Stability and bifurcation of difference equations from stochastic logistic models

Haiyan Wang; Emily Wang; Haiyan Wang; Emily Wang

doi:10.3934/mbe.2026018

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 2: 449-473. doi: 10.3934/mbe.2026018

Previous Article Next Article

Research article Special Issues

Stability and bifurcation of difference equations from stochastic logistic models

Haiyan Wang ^{1
,
,},
Emily Wang ²

1.
School of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069, USA
2.
LinkedIn, San Francisco, CA 94105, USA

Received: 10 November 2025 Revised: 15 December 2025 Accepted: 23 December 2025 Published: 07 January 2026

This paper investigates the stochastic logistic difference equation, $ X_{n+1} = r X_n (1 - X_n)\varepsilon_n $, where $ X_n $ is a random variable of population size, and $ \{\varepsilon_n\} $ represents independent random perturbations with $ E[\varepsilon_n] = 1 $ and $ E[\varepsilon_n^2] = v > 1 $. Under the Gaussian moment-closure approximation, we derived a closed system of difference equations for the mean and variance of $ X_n $. The analysis of the system of difference equations identified two classes of equilibria: a trivial equilibrium $ (0, 0) $ representing extinction, and nontrivial equilibria corresponding to positive steady population levels. Explicit conditions for the existence and local stability of these equilibria were obtained, showing that the extinction state is stable when $ r^2v < 1 $, whereas nontrivial equilibria arise for $ r > 1 $ with stability dependent on the stochastic intensity $ v $. The saddle-node (fold) bifurcation induced by variations in the stochastic intensity $ v $ was explicitly formulated. Monte Carlo simulations confirmed the analytical analysis.
- logistic difference equation,
- stochastic population equations,
- Gaussian moment-closure approximation,
- equilibrium,
- stability,
- bifurcation
Citation: Haiyan Wang, Emily Wang. Stability and bifurcation of difference equations from stochastic logistic models[J]. Mathematical Biosciences and Engineering, 2026, 23(2): 449-473. doi: 10.3934/mbe.2026018

Related Papers:

Abstract

This paper investigates the stochastic logistic difference equation, $ X_{n+1} = r X_n (1 - X_n)\varepsilon_n $, where $ X_n $ is a random variable of population size, and $ \{\varepsilon_n\} $ represents independent random perturbations with $ E[\varepsilon_n] = 1 $ and $ E[\varepsilon_n^2] = v > 1 $. Under the Gaussian moment-closure approximation, we derived a closed system of difference equations for the mean and variance of $ X_n $. The analysis of the system of difference equations identified two classes of equilibria: a trivial equilibrium $ (0, 0) $ representing extinction, and nontrivial equilibria corresponding to positive steady population levels. Explicit conditions for the existence and local stability of these equilibria were obtained, showing that the extinction state is stable when $ r^2v < 1 $, whereas nontrivial equilibria arise for $ r > 1 $ with stability dependent on the stochastic intensity $ v $. The saddle-node (fold) bifurcation induced by variations in the stochastic intensity $ v $ was explicitly formulated. Monte Carlo simulations confirmed the analytical analysis.

References

[1]	J. D. Murray, Mathematical Biology Ⅰ: An Introduction, Springer, 2002.
[2]	R. M. May, Simple mathematical models with very complicated dynamics, Nature, 261 (1976), 459–467. https://doi.org/10.1038/261459a0 doi: 10.1038/261459a0
[3]	M. Kot, Elements of Mathematical Ecology, Cambridge University Press, 2001.
[4]	M. Aktar, U. Karim, V. Aithal, A. R. Bhowmick, Random variation in model parameters: A comprehensive review of stochastic logistic growth equation, Ecol. Modell., 484 (2023), 110475. https://doi.org/10.1016/j.ecolmodel.2023.110475 doi: 10.1016/j.ecolmodel.2023.110475
[5]	L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Chapman and Hall/CRC, 2010.
[6]	S. J. Schreiber, S. Huang, J. Jiang, H. Wang, Extinction and quasi-stationarity for discrete-time, endemic SIS and SIR models, SIAM J. Appl. Math., 81 (2021), 2195–2217. https://doi.org/10.1137/20M1339015 doi: 10.1137/20M1339015
[7]	E. Braverman, A. Rodkina, Difference equations of Ricker and logistic types under bounded stochastic perturbations with positive mean, Comput. Math. Appl., 66 (2013), 2281–2294. https://doi.org/10.1016/j.camwa.2013.06.014 doi: 10.1016/j.camwa.2013.06.014
[8]	Y. Kang, P. Chesson, Relative nonlinearity and permanence, Theor. Popul. Biol., 78 (2010), 26–35. https://doi.org/10.1016/j.tpb.2010.04.002 doi: 10.1016/j.tpb.2010.04.002
[9]	H. Makarem, H. N. Pishkenari, G. R. Vossoughi, A modified Gaussian moment closure method for nonlinear stochastic differential equations, Nonlinear Dyn., 89 (2017), 2609–2620. https://doi.org/10.1007/s11071-017-3608-9 doi: 10.1007/s11071-017-3608-9
[10]	L. Marrec, C. Bank, T. Bertrand, Solving the stochastic dynamics of population growth, Ecol. Evol., 13 (2023), e10295. https://doi.org/10.1002/ece3.10295 doi: 10.1002/ece3.10295
[11]	J. H. Matis, T. R. Kiffe, On approximating the moments of the equilibrium distribution of a stochastic logistic model, Biometrics, 52 (1996), 980–991. https://doi.org/10.2307/2533059 doi: 10.2307/2533059
[12]	L. Socha, Moment equations for nonlinear stochastic dynamic systems (NSDS), in Linearization Methods for Stochastic Dynamic Systems, Springer, (2008), 85–102. https://doi.org/10.1007/978-3-540-72997-6_4
[13]	H. Wang, A. Tsiairis, J. Duan, Bifurcation in mean phase portraits for stochastic dynamical systems with multiplicative gaussian noise, Int. J. Bifurcation Chaos, 30 (2020), 2050216. https://doi.org/10.1142/S0218127420502168 doi: 10.1142/S0218127420502168
[14]	H. Wang, Equilibrium analysis of discrete stochastic population models with gamma distribution, Math. Biosci., 381 (2025), 109398. https://doi.org/10.1016/j.mbs.2025.109398 doi: 10.1016/j.mbs.2025.109398
[15]	H. Wang, Analysis of discrete stochastic population models with normal distribution, preprint, arXiv: 2504.14296. https://doi.org/10.48550/arXiv.2504.14296
[16]	H. Wang, Y. Wang, Detecting transitions from steady states to chaos with gamma distribution, Int. J. Bifurcation Chaos, 35 (2025), 2550163. https://doi.org/10.1142/S0218127425501639 doi: 10.1142/S0218127425501639
[17]	E. Lakatos, A. Ale, P. D. Kirk, M. P. Stumpf, Multivariate moment closure techniques for stochastic kinetic models, J. Chem. Phys., 143 (2015), 094107. https://doi.org/10.1063/1.4929837 doi: 10.1063/1.4929837
[18]	Wikipedia Contributors, Normal distribution, Wikipedia, The Free Encyclopedia, 2024. Available from: https://en.wikipedia.org/wiki/Normal_distribution.
[19]	S. N. Elaydi, An Introduction to Difference Equations, 3rd edition, Springer, New York, 2005.
[20]	Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd edition, Springer-Verlag, New York, 2004.
[21]	J. M. Grandmont, Nonlinear difference equations, bifurcations and chaos: An introduction, Res. in Econ., 62 (2008), 122–177. https://doi.org/10.1016/j.rie.2008.06.003 doi: 10.1016/j.rie.2008.06.003

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)