Organismal movement can take on a variety of spatial and temporal forms. These forms depend in part on the type and scale of environment experienced as well as the internal state of the individual. However, individuals experiencing seemingly the same environment on the same time scale can display different movement strategies. While theorists have mathematically analyzed patch models and simulated spatially-explicit models, few studies have provided a mathematical analysis of migration in spatially-explicit models. Here, we consider a spatially explicit one-dimensional model where movement is costly and individuals must return to a common breeding ground annually to reproduce. We derive the optimal movement strategy, given specific movement costs and environmental resource distributions, obtaining closed-form solutions and results in several important special cases. We find, intuitively, that steep resource clines favor migratory behavior and shallow resource clines favor residential behavior, while lower movement efficiencies and shorter breeding cycles favor residency. However, we also show that when resource clines are sharp, migrants and residents can coinvade with each exploiting a locally optimal behavior. This can be interpreted as an example of partial migration (if migrants and residents are members of the same species). Alternatively, this can also be interpreted as two recently divergent species coinvading on a single resource, using different movement strategies to share the niche. We conclude with a discussion of density-dependent pressures on movement, including local resource depletion, and show that the density-independent results are relevant to density-dependent situations by calculating some stable strategy allocations analogous to ideal free distributions.
Bibliographical noteFunding Information:
Thanks to the editor and two anonymous reviewers for helpful criticism in revising our manuscript. This research was supported by NSF grants DMS-0920822 to TCR and OISE-1159097 to AKS. The calculations in this paper were obtained using the sympy and scipy libraries for python (SymPy Development Team 2013; Jones et al. 2001; Python Software Foundation 2010). Graphics were prepared using Gnuplot and Matplotlib. (Williams et al. 2010; Hunter 2007).
- Optimal control
- Partial migration