Environments for living organisms are often fragmented by natural or artificial habitat destructions. In such heterogeneous environments, some animals are influenced by the senses of sight, hearing or smell to undergo directed movement toward favorable habitats. Here we consider a single-species invasion in a periodically varying environment, in which favorable and unfavorable patches are alternately arranged in a one-dimensional space. To incorporate the spatial heterogeneity and directed movement, we propose an advection-reaction-diffusion equation in which the advection velocity and the intrinsic growth rate vary depending on habitat properties. By using a heuristic method as we presented before (Shigesada et al. 1986), we derive an analytical formula for the speed of the traveling periodic wave. Based on the formula, we examine how the rate of spread is influenced by the advection velocity, habitat qualities and the scale of habitat fragmentation. Of particular interest is that the advection to the favorable habitat does not always enhance the rate of spread.
Plants and animals often exhibit strong and persistent growth variation among individuals within a species. Persistently fast-growing individuals have a higher chance of reaching reproductive size, do so at a much younger age and may therefore contribute disproportionately to population growth (λ). Here we introduce a new approach to quantify this 'fast-growth effect'. We propose to apply age-size structured matrix models in which persistent fast and slow growers can be distinguished as they occur in relatively young and old age classes, respectively. Elasticities of young classes then quantify the relative contribution of fast growers to λ. Loop elasticities are a better estimate as they represent the lifetime contribution of fast growers to λ. We applied this approach to an example species, the tropical rainforest tree Cedrela odorata. Fast growers had a 12% higher elasticity than slow growers. Using loop analysis, this 'fast-growth effect' was much larger: juvenile trees that reached 10 cm diameter by persistent fast growth had a 2.3 times higher lifetime contribution to λ than slow growers. Fast growth to larger size categories also resulted in disproportionate contributions to λ. Fast-growth effects are likely common among long-lived species, and should be accommodated for in demographic models and life history studies.