p-spin models and models of random recurrent neural networks where it is possible to obtain from the get go explicit partial differential equations for correlation and response functions). We therefore expect that these technical aspects will be of interest for the community of physicists working with Dynamical Mean-Field Theory equations, beyond the sole Lotka-Volterra model. Similarly, Sec. 4 has also been extensively rewritten in a way that highlights the specifics of this calculation. To ensure that the paper is self-contained, we have also added more details about the derivation of the effective dynamics used to describe the dynamical phases when $\lambda=0$ and $\lambda \to 0^{+}$.

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Response to the Second Referee
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General response: We thank the referee for their insightful comments which helped us improving the presentation of our manuscript. We were glad to read the referee’s positive assessments of the manuscript’s content. We have made considerable changes to the manuscript, with the aim of improving readability, and making it more accessible to both experts and readers from a wider background, as detailed below.

Point-by-point responses:

1) The referee is right: In the presence of migration $\lambda>0$ and for symmetric interactions, aging dynamics are also observed. We cite the relevant literature in the new version of the introduction. Our aging behavior is nonetheless different in that the system remains trapped for long times in the vicinity of unstable fixed points, which it can approach very closely following some stable directions. This is, in a way, a high-dimensional generalization of the notion of heteroclinic cycles. This mechanism was uncovered in some earlier work of ours [22], which is why we cited the reference there. It is different from the symmetric case, where the dynamics slowly evolve following marginal directions of the underlying “energy” landscape. The difference between these two aging regimes is now explained in the introduction.

2) The referee is right in that this work assumes a regime of parameter such that population sizes do not blow-up in time, and further works discussing the unbounded growth phase are now cited in Sec. V. 5. Since we are interested in the near-critical regime, this assumption is perfectly safe as long as we are not sitting at the point where the fixed point phase, the fluctuating phase, and the unbounded growth phase meet. The critical exponents we are interested in are then independent of the mean interaction strength $\mu$. On the other hand, the non-universal prefactors $Q_{c}(\lambda,\mu)$ and $\tau_{c}(\lambda,\mu)$, as well as the position of the critical point $\sigma_{c}(\lambda,\mu)$, exhibit some non-dramatic dependence on the mean interaction strength. For instance, as we have now stressed in Sec. 4, the amplitude of the critical fluctuations decreases linearly with $1/\mu$ when $\lambda=0$ and $\lambda\to0^{+}$. The most interesting feature regarding the $\mu$-dependence, is that there exists a critical $\lambda_{c}(\mu)$ above which the chaotic phase is not observed, as the system directly jumps to the unbounded growth phase.

3) The referee is right in that the correlation function of the noise does not decay to zero at infinity. In fact, since by construction $\left\langle \xi(t)\xi(t')\right\rangle =\left\langle N(t)N(t')\right\rangle$ and $N(t)$ is a positive variable, we are bound to have $\lim_{\tau\to\infty}\left\langle \xi(t)\xi(t+\tau)\right\rangle >0$. Therefore, in the steady-state, we decompose the correlation function as $

\[ \left\langle \xi(t)\xi(t')\right\rangle =w^{2}+Q\,\delta C(t,t') \]

where $\lim_{\tau\to\infty}\delta C(t,t+\tau)=0$ and $\delta C(t,t)=1$, which then uniquely prescribes the values of $w^{2}$ and $Q$. This decomposition tells us that the noise $\xi(t)$ is statistically equivalent to $\bar{\xi}+\sqrt{Q}\delta\xi(t)$, where $\bar{\xi}$ is a zero-mean Gaussian variable sampled with variance $w^{2}$ which is independent from the zero-mean Gaussian process $\delta\xi(t)$ whose correlations are given by $\left\langle \delta\xi(t)\delta\xi(t')\right\rangle =\delta C(t,t')$. Even though the dynamics has some finite long-term memory accounted for by the frozen noise $\bar{\xi}$, they are initialized with completely random initial conditions, meaning that the population sizes are all sampled independently at the initial time (equivalent to initializing the p-spin model with infinite temperature). And in fact, as for the mixed p-spin model initialized at infinite temperature, the memory of the initial state is lost

\[ \lim_{t\to\infty}\left\langle N(t)N(0)\right\rangle -\left\langle N(t)\right\rangle \left\langle N(0)\right\rangle =0 \]

Since there is no energy landscape, there is no immediate way to probe the emergence of memory of the initial condition as a function of the temperature with which the initial state is sampled. Following the simple idea that energy goes down in time after a quench from infinite temperature, an interesting follow up could be to observe the emergence of memory in the transient dynamics of the Lotka-Volterra model, namely by looking at the behavior of

\[ \lim_{t\to\infty}\left\langle N(t)N(t_{0})\right\rangle -\left\langle N(t)\right\rangle \left\langle N(t_{0})\right\rangle =0 \]

as a function of $t_{0}$, after starting at a completely random initial condition. Does memory emerge continuously as a function of $t_{0}$ or is there a critical time after which memory starts to build in? These are questions that go beyond the scope of the present work which is only interested in the long-time dynamics of the system, and which can be successfully described within the ansatz written above.

4) The dual limit $s\to\infty$ and $z\to 0$ is indeed singular. The validity of the procedure used to handle this limit was demonstrated in [21], and we use results from this earlier work to solve the Dynamical Mean-Field Theory equation near the critical point. In order to ensure that the manuscript is self-contained, we have provided in Sec. 4 more details about the derivations of these effective equations. Regarding the applicability of our model to describe real-world ecosystems, we have now stressed in the introduction what is one of its most striking predictions: in a homogeneous system, the timescale over which population size fluctuates is extremely sensitive to migration and the way the system is coupled to its environment (compared, say, to a timescale that would mainly depend on the interaction coefficients, as it is often the case in low-dimensional systems). In fact, we have extensively rewritten the introduction to provide a better physical and ecological motivation for the present work.

We note that this work deals with the regime close to the transition from the fixed point to the fluctuations. Indeed, if one continues to increase the value of the interaction heterogeneity $\sigma$ one reaches a situation where the quadratic self-regulation term leads to unbounded growth, which may be relevant in other situations and treated with other self-regulation terms. As we now discuss in the revised version of the conclusion, we expect the general picture (namely the existence of three universality classes corresponding to different regimes of migration) to hold for a variety of models, but we believe that the value of the critical exponents might be different.

5) Following the referee's comment, we have extensively rewritten Sec. 5. This includes putting some of the algebra to the appendix and rephrasing what stayed in the main text to make the flavor of the calculation much more transparent. We believe technical aspects nonetheless deserve to be part of the main text since this is (to our knowledge) one of the first time DMFT equations are explicitly and enterely solved in a model where the effective dynamics of the individual degrees of freedom is non-Gaussian (unlike the sphericalp-spin model and random recurrent neural network where this gaussianity allows to obtain explicit coupled equations for correlation and response functions). In order to improve the presentation, we have also extensively rewritten Sec. 4.

6) This would indeed be very interesting. At the moment, we unfortunately are not able to make quantitative statements about the crossover from fixed point to persistent fluctuations in finite size systems.

7) Following the referee's suggestion, we have expanded the part on numerical simulations. We have better explained the various steps and put emphasis on the time integration of the population size dynamics. In particular, for $\lambda>0$, we use an integration scheme that allows us to scale the timestep with the correlation time of the noise. This allows efficient numerical simulations close to the critical point where the correlation time of the noise diverges.