Local and Ensemble Dynamics of Linked Populations: Reply to Ruxton

Abstract

We wish to make two brief comments on Ruxton's note about our paper (Bascompte & Sole 1994). His first and more important point concerns the apparent contradiction between our finding (i.e. that the coupling of different patches induces local chaos) and the results obtained in recent papers mentioned in his note (i.e. that the linkage between different populations tends to decrease the likelihood of chaotic dynamics). We believe that these latter conclusions are not as consistent as our own for two reasons. 1. The model described by Gonzalez-Andujar & Perry (1993) is just a trivial linear change of coordinates in the original map, which reduces the value of the growth rate, and consequently stabilizes the dynamics, as shown by Ruxton (1993). 2. Both McCallum (1992) and Stone (1993) consider immigration as a density-independent constant added to the map. If this constant parameter is large enough to dominate the density-dependent term, the resultant dynamics will fluctuate near this fixed value. This is not a clear physical diffusion. This point is developed elsewhere (Sole & Bascompte 1994; Bascompte & Sol6, unpublished; see, however, Stone 1994). Ruxton's conclusion is intermediate: diffusioninduced chaos is observed for low levels of immigration, while higher levels show simpler periodic cycles. This can be reconciled with our result because, as seen in Fig. 6 of our paper which shows a bifurcation diagram for increasing diffusion rates, periodic windows are obtained for high levels. Perhaps Ruxton's high immigration rates may be located in a similar periodic window, which would not be at odds with our conclusion. Furthermore, if the lattice size is small enough, a high diffusion rate can homogenize the dynamics, avoiding the asynchronous fluctuations. We strongly believe that our approach is a better one, because diffusion is natural and we can define local as well as global scales. As pointed out in our paper, the effect of enlarging the lattice size is the opposite, depending on the scale. In the McCallumStone diffusion term, on the other hand, we have no local or global scale to compare with our result. Furthermore, diffusion-induced chaos is a well-known process in a wide range of real systems, where there is a coupling of periodic oscillators (Kuramoto 1984). Such a result is confirmed by the theoretical modelling of this kind of system, both with a continuum approach (reaction-diffusion mathematical models), and with a discrete approach (coupled map lattices). As additional evidence, diffusion-induced chaos has been nicely decribed in a continuous prey-predator model (Pascual 1993). In his second comment, Ruxton emphasizes the different global behaviour obtained depending on whether the linkage is local or global. This difference has already been pointed out in our paper. It seems evident that if we are interested in studying the role of space, we must maintain the coupling to the nearest patches. If there is a global mixing of populations, space does not have a role. However, we agree with Ruxton that the effect of space on the population dynamics can be complex, and that much more study is required in order to obtain a complete picture. In this respect, his comments are well received.