Traveling waves in discrete models of biological populations with sessile stages

A common approach to describing invasions of non-native species into previously unoccupied habitat is to consider the speed of population expansion and the existence of traveling waves. Typical existence theorems for traveling waves require some compactness properties of the next-generation operator. Many realistic modeling assumptions, however, give rise to non-compact operators; for example the occurrence of sessile stages during the life cycle of an individual. Recent results have extended the existence theory of traveling waves to a large class of weakly compact operators, but conditions can be difficult to check and not easily accessible to theoretical ecologists. In this paper, we give a new proof for the existence of traveling waves in a large class of equations where the next generation operator is not compact, but rather the sum of an integral operator and a contraction. We illustrate our proof with a model for the dispersal of a plant species with a seed-bank and a model for dispersal of stream insects with larval stages.     

Competition of three species in an advective environment

Individuals in advective environments, for example rivers, coastlines, or the gut, are subject to movement with directional bias. We study how this movement bias shapes community composition by considering how the strength of movement bias affects the outcome of competition among three species. Our model is a system of three reaction-advection-diffusion equations with Danckwerts boundary conditions. Our key tool in this study is to use the dominant eigenvalue of the diffusion-advection operator in order to reduce the spatially explicit model to a spatially implicit ordinary differential equation model. After an in-depth analysis of the implicit model, we use numerical simulations of the explicit model to test the predictions obtained from the analysis. In general, we find a good qualitative agreement between the explicit and the implicit model. We find that varying the strength of advection can fundamentally alter the outcome of competition between the three species, and we characterize the possible transitions. In particular, water extraction and flow control can destabilize existing species communities or facilitate invasions of non-native species.     

Emerging Patterns in a Hyperbolic Model for Locally Interacting Cell Systems

Summary. Morphogenetic processes such as neurulation and gastrulation involve co- ordinated movements of cells. It is assumed that these processes happen due to long-range signaling, although the detailed mechanisms are not completely understood. Therefore, one is interested in biological ``model-systems' where self-organization of cells and in particular the mechanisms of signaling can be analyzed in greater detail. A major question is whether or not short-range signaling or local interaction of cells can also be the cause of coordinated movement and morphogenetic processes. As a model problem we analyze ripple formation of myxobacteria due to purely local interaction, a hypothesis which is discussed in the biological literature. These ripples can be observed before the final aggregation of the bacteria and fruiting body formation take place. Our basic mathematical model is a one-dimensional hyperbolic system of Goldstein-Kac type with density-dependent coefficients. Conditions for the existence of traveling waves are discussed by means of linear analysis and the construction of invariant domains.