Dynamics of a non-autonomous stochastic mutualism model with jumps
DOI:
https://doi.org/10.17721/1812-5409.2026/1.3Keywords:
stochastic mutualism model, global solution, stochastic ultimate boundedness, stochastic permanence, extinction, non-persistence in the mean, weak and strong persistence in the meanAbstract
In nature, we can find many examples where the interaction of two or more species is to the advantage of all. Mutualism occurs when one species provides some benefit in exchange for some benefit, for example, ants and aphids, in which the ants obtain honeydew food resources excreted by aphids while the aphids obtain increased survival by the non-trophic service of ant defense against natural enemies of the aphids. Mathematical deterministic mutualism models are widely used to study the dynamics of such population systems. In real life, there is a lot of randomness and stochasticity, such as environmental noise, so using stochastic models is more suitable. Suppose we want to take into account abrupt environmental perturbations, such as epidemics, fires, earthquakes, etc., in the considered models. In that case, we must introduce Poisson noises into population models to describe such discontinuous systems. So, we take into account not only “small” jumps, corresponding to the centered Poisson measure, but also the “large” jumps, corresponding to the non-centered Poisson measure. The existence and uniqueness of the global positive solution are proved for the system of stochastic differential equations describing a non-autonomous mutualism model disturbed by white noise, centered, and non-centered Poisson noises. We obtain sufficient conditions of stochastic ultimate boundedness, stochastic permanence, non-persistence in the mean, weak and strong persistence in the mean, and extinction of the solution to the considered system.
Pages of the article in the issue: 17 - 26
Language of the article: English
References
Borysenko, O., & Borysenko, D. (2019). Persistence and extinction in stochastic nonautonomous logistic model of population dynamics. Theory of Probability and Mathematical Statistics, 99, 67–75. https://doi.org/10.1090/tpms/1080
Borysenko, O., & Borysenko, V. (2023). Extinction and persistence in stochastic predator population density-dependent predator-prey model with jumps. Bulletin of Taras Shevchenko National University of Kyiv Series: Physics & Mathematics, 1, 30–36. https://doi.org/10.17721/1812-5409.2023/1
Gikhman, I. I., & Skorokhod, A. V. (1982). Stochastic differential equations and its applications. Naukova Dumka [in Russian].
Li, M., Gao, H., Sun, C., & Gong, Y. (2015). Analysis of a mutualism model with stochastic perturbations. International Journal of Biomathematics, 08(06), 1550072. https://doi.org/10.1142/S1793524515500710
Li, M., Gao, H., & Wang, B. (2016). Analysis of a non-autonomous mutualism model driven by lévy jumps. Discrete & Continuous Dynamical Systems - B, 21(4), 1189–1202. https://doi.org/10.3934/dcdsb.2016.21.1189
Li, X., & Mao, X. (2009). Population dynamical behavior of non-autonomous lotka-volterra competitive system with random perturbation. Discrete & Continuous Dynamical Systems, 24(2), 523–545. https://doi.org/10.3934/dcds.2009.24.523
Lipster, R. (1980). A strong law of large numbers for local martingales. Stochastics, 3, 217–228. https://doi.org/10.1080/17442508008833146
May, R. M. (1976). Models of two interacting populations. In May (Ed.), Theoretical ecology: Principles and application (pp. 78–104). PA: Saunders.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Oleksandr Borysenko, Olga Borysenko
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
