# American Institute of Mathematical Sciences

• Previous Article
On weak martingale solutions to a stochastic Allen-Cahn-Navier-Stokes model with inertial effects
• DCDS-B Home
• This Issue
• Next Article
Upper semi-continuity of non-autonomous fractional stochastic $p$-Laplacian equation driven by additive noise on $\mathbb{R}^n$
doi: 10.3934/dcdsb.2022023
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Stochastic persistence in degenerate stochastic Lotka-Volterra food chains

 1 Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, 2000 Neuchâtel, Switzerland 2 Department of Mathematics, University of Alabama, Tuscaloosa, Al 35487-0350, USA

*Corresponding author: Michel Benaïm

Received  April 2021 Revised  November 2021 Early access February 2022

Fund Project: Michel Benaim and Antoine Bourquin are supported in part by the SNF grant 200020-196999. Dang H. Nguyen is supported in part by NSF through the grant DMS-1853467

We consider a Lotka-Volterra food chain model with possibly intra-specific competition in a stochastic environment represented by stochastic differential equations. In the non-degenerate setting, this model has already been studied by A. Hening and D. Nguyen in [9, 10] where they provided conditions for stochastic persistence and extinction. In this paper, we extend their results to the degenerate situation in which the top or the bottom species is subject to random perturbations. Under the persistence condition, there exists a unique invariant probability measure supported by the interior of ${{\mathbb R}}_+^n$ having a smooth density.

Moreover, we study a more general model, in which we give new conditions which make it possible to characterize the convergence of the semi-group towards the unique invariant probability measure either at an exponential rate or at a polynomial one. This will be used in the stochastic Lotka-Volterra food chain to see that if intra-specific competition occurs for all species, the rate of convergence is exponential while in the other cases it is polynomial.

Citation: Michel Benaïm, Antoine Bourquin, Dang H. Nguyen. Stochastic persistence in degenerate stochastic Lotka-Volterra food chains. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022023
##### References:
 [1] E. Allen, Environmental variability and mean-reverting processes, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2073-2089.  doi: 10.3934/dcdsb.2016037. [2] M. Benaïm, Stochastic persistence, arXiv preprint arXiv: 1806.08450. [3] M. Benaïm, D. N. Nguyen and N. Nguyen, Stochastic Kolmogorov systems under regime-switching: Coexistence and extinction, preprint. [4] B. Dennis, P. L. Munholland and J. M. Scott, Estimation of growth and extinction parameters for endangered species, Ecological Monographs, 61 (1991), 115-143. [5] S. N. Evans, P. L. Ralph, S. J. Schreiber and A. Sen, Stochastic population growth in spatially heterogeneous environments, J. Math. Biol., 66 (2013), 423-476.  doi: 10.1007/s00285-012-0514-0. [6] P. Foley, Predicting extinction times from environmental stochasticity and carrying capacity, Conservation Biology, 8 (1994), 124-137.  doi: 10.1046/j.1523-1739.1994.08010124.x. [7] T. C. Gard and T. G. Hallam, Persistence in food webs. I. Lotka-Volterra food chains, Bull. Math. Biol., 41 (1979), 877-891. [8] A. Hening and D. H. Nguyen, Coexistence and extinction for stochastic Kolmogorov systems, Ann. Appl. Probab., 28 (2018), 1893-1942.  doi: 10.1214/17-AAP1347. [9] A. Hening and D. H. Nguyen, Persistence in stochastic Lotka-Volterra food chains with intraspecific competition, Bull. Math. Biol., 80 (2018), 2527-2560.  doi: 10.1007/s11538-018-0468-5. [10] A. Hening and D. H. Nguyen, Stochastic Lotka-Volterra food chains, J. Math. Biol., 77 (2018), 135-163.  doi: 10.1007/s00285-017-1192-8. [11] J. Hofbauer and K. Sigmund, Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179. [12] K. Ichihara and H. Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characterization, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 30 (1974), 235-254.  doi: 10.1007/BF00533476. [13] S. r. F. Jarner and G. O. Roberts, Polynomial convergence rates of {M}arkov chains, Ann. Appl. Probab., 12 (2002), 224-247.  doi: 10.1214/aoap/1015961162. [14] R. Lande, S. Engen, B.-E. Saether, et al., Stochastic Population Dynamics in Ecology and Conservation, Oxford University Press on Demand, 2003. [15] S. P. Meyn and R. L. Tweedie, Stability of Markovian processes. I. Criteria for discrete-time chains, Adv. in Appl. Probab., 24 (1992), 542-574.  doi: 10.2307/1427479. [16] S. J. Schreiber, The evolution of patch selection in stochastic environments, The American Naturalist, 180 (2012), 17-34.  doi: 10.1086/665655. [17] S. J. Schreiber, M. Benaïm and K. A. S. Atchadé, Persistence in fluctuating environments, J. Math. Biol., 62 (2011), 655-683.  doi: 10.1007/s00285-010-0349-5.

show all references

##### References:
 [1] E. Allen, Environmental variability and mean-reverting processes, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 2073-2089.  doi: 10.3934/dcdsb.2016037. [2] M. Benaïm, Stochastic persistence, arXiv preprint arXiv: 1806.08450. [3] M. Benaïm, D. N. Nguyen and N. Nguyen, Stochastic Kolmogorov systems under regime-switching: Coexistence and extinction, preprint. [4] B. Dennis, P. L. Munholland and J. M. Scott, Estimation of growth and extinction parameters for endangered species, Ecological Monographs, 61 (1991), 115-143. [5] S. N. Evans, P. L. Ralph, S. J. Schreiber and A. Sen, Stochastic population growth in spatially heterogeneous environments, J. Math. Biol., 66 (2013), 423-476.  doi: 10.1007/s00285-012-0514-0. [6] P. Foley, Predicting extinction times from environmental stochasticity and carrying capacity, Conservation Biology, 8 (1994), 124-137.  doi: 10.1046/j.1523-1739.1994.08010124.x. [7] T. C. Gard and T. G. Hallam, Persistence in food webs. I. Lotka-Volterra food chains, Bull. Math. Biol., 41 (1979), 877-891. [8] A. Hening and D. H. Nguyen, Coexistence and extinction for stochastic Kolmogorov systems, Ann. Appl. Probab., 28 (2018), 1893-1942.  doi: 10.1214/17-AAP1347. [9] A. Hening and D. H. Nguyen, Persistence in stochastic Lotka-Volterra food chains with intraspecific competition, Bull. Math. Biol., 80 (2018), 2527-2560.  doi: 10.1007/s11538-018-0468-5. [10] A. Hening and D. H. Nguyen, Stochastic Lotka-Volterra food chains, J. Math. Biol., 77 (2018), 135-163.  doi: 10.1007/s00285-017-1192-8. [11] J. Hofbauer and K. Sigmund, Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179. [12] K. Ichihara and H. Kunita, A classification of the second order degenerate elliptic operators and its probabilistic characterization, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 30 (1974), 235-254.  doi: 10.1007/BF00533476. [13] S. r. F. Jarner and G. O. Roberts, Polynomial convergence rates of {M}arkov chains, Ann. Appl. Probab., 12 (2002), 224-247.  doi: 10.1214/aoap/1015961162. [14] R. Lande, S. Engen, B.-E. Saether, et al., Stochastic Population Dynamics in Ecology and Conservation, Oxford University Press on Demand, 2003. [15] S. P. Meyn and R. L. Tweedie, Stability of Markovian processes. I. Criteria for discrete-time chains, Adv. in Appl. Probab., 24 (1992), 542-574.  doi: 10.2307/1427479. [16] S. J. Schreiber, The evolution of patch selection in stochastic environments, The American Naturalist, 180 (2012), 17-34.  doi: 10.1086/665655. [17] S. J. Schreiber, M. Benaïm and K. A. S. Atchadé, Persistence in fluctuating environments, J. Math. Biol., 62 (2011), 655-683.  doi: 10.1007/s00285-010-0349-5.
 [1] Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 245-263. doi: 10.3934/dcdss.2020468 [2] P. Auger, N. H. Du, N. T. Hieu. Evolution of Lotka-Volterra predator-prey systems under telegraph noise. Mathematical Biosciences & Engineering, 2009, 6 (4) : 683-700. doi: 10.3934/mbe.2009.6.683 [3] Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559 [4] Rui Wang, Xiaoyue Li, Denis S. Mukama. On stochastic multi-group Lotka-Volterra ecosystems with regime switching. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3499-3528. doi: 10.3934/dcdsb.2017177 [5] Fuke Wu, Yangzi Hu. Stochastic Lotka-Volterra system with unbounded distributed delay. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 275-288. doi: 10.3934/dcdsb.2010.14.275 [6] Dejun Fan, Xiaoyu Yi, Ling Xia, Jingliang Lv. Dynamical behaviors of stochastic type K monotone Lotka-Volterra systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2901-2922. doi: 10.3934/dcdsb.2018291 [7] Yuzo Hosono. Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator. Discrete and Continuous Dynamical Systems - B, 2015, 20 (1) : 161-171. doi: 10.3934/dcdsb.2015.20.161 [8] Rui Xu, M.A.J. Chaplain, F.A. Davidson. Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 823-831. doi: 10.3934/dcdsb.2004.4.823 [9] Michael Y. Li, Xihui Lin, Hao Wang. Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 747-760. doi: 10.3934/dcdsb.2014.19.747 [10] S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system. Mathematical Biosciences & Engineering, 2006, 3 (1) : 173-187. doi: 10.3934/mbe.2006.3.173 [11] Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195 [12] Vladimir Kazakov. Sampling - reconstruction procedure with jitter of markov continuous processes formed by stochastic differential equations of the first order. Conference Publications, 2009, 2009 (Special) : 433-441. doi: 10.3934/proc.2009.2009.433 [13] Xiaoling Zou, Ke Wang. Optimal harvesting for a stochastic N-dimensional competitive Lotka-Volterra model with jumps. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 683-701. doi: 10.3934/dcdsb.2015.20.683 [14] Pankaj Kumar, Shiv Raj. Modelling and analysis of prey-predator model involving predation of mature prey using delay differential equations. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021035 [15] Tian Zhang, Chuanhou Gao. Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3725-3747. doi: 10.3934/dcdsb.2021204 [16] Felix X.-F. Ye, Yue Wang, Hong Qian. Stochastic dynamics: Markov chains and random transformations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (7) : 2337-2361. doi: 10.3934/dcdsb.2016050 [17] Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 [18] H. Malchow, F.M. Hilker, S.V. Petrovskii. Noise and productivity dependence of spatiotemporal pattern formation in a prey-predator system. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 705-711. doi: 10.3934/dcdsb.2004.4.705 [19] Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203 [20] Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244

2021 Impact Factor: 1.497