Population size in stochastic discrete-time ecological dynamics

Alexandru Hening; Siddharth Sabharwal

REVIEW 2 major objections 6 minor 6 references

Population size in stochastic discrete-time ecological dynamics

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper establishes that environmental stochasticity can decrease, increase, or leave unchanged the expected population size, depending on which parameters are random and on which deterministic system is taken as the no-noise baseline.

desk verdict Large-noise results are solid and novel; small-noise section has a load-bearing unproved assumption and some typos, but the paper deserves peer review. read the letter →

arxiv 2507.09325 v1 pith:OHP6PT5Y submitted 2025-07-12 q-bio.PE math.PR

Alexandru Hening , Siddharth Sabharwal This is my paper · ORCID

classification q-bio.PEmath.PR MSC 92D2592D4060J0537N25

keywords stochasticpopulationdynamicsexpectedsizeinvariantprobabilitymeasureBeverton-HoltmodelRickerHassellpredator-preysmall-noiseexpansion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Environmental randomness does not have a single built-in effect on population abundance. This paper studies discrete-time population models at stationarity and shows that long-run expected population size can fall, rise, or stay equal to the deterministic equilibrium, depending on the model, on which parameters fluctuate, and on which deterministic system is chosen as the no-noise baseline. The proofs exploit the fact that per-capita growth rates are zero at stationarity, which turns the problem of comparing population sizes into inequalities like Jensen's inequality, and small-noise expansions are used where exact formulas are unavailable. The concrete payoffs include a strict decrease for Beverton-Holt and Ricker models with random carrying capacity, no change for Ricker with random growth rate, an increase for the Hassell model with random competition, and an increase in expected prey abundance for a predator-prey model.

What carries the argument

The load-bearing mechanism is the stationarity condition $r_i(\mu)=0$ for the realized per-capita growth rate (invasion rate) of any species supported by an invariant probability measure $\mu$. Since $r_i$ is the expectation of $\ln f_i$ under the stationary distribution, setting it to zero yields equations that connect the stationary population mean to the moments of the random parameters; Jensen's inequality then converts these equations into inequalities between the stochastic mean and the deterministic equilibrium. For models where the stationary distribution cannot be solved, the paper applies a small-noise Taylor expansion around a globally attracting fixed point, which expresses the change in expected abundance as an explicit second-order correction involving the Jacobian, Hessian, and noise covariance of the dynamics.

What would settle it

Simulate the stochastic predator-prey model with initial prey density far above the carrying capacity and parameters where only local stability of the interior equilibrium is guaranteed; if the long-run sample mean of the prey does not approach the predicted $d/\gamma$ plus the second-order correction, the global-attraction premise behind the small-noise result would be refuted.

Watch

Extended reading notes

Core claim

The paper's central claim is that environmental stochasticity can push the stationary expected population size in any direction, and the direction is controlled by the model structure, by the identity of the fluctuating parameters, and by the comparison baseline. For the Beverton-Holt model with i.i.d. random carrying capacity $K_t$, the stationary mean satisfies $E X_\infty < K = E K_1$, so noise is always detrimental; for the Ricker model the same is true when the carrying capacity is random, but when only the growth rate $r_t$ fluctuates one gets $E X_\infty = K$, so noise is neutral. For the Hassell model with random $K_t$, the stationary mean satisfies $E N_\infty > \bar N$, so noise is beneficial. The same reasoning applied to a two-species discrete Lotka-Volterra system shows the stationary means equal the deterministic stable fixed point, while for a predator-prey model the expected prey abundance satisfies $E X_\infty > d/\gamma$, exceeding its deterministic value $d/\gamma$.

Load-bearing premise

The small-noise predator-prey expansions require the deterministic model to converge to its interior equilibrium from every starting population, whereas the underlying model only establishes local convergence near that equilibrium; if far-away trajectories escape to another attractor, the predicted mean shifts need not hold.

Editorial extensions

If this is right

In single-species models with a concave log-growth factor, random carrying capacity lowers expected abundance below the deterministic equilibrium, so management targets based on deterministic models may overestimate abundance.
In the Ricker model, the effect of noise disappears if only the growth rate fluctuates: expected abundance stays equal to the deterministic carrying capacity.
In the Hassell model, random competition raises expected abundance above the deterministic equilibrium, so noise can be beneficial for population size.
For two-species discrete Lotka-Volterra dynamics, the stationary expected abundances coincide with the deterministic globally stable fixed point, so noise has no effect on mean abundance.
In the predator-prey model, stochasticity increases expected prey abundance above $d/\gamma$; whether the predator mean rises or falls depends on parameter values.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the zero-growth-at-stationarity trick suggests a general criterion for when noise helps or hurts in single-species maps, without solving for the invariant measure.
The small-noise results imply that the sign of the correlation between fluctuations in growth rate and carrying capacity is a measurable predictor of whether noise raises or lowers abundance; field studies could test this by estimating that covariance.
The baseline-dependence result suggests that empirical comparisons of stochastic and deterministic predictions should always report whether the deterministic baseline uses the arithmetic or harmonic mean of the fluctuating parameter.
For the predator-prey model, if global convergence to the interior fixed point ever fails, the predicted prey increase may only hold near equilibrium; the large-noise simulations in the paper already hint that the predator's response can reverse as noise grows.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The thing to know: the large-noise part of this paper is the real contribution, and it holds up. The small-noise section is conditional in places and has a few mechanical slips, but nothing that sinks the central message.

What's new: using the zero-growth condition r_i(µ)=0 plus Jensen, they prove directional statements for expected stationary population size that were not in the literature. Specifically, stochastic Hassell with random K gives EN∞ > N̄ (noise helps), Ricker with random r gives EX∞ = K exactly (noise neutral), while Beverton-Holt with random K gives EX∞ < K (noise harmful). The Ricker random-r result and the Hassell increase are new as far as I can tell. The Hassell example is nicely framed as an artifact of baseline choice: compare with EK or E(1/K) and you reverse the sign. That's a clean and honest point, and they credit Chesson for the baseline ambiguity.

The two-species Ricker calculation is a neat application: linearity of log-fitness gives exact expectations and shows noise does not change the deterministic fixed point. The predator-prey large-noise bound X∞ > d/γ is simple and uses only Jensen. These results are transparent and reproducible—there's no hidden fitting.

Where it's soft: the small-noise predator-prey section (4.4) rests on Theorem 4.1's global-attracting fixed point assumption. But the deterministic model from Streipert et al. has only a proved locally stable interior fixed point under d<γK<1+2d; global stability is explicitly a conjecture in that paper. The authors cite it as a conjecture but then apply the global result as if it were proved. That makes the small-noise formulas around E* conditional on an unproved premise. They could fix this by stating the result as local (convergence from the basin of E*) or by citing a global proof if one exists. There are also smaller mechanical issues: in the two-species Ricker section, r_i(δ0) is written as E ln b_i(1) when it should be E[b_i(1)]; and Corollary 4.2's expression for Ã is missing a factor of 2 on the cross term relative to Theorem 4.1. These are typos, but the paper's small-noise section is largely asserted formulas without derivations or code, so the typos are harder to catch.

The simulations are illustrative but not fully reproducible: no code, no data, no error bars. That's not fatal for the theory but lowers confidence in the unproven simulation claims about large-noise predator-prey behavior.

Bottom line: the large-noise theorems are solid and worth citing. The small-noise section is not load-bearing for the abstract's main claim—that claim is independently supported by the Hassell and Ricker results. For a theoretical ecology audience, this deserves referee time. I'd advise a conditional accept path: require the authors to either prove or clearly localize the global-stability assumption in Section 4.4 and fix the algebraic slips.

Referee Report

2 major / 6 minor

Summary. The paper studies expected population sizes at stationarity for stochastic discrete-time ecological models. Two tools are used: (i) the stationarity condition r_i(mu)=0 for per-capita growth rates under an invariant measure, combined with Jensen's inequality, to obtain inequalities for large environmental noise; and (ii) small-noise Taylor expansions from Cuello (2019) around deterministic fixed points to obtain O(rho^2) corrections to the mean. The large-noise results give: Beverton-Holt with random carrying capacity satisfies E X_infty < K together with a lower bound; Ricker with random K satisfies E X_infty = 1/E(1/K_1) < K; Ricker with random r satisfies E X_infty = K; Hassell with random K satisfies E N_infty > N-bar; two-species Ricker preserves the deterministic means; and a Streipert-type predator-prey model gives prey mean > d/gamma. The small-noise results treat Beverton-Holt and Hassell with two random parameters, showing that noise correlation can reverse the sign of the effect, and give an expansion for the predator-prey model. Simulations illustrate the main trends.

Significance. If the results are correct, the paper makes a useful contribution by showing rigorously that environmental fluctuations can decrease, increase, or leave unchanged the expected population size, depending on the model, on which parameters are randomized, and on the deterministic baseline chosen. The large-noise inequalities are elegant, essentially parameter-free consequences of stationarity and Jensen's inequality, and they identify the Hassell model as a counterpoint to the Cushing-Henson intuition. The paper also gives a careful discussion, following Chesson, of how the choice of no-noise baseline affects conclusions. The small-noise expansions provide explicit, falsifiable formulas for correlation effects. The main weaknesses are the unverified global-stability premise in the predator-prey small-noise application and an internal factor-of-two error in Corollary 4.2.

major comments (2)

[Section 4.4 and Theorem 4.1] The small-noise predator-prey expansion is applied to the interior fixed point E* = (d/gamma, r(gamma K - d)/(alpha gamma K)), but Theorem 4.1 assumes that the deterministic system has a globally attracting fixed point in R^n_{++}. In Section 3.3 the authors state that Streipert et al. (2022) proved only local asymptotic stability under d < gamma K < 1 + 2d and conjectured global stability; Appendix A, Assumption B3, similarly requires every deterministic trajectory to converge to an equilibrium. The paper neither proves this conjecture nor rules out other attractors in the interior. Therefore the formulas in Section 4.4 are justified, at best, for initial conditions in the local basin of E* (as in Cuello's local Theorem A.1), not for every positive initial condition as Theorem 4.1 promises. The authors should either prove the global-stability conjecture, explicitly restrict the application to the local basin, or present the Section 4.4 formulas as conditional on Streipert et al.'s conjecture.
[Corollary 4.2] The displayed expression for \tilde A in Corollary 4.2 contains the cross term (partial F/partial xi_1)(partial F/partial xi_2) E(xi_1 xi_2) without a factor of 2. In Theorem 4.1, vec(A) = (I - D_x F \otimes D_x F)^{-1}(D_e F \otimes D_e F) vec(Cov(e)) sums over all ordered pairs (j1,j2), so the off-diagonal covariance contributes 2(partial F/partial xi_1)(partial F/partial xi_2) Cov(xi_1, xi_2). The direct second-derivative term in the same corollary has the correct factor 2, making the corollary internally inconsistent. The applications in Sections 4.1 and 4.2 appear to use the correct expression, but the stated corollary should be corrected.

minor comments (6)

[Section 4, after Eq. (27)] The sentence 'the change in mean is O(rho^3)' should read O(rho^2), since Eq. (27) has an explicit rho^2 term; the following phrase also contains a typo ('te mean').
[Section 3.2.1, before Eq. (18)] The formula 'r_i(delta_0) = E ln b_i(1)' should be 'E[b_i(1)]', since the log of the Ricker fitness is b_i(1), not the log of the coefficient.
[Section 4, introduction] The sentence 'Here we see how the correlations play an important role' is duplicated immediately before Section 4.1.
[Section 4.4] The displayed expansion is for E(X(infinity)) + E(Y(infinity)), but the biological interpretation refers to the expected prey population. Since the expansion is for the sum, it does not by itself show that the prey marginal mean increases; the prey-increase claim is supported by Eq. (22) and by simulations, but the presentation should be clarified, and the missing parenthesis in 'E(Y (infinity)' should be fixed.
[Section 3.1.2] The sentence 'If r > 1 we get that as t tends to infinity one has X_t to X_infinity in distribution' is too terse: the deterministic Ricker map is not globally stable for all r > 1, so the relevant range of r (or the precise stochastic persistence theorem) should be stated.
[Section 5.1] The noise strength is called t and is also used as the time index elsewhere; this is confusing and should be renamed, for example to epsilon or sigma.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reduction found: the claims follow from the stationarity identity r_i(mu)=0 plus Jensen and Cuello's expansion; the predator-prey small-noise application is conditional on Streipert et al.'s unproved global-stability conjecture, which is a rigor caveat, not circularity.

full rationale

Walk-through of the derivation chain: the large-noise results use Proposition 2.1, which states that at stationarity the per-capita growth rate is zero, together with the explicit fitness functions and Jensen's inequality. For Beverton-Holt with random K this gives the lower bound (7) and, via the cited Haskell-Sacker proof of the stochastic Cushing-Henson conjecture, the upper bound (8). For Ricker with random r it gives EX_infinity = K, for Ricker with random K it gives EX_infinity = 1/E(1/K1) < K, for Hassell with random K it gives EN_infinity > N-bar, and for the predator-prey model it gives EX_infinity > d/gamma. None of these steps fits a parameter to the target quantity or defines a prediction in terms of itself; the model parameters are inputs and the invariant-measure identity is a mathematical consequence of stationarity. The small-noise results are applications of Cuello's published expansion, quoted in Appendix A as Theorem A.2, with assumptions (B1)-(B4) stated; no ansatz is smuggled in via a self-citation. The self-citations to Hening et al. (2021) provide existence and uniqueness of invariant measures and are published theorems with independent standing, not the claims being derived. Two caveats are correctness or rigor issues rather than circularity: Section 4.4 applies Theorem 4.1 at the interior predator-prey fixed point even though Section 3.3 explicitly states that Streipert et al. only proved local stability and 'conjectured that E* is actually globally asymptotically stable', so that expansion is conditional on an unproved premise; and Corollary 4.2's displayed cross term appears to miss the factor 2 that Theorem 4.1's vec(A) formula would produce. Neither caveat makes the central claims self-referential. Therefore no circular step is present.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim does not fit any empirical constants; all parameters are model inputs from the cited ecological literature. The main axioms are external theorems, namely Theorem 2.1, Proposition 2.1, and Cuello's expansion, plus the iid-noise and compactness assumptions, and one unproved global-stability condition for the predator-prey model. The hand-chosen simplifications in Sections 4.2 and 4.4 restrict scope but are not fitted to data. No new entities are postulated.

free parameters (2)

Hassell small-noise specialization c̃=1/2, i.e. α=(2c/(2c-1))^c
Hand-chosen simplification in Section 4.2 that makes the small-noise formulas tractable; it restricts the analysis to c in (1/2,1) and does not fit data.
Predator-prey simplifications α/γ=p, d=r, r - rd/(Kγ)=r/2, Cov(ξ)=I
Hand-chosen parameter reduction in Section 4.4 used to compute the explicit small-noise formula for the predator-prey model; not estimated from data.

assumptions (6)

domain assumption Theorem 2.1: stochastic persistence and convergence to a unique invariant probability measure under max_i r_i(μ)>0 and irreducibility (Hening et al. 2021, quoted in Section 2.2).
Used to justify existence of stationary distributions in Sections 3.1 through 3.3.
standard math Proposition 2.1: for an ergodic invariant measure μ, any supported species has zero realized per-capita growth rate r_i(μ)=0.
Core identity used throughout Section 3; proof sketch given in Section 2.2 using Birkhoff and the strong law of large numbers.
domain assumption Cuello's small-noise expansion theorem (Theorem A.2, quoted from Cuello 2019): for small ρ and an internally stable fixed point, the stationary expectation has the stated Taylor expansion.
Basis for all Section 4 small-noise formulas; the theorem is quoted in Appendix A but not proved in this paper.
domain assumption Assumptions (A1), (A2), and (A3): iid environmental noise, continuous positive fitness functions, and exponential return to compact sets.
Standing assumptions from Section 2.1 ensuring the Markov chain is Feller and invariant measures exist.
domain assumption Independence of the noise variable at the next transition from the current population state, for example K1 independent of X∞.
Used to factor expectations like E[X∞/K1]=E[X∞]E[1/K1] in the Jensen steps of Sections 3.1 and 3.3.
ad hoc to paper Global asymptotic stability of the deterministic predator-prey fixed point E* in Section 4.3; Streipert et al. (2022) proved local stability and conjectured global stability.
Theorem 4.1's expansion requires a globally attracting fixed point; this premise is unproved and load-bearing for the small-noise predator-prey formulas.

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Pith. "Pith review of Population size in stochastic discrete-time ecological dynamics." pith.science (2026). https://pith.science/paper/OHP6PT5Y

@misc{pith2026250709325,
  author       = {Pith},
  title        = {Pith review of: Population size in stochastic discrete-time ecological dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OHP6PT5Y}},
  note         = {Machine review of arXiv:2507.09325}
}

read the original abstract

We study how environmental stochasticity influences the long-term population size in certain one- and two-species models. The difficulty is that even when one can prove that there is persistence, it is usually impossible to say anything about the invariant probability measure which describes the persistent species. We are able to circumvent this problem for some important ecological models by noticing that the per-capita growth rates at stationarity are zero, something which can sometimes yield information about the invariant probability measure. For more complicated models we use a recent result by Cuello to explore how small noise influences the population size. We are able to show that environmental fluctuations can decrease, increase, or leave unchanged the expected population size. The results change according to the dynamical model and, within a fixed model, also according to which parameters (growth rate, carrying capacity, etc) are affected by environmental fluctuations. Moreover, we show that not only do things change if we introduce noise differently in a model, but it also matters what one takes as the deterministic `no-noise' baseline for comparison.

Figures

Figures reproduced from arXiv: 2507.09325 by the authors.

**Figure 1.** Expected population size versus noise strength, t, in a Beverton-Holt model. We suppose the random variables to be uncorrelated: Cov((s(t), K(t))) = tI. A: K = 5.75 and r = 5. B: K = 7 and r = 5. We see a clear decreasing trend between the expected population and the noise strength parameter. The analytical results show a decrease in the expected population, if the noise is small. The simulation shows the same behav… view at source ↗

**Figure 2.** Here we consider the random variables from the single-species Beverton Holt model, with r = 5, and K = 7 to be correlated. We fix E(r 2 (t)) = 0.5, E(K2 (t)) = 0.5 and E(r(t)K(t)) = 0.25b and plot the sample mean of the population versus b which goes from -1 to 1. Since our analytical result for small noise shows that negative correlation between the random variables affects the average population negatively, and … view at source ↗

**Figure 3.** Prey/Predator population versus r in the predator-prey model, with K = 1000, γ = 1, α = 2γ, d = 2r. We consider the random variables to be uncorrelated: Σ = diag(100s, s). We see the plot for the prey population on the left, and predator population on the right. The blue curves indicate how the equilibrium population changes with r, and the red curves indicate the sample mean population with s = 5 × 10−4 , the yello… view at source ↗

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Reference graph

Works this paper leans on

6 extracted references · 5 canonical work pages

[1]

Selfimprovemvent of the inequality between arithmetic and geometric means

Abrahams, M. V., Mangel, M. & Hedges, K. (2007), ‘Predator–prey interactions and changing environments: who benefits?’, Philosophical Transactions of the Royal Society B: Biological Sciences 362(1487), 2095–2104. Aldaz, J. (2008), ‘Selfimprovemvent of the inequality between arithmetic and geometric means’, arXiv preprint arXiv:0807.1788 . Alstad, D. & And...

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[3]

Hening, A. & Li, Y. (2021), ‘Stationary distributions of persistent ecological systems’, Journal of Mathematical Biology 82(7),

work page 2021
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Hening, A., Nguyen, D. H. & Schreiber, S. J. (2022), ‘A classification of the dynamics of three- dimensional stochastic ecological systems’, The Annals of Applied Probability 32(2), 893–931. Hofbauer, J., Hutson, V. & Jansen, W. (1987a), ‘Coexistence for systems governed by difference equations of lotka-volterra type’, Journal of mathematical biology 25, ...

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The competitive exclusion principle in stochastic environments

Hening, A. & Nguyen, D. H. (2018 a), ‘Coexistence and extinction for stochastic Kolmogorov systems’, Ann. Appl. Probab. 28(3), 1893–1942. Hening, A. & Nguyen, D. H. (2018 b), ‘The competitive exclusion principle in stochastic envi- ronments’, arXiv preprint arXiv:1810.00954 . Hening, A., Nguyen, D. H. & Chesson, P. (2021), ‘A general theory of coexistence...

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(2013), Population dynamics in variable environments , Vol

Tuljapurkar, S. (2013), Population dynamics in variable environments , Vol. 85, Springer Science & Business Media. Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843-3368, United States Email address: ahening@tamu.edu Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843-3368, Unit...

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arXiv 1957

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Selfimprovemvent of the inequality between arithmetic and geometric means

Abrahams, M. V., Mangel, M. & Hedges, K. (2007), ‘Predator–prey interactions and changing environments: who benefits?’, Philosophical Transactions of the Royal Society B: Biological Sciences 362(1487), 2095–2104. Aldaz, J. (2008), ‘Selfimprovemvent of the inequality between arithmetic and geometric means’, arXiv preprint arXiv:0807.1788 . Alstad, D. & And...

work page Pith review arXiv 2007

[2] [3]

Hening, A. & Li, Y. (2021), ‘Stationary distributions of persistent ecological systems’, Journal of Mathematical Biology 82(7),

work page 2021

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Hening, A., Nguyen, D. H. & Schreiber, S. J. (2022), ‘A classification of the dynamics of three- dimensional stochastic ecological systems’, The Annals of Applied Probability 32(2), 893–931. Hofbauer, J., Hutson, V. & Jansen, W. (1987a), ‘Coexistence for systems governed by difference equations of lotka-volterra type’, Journal of mathematical biology 25, ...

work page 2022

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The competitive exclusion principle in stochastic environments

Hening, A. & Nguyen, D. H. (2018 a), ‘Coexistence and extinction for stochastic Kolmogorov systems’, Ann. Appl. Probab. 28(3), 1893–1942. Hening, A. & Nguyen, D. H. (2018 b), ‘The competitive exclusion principle in stochastic envi- ronments’, arXiv preprint arXiv:1810.00954 . Hening, A., Nguyen, D. H. & Chesson, P. (2021), ‘A general theory of coexistence...

work page Pith review arXiv 2021

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(2013), Population dynamics in variable environments , Vol

Tuljapurkar, S. (2013), Population dynamics in variable environments , Vol. 85, Springer Science & Business Media. Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843-3368, United States Email address: ahening@tamu.edu Department of Mathematics, Texas A&M University, Mailstop 3368, College Station, TX 77843-3368, Unit...

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Bartlett, M. S. (1957), ‘On theoretical models for competitive and predatory biological systems’, Biometrika 44(1/2), 27–42. Benaim, M. (2018), ‘Stochastic persistence’, arXiv preprint arXiv:1806.08450 . Bena ¨ ım, M. & Lobry, C. (2016), ‘Lotka Volterra in fluctuating environment or “how switching between beneficial environments can make survival harder”’...

arXiv 1957