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

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

 

 

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:

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.