Probabilistic analysis of a class of impulsive linear random differential equations via density functions

López, Juan Carlos Cortés; Delgadillo-Aleman, Sandra E.; Kú-Carrillo, Roberto A.; Micó, Rafael Jacinto Villanueva

doi:10.1016/j.aml.2021.107519

Cited by 9 publications

(5 citation statements)

References 9 publications

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“…, and therefore by (18), the positive equilibrium x * is locally asymptotically stable, as (18) implies (20).…”

Section: Stability With Delay Impulsive Harvestingmentioning

confidence: 99%

“…For logistic and other simple population models, such as Gompertz, incorporating stochastic fluctuations or random differential equations created an additional challenge but reflected unpredictable changes in the environment, see [18,19,20] and the references therein. Impulsive harvesting of a stochastic Gompertz model was a focus of [19].…”

Section: Introductionmentioning

confidence: 99%

“…Note that p 0 < p k + 1, or e −rT < Ee −rT + (1 − E) 2 is equivalent to e −rT < 1 − E, which is satisfied due to (18). The first inequality is the same as in (22), while computing the squares in the second one gives…”

mentioning

confidence: 99%

“…Proof. If (25) holds, then so does (18), and the positive equilibrium exists. Reducing (1) to difference equation ( 11), we once again linearize (11) around x * and obtain (24), where by (25),…”

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confidence: 99%

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Optimality and sustainability of delayed impulsive harvesting

Lawson,

Braverman

2022

Preprint

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We consider a logistic differential equation subject to impulsive delayed harvesting, where the deduction information is a function of the population size at the time of one of the previous impulses. A close connection to the dynamics of high-order difference equations is used to conclude that while the inclusion of a delay in the impulsive condition does not impact the optimality of the yield, sustainability may be highly affected and is generally delay-dependent. Maximal and other types of yields are explored, and sharp stability tests are obtained for the model, as well as explicit sufficient conditions. It is also shown that persistence of the solution is not guaranteed for all positive initial conditions, and extinction in finite time is possible, as is illustrated in the simulations.

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“…, and therefore by (18), the positive equilibrium x * is locally asymptotically stable, as (18) implies (20).…”

Section: Stability With Delay Impulsive Harvestingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Optimality and sustainability of delayed impulsive harvesting

Lawson,

Braverman

2022

Preprint

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“…of the solution has been addressed for different classes of equations including equations with delay and fractional derivatives [5,6,8,13,14,15,21]. In recent contributions, we have extended the study to a class of first-order linear impulsive RDEs with finite [11] and infinite jumps [10]. In both cases, the impulses represent controls and they are evenly applied over time via the Heaviside and Dirac's delta functions, respectively.…”

mentioning

confidence: 99%

Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions

López

Delgadillo-Aleman

Kú-Carrillo

et al. 2022

DCDS-S

Self Cite

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<p style='text-indent:20px;'>We study a full randomization of the complete linear differential equation subject to an infinite train of Dirac's delta functions applied at different time instants. The initial condition and coefficients of the differential equation are assumed to be absolutely continuous random variables, while the external or forcing term is a stochastic process. We first approximate the forcing term using the Karhunen-Loève expansion, and then we take advantage of the Random Variable Transformation method to construct a formal approximation of the first probability density function (1-p.d.f.) of the solution. By imposing mild conditions on the model parameters, we prove the convergence of the aforementioned approximation to the exact 1-p.d.f. of the solution. All the theoretical findings are illustrated by means of two examples, where different types of probability distributions are assumed to model parameters.</p>

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Forward uncertainty quantification in random differential equation systems with delta‐impulsive terms: Theoretical study and applications

Bevia

López

Micó

2023

Math Methods in App Sciences

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This contribution aims at studying a general class of random differential equations with Dirac‐delta impulse terms at a finite number of time instants. Our approach directly addresses calculating the so‐called first probability density function, from which all the relevant statistical information about the solution, a stochastic process, can be extracted. We combine the Liouville partial differential equation and the random variable transformation method to conduct our study. Finally, all our theoretical findings are illustrated on two stochastic models, widely used in mathematical modeling, for which numerical simulations are carried out.

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Probabilistic analysis of a class of impulsive linear random differential equations via density functions

Cited by 9 publications

References 9 publications

Optimality and sustainability of delayed impulsive harvesting

Optimality and sustainability of delayed impulsive harvesting

Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions

Forward uncertainty quantification in random differential equation systems with delta‐impulsive terms: Theoretical study and applications

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