Low-assumption fitting of interaction kernels: Spline models for neighborhood competition
An important part of understanding the interaction between two organisms is knowing how the interaction strength varies with the distance between the. We expect that competition between plants is weaker when the plants are further apart - but how much weaker at any given distance? We characterize this relationship between distance and interaction strength by fitting the ``interaction kernel'', which describes relative interaction strength as individuals are moved further apart. However, the shape and fit of an interaction kernel (and thus our inferences about interactions between individuals) are strongly influenced by our assumptions of the underlying shape (linear, exponential, Gaussian, etc). We have developed methods to fit interaction kernels with splines, which are extremely flexible and can take many different shapes. By using splines we prevent our initial assumptions from determining our results, allowing us to reasonably determine the actual underlying relationship between distance and interaction strength. Our methods improve on standard spline-fitting techniques, allowing us to analyze data for which interaction strength is inferred from fitness or survival, which are functions of the interaction as well as other covariates.
We generated sets of simulated data from various known interaction kernels with noise and covariates drawn from real data, and found that our spline-fitting methods were able to recover the underlying interaction kernels. We then used our methods to analyze a set of data for competing perennial plants in eastern Idaho. For near distances (<5cm), strength of competition decreased with distance as either a negative exponential or Gaussian function (depending on plant species), but for each species there was a distinct plateau in intermediate distances and no effect of competition for plants beyond a moderate distance (>15cm). In addition to informing our understanding of spatial patterns of plant competition in this system, these results demonstrate the importance of using spline fitting to determine interaction kernels, as traditional statistical methods would not have found these relationships.