Estimation of total income with discounting for probabilistic models of population dynamics

A. A. Bazulkina

Vestnik rossijskih universitetov. Matematika, Tome 28 (2023) no. 143, pp. 217-226

Résumé

Models of homogeneous and structured populations given by differential equations depending on random parameters are considered. A population is called homogeneous if it consists of only one animal or plant species, and structured if it contains $n\geqslant 2$ different species or age classes. We assume that in the absence of exploitation, the dynamics of the population is given by the system of differential equations \begin{equation*} \dot{x}=g(x), \quad x\in\mathbb R^{n}_{+}\doteq\left\{x\in \mathbb R^{n}: x^1\geqslant 0 ,\ldots,x^n\geqslant 0\right\}. \end{equation*} At times $\tau_{k}=kd,$ where $d>0,$ $k=1,2,\ldots,$ random shares of the resource $\omega_{k}=(\omega_{k }^1,\ldots,\omega_{k}^n)\!\in\Omega\subseteq [0,1]^n$ are extracted from this population. If $\omega_{k}^i$ is greater than some value $u_{k}^i\in[0,1),$ then the collection of the resource of the $i$-th type stops at the moment $\tau_{k}$ and the share of the extracted resource turns out to be equal to $\ell_{k}^i\doteq\min(\omega_{k}^i,u_{k}^i).$ Let $C^{i}\geqslant 0$ be the cost of the resource of the $i$-th type, $i=1,\ldots,n,$ $X_k^{i}=x^{i}(kd-0)$ the quantity of the $i$-th type of resource at the time $\tau_k$ before collection; then the amount of income at the moment equals $Z_k\doteq\displaystyle\sum_{i=1}^n{C^{i}X_k^{i}\ell_{k}^i}.$ The properties of the characteristic of the total income, which is defined as the sum of the series of income values at the time $\tau_k,$ taking into account the discounting factor $\alpha>0$ are investigated: \begin{equation*} H_{\alpha}\bigl(\overline{\ell},x_{0}\bigr)=\sum_{k=1}^\infty{Z_k e^{-\alpha{k}}}=\sum_{k=1}^{\infty}e^{-\alpha{k}} \sum_{i=1}^{n}C^{i}X_k^{i}\ell_{k}^i, \end{equation*} where $\overline{\ell}\doteq(\ell_{1},\ldots,\ell_{k},\ldots),$ $x_0$ is the initial population size. The value of $\alpha$ indicates that the value of the income received later decreases. Estimates of the total income, taking into account discounting, made with probability one are obtained.

Détail

Keywords: structured population, total income estimate.

Bibliographie

[1] D. D. Bainov, “Population dynamics control in regard to minimizing the time necessary for the regeneration of a biomass taken away from the population”, Applied Mathematics and Computation, 39:1 (1990), 37–48 | MR | Zbl

[2] G. P. Neverova, O. L. Zhdanova, E. Ya. Frisman, “Dynamics of predator-prey community with age structures and its changing due to harvesting”, Mathematical Biology and Bioinformatics, 15:1 (2020), 73–92 (In Russian) | DOI

[3] A. I. Abakumov, Yu. G. Izrailsky, “The harvesting effect on a fish population”, Mathematical Biology and Bioinformatics, 11:2 (2016), 191–204 (In Russian) | DOI

[4] G. P. Neverova, A. I. Abakumov, E. Ya. Frisman, “Dynamic modes of exploited limited population: results of modeling and numerical study”, Mathematical Biology and Bioinformatics, 11:1 (2016), 1–13 (In Russian) | DOI

[5] A. O. Belyakov, A. A. Davydov, “Optimizatsiya effektivnosti tsiklicheskogo ispolzovaniya vozobnovlyaemogo resursa”, Trudy Instituta matematiki i mekhaniki UrO RAN, 22, no. 2, 2016, 38–46 ; A. O. Belyakov, A. A. Davydov, “Efficiency optimization for the cyclic use of a renewable resource”, Proc. Steklov Inst. Math. (Suppl.), 299, no. suppl. 1 (2017), 14–21 | MR

[6] A. A. Davydov, “Suschestvovanie optimalnykh statsionarnykh sostoyanii ekspluatiruemykh populyatsii s diffuziei”, Izbrannye voprosy matematiki i mekhaniki, Sbornik statei. K 70-letiyu so dnya rozhdeniya akademika Valeriya Vasilevicha Kozlova, Trudy MIAN, 310, MIAN, M., 2020, 135–142 | DOI

[7] A. V. Egorova, L. I. Rodina, “On optimal harvesting of renewable resource from the structured population”, The Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 29:4 (2019), 501–517 (In Russian) | MR | Zbl

[8] A. V. Egorova, “Optimization of discounted income for a structured population exposed to harvesting”, Vestnik rossiyskikh universitetov. Matematika = Russian Universities Reports. Mathematics, 26:133 (2021), 15–25 (In Russian) | Zbl

[9] Yu. V. Masterkov, L. I. Rodina, “Estimation of average time profit for stochastic structured population”, Izv. IMI UdGU, 56 (2020), 41–49 (In Russian) | MR | Zbl

[10] L. I. Rodina, “Optimization of average time profit for a probability model of the population subject to a craft”, The Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 28:1 (2018), 48–58 (In Russian) | MR | Zbl

[11] L. I. Rodina, A. H. Hammadi, “Optimization problems for models of harvesting a renewable resourse”, Journal of Mathematical Sciences, 25:1 (2020), 113–122 | MR

[12] O. A. Kuzenkov, E. A. Ryabova, Mathematical Modelling of Selection Processes, Nizhny Novgorod University Press, Nizhnii Novgorod, 2007, 324 pp. (In Russian)

[13] A. N. Shiryaev, Probability-1, Nauka Publ., Moscow, 1975, 580 pp. (In Russian)

[14] M. S. Woldeab, “Properties of the averadge time benefit for probabilistic models of exploited populations”, Vestnik rossiyskikh universitetov. Matematika = Russian Universities Reports. Mathematics, 28:141 (2023), 26–38 (In Russian)