Uncertainty quantification for hybrid random logistic models with harvesting via density functions. (February 2022)

• Record Type: Journal Article
• Authors: Cortés, J.-C.; Moscardó-García, A.; Villanueva, R.-J.
• Is Part Of: Chaos, solitons and fractals. Volume 155(2022)
• Publication Date: 2022-02
• Subjects: Hybrid random differential equation -- Uncertainty quantification -- First probability density function -- Real-world application -- Random variable transformation technique; Chaotic behavior in systems -- Periodicals; Solitons -- Periodicals; Fractals -- Periodicals; Chaotic behavior in systems; Fractals; Solitons; Periodicals; 003.7
• DOI: 10.1016/j.chaos.2021.111762
• Languages: English
• ISSNs: 0960-0779

Abstract

Abstract: The so-called logistic model with harvesting, p ′ ( t ) = r p ( t ) ( 1 − p ( t ) K ) − c ( t ) p ( t ), p ( t 0 ) = p 0, is a classical ecological model that has been extensively studied and applied in the deterministic setting. It has also been studied, to some extent, in the stochastic framework using the Itô Calculus by formulating a Stochastic Differential Equation whose uncertainty is driven by the Gaussian white noise. In this paper, we present a new approach, based on the so-called theory of Random Differential Equations, that permits treating all model parameters as a random vector with an arbitrary join probability distribution (so, not just Gaussian). We take extensive advantage of the Random Variable Transformation method to probabilistically solve the full randomized version of the above logistic model with harvesting. It is done by exactly computing the first probability density function of the solution assuming that all model parameters are continuous random variables with an arbitrary join probability density function. The probabilistic solution is obtained in three relevant scenarios where the harvesting or influence function is mathematically described by discontinuous parametric stochastic processes having a biological meaning. The probabilistic analysis also includes the computation of the probability density function of the nontrivial equilibrium state, as well as the probability that stability is reached. All these results are new and extend their deterministic counterpart under very general assumptions. The theoretical findings are illustrated via two numerical examples. Finally, we show a detailed example where results are applied to describe the dynamics of stock of fishes over time using real data.