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doi: 10.3934/dcdsb.2022023
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Stochastic persistence in degenerate stochastic Lotka-Volterra food chains

Michel Benaïm 1,, , Antoine Bourquin 1, and  Dang H. Nguyen 2,

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.

Keywords: Markov processes, stochastic differential equations, stochastic persistence, Lotka-Volterra food chains, prey-predator, Hörmander condition, rate of convergence, degenerate noise.
Mathematics Subject Classification: Primary: 60J60, 60H10; Secondary: 37H15, 92D25.
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.

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