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

Evolution of dispersal in river networks

Evolution of dispersal is a fascinating topic at the intersection of ecology and evolutionary dynamics that has generated many challenging problems in the analysis of reaction-diffusion equations. Early results indicated that lower random diffusion rates are generally beneficial. However, in riverine environments with downstream drift, high diffusion may be optimal, depending on downstream boundary conditions. Most of these results were obtained from modeling a single river reach, yet many rivers form intricate tree-shaped networks. We study the evolution of dispersal on a metric graph representing the simplest such possible network: two upstream segments joining to form one downstream segment. We first show that the shape of the positive steady state of a single population depends crucially on the geometry of the network, here considered as the relative length of the three segments. We then study the evolution of dispersal by considering the possibility of "invasion" of a second type (invader) at the steady state of the first type (resident). We show that the geometry of the network determines whether higher or intermediate dispersal is favored.

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).

Impact of resource distributions on the competition of species in stream environment

Our earlier work in Nguyen et al. (Maximizing metapopulation growth rate and biomass in stream networks. arXiv preprint arXiv:2306.05555 , 2023) shows that concentrating resources on the upstream end tends to maximize the total biomass in a metapopulation model for a stream species. In this paper, we continue our research direction by further considering a Lotka-Volterra competition patch model for two stream species. We show that the species whose resource allocations maximize the total biomass has the competitive advantage.

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.