Population Dynamics in River Networks

Abstract

Natural rivers connect to each other to form networks. The geometric structure of a river network can significantly influence spatial dynamics of populations in the system. We consider a process-oriented model to describe population dynamics in river networks of trees, establish the fundamental theories of the corresponding parabolic problems and elliptic problems, derive the persistence threshold by using the principal eigenvalue of the eigenvalue problem, and define the net reproductive rate to describe population persistence or extinction. By virtue of numerical simulations, we investigate the effects of hydrological, physical, and biological factors, especially the structure of the river network, on population persistence. physical and hydrological features in a river network, can greatly influence the spatial distribution of the flow profile (including the flow velocity, water depth, etc.) in branches of the network. Moreover, the population dispersal vectors may be constrained by the network configuration and the flow profile, and species life history traits may depend on varying habitat conditions in the network. As a result, population distribution and persistence in river systems can be significantly affected by the network topology or structure, see e.g., [8,11,12,[14][15][16]46]. Then there arise interesting questions such as whether a population can persist in the desert streams of the southwestern United State while the streams are experiencing substantial natural drying trends [11], or whether dendritic geometry enhances dynamic stability of ecological systems [15] etc. Furthermore, other related dynamics in the network, such as the dynamics of water-born infectious diseases like cholera may also be greatly affected by the river network geometry (see e.g., [6]).Branches in a river network have been modeled as point nodes in a network of habitats in individual based models [11,16] and matrix population models for stage-structured populations [14,39]. However this oversimplifies the spatial heterogeneity of river networks. In a real river ecosystem, organisms mainly live in the branches of the network and the connections between branches (e.g., the network nodes) are mainly for population transitions from one branch to another. To take into account this realistic situation, in recent works [48-50], integro-differential equations and reactiondiffusion-advection equations were used to model population dynamics in river networks where the network branches, instead of the network nodes, are the main habitat for organisms. Here the river networks are modeled under the framework of metric graphs (or metric networks). A metric graph is a graph G = (V, E) with a set V of vertices and a set E of edges, such that each edge e ∈ E is associated with either a closed bounded interval. Mathematic notion of metric graphs was first introduced in the context of wave propagation on thin graph-like domains [5,25], and they are also called quantum graphs.The theories of parabolic and elliptic equations as well as the corr...

The role of protection zone on species spreading governed by a reaction–diffusion model with strong Allee effect

“…The network topology (i.e., the topological structure of a river network) can greatly influence the species persistence and extinction. Therefore, as in [25,42,43], it would be interesting to consider a more general river habitat (bounded or unbounded) consisting of more than one branch. Moreover, if a branch is bounded, the works in [29,30,35] have shown that different boundary conditions could be vital in the population dynamics.…”

“…Given u 0 ∈ X, it is known that (1.6) admits a unique nonnegative solution u ∈ C 1,2 ((0, ∞) × (R \ {±L})) ∩ C α/2,1+α ((0, ∞) × R) for any α ∈ (0, 1), and u exists for all time t > 0; refer to [16,25,46]. To further simplify (1.6), we consider the scenario that the initial data u 0 are symmetric with respect to the origin in the sense that u 0 (x) = u 0 (−x) a.e.…”

The Fisher-KPP equation over simple graphs: varied persistence states in river networks

“…It turns out that this trichotomy behavior is also different from the finite graph case considered in [8,13,14,15] (see Remark 1.9 below), where the unique persistence state does not seem easily distinguishable when the parameters are varied.…”

“…Partly motivated by these questions, population models in rivers or streams have gained increasing attention recently. A brief account of these efforts can be found in the introduction of [8]. For example, in [6,7,9,10,16], the rivers and streams are treated as an interval on the real line with finite or infinite length, and questions on persistence and vanishing are examined via various advection-diffusion models over such an interval.…”

“…However, as argued in [2], the topological structure of a river network may also greatly influence the population growth and spread of organisms living in it. Several recent papers (see, for example, [8,13,14,15]) use suitable metric graphs to represent the topological structures of a river network, and study the persistence and vanishing problem by models of advection-diffusion equations over such graphs. The graphs in these works are all finite: They contain finitely many edges and vertices, and every edge has finite length.…”