Evolutionary Systems in an Asymptotic Split Phase Space

Korolyuk, V. S.; Limnios, Nikolaos

doi:10.1007/978-1-4612-1384-0_10

Cited by 11 publications

(11 citation statements)

References 7 publications

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“…In the present paper, we compensate the initial process by an averaging deterministic function instead of an equilibrium point (see, e.g., [6]) and obtain an inhomogeneous diffusion approximation result.…”

Section: Introductionmentioning

confidence: 99%

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Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes

Korolyuk

Limnios

2005

Ukr Math J

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Section: Introductionmentioning

confidence: 99%

“…Another diffusion approximation can be obtained by considering a fluctuation with respect to the average process. In a previous work, we studied evolutionary systems with Markov switching in two cases [6] (the case where the average process is a deterministic function and the case where the average is a stochastic process).…”

Section: Introductionmentioning

confidence: 99%

Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes

Korolyuk

Limnios

2005

Ukr Math J

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“…In the paper, we will prove that limit theorems can be used for stability analysis as is the case with smooth stochastic systems [10][11][12][13]. In the second and third sections, an averaged system and diffusion approximation are set up; their analysis with the use of modern computer technologies allows impruving the computational efficiency by a factor of tens of millions.…”

Section: Introductionmentioning

confidence: 99%

“…For clarity, let us first present the well-known results from [5,6,14]. In contrast to most studies on the averaging method of impulsive systems, we will consider impulsive systems with Markov switching.In the paper, we will prove that limit theorems can be used for stability analysis as is the case with smooth stochastic systems [10][11][12][13]. In the second and third sections, an averaged system and diffusion approximation are set up; their analysis with the use of modern computer technologies allows impruving the computational efficiency by a factor of tens of millions.…”

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confidence: 99%

Stability in impulsive systems with Markov perturbations in averaging scheme. I. Averaging principle for impulsive Markov systems

2010

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The Bogoliubov-Mitropolsky small parameter method is used to study the behavior of stochastic differential systems in the analysis of the corresponding properties of solutions of averaged systems. INTRODUCTIONAveraging over explicit time [4] is one of the most popular methods for studying dynamic systems. This method also performs well in the theory of random perturbations [1, 2, 5, 6, 10-13] (see the references therein for a more detailed bibliography).This method allows not only setting up an averaged system that approximately describes the dynamics of the original model, but also writing stochastic differential equations (SDEs) for normalized deviations of the solution of the original equation from the corresponding solutions of averaged motion.The Bogoliubov-Mitropolsky principle of averaging is proved in the monograph [5] for impulsive systems and in [6] for the case of random perturbations. For clarity, let us first present the well-known results from [5,6,14]. In contrast to most studies on the averaging method of impulsive systems, we will consider impulsive systems with Markov switching.In the paper, we will prove that limit theorems can be used for stability analysis as is the case with smooth stochastic systems [10][11][12][13]. In the second and third sections, an averaged system and diffusion approximation are set up; their analysis with the use of modern computer technologies allows impruving the computational efficiency by a factor of tens of millions.

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“…guarantees Markov property of these random processes. This makes it possible to advance the method proposed in Anisimov (1995) and Korolyuk and Limnios (2005) of the diffusion approximation for switching Markov processes and construct a stochastic approximation procedure for impulse type differential equations with fast Markov impulse switching (2)-(4). The resulting differential equation for the mean value of population size and stochastic differential equations for deviations on the mean permit us to describe the probability characteristics of population dynamics.…”

mentioning

confidence: 99%

Dynamics of a Population Subject to Impulse Type Random Loss

Carkovs

Šadurskis

2017

Proceedings of the Latvian Academy of Sciences. Section B. Natural, Exact, and Applied Sciences.

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The logistic growth model is the most popular one in population dynamics, and considers the birth and death rates typical to the species as well as the effects of different mechanisms of consumption of resources on the dynamics of the biological system. For qualitative analysis of the behaviour of an ecosystem the ordinary differential equation (Otto and Day, 1995):is commonly used, where the parameter a describes growth rate, i.e. fertility, and the parameter h determines the intensity of extraction of specimens from ecosystem (as an example, the intensity of fishing in the case of fish populations in fisheries, which we use here, and further as an illustrative example scenario). Parameter K corresponds to stationary population size in the case of an absence of fishing, i.e., stationary solution of the equation (1) if h = 0. As K -1 describes the rate of reduction of the population size, this coefficient may be considered as the death rate of the population. Let us note that the stationary solution of the equation (1)differs from K, and solutions of (1) tend exponentially to x st only if a > h. If this inequality is not true, the solution of (1) tends exponentially to zero. This means that the population dies out under such a fishing regime. If the fishing rate h is equal to the birth rate a, solutions of (1) also tend to zero, albeit not with an exponential velocity. This means that in order for fishing to be sustainable, it is necessary for the birth rate a to be greater than the fishing rate h. It should be noted that it is not just the size of the population that determines the fishing regime. Using the finite difference method, we see that the amount of fishing at time t should be approximately equal to hx(t)Dt. However, it may be the case that the entire volume of the population x(t) may not be suitable for fishing at the time t, and we can only rely on the average value of fishing hx(t)Dt. Sometimes it is possible to use a different fishing strategy: wait for a time period d, which, generally speaking, is a random time dependent variable, and only then remove the necessary amount of the population. Therefore, deterministic fishery strategies do not allow us to predict the dynamics of the population size and/or to determine fishing mode, especially if a » h. In this case, construction and exploration of mathematical models PROCEEDINGS OF THE LATVIAN ACADEMY OF SCIENCES. Section B, Vol. 71 (2017) ) the population disappears with probability one, otherwise the distribution of the scaled population size with increasing time tends to the Gamma-distribution G(k,q) with the shape k = 2c/l(h 2 + b 2 ) and the scale q = l(h 2 + b 2 )/2c.

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Evolutionary Systems in an Asymptotic Split Phase Space

Cited by 11 publications

References 7 publications

Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes

Diffusion Approximation with Equilibrium for Evolutionary Systems Switched by Semi-Markov Processes

Stability in impulsive systems with Markov perturbations in averaging scheme. I. Averaging principle for impulsive Markov systems

Dynamics of a Population Subject to Impulse Type Random Loss

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