In this modelling exercise, we are interested in developing a pair of models which best approximate the observed patterns in two “real” species.
The pattern-oriented modelling approach requires starting with an existing pattern of which we want to understand the internal dynamics. The utility of the basic approach is that it requires alternative hypotheses concerning the underlying reality of the observed pattern. This explicit conceptual model is rigorous in that respect and leads directly to testable modelling “experiments” (falsifiability). In this case, we use three agent behavior model types – random walks, flocking, and foraging – to describe the two models. Thus we have the rudimentary setup for a modelling experiment, where we can propose multiple models to explain the observed pattern and then run those models to determine whether model behavior resembles the patterns we have seen in the given graphics (below).
The three models we explore here have distinct patterns in space and time, in order to evaluate these changes we track the temporal pattern of nearest-neighbor-distance (NND) for the population of turtles. In the random walks model, the spatial pattern is random. Agents release at t=0 in a single location and move at some random angle between 0 and 360⁰ at every time step. As time progresses, the turtles gradually cover the screen with their randomly distributed pattern. The random pattern does not spread over the entire screen until ~1500-2000 ticks, and over that same temporal scale NND increases toward an asymptote. The temporal pattern could be considered emergent, but the value of the asymptote is a feature of the toroidal space the model inhabits, rather than some underlying reality of the movement style.
Random
The foraging model produces a spatial pattern that is more traditionally emergent. Within this model, agents move at some random angle between 90⁰ left and 90⁰ right, with values near 0⁰ more likely (angle=random-right + random-left). At each tick they use a unit of energy and move one cell. However, they are shuttled to the nearest patch of grass when their energy turns to 0, leading to a clustered spatial pattern through time with fairly predictable increases and decreased in NND. As such, the spatial pattern is based on the spatial relationship between the agents and their energy source exclusively. When the agents are mostly returned to a patch of grass (food-energy source) the NND goes to near zero, and then agents gradually move away from the grass and each other on average until the next time they are pulled back. An interesting novelty is that some turtles return to the grass before their energy turns to 0 and re-energize, such that they are then out of phase with the larger population.
Foraging
The flocking model also has a distinct spatial pattern becoming emergent during runs. This model begins with all agents in a single location and radiating randomly. Subsequently, each time step includes a choice of movement angle which largely contingent on the relationship of the agents to one another. Agents prioritize aligning with their neighbors and keeping some within some minimum NND, thus aggregating into a “flock” within a few hundred ticks at this spatial scale. This is a sharp contrast to the previous models in which agents do not determine their movements based on the position of other agents. The temporal pattern of NND that emerges from this system is nearly flat. The average distance between agents begins near zero and never increases substantially thereafter.
Flocking
For each species we now have three potential models, and we would be able to test hypotheses about which model fit the spatial and temporal patterns of the original observations if we were interested in statistical approaches. The model that fits the pattern we expected to observe for species A is the foraging model. In the sample data graphic, NND increases rapidly for a brief period and then oscillates over time. In addition, the spatial arrangement of agents is clustered, and the foraging model leads to clustering due to the resource constraint on agent behavior. An important distinction in this case is that the oscillation of NND range covers a substantial (majority) of the total range of the data.
Comparing this pattern to the other two, it seems likely that this is the best model for approximating the pattern provided. The random pattern shows a more gradual increase and less pronounced oscillation and the flocking model does not have any substantial variability in NND. The distribution of agents is also different, with the random model showing uniform distribution and the flocking showing one or two clusters rather than many.
The model that best suites the pattern of NND for species B is the random-walk model. Again, at this qualitative level of analysis, we rely on the visual appearance of a quantitative graphic. In the case of species B we are interested in a pattern that shows rapid initial growth, which slows over time and may even appear to approach some number less than infinity. In the provided detail of the spatio-temporal pattern, this is shown as a smooth curve, but the data from the random model is not quite as well behaved. In fact, it is easiest to see the underlying pattern after 2000 time steps.
For reasons similar to the previous example, the other two models show NND-time relationships that do not match the pattern we observe for species B. The foraging model is too clustered and only increases the NND value for a brief period, while the flocking is too clustered and shows no variability in NND at all.
The value of this approach to modelling is that we are comparing model options based on the performance. The use of strong inference is critical. By proposing multiple possible models, which ideally all reflect aspects of the true system under study, we reduce the potential for equifinality to confound any potential conclusions about the nature of a system. Thus we made use of part of the pattern-orient modelling approach. Importantly, there was no model development stage in this exercise. The other aspect of pattern-oriented modelling is using an existing understanding of the system of interest to craft models that are robust. Because we did not engage this aspect it is less important here. However, implied in our choices about the most correct model based on multiple patterns was a recognition of mechanisms. The use of multiple patterns in this experiment is an important feature as well, and judging on two criteria prevented more difficult distinctions between patterns.