finite constants $C_1, C_2$, independent from $t$ and $f$, such that

$$ |\langle \mathbf{P}_t^1 - \mathbf{P}_t^2, f \rangle| = \left| \int_0^t \int_0^1 f(x) [g^P(c\phi \star \mathbf{A}_s^1(x)) - k \star \mathbf{P}_s^1] d\mathbf{P}_s^1(x) ds - \int_0^t \int_0^1 f(x) [g^P(c\phi \star \mathbf{A}_s^2(x)) - k \star \mathbf{P}_s^2] d\mathbf{P}_s^2(x) ds \right| \\ \le C_1 \int_0^t \rho(\mathbf{P}_s^1, \mathbf{P}_s^2) ds + \int_0^t \int_0^1 (L^P |c\phi \star (\mathbf{A}_s^1 - \mathbf{A}_s^2(x))| + |k \star (\mathbf{P}_s^1 - \mathbf{P}_s^2)(x)|) d\mathbf{P}_s^2(x) ds \\ \le C_2 \int_0^t (\rho(\mathbf{P}_s^1, \mathbf{P}_s^2) + \rho(\mathbf{A}_s^1, \mathbf{A}_s^2)) ds, $$

since $x \mapsto f(x)(g^P(c\phi \star \mathbf{A}_s^1(x)) - k \star \mathbf{P}_s^1(x))$ are continuous bounded functions on $[0, 1]$ for all $s \in [0, T]$, and $k$ and $c\phi$ are continuous bounded functions on $[0, 1]^2$. Same computations can be done with $\mathbf{A}^1$ and $\mathbf{A}^2$, then taking the supremum over all $f$, we find a constant $C_3 > 0$ such that

$$ \rho(\mathbf{P}_t^1, \mathbf{P}_t^2) + \rho(\mathbf{A}_t^1, \mathbf{A}_t^2) \le C_2 \int_0^t (\rho(\mathbf{P}_s^1, \mathbf{P}_s^2) + \rho(\mathbf{A}_s^1, \mathbf{A}_s^2)) ds. $$

We conclude with Gronwall inequality that $\rho(\mathbf{P}_t^1, \mathbf{P}_t^2) + \rho(\mathbf{A}_t^1, \mathbf{A}_t^2) = 0$ for all $t \in [0, T]$. In other words, $(\mathbf{P}^1, \mathbf{A}^1)$ and $(\mathbf{P}^2, \mathbf{A}^2)$ are identical, and the solution to (3.4) is unique. This ends the proof. ■

## References

[1] E. Abbe. Community detection and stochastic block models: recent development. Journal of Machine Learning Research, 18(177):1–86, 2018.

[2] I. Akjouj, M. Barbier, M. Clénet, W. Hachem, M. Maïda, F. Massol, J. Najim, and V.C. Tran. Complex systems in ecology: A guided tour with the lotka-volterra model. review in progress, 2021.

[3] G. Ben Arous, Y.V. Fyodorov, and B.A. Khoruzhenko. Counting equilibria of large complex systems by instability index. Proceedings of the National Academy of Sciences, 118(34):e2023719118, August 2021.

[4] Blanca Arroyo-Correa, Ignasi Bartomeus, and Pedro Jordano. Individual-based plant-pollinator networks are structured by phenotypic and microsite plant traits. Journal of Ecology, 109(8):2832–2844, 2021.

[5] V. Bansaye, S. Billiard, and J.R. Chazottes. Rejuvenating functional responses avec renewal theory. Journal of the Royal Society Interface, 15(146):20180239, 2018.

[6] V. Bansaye and S. Méléard. Stochastic models for structured populations. Scaling limits and long time behavior. MBI Lecture Series 1.4. Springer, 2015.

[7] M. Barbier, J.F. Arnoldi, G. Bunin, and M. Loreau. Generic assembly patterns in complex ecological communities. Proceedings of the National Academy of Sciences, 115(9):2156–2161, 2018.

[8] J. Bascompte, P. Jordano, C. Melian, and J.M. Olesen. The nested assembly of plant-animal mutualistic networks. Proceedings of the National Academy of Sciences, 100(16):9383–9387, 2003.

[9] Nicola Bellomo, Ahmed Elaiw, Abdullah M Althiabi, and Mohammed Ali Alghamdi. On the interplay between mathematics and biology: Hallmarks toward a new systems biology. Physics of life reviews, 12:44–64, 2015.

[10] Cecilia Berardo, Stefan Geritz, Mats Gyllenberg, and Gaël Raoul. Interactions between different predator–prey states: a method for the derivation of the functional and numerical response. Journal of Mathematical Biology, 80:2431–2468, 2020.

[11] P. Billingsley. Convergence of Probability Measures. Wiley Series in Probability and Statistics. John Wiley & Sons, Inc., New York, second edition edition, 1999.

[12] B. Bollobás. Random graphs. Cambridge University Press, 2 edition, 2001.

[13] C. Borgs, J. Chayes, L. Lovász, V. Sós, and K. Vesztergombi. Limits of randomly grown graph sequences. European Journal of Combinatorics, 32(7):985–999, 2011.

[14] G. Bunin. Ecological communities with Lotka-Volterra dynamics. Physical Review E, 95(4):042414, 2017.

[15] S.A. Chamberlain, R.V. Cartar, A.C. Worley, S.J. Semmler, G. Gielens, S. Elwell, M.E. Evans, J.C. Vamosi, and E. Elle. Traits and phylogenetic history contribute to network structure across canadian plant-pollinator communities. Oecologia, 176:545–556, 2014.
[16] N. Champagnat, R. Ferrière, and S. Méléard. Unifying evolutionary dynamics: from individual stochastic processes to macroscopic models via timescale separation. *Theoretical Population Biology*, 69:297–321, 2006.

[17] J.E. Cohen and C.M. Newman. When will a large complex system be stable? *Journal of Theoretical Biology*, 113:153–156, 1985.

[18] Eva Delmas, Mathilde Besson, Marie-Hélène Brice, Laura A. Burkle, Giulio V. Dalla Riva, Marie-Josée Fortin, Dominique Gravel, Paulo R. Guimarães Jr., David H. Hembry, Erica A. Newman, Jens M. Olesen, Mathias M. Pires, Justin D. Yeakel, and Timothée Poisot. Analysing ecological networks of species interactions. *Biological Reviews*, 94(1):16–36, 2019.

[19] R. Van der Hofstad. *Random Graphs and Complex Networks*, volume 1 of *Cambridge Series in Statistical and Probabilistic Mathematics*. Cambridge University Press, Cambridge, 2017.

[20] Laurent Desvillettes, Pierre Emmanuel Jabin, Stéphane Mischler, and Gaël Raoul. On selection dynamics for continuous structured populations. *Communications in Mathematical Sciences*, 6(3):729–747, 2008.

[21] R. Durrett. *Random graph dynamics*. Cambridge University Press, New York, 2007.

[22] S.N. Ethier and T.G. Kurtz. *Markov Processes, Characterization and Convergence*. John Wiley & Sons, New York, 1986.

[23] R. Ferriere and V.C. Tran. Stochastic and deterministic models for age-structured populations with genetically variable traits. *ESAIM: Proceedings, EDP Sciences*, 27:289–310, 2009.

[24] C. Fontaine, P.R. Guimaraes Jr., S. Kéfi, and E. Thébault. The ecological and evolutionary implications of merging different types of networks. *Ecology Letters*, 14(11):1170–1181, 2011.

[25] N. Fournier and S. Méléard. A microscopic probabilistic description of a locally regulated population and macroscopic approximations. *Ann. Appl. Probab.*, 14(4):1880–1919, 2004.

[26] Y.V. Fyodorov and B.A. Khoruzhenko. Nonlinear analogue of the May-Wigner instability transition. *Proceedings of the National Academy of Sciences*, 113(25):6827–6832, 2016.

[27] D.T. Gillespie. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. *Journal of Computational Physics*, 22(4):403–434, 1976.

[28] D.T. Gillespie. Exact stochastic simulation of coupled chemical reactions. *Journal of Physical Chemistry*, 81(25):2340–2361, 1977.

[29] Paulo R. Guimarães. The structure of ecological networks across levels of organization. *Annual Review of Ecology, Evolution, and Systematics*, 51(1):433–460, 2020.

[30] J.N. Holland and J.L. Bronstein. Mutualism. In *Sven Erik Jørgensen and Brian D. Fath, editors, Encyclopedia of Ecology*, pages 2485–2491. Academic Press, Oxford, 2008.

[31] JN Holland and Judith L Bronstein. Mutualism. In *Encyclopedia of Ecology, Five-Volume Set*, pages 2485–2491. Elsevier Inc., 2008.

[32] J.N. Holland and D.L DeAngelis. A consumer-resource approach to the density-dependent population dynamics of mutualism. *Ecology*, 91:1286–1295, 2010.

[33] P. Holland, K. Laskey, and S. Leinhardt. Stochastic blockmodels: some first steps. *Social Networks*, 5:109–137, 1983.

[34] N. Ikeda and S. Watanabe. *Stochastic Differential Equations and Diffusion Processes*, volume 24. North-Holland Publishing Company, 1989. Second Edition.

[35] Pierre-Emmanuel Jabin and Gaël Raoul. On selection dynamics for competitive interactions. *Journal of Mathematical Biology*, 63(3):493–517, 2011.

[36] A. Joffe and M. Métivier. Weak convergence of sequences of semimartingales with applications to multitype branching processes. *Advances in Applied Probability*, 18:20–65, 1986.

[37] E. Knop, L. Zoller, R. Ryser, C. Gerpe, M. Hörler, and C. Fontaine. Artificial light at night as a new threat to pollination. *Nature*, 548(7666):206–209, 2017.

[38] J. Jelle Lever, Egbert H. van Nes, Marten Scheffer, and Jordi Bascompte. The sudden collapse of pollinator communities. *Ecology Letters*, 17(3):350–359, 2014.

[39] L. Lovász. Large networks and graph limits. (60), 2012.

[40] R.M. May. Will a large complex system be stable? *Nature*, 238:413–414, 1972.

[41] R.M. May. Qualitative stability in model ecosystems. *Ecology*, 54(3):638–641, 1973.

[42] J.A.J. Metz and V.C. Tran. Daphnias: from the individual based model to the large population equation. *Journal of Mathematical Biology*, 66(4-5):915–933, 2013. Special issue in honor of Odo Diekmann.
[43] Camille Pouchol and Emmanuel Trélat. Global stability with selection in integro-differential lotka-volterra systems modelling trait-structured populations. *Journal of Biological Dynamics*, 12(1):872-893, 2018.

[44] Jonathan Roughgarden. *Theory of Population Genetics and Evolutionary Ecology: An Introduction*. Macmillan, 1979.

[45] Y. Takeuchi. *Global dynamical properties of Lotka-Volterra systems*. World Scientific, 1996.

[46] S. Tang and S. Allesina. Reactivity and stability of large ecosystems. *Frontiers in Ecology and Evolution, 2*(21), 2014.

[47] E. Thébault and C. Fontaine. Stability of ecological communities and the architecture of mutualistic and trophic networks. *Science*, 329(5993):853-856, 2010.

[48] S. Villalobos, J.M. Sevenello-Montagner, and J.C. Vamosi. Specialization in plant-pollinator networks: insights from local-scale interactions in glenbow ranch provincial park in alberta, canada. *BMC Ecology*, 19:34, 2019.

[49] C. Villani. Optimal transport: old and new. Springer, 338, 2009.

[50] Stella Watts, Carsten F. Dormann, Ana M. Martín González, and Jeff Ollerton. The influence of floral traits on specialization and modularity of plant-pollinator networks in a biodiversity hotspot in the Peruvian Andes. *Annals of Botany*, 118(3):415-429, 07 2016.