Simulations:

Simulation 1: foraging agents in variable food densities

This is a simulation used for a reference point for basic agent behaviour, food consumption and metabolism. This simulation involved removing all obstacles and voids, disabling agent death and the agent energy cap. Twenty agents were placed in the environment with food densities ranging from 0.04 to 1.6 (units/unit area). The agents did a random walk to simulate basic foraging for food. Each simulation run-length was 20000 time-steps. Raw results are here.

The plot shows that the net energy gain is a linear function of the food density. The gradient calculated from the plot was calculated to be 240 (units of area). The gradient can be analytically calculated by taking the area an agent searches each time step and multiplying it the length of the simulation run. The analytic calculation gave the result of 224 (units of area). From inspecting the plot it was determined the food density the agents break even at was approximately 0.052 food spheres/unit area. The average amount of food spheres consumed per simulation run as a function of food density (fd) was determined to be: no. units consumed = 78.8*fd - 0.75.

Doing a random walk, an agent travels a distance of 400 (unit lengths) per simulation run. So each agent needs to find a food sphere for approximately every 95 unit lengths traveled.

Simulation 2: foraging agents in variable obstacle densities

This simulation was done to determine how obstacle density affected the agents' foraging efficiency. In a natural system, when a creature is searching for food and encounters an obstacle, it must pause its search while the obstacle is being avoided. Therefore it is expected that for increased obstacle density the agent's foraging efficiency should decrease. For this simulation the voids were removed and nine fixtures were placed evenly on the platform. The percentage of platform fixture coverage was varied for each simulation run. The results were averaged over 40 agents. Food density was set to a consent of 0.52 (units/unit area), the break-even point. Each simulation run length was 20000 time-steps.

A linear relationship with a negative gradient is present in plot 2, showing that increasing obstacle density does have a negative effect on foraging efficiency when a random walk strategy is used. Which agrees with the expected results.

Simulation 3: the optimal foraging techniques of the Canadian northwestern crows

The optimal foraging techniques of the Canadian northwestern crows were chosen because they were the subject of several extensive field studies * and the crow-whelk systems could be easily mapped to the simulation.

The crows' behaviour has been observed by R. Zach * -top perch to the beach, where they search for a whelk. When a suitable whelk is located they collect it and fly to a rocky area. At the rocky area, the crows fly to an average hight of 5 meters, where they drop the whelk, attempting to break its shell. The crow then inspects the whelk. If the whelk's shell is broken, the crow extracts and eats the animal. If the whelk's shell is not broken the crow re-attempts to break the shell by flying up and dropping it again. The crow repeatedly continues to break the whelk's shell until it is broken. It has been observed that if a crow has a whelk that is hard to break it does not discard the whelk to search for another whelk which is easier to break.

The whelks were found to have the following properties: the whelk's shell does not weaken after a break attempt, so has the same probability of breaking each drop. The probability of a whelk breaking was about one in four drops. On average, it was determined that a crow uses 25% of the energy gained by eating the whelk to break the whelk's shell~\cite{Alcock:1998}.

R. Zach * asked the question: ``Why do crows not give up and search for a new whelk if a particular one does not break readily?'' This leads to the hypothesis: that the crows are behaving optimally. This must be because the crows use less energy continuing to try to break the whelk they have, rather then discarding the whelk and searching for another easier to break whelk. This hypothesis is tested in the following simulation.

The components of the crow-whelk system were mapped into the simulation as follows: The crows were represented by agents. The agents did a random walk, searching for food spheres (whelks). When an agent finds a food sphere it picks it up with its food actuator. The agent then attempts to 'break the food sphere's shell' by applying some of its stored energy to the food sphere. The energy applied to the shell was calculated to be 1875 (energy units). This figure was calculated as follows. When four break attempts were made, the energy expended by the agent would be equal to 25% of the food sphere's total energy. Each food sphere is worth 30000 (energy units), so 30000*0.25*0.25 = 1875 (energy units). Each food sphere has a 'resistance to breaking' that is represented by a probability of breaking on a drop. This parameter was ranged from 0.05 to 0.45 with a mean of 0.25. This was done to correspond to Zach's finding that a whelks shell had a probability of breaking of one in four on each drop.

An agent changes its mind based on a `tolerance' of hard to break the food spheres. This tolerance was represented by a probability of discarding a food sphere after each failed break attempt, that was varied each simulation run. So if an agent had a tolerance of 0.1, it had a 10% chance of discarding the food sphere after each failed break attempt. Hence a tolerance of 1 meant only a single attempt to break the shell was made, and a tolerance of 0 meant that the agent never discards the food, continually trying to break its shell until it either drains all energy (dies) or breaks the food shell.

The plot shows for the lower food densities of 0.05-0.2 (units/unit area) there is a clear loss of efficiency for increasing probability of dropping the food. For the higher food densities from 0.4-1.6 (units/unit area) the efficiency levels out. This is to be expected, because when food is abundant it makes little difference if a unit is discarded, as the agent has only to expend a little energy to find another unit of food.

The plot shows the agents behave most optimally when they never discard a unit of food. This remains true even when they have a unit that is very hard to break. This agrees with the observed behaviour of the crows.

Results were plotted as: average number of food consumed vs the probability of discarding a food sphere. Each simulation run length was 20000 time-steps.

Another interesting foraging technique of the crows is that they only select large whelks for dropping. This raises the question: why is it that the crows do not select a small whelk rather then expending further energy searching for a large whelk? An explanation for this question is the hypothesis: that the crows are behaving optimally. This is because they will be unlikely to get the energy back they invest in trying to break a small whelk.

The properties of different sized whelks are: the larger whelks were found not only to contain a larger food source (hence more energy), but they were also found to be easier to break. The whelks' size ranged from 14-50 (mm) and crows only selected whelks greater then 32 (mm).

These properties were mapped to the food spheres by: each food sphere was given a random size between 25--40 (mm). This range was selected because, the crows only selected whelks greater then 32mm. Hence there was little point running the simulation over the entire whelk size range. Probability of breaking a food sphere was ranged linearly from 0.3 (size 40mm) to 0.05 (size 25 (mm)).

For the first simulation all the agents' energy consumption was disabled, except for energy used when attempting to break a food sphere. So the agents could move without any energy usage penalty. This was done to determine the energy break-even point for different sized food sphere. For each simulation run, the agents changed their minds about what different sized whelks they liked (varied by 10%). Results were plotted net energy gain vs. food sized selected. Each simulation run length was 20000 time-steps. Food density was kept constant at 0.3 (units/unit area).

Plot 1 shows efficiency increasing with food size. The energy break even point for food size is approximately 28 (mm). The dip at 38 (mm) can be explained by there being less food in the range at 38 because 38 (mm) + 10\% = 41.8. This is outside the food range available to the agent. This plot shows that even when no energy is expended foraging there is a net energy loss when the smaller food size of less then 28 (mm) was selected.

For the next simulation, foraging energy expense was enabled and the simulation re-run with the same parameters as the previous simulation. Results shown in Figure~\ref{plot-sizeenergy}.

Plot2 shows the break-even point only marginally increased to approximately 30 (mm). However the energy loss for eating small food is much more severe. Also the energy gain for eating large food is much less. This plot shows us that only selecting large food is profitable, because when small sized food is selected the energy invested breaking that food's shell is never recovered.

Simulation 4: dual nest foraging for varying food densities

This simulation was based on colonies of insects that utilize a large communal nest. Ants and Bees have been observed sending foragers out to locate food sources which then return to the nest carrying the food. This builds a food surplus that can feed the entire nest. The advantage of this strategy is that when food is scarce, the nest can feed off its surplus. It also has the advantage of enabling the colony to divide up its population for different tasks (foraging, nest maintenance, building and defense), allowing the nest to function as efficiently as possible.

Mapping components of the nesting insects to the simulation space is as follows. Two different types of agent were placed on the platform. The first group is a reference group (the blue agents). They do not nest at all. The second group is a group of nesting agents. They work as follows. They do a random walk, when they will locate a food sphere. This food sphere is collected by the food actuator. If the agent is not starving they carry the food sphere back to their base, placing it in storage. If they become starving while they are carrying the food back to the base, they eat the food and continue searching. While searching, if they become starving, they check if the base contains any food. If it does, they then return to the base and eat a unit of surplus food. They then continue searching. If the base has no food stored, then the agent must try its luck searching by itself for a food sphere to survive. Results were plotted as food density vs average life span (capped at the run-length of the simulation (20000 time-steps)).

The plotshows that for these low food densities, the nesting agents are better off. They reach a point of sustainable foraging at a food density of 0.1, while the non-nesters only reach that point at a food density greater then 0.2. This shows that for low food densities the agents are better off working together to build a communal food source.

* Zach, R. 1978. Selection and Dropping of Whelks by Northwestern Crows. Behavior, 67: 134-147.
Zach, R.. 1978. Shell Dropping: Decision Making and Optimal Foraging in Northwestern Crows. Behavior, 68: 106-117.

Andrew R. Jones: jones@csse.monash.edu.au.