Optimization of size-structured population with interacting species

Panesh, A. A.; Platov, A. S.

doi:10.1007/s10958-012-1128-1

Cited by 7 publications

(2 citation statements)

References 3 publications

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“…In accordance with this formula, the development of individuals of a given size is only affected by the individuals of the same size or larger. This distinguishes our paper from [1]- [3], where the integral was taken over the whole of [0, L] and the level of competition was the same for all sizes of individuals. The boundary condition has the form…”

Section: And a F Nassarmentioning

confidence: 99%

On a stationary state in the dynamics of a population with hierarchical competition

Davydov¹,

Nassar²

2014

Russ. Math. Surv.

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We shall show that for a given control u there exists a non-trivial stationary state in the dynamics of a size-structured population with hierarchical competition. More precisely, we assume that the dynamics is described by the equationwhere x(t, l) is the density of individuals of size l at time t, the functions g and µ characterize the growth and mortality of these individuals for level of competition E, and the control u characterizes the exploitation of the population. The competition function E is defined bywhere χ is a continuous function which is positive for positive values of the argument, and the interval [0, L] with L > 0 is the range of sizes on which the population is exploited. In accordance with this formula, the development of individuals of a given size is only affected by the individuals of the same size or larger. This distinguishes our paper from [1]-[3], where the integral was taken over the whole of [0, L] and the level of competition was the same for all sizes of individuals. The boundary condition has the formand it is responsible for the influx of individuals of size l = 0: this influx is a combination of natural reproduction and industrial regeneration p0 = const 0. The function r and the exponent β ∈ (0, 1) reflect the reproductiveness of the biomass and the non-linear dependence of the reproduction on the density, respectively. The model (1)-(3) is similar to well-known models (see, for instance, [4], [5]). We assume that all the functions involved, apart from u, are continuous and satisfy the following natural conditions: (a) for each l ∈ [0, L] the functions g and r are non-increasing with respect to the level of competition E, the function g is positive everywhere, and r is non-negative everywhere and positive on some non-empty subinterval of [0, L];(b) µ is a positive function which is non-decreasing with respect to E for each l ∈ [0, L]; (c) for 0 l1 < l2 L the ratio g(l1, · )/g(l2, · ) is non-increasing with respect to E. The conditions (a)-(c) express that the development of the population does not improve with an increase of competition and that this increase does not affect smaller-sized individuals any less than larger-sized ones.By an admissible measurable control u we mean one satisfying the condition 0 u1 u u2 on [0, L], where u1 and u2 are some piecewise continuous functions.

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Section: And a F Nassarmentioning

confidence: 99%

On a stationary state in the dynamics of a population with hierarchical competition

Davydov¹,

Nassar²

2014

Russ. Math. Surv.

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“…где χ i непрерывная функция, а отрезок [0, L i ], L i > 0, диапазон размеров, на котором мы эксплуатируем i-ю популяцию. В работе [3] предполагалось, что все коэффициенты модели зависели от суммарной конкуренции, которая имела вид…”

Section: Introductionunclassified

Optimal exploitation of two competing size-structured populations

Davydov

Platov

2014

Proc. Steklov Inst. Math.

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Для модели, описывающей динамику двух структурированных по размеру конкурирующих популяций при выбранных интенсивностях их эксплуатации, доказано существование и единственность стационарного решения. Показано, что существуют интенсивности эксплуатации, доставляющие максимум заданного функционала выгоды на соответствующем им стационарном решении.Ключевые слова: Структурированная по размеру популяция, оптимальная эксплуатация, стационарное решение.A. A. Davydov, А. S. Platov. Optimal exploitation of two competing size-structured populations.For a model describing the dynamics of two competing size-structured populations under chosen intensities of their exploitation, the existence and uniqueness of a stationary solution is proved. It is shown that there exist exploitation intensities that maximize a given profit functional on the stationary solution corresponding to them.

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Stationary Regime of Exploitation of Size-Structured Population with Hierarchical Competition

Davydov

Nassar

2015

J Math Sci

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Optimization of size-structured population with interacting species

Cited by 7 publications

References 3 publications

On a stationary state in the dynamics of a population with hierarchical competition

On a stationary state in the dynamics of a population with hierarchical competition

Optimal exploitation of two competing size-structured populations

Stationary Regime of Exploitation of Size-Structured Population with Hierarchical Competition

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