Depletion of resources by a population of diffusing species

Depletion of natural and artificial resources is a fundamental problem and a potential cause of economic crises, ecological catastrophes, and death of living organisms. Understanding the depletion process is crucial for its further control and optimized replenishment of resources. In this paper, we investigate a stock depletion by a population of species that undergo an ordinary diffusion and consume resources upon each encounter with the stock. We derive the exact form of the probability density of the random depletion time, at which the stock is exhausted. The dependence of this distribution on the number of species, the initial amount of resources, and the geometric setting is analyzed. Future perspectives and related open problems are discussed.

Figure 1

Schematic illustration of a stock depletion problem. (a) Random trajectories of three species diffusing in a bounded domain with the reflecting boundary (shown in gray); at each encounter with the stock region (black circle), one unit of resources is consumed; here, the species are released at different starting points (indicated by small black disks) for a better visualization. (b) The number of consumed resources (thick solid red curve), ${\ell }_t$, as a function of time, and a prescribed threshold (thick dotted black horizontal line), $\ell$, of initially available resources on the stock region; the arrow indicates the first-crossing time ${\mathcal{T}}_{\ell ,N}$ when the stock is depleted. Thin curves show the resources ${\ell }_t^i$ consumed by individual species.

Figure 2

Probability density function $U_N(\ell ,t|{\mathit{x}}_0)$ of the first-crossing time ${\mathcal{T}}_{\ell ,N}$ for $N$ species diffusing in the exterior of a spherical stock region of radius $R$, with $\ell =R$, and $|{\mathit{x}}_0|=2R$ (a) and $|{\mathit{x}}_0|=R$ (b). Symbols present the explicit form (30) for a single species, whereas thick lines show the result of numerical integration in Eq. (B22), see Appendix pp3. Thin solid lines indicate the long-time asymptotic relation (28), with $\alpha =1/2$ and ${\mathrm{\Psi }}_N({\mathit{x}}_0,\ell )$ given by Eq. (B24); in turn, thin dashed lines present the short-time behavior in Eq. (22) for (a) and Eq. (24) for (b).

Figure 3

Probability density function ${\rho }_N(\ell ,t|x_0)$ of the total boundary local time ${\ell }_t$ for $N$ particles diffusing on the half line, with $t=1$, $D=1$, and $x_0=1$ (a) and $x_0=0$ (b). Symbols present the explicit form (A7) for a single particle, whereas thick lines show the numerical integration in Eqs. (A11) and (A12). Thin dashed lines present the large-$\ell$ asymptotic relation (A20), while thin solid lines indicate the small-$\ell$ asymptotic relation (A14) for $x_0=1$ and (A18) for $x_0=0$, respectively. In (a), only the regular part is presented, whereas the explicit term with $\delta \left(\ell \right)$ is excluded.

Figure 4

(a) Schematic illustration of crossing of a threshold $\ell$ by ${\ell }_t={\ell }_t^1+{\ell }_t^2$ for two particles that corresponds to crossing the gray line. Red filled circle indicates the closest point, through which the crossing is most probable at short times. (b) An equivalent view onto this problem in terms of two-dimensional Brownian motion that starts from the origin (blue circle) and searches to exit the rotated square. Four red-filled circles indicate the closest points through which the exit is most probable at short times.

Figure 5

Probability density function $U_N(\ell ,t|x_0)$ of the first-crossing time ${\mathcal{T}}_{\ell ,N}$ for $N$ particles diffusing on the half line, with $\ell =1$, $D=1$, and $x_0=1$ (a) and $x_0=0$ (b). Symbols present the explicit form (A8) for a single particle, whereas thick lines show the numerical integration in Eqs. (A26) and (A12). Thin solid lines indicate the long-time asymptotic relation (A29) for $x_0=1$ and (A30) for $x_0=0$, respectively. Thin dashed lines present the short-time asymptotic relation (A33) for $x_0=0$ and (A31) for $x_0=1$, respectively.

Figure 6

(a) Mean first-crossing time ${\mathbb{E}}_{x_0}\left\{{\mathcal{T}}_{\ell ,N}\right\}$ as a function of the number $N$ of particles diffusing on the half line, with $\ell =1$, $D=1$, and several values of $x_0$ as indicated in the legend. Lines show the result of numerical integration in Eq. (A34), with the probability density $U_N(\ell ,t|x_0)$ given by Eqs. (A26) and (A12). Symbols present the result of numerical integration in Eq. (A41) for the case $x_0=0$. Thin black line indicates the asymptotic behavior (A45). (b) Another representation of the mean first-crossing time ${\mathbb{E}}_{x_0}\left\{{\mathcal{T}}_{\ell ,N}\right\}$, multiplied by $lnN$ and shown as a function of $1/lnN$, for $x_0>0$.

Figure 7

Crossing probability $Q_N(\ell ,\infty |{\mathit{x}}_0)$ for $N$ particles diffusing in the exterior of a ball of radius $R=1$, with three values of $\ell$ indicated in the legend, and $|{\mathit{x}}_0|=2R$ (a) and $|{\mathit{x}}_0|=R$ (b).

Figure 8

Cumulative distribution function $Q_N(\ell ,t|{\mathit{x}}_0)$ of the first-crossing time ${\mathcal{T}}_{\ell ,N}$ for $N$ particles diffusing in the exterior of a ball of radius $R=1$, with $\ell =1$, $D=1$, $|{\mathit{x}}_0|=2$ (a) and $|{\mathit{x}}_0|=1$ (b). Symbols present the explicit form (B9) for a single particle, whereas thick lines show the numerical integration in Eq. (C5). Thin lines indicate the long-time asymptotic relation (29), while thin dashed lines present the short-time behavior in Eq. (25) for $|{\mathit{x}}_0|=2$ and Eq. (26) for $|{\mathit{x}}_0|=1$, respectively.

Figure 9

Probability density function $U_N(\ell ,t|{\mathit{x}}_0)$ of the first-crossing time for $N$ particles diffusing in the exterior of a ball of radius $R=1$, with $\ell =0.1$, $D=1$, $|{\mathit{x}}_0|=2$ (a) and $|{\mathit{x}}_0|=1$ (b). Symbols present the explicit form (30) for a single particle, whereas thick lines show the numerical integration in Eq. (B22). Thin lines indicate the long-time asymptotic relation (B25), while thin dashed lines present the short-time behavior in Eq. (22) for $|{\mathit{x}}_0|=2$ and Eq. (24) for $|{\mathit{x}}_0|=1$.