We study homoclinic snaking in the cubic-quintic Swift-Hohenberg equation (SHE) close to the onset of a subcritical pattern-forming instability. Application of the usual multiple-scales method produces a leading-order stationary front solution, connecting the trivial solution to the patterned state. A localized pattern may therefore be constructed by matching between two distant fronts placed back-to-back. However, the asymptotic expansion of the front is divergent, and hence should be truncated. By truncating optimally, such that the resultant remainder is exponentially small, an exponentially small parameter range is derived within which stationary fronts exist. This is shown to be a direct result of the 'locking' between the phase of the underlying pattern and its slowly varying envelope. The locking mechanism remains unobservable at any algebraic order, and can only be derived by explicitly considering beyond-allorders effects in the tail of the asymptotic expansion, following the method of Kozyreff and Chapman as applied to the quadratic-cubic SHE Kozyreff 2009 Physica D 238 319-54, Kozyreff and Chapman 2006 Phys. Rev. Lett. 97 44502). Exponentially small, but exponentially growing, contributions appear in the tail of the expansion, which must be included when constructing localized patterns in order to reproduce the full snaking diagram. Implicit within the bifurcation equations is an analytical formula for the width of the snaking region. Due to the linear nature of the beyond-all-orders calculation, the bifurcation equations contain an analytically indeterminable constant, estimated in the previous work by Chapman and Kozyreff using a best fit approximation. A more accurate estimate of the equivalent constant in the cubic-quintic case is calculated from the iteration of a recurrence relation, and the subsequent analytical bifurcation diagram compared with numerical simulations, with good agreement.

Photosymbiosis is one of the most important evolutionary trajectories, resulting in the chloroplast and the subsequent development of all complex photosynthetic organisms. The ciliate Paramecium bursaria and the alga Chlorella have a well established and well studied light dependent endosymbiotic relationship. Despite its prominence, there remain many unanswered questions regarding the exact mechanisms of the photosymbiosis. Of particular interest is how a host maintains and manages its symbiont load in response to the allocation of nutrients between itself and its symbionts. Here we construct a detailed mathematical model, parameterised from the literature, that explicitly incorporates nutrient trading within a deterministic model of both partners. The model demonstrates how the symbiotic relationship can manifest as parasitism of the host by the symbionts, mutualism, wherein both partners benefit, or exploitation of the symbionts by the hosts. We show that the precise nature of the photosymbiosis is determined by both environmental conditions (how much light is available for photosynthesis) and the level of control a host has over its symbiont load. Our model provides a framework within which it is possible to pose detailed questions regarding the evolutionary behaviour of this important example of an established light dependent endosymbiosis; we focus on one question in particular, namely the evolution of host control, and show using an adaptive dynamics approach that a moderate level of host control may evolve provided the associated costs are not prohibitive.

Despite numerous negative impacts, nonnative trout are still being stocked to provide economically and socially valuable sport fisheries in western mountain lakes. We evaluated relative performance and potential differences in feeding strategy and competitive ability of triploid versus diploid brook trout Salvelinus fontinalis in alpine lakes, as well as behavioral and performance differences of diploid and triploid brook trout in two controlled experimental settings: behavioral experiments in the laboratory and performance evaluations in ponds. Across lakes, catch per unit effort (CPUE) and relative weight (Wr) were not significantly different between ploidy levels. Mean sizes were also similar between ploidy levels except in two of the larger lakes where diploids attained slightly larger sizes (approximately 20 mm longer). We observed no significant differences between diploids and triploids in diet, diet preference, or trophic structure. Similarly, growth and condition did not differ between ploidy levels in smaller‐scale pond experiments, and aggressive behavior did not differ between ploidy levels (fed or unfed fish trials) in the laboratory. Independent of ploidy level, the relative performance of brook trout varied widely among lakes, a pattern that appeared to be a function of lake size or a factor that covaries with lake size such as temperature regime or carrying capacity. In summary, we observed no significant differences in the relative performance of brook trout from either ploidy level across a number of indices, systems, and environmental conditions, nor any indication that one group is more aggressive or a superior competitor than the other. Collectively, these results suggest that triploid brook trout will offer a more risk‐averse and promising management opportunity when they are stocked to these lakes and elsewhere to simultaneously meet the needs for the sport fishery and conservation objectives. Received February 6, 2012; accepted August 27, 2012