Optimal exploitation of a periodic Gompertz equation with random fluctuations

Weiming Ji; Meng Liu; Weiming Ji; Meng Liu

doi:10.3934/math.2025838

AIMS Mathematics

2025, Volume 10, Issue 8: 18770-18783. doi: 10.3934/math.2025838

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Optimal exploitation of a periodic Gompertz equation with random fluctuations

Weiming Ji ,
Meng Liu ^,

School of Mathematical Science, Huaiyin Normal University, Huaian 223300, Jiangsu, China

Received: 29 April 2025 Revised: 29 July 2025 Accepted: 08 August 2025 Published: 19 August 2025
MSC : 60H10, 92D25

Taking white noise, Lévy jumps, and periodic factors in the environments into account, we formulated a periodic stochastic Gompertz model with impulsive harvesting and investigated its optimal impulsive exploitation problem. For periodic stochastic ecosystems with Lévy jumps and impulsive harvesting, two traditional approaches, the Fokker-Planck equation approach and ergodicity-based approach used to study the optimal exploitation problems were invalid because for almost all such ecosystems, one could not solve the associated Fokker-Planck equations. In addition, the ecosystems have no traditional non-boundary invariant measures. The present study utilized a different method. By exploring the associated Hamilton function, we derived an optimal exploitation strategy. An example was also introduced to illustrate the theoretical results.
- optimal exploitation,
- random environments,
- impulsive,
- periodic coefficients
Citation: Weiming Ji, Meng Liu. Optimal exploitation of a periodic Gompertz equation with random fluctuations[J]. AIMS Mathematics, 2025, 10(8): 18770-18783. doi: 10.3934/math.2025838

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Abstract

Taking white noise, Lévy jumps, and periodic factors in the environments into account, we formulated a periodic stochastic Gompertz model with impulsive harvesting and investigated its optimal impulsive exploitation problem. For periodic stochastic ecosystems with Lévy jumps and impulsive harvesting, two traditional approaches, the Fokker-Planck equation approach and ergodicity-based approach used to study the optimal exploitation problems were invalid because for almost all such ecosystems, one could not solve the associated Fokker-Planck equations. In addition, the ecosystems have no traditional non-boundary invariant measures. The present study utilized a different method. By exploring the associated Hamilton function, we derived an optimal exploitation strategy. An example was also introduced to illustrate the theoretical results.

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