Population persistence in river networks

Dynamics of Lotka-Volterra Competition Patch Models in Streams with Two Branches

Streams may have many branches and form complex river networks. We investigate two competition patch models associated with two different river network modules, where one is a distributary stream with two branches at the downstream end, and the other is a tributary stream with two branches at the upstream end. Treating one species as resident species and the other one as mutant species, it is shown that, for each model, there exists a invasion curve such that the mutant species can invade when rare if and only if its dispersal strategy is below this curve, but the shapes of the invasion curves are different. Moreover, we show that the global dynamics of the two models can be similar or different depending on river networks. Especially, if the drift rates of the two species are equal, then the global dynamics are similar for small drift rate and different for large drift rate. Our results also confirm a conjecture in Jiang et al. (Bull Math Biol 82:131, 2020).

Three-patch Models for the Evolution of Dispersal in Advective Environments: Varying Drift and Network Topology

We study the evolution of dispersal in advective three-patch models with distinct network topologies. Organisms can move between connected patches freely and they are also subject to the passive, directed drift. The carrying capacity is assumed to be the same in all patches, while the drift rates could vary. We first show that if all drift rates are the same, the faster dispersal rate is selected for all three models. For general drift rates, we show that the infinite diffusion rate is a local Convergence Stable Strategy (CvSS) for all three models. However, there are notable differences for three models: For Model I, the faster dispersal is always favored, irrespective of the drift rates, and thus the infinity dispersal rate is a global CvSS. In contrast, for Models II and III a singular strategy will exist for some ranges of drift rates and bi-stability phenomenon happens, i.e., both infinity and zero diffusion rates are local CvSSs. Furthermore, for both Models II and III, it is possible for two competing populations to coexist by varying drift and diffusion rates. Some predictions on the dynamics of n-patch models in advective environments are given along with some numerical evidence.

The geography of metapopulation synchrony in dendritic river networks

Dendritic habitats, such as river ecosystems, promote the persistence of species by favouring spatial asynchronous dynamics among branches. Yet, our understanding of how network topology influences metapopulation synchrony in these ecosystems remains limited. Here, we introduce the concept of fluvial synchrogram to formulate and test expectations regarding the geography of metapopulation synchrony across watersheds. By combining theoretical simulations and an extensive fish population time‐series dataset across Europe, we provide evidence that fish metapopulations can be buffered against synchronous dynamics as a direct consequence of network connectivity and branching complexity. Synchrony was higher between populations connected by direct water flow and decayed faster with distance over the Euclidean than the watercourse dimension. Likewise, synchrony decayed faster with distance in headwater than mainstem populations of the same basin. As network topology and flow directionality generate fundamental spatial patterns of synchrony in fish metapopulations, empirical synchrograms can aid knowledge advancement and inform conservation strategies in complex habitats.

Are Two-Patch Models Sufficient? The Evolution of Dispersal and Topology of River Network Modules

We study the dynamics of two competing species in three-patch models and illustrate how the topology of directed river network modules may affect the evolution of dispersal. Each model assumes that patch 1 is at the upstream end, patch 3 is at the downstream end, but patch 2 could be upstream, or middle stream, or downstream, depending on the specific topology of the modules. We posit that individuals are subject to both unbiased dispersal between patches and passive drift from one patch to another, depending upon the connectivity of patches. When the drift rate is small, we show that for all models, the mutant species can invade when rare if and only if it is the slower disperser. However, when the drift rate is large, most models predict that the faster disperser wins, while some predict that there exists one evolutionarily singular strategy. The intermediate range of drift is much more complex: most models predict the existence of one singular strategy, but it may or may not be evolutionarily stable, again depending upon the topology of modules, while one model even predicts that for some intermediate drift rate, singular strategy does not exist and the faster disperser wins the competition.

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

In this article, we study the dynamical behaviour of a new species spreading from a location in a river network where two or three branches meet, based on the widely used Fisher-KPP advection-diffusion equation. This local river system is represented by some simple graphs with every edge a half infinite line, meeting at a single vertex. We obtain a rather complete description of the long-time dynamical behaviour for every case under consideration, which can be classified into three different types (called a trichotomy), according to the water flow speeds in the river branches, which depend crucially on the topological structure of the graph representing the local river system and on the cross section areas of the branches. The trichotomy includes two different kinds of persistence states, and the state called "persistence below carrying capacity" here appears new.

Population dynamics in river networks: analysis of steady states

We study the population dynamics of an aquatic species in a river network. The habitat is viewed as a binary tree-like metric graph with the population density satisfying a reaction-diffusion-advection equation on each edge, along with the appropriate junction and boundary conditions. In the case of a linear reaction term and hostile downstream boundary condition, the question of persistence in such models was studied by Sarhad, Carlson and Anderson. We focus on the case of a nonlinear (logistic) reaction term and use an outflow downstream boundary condition. We obtain necessary and sufficient conditions for the existence and uniqueness of a positive steady state solution for a simple Y-shaped river network (with a single junction). We show that the existence of a positive steady state is equivalent to the persistence condition for the linearized model. The method can be generalized to a binary tree-like river network with an arbitrary number of segments.

Metacommunity theory meets restoration: isolation may mediate how ecological communities respond to stream restoration

An often-cited benefit of river restoration is an increase in biodiversity or shift in composition to more desirable taxa. Yet, hard manipulations of habitat structure often fail to elicit a significant response in terms of biodiversity patterns. In contrast to conventional wisdom, the dispersal of organisms may have as large an influence on biodiversity patterns as environmental conditions. This influence of dispersal may be particularly influential in river networks that are linear branching, or dendritic, and thus constrain most dispersal to the river corridor. As such, some locations in river networks, such as isolated headwaters, are expected to respond less to environmental factors and less by dispersal than more well-connected downstream reaches. We applied this metacommunity framework to study how restoration drives biodiversity patterns in river networks. By comparing assemblage structure in headwater vs. more well-connected mainstem sites, we learned that headwater restoration efforts supported higher biodiversity and exhibited more stable ecological communities compared with adjacent, unrestored reaches. Such differences were not evident in mainstem reaches. Consistent with theory and mounting empirical evidence, we attribute this finding to a relatively higher influence of dispersal-driven factors on assemblage structure in more well-connected, higher order reaches. An implication of this work is that, if biodiversity is to be a goal of restoration activity, such local manipulations of habitat should elicit a more profound response in small, isolated streams than in larger downstream reaches. These results offer another significant finding supporting the notion that restoration activity cannot proceed in isolation of larger-scale, catchment-level degradation.

Geometric indicators of population persistence in branching continuous-space networks

We study population persistence in branching tree networks emulating systems such as river basins, cave systems, organisms on vegetation surfaces, and vascular networks. Population dynamics are modeled using a reaction-diffusion-advection equation on a metric graph which provides a continuous, spatially explicit model of network habitat. A metric graph, in contrast to a standard graph, allows for population dynamics to occur within edges rather than just at graph vertices, subsequently adding a significant level of realism. Within this framework, we stochastically generate branching tree networks with a variety of geometric features and explore the effects of network geometry on the persistence of a population which advects toward a lethal outflow boundary. We identify a metric (CM), the distance from the lethal outflow point at which half of the habitable volume of the network lies upstream of, as a promising indicator of population persistence. This metric outperforms other metrics such as the maximum and minimum distances from the lethal outflow to an upstream boundary and the total habitable volume of the network. The strength of CM as a predictor of persistence suggests that it is a proper "system length" for the branching networks we examine here that generalizes the concept of habitat length in the classical linear space models.

Connectivity, passability and heterogeneity interact to determine fish population persistence in river networks

The movement of fish in watersheds is frequently inhibited by human-made migration barriers such as dams or culverts. The resulting lack of connectivity of spatial subpopulations is often cited as a cause for observed population decline. We formulate a matrix model for a spatially distributed fish population in a watershed, and we investigate how location and other characteristics of a single movement barrier impact the asymptotic growth rate of the population. We find that while population growth rate often decreases with the introduction of a movement obstacle, it may also increase due to a 'retention effect'. Furthermore, obstacle mortality greatly affects population growth rate. In practice, different connectivity indices are used to predict population effects of migration barriers, but the relation of these indices to population growth rates in demographic models is often unclear. When comparing our results with the dentritic connectivity index, we see that the index captures neither the retention effect nor the influences of obstacle mortality. We argue that structural indices cannot entirely replace more detailed demographic models to understand questions of persistence and extinction. We advocate the development of novel functional indices and characteristics.

Analysis of spread and persistence for stream insects with winged adult stages

Species such as stoneflies have complex life history details, with larval stages in the river flow and adult winged stages on or near the river bank. Winged adults often bias their dispersal in the upstream direction, and this bias provides a possible mechanism for population persistence in the face of unidirectional river flow. We use an impulsive reaction-diffusion equation with non-local impulse to describe the population dynamics of a stream-dwelling organism with a winged adult stage, such as stoneflies. We analyze this model from a variety of perspectives so as to understand the effect of upstream dispersal on population persistence. On the infinite domain we use the perspective of weak versus local persistence, and connect the concept of local persistence to positive up and downstream spreading speeds. These spreading speeds, in turn are connected to minimum travelling wave speeds for the linearized operator in upstream and downstream directions. We show that the conditions for weak and local persistence differ, and describe how weak persistence can give rise to a population whose numbers are growing but is being washed out because it cannot maintain a toe hold at any given location. On finite domains, we employ the concept of a critical domain size and dispersal success approximation to determine the ultimate fate of the populations. A simple, explicit formula for a special case allows us to quantify exactly the difference between weak and local persistence.