Impact of food distribution on lifetime of a forager with or without sense of smell

Modeling foraging via basic models is a problem that has been recently investigated from several points of view. However, understanding the effect of the spatial distribution of food on the lifetime of a forager has not been achieved yet. We explore here how the distribution of food in space affects the forager's lifetime in several different scenarios. We analyze a random forager and a smelling forager in both one and two dimensions. We first consider a general food distribution, and then analyze in detail specific distributions including constant distance between food, certain probability of existence of food at each site, and power-law distribution of distances between food. For a forager in one dimension without smell we find analytically the lifetime, and for a forager with sense of smell we find the condition for immortality. In two dimensions we find based on analytical considerations that the lifetime ($T$) scales with the starving time ($S$) and food density ($f$) as $T\sim S^4{f}^{3/2}$.

Figure 1

Illustration of forager in one dimension. (a) Here all food is on the right of forager (filled circles), whereas on the left there is a semi-infinite desert. Between food there are distances, $l$, distributed according to $P\left(l\right)$. The forager walks randomly. We assume that the forager just ate the food in its initial position, hence it can walk $S$ steps without eating. This case is studied in Sec. 2. (b) In this scenario food is located in both directions. Here we study in detail the case of forager with long-range smell in Sec. 3.

Figure 2

Random forager in one dimension. Comparison between constant distance between food and random food distribution. (a) Results of the forager's lifetime for random spread of food, $T_f$, from theory (lines), Eqs. (4), (8) and (18), and simulations (symbols) averaged over ${10}^3$ realizations, show good agreement. In our simulations, whenever the forager approaches the edge of the system we add new sites in this direction, therefore we obtain identical behavior as an infinite lattice. (b) Results of forager's lifetime for a constant distance between food units, $T_L$. Lines represent the theory, Eqs. (4), (8), and (16), and symbols are simulation results. Good agreement can be observed. (c) Phase diagram that compares which food distribution for the same $\langle l\rangle$ (same amount of food) is better and yields longer lifetime of the forager. Above the black line the random spread gives longer life, $T_f$, while below the line, where $S$ is significantly larger than $\langle l\rangle$, the constant distance enables longer lifetime, $T_L$. In (c) the lifetimes are calculated excluding the last walk which is always $S$ steps for any food distribution.

Figure 3

Random forager in one dimension with random (uniform) and power-law food distributions. In the upper panels we show results of theory for random uniform spread of food, Eqs. (4),(6, 7, 8), and (18). One can see that the scaling relations for large $S$ in Eq. (10) are valid, (a) $\mathcal{N}\sim f{S}^{1/2}$, (b) $\tau \sim f^{-1}{S}^{1/2}$, and (c) $T\sim S$. Analysis of the behavior for small values of $f$ is presented in the Appendix in Fig. 9. In the lower panels, (d),(e),(f), we show results of simulations for power-law distribution of distance between food, $P\left(l\right)=A{l}^{-(1+\beta )}$. The results agree well with the theoretical scaling (dashed lines) found in Eq. (12), $\mathcal{N}\sim S^{\beta /2}$, $\tau \sim S^{1-\beta /2}$, $T\sim S$, for $\beta <1$, and with the scaling in Eq. (10), $\mathcal{N}\sim S^{1/2},\tau \sim S^{1/2},T\sim S$, for $\beta >1$. The results of simulations have been averaged over ${10}^3$ realizations. Note that in (f) for $\beta =0.6$ the factor $T/S$ is close to 1.5 instead of 1.99 as found in Eq. (12). This is since the analytical approximation in Eq. (12) should be valid only for much larger S.

Figure 4

Smelling forager with a constant distance between food and power law decay of smell. (a) Results of theory (line between two phases) and simulations (symbols) for forager in one dimension with power-law decay of smell, and food is distributed with a constant distance, $L=3$, between food units. The theory is taken from Eq. (33), and the simulations have been performed over ${10}^8$ realizations, where ${\alpha }_c$ is determined by the maximal value of $\alpha$ for which there was no forager which lived forever in any realization, i.e., in all ${10}^8$ realizations the forager died. The deviation between theory and simulations is reasonable because the theory finds when the probability to live forever is completely zero, whereas the simulations find when the probability is small enough such that it does not appear in the finite number of realizations, and thus it happens for a slightly larger value of $\alpha$. (b) Shows the dependence of ${\alpha }_c$ on $L$ and $S$ according to Eq. (33). Where $L>S$ of course the forager dies after one walk and it is mortal for any value of $\alpha$. The color represents the value of ${\alpha }_c$ required for immortality which changes with $L$ and $S$.

Figure 5

Sketches of theory results for lifetime of smelling forager in one dimension. (a) If the maximal possible distance between food ($l^{*}$) is not greater than the starving time ($S$), then there is ${\alpha }_c$ above which the average lifetime is infinite. Its value depends on $l^{*}$ according to Eq. (32). (b) If the distance between food can be larger than $S$, there is no immortality regime. However, the lifetime has a saturation for large $\alpha$, and in this region the scaling of $T\left(S\right)$ is found for power-law distribution of distance between food, Eq. (44), and for random spread of food, Eq. (39). For $\alpha <1$ the total smell diverges, and hence the walk is random, thus the scaling is as in Eqs. (10) and (12).

Figure 6

Illustrations of forager in two dimensions. (a) The forager dies when it creates a loop and goes inside. The loop should be large enough such that after it eats most of the food inside and find itself at the middle of desert, it does not succeed escaping the desert in $S$ steps. At this situation, the long-range smelling forager and one with short range are very different but can be mapped. While the long- range smelling forager goes directly towards the closest food (blue straight line), the short-range one walks randomly (red random path). Typically the random walker reaches a distance of $\sim \sqrt{S}$ in $S$ steps, while the direct walk reaches a distance $S$. Hence the short-range smelling forager should have starving time of $S^2$ to die at the same trap as the long-range smeller forager with starving time of $S$. Therefore, we obtain Eq. (45). (b) Illustration of a constant distance between food in the 2D lattice where $L=2$. The filled circles represent food, while the white empty ones represent empty sites. Theory for this case is given in Eqs. (48) and (52). Simulation results of this case are shown in Fig. 8. (c) Illustration of uniform random spread of food in two dimensions with density $f\approx 1/2$. Simulation results for this scenario are presented in Fig. 8.

Figure 7

Lifetime in two dimensions where the space is full with food. We show the scaling between lifetime, $T$, and starving time, $S$, at three cases: (a) random walk, (b) short-range smell (steps towards food only if distance is one), and (c) perfect smelling forager (steps towards the closest food). It can be seen that for random forager and short-range smelling forager the scaling is about $T\sim S^2$ when $S$ is getting large, while perfect smelling forager has scaling of about $T\sim S^4$. This confirms our theoretical argument that the transformation from short-range smell to perfect smelling should be expressed by $S\to S^2$, see Fig. 6 and Eq. (45). The insets suggest that the exponents smaller than 2 and 4 are due to finite size systems, and when $S$ increases they reach the values 2 and 4, respectively.

Figure 8

Forager in two dimensions where space is not full with food. (a) Here the distance between food units is constant, $L$, and the forager walks perfectly according to smell, i.e., towards the closest food. We show that the scaling with the distance $L$ is $T/L$ and $S/L$ when $S/L$ is large, hence all points lay approximately on one curve. This supports Eq. (48) derived from theoretical considerations. This scaling allows us to obtain $T\sim L^{-3}$, see Eq. (52). The dashed line represents the expected approximated slope according to the result of Fig. 7. (b) Here food is distributed randomly with density $f$, see Fig. 6. The forager walks perfectly according to smell. We examine in this case the scaling of $T$ vs $f$. One can see that the approximated scaling in Eq. (53), $T\sim f^{3/2}$, found for constant distances, is approximately valid also for a uniform random food distribution.

Figure 9

Random forager in one dimension with uniform random food distribution. (a)–(c) We can see the two limits of small densities $f$ and large ones. (a) The quantity $\mathcal{N}$ is linear with $f$ while (b) $\tau$ approaches to a constant for small $f$ and then scales as $1/f$ when $f$ tends to 1, and (c) $T$ is linear with $f$ for small $f$ and then approaches to a constant. (d) The linear scaling of $T$ versus $S$ is valid for large $S$ where it is large relative to the average distance between food units such that $\sqrt{S}\gg \langle l\rangle$, which in terms of $f$ becomes $S\gg {(1/f)}^2$. Therefore $T/S$ is getting constantly slower for small values of $f$. However, this figure shows the nonleading correction while in Fig. 3 one can see only the effect of the leading term.