To connect or not to connect isolated patches

One way or another: Combined effect of dispersal and asymmetry on total realized asymptotic population abundance

Understanding the consequences on population dynamics of the variability in dispersal over a fragmented habitat remains a major focus of ecological and environmental inquiry. Dispersal is often asymmetric: wind, marine currents, rivers, or human activities produce a preferential direction of dispersal between connected patches. Here, we study how this asymmetry affects population dynamics by considering a discrete-time two-patch model with asymmetric dispersal. We conduct a rigorous analysis of the model and describe all the possible response scenarios of the total realized asymptotic population abundance to a change in the dispersal rate for a fixed symmetry level. In addition, we discuss which of these scenarios can be achieved just by restricting mobility in one specific direction. Moreover, we also report that changing the order of events does not alter the population dynamics in our model, contrary to other situations discussed in the literature.

The effect of dispersal on asymptotic total population size in discrete- and continuous-time two-patch models

Many populations occupy spatially fragmented landscapes. How dispersal affects the asymptotic total population size is a key question for conservation management and the design of ecological corridors. Here, we provide a comprehensive overview of two-patch models with symmetric dispersal and two standard density-dependent population growth functions, one in discrete and one in continuous time. A complete analysis of the discrete-time model reveals four response scenarios of the asymptotic total population size to increasing dispersal rate: (1) monotonically beneficial, (2) unimodally beneficial, (3) beneficial turning detrimental, and (4) monotonically detrimental. The same response scenarios exist for the continuous-time model, and we show that the parameter conditions are analogous between the discrete- and continuous-time setting. A detailed biological interpretation offers insight into the mechanisms underlying the response scenarios that thus improve our general understanding how potential conservation efforts affect population size.

The role of the spatial topology in trophic metacommunities: Species with reduced mobility and total population size

A central question in ecology is understanding the influence of the spatial topology on the dynamics of a metacommunity. This is not an easy task, as most fragmented ecosystems have trophic interactions involving many species and patches. Recent attempts to solve this challenge have introduced certain simplifying assumptions or focused on a limited set of examples. These simplifications make the models mathematically tractable but keep away from real-world problems. In this paper, we provide a novel methodology to describe the influence of the spatial topology on the total population size of the species when the dispersal rates are small. The main conclusion is that the influence of the spatial topology is the result of the influence of each path in isolation. Here, a path refers to a pairwise connection between two patches. Our framework can be readily used with any metacommunity, and therefore represents a unification of biological insights. We also discuss several applications regarding the construction of ecological corridors.

Vulnerability assessment of freeway network considering the probabilities and consequences from a perspective based on network cascade failure

Freeway networks are vulnerable to natural disasters and man-made disruptions. The closure of one or more toll stations of the network often causes a sharp decrease in freeway performance. Therefore, measuring the probability and consequences of vulnerability to identify critical parts in the network is crucial for road emergency management. Most existing techniques only measure the consequences of node closure and rarely consider the probability of node closure owing to the lack of an extensive historical database; moreover, they ignore highways outside the study area, which can lead to errors in topological analysis and traffic distribution. Furthermore, the negative effects produced by the operation of freeway tunnels in vulnerability assessment have been neglected. In this study, a framework for freeway vulnerability assessment that considers both the probability and consequences of vulnerability is proposed, based on the perspective of network cascade failure analysis. The cascade failure analysis is conducted using an improved coupled map lattice model, developed by considering the negative effects of tunnels and optimizing the rules of local traffic redistribution. The perturbation threshold and propagation time step of network cascade failure are captured to reflect the probabilities and consequences of vulnerability. A nodal vulnerability index is established based on risk assessment, and a hierarchical clustering method is used to identify the vulnerability classification of critical nodes. The freeway network of Fuzhou in China is utilized to demonstrate the effectiveness of the proposed approach. Specifically, the toll stations in the study area are classified into five clusters of vulnerability: extremely high, high, medium, low, and extremely low. Approximately 31% of the toll stations were classified as the high or extremely high cluster, and three extremely vulnerable freeway sections requiring different precautions were identified. The proposed network vulnerability analysis method provides a new perspective to examine the vulnerability of freeway networks.

Effects of Prey's Diffusion on Predator-Prey Systems with Two Patches

This paper considers predator-prey systems in which the prey can move between source and sink patches. First, we give a complete analysis on global dynamics of the model. Then, we show that when diffusion from the source to sink is not large, the species would coexist at a steady state; when the diffusion is large, the predator goes to extinction, while the prey persists in both patches at a steady state; when the diffusion is extremely large, both species go to extinction. It is derived that diffusion in the system could lead to results reversing those without diffusion. That is, diffusion could change species' coexistence if non-diffusing, to extinction of the predator, and even to extinction of both species. Furthermore, we show that intermediate diffusion to the sink could make the prey reach total abundance higher than if non-diffusing, larger or smaller diffusion rates are not favorable. The total abundance, as a function of diffusion rates, can be both hump-shaped and bowl-shaped, which extends previous theory. A novel finding of this work is that there exist diffusion scenarios which could drive the predator into extinction and make the prey reach the maximal abundance. Diffusion from the sink to source and asymmetry in diffusion could also lead to results reversing those without diffusion. Meanwhile, diffusion always leads to reduction of the predator's density. The results are biologically important in protection of endangered species.

Population abundance of two-patch competitive systems with asymmetric dispersal

This paper considers two-species competitive systems with two patches, in which one of the species can move between the patches. One patch is a source where each species can persist alone, but the other is a sink where the mobile species cannot survive. Based on rigorous analysis on the model, we show global stability of equilibria and bi-stability in the first octant Int[Formula: see text]. Then total population abundance of each species is explicitly expressed as a function of dispersal rates, and the function of the mobile species displays a distorted surface, which extends previous theory. A novel prediction of this work is that appropriate dispersal could make each competitor approach total population abundance larger than if non-dispersing, while the dispersal could reverse the competitive results in the absence of dispersal and promote coexistence of competitors. It is also shown that intermediate dispersal is favorable, large or small one is not good, while extremely large or small dispersal will result in extinction of species. These results are important in ecological conservation and management.

Multiple Attractors and Long Transients in Spatially Structured Populations with an Allee Effect

We present a discrete-time model of a spatially structured population and explore the effects of coupling when the local dynamics contain a strong Allee effect and overcompensation. While an isolated population can exhibit only bistability and essential extinction, a spatially structured population can exhibit numerous coexisting attractors. We identify mechanisms and parameter ranges that can protect the spatially structured population from essential extinction, whereas it is inevitable in the local system. In the case of weak coupling, a state where one subpopulation density lies above and the other one below the Allee threshold can prevent essential extinction. Strong coupling, on the other hand, enables both populations to persist above the Allee threshold when dynamics are (approximately) out of phase. In both cases, attractors have fractal basin boundaries. Outside of these parameter ranges, dispersal was not found to prevent essential extinction. We also demonstrate how spatial structure can lead to long transients of persistence before the population goes extinct.

Optimal Network Architectures for Spatially Structured Populations with Heterogeneous Diffusion

The motivation of this article is to derive new management guidelines to maximize the overall population size using popular management and conservation strategies, such as protected marine areas and ecological corridors. These guidelines are based on the identification of the network architectures for which the total population size is maximized. Describing the biological roles of the typical network variables in the fate of the population is a classic problem with many practical applications. This article suggests that the optimal network architecture relies heavily on the degree of mobility of the population. The recommended network architecture for populations with reduced mobility (in the absence of cost of dispersal and landscapes made up of many sources) is a graph with a patch that has routes toward any other patch with a lower growth rate. However, for highly mobile populations there are many possible network architectures for which the total population size is maximized (e.g., any cyclic graph). We have paid special attention to species with symmetric movement in heterogeneous landscapes. A striking result is that the network architecture does not have any influence on the total population size for highly mobile populations when any pair of different patches can be connected by a sequence of paths.

Management guidelines in disturbance-Prone populations: The importance of the intervention time

The use of conservation and management practices to buffer possible damages after disturbance events is growing to become popular worldwide. However, little is known about their efficacy in real-life situations. To fill this gap, we will derive management guidelines in disturbance-prone populations regarding the external introduction of individuals and the ecological restoration. We will also discuss the efficacy of these practices in the population dynamics of three species (a fast life-cycle mayfly, a slow life-cycle dragonfly and an ostracod) when their habitat suffers from periodic controlled flooding. One of the main messages of this paper is that the interplay between the inherited parameters of the population and disturbance events is a source of rich and unexpected behaviours. More importantly, intervention time plays a critical role in the performance of some management strategies.

Dispersal asymmetry in a two-patch system with source-sink populations

This paper analyzes source-sink systems with asymmetric dispersal between two patches. Complete analysis on the models demonstrates a mechanism by which the dispersal asymmetry can lead to either an increased total size of the species population in two patches, a decreased total size with persistence in the patches, or even extinction in both patches. For a large growth rate of the species in the source and a fixed dispersal intensity, (i) if the asymmetry is small, the population would persist in both patches and reach a density higher than that without dispersal, in which the population approaches its maximal density at an appropriate asymmetry; (ii) if the asymmetry is intermediate, the population persists in both patches but reaches a density less than that without dispersal; (iii) if the asymmetry is large, the population goes to extinction in both patches; (iv) asymmetric dispersal is more favorable than symmetric dispersal under certain conditions. For a fixed asymmetry, similar phenomena occur when the dispersal intensity varies, while a thorough analysis is given for the low growth rate of the species in the source. Implications for populations in heterogeneous landscapes are discussed, and numerical simulations confirm and extend our results.

Migration alters oscillatory dynamics and promotes survival in connected bacterial populations

Migration influences population dynamics on networks, thereby playing a vital role in scenarios ranging from species extinction to epidemic propagation. While low migration rates prevent local populations from becoming extinct, high migration rates enhance the risk of global extinction by synchronizing the dynamics of connected populations. Here, we investigate this trade-off using two mutualistic strains of E. coli that exhibit population oscillations when co-cultured. In experiments, as well as in simulations using a mechanistic model, we observe that high migration rates lead to synchronization whereas intermediate migration rates perturb the oscillations and change their period. Further, our simulations predict, and experiments show, that connected populations subjected to more challenging antibiotic concentrations have the highest probability of survival at intermediate migration rates. Finally, we identify altered population dynamics, rather than recolonization, as the primary cause of extended survival.

Age-structure density-dependent fertility and individuals dispersal in a population model

In this work, we analyze the interplay between general age structured density-dependent fertility functions and age classes dispersal in a patchy environment. As novelties, (i) the fertility function depends on age classes (instead of on the total population size) and (ii) dispersal patterns are also allowed to be different for individuals belonging to different age classes. Our results highlight the interplay between the shape of the age structured density-dependent fertility function and the age classes dispersal patterns. We analyze this interaction from an environmental management point of view by exploring the consequences of connecting patches that can sustain a population (source patch) or cannot (sink patch), as well as its relation to component Allee effects and strong Allee effects. In particular, we have found scenarios such that the metapopulation goes extinct when two isolated source patches are connect due to heterogeneous age classes distribution. On the contrary, there are settings such that heterogeneous age classes distribution enables two isolated sink patches to be sustainable when connected. Besides, we discuss what kind of local interventions are helpful to manage component Allee effect and its impact at the metopopulation level. The source code used to simulations is fully available. The code is presented as a knitr reproducible document in the open source R computing system. Thus, free access and usability of the code are granted.

Effects of diffusion on total biomass in simple metacommunities

This paper analyzes the effects of diffusion on the overall population size of the different species of a metacommunity. Depending on precise thresholds, we determine whether increasing the dispersal rate of a species has a positive or negative effect on population abundance. These thresholds depend on the interaction type of the species and the quality of the patches. The motivation for researching this issue is that spatial structure is a source of new biological insights with management interest. For instance, in a metacommunity of two competitors, the movement of a competitor could lead to a decrease of the overall population size of both species. On the other hand, we discuss when some classic results of metapopulation theory are preserved in metacommunities. Our results complement some recent experimental work by Zhang and collaborators.

A quantitative approach to the stabilizing role of dispersal in metapopulations

We study a classical model for a population that reproduces and disperses in a landscape of heterogeneous patches. Under symmetrical dispersal, we provide a sufficient condition to ensure the existence of a globally attracting fixed point. This condition is used in order to prove that certain patches with complex dynamics can be stabilized by the combination with stable patches. Specifically, given a patch with complex dynamics, we estimate the necessary number of patches with simple dynamics so that the whole metapopulation has a globally attracting equilibrium.

Bad neighbors: how spatially disjunct habitat degradation can cause system-wide population collapse

Movement of individuals links the effects of local variation in habitat quality with growth and persistence of populations at the landscape scale. When the populations themselves are linked by interspecific interactions, such as predation, differential movement between habitats may lead to counterintuitive system-wide dynamics. Understanding the interaction between local drivers and dynamics of widely dispersed species is necessary to predict the impacts of habitat fragmentation and degradation, which may be transmitted across habitat boundaries by species' movements. Here we model predator-prey interactions across unaltered and degraded habitat areas, and we explore the additional effects of adaptive habitat choice by predators on the resilience of prey populations. We show how movement between habitats can produce the "bad neighbor effect," in which predators' response to localized habitat degradation causes system-wide loss of prey populations. This effect arises because adaptive foraging results in the concentration of predators in the more productive unaltered habitat, even when this habitat can not support the increased prey mortality. The mechanisms underlying this effect are especially sensitive to prey dispersal rate and adaptive predator behavior.

Adaptive threshold harvesting and the suppression of transients

Fluctuations in population size are in many cases undesirable, as they can induce outbreaks and extinctions or impede the optimal management of populations. We propose the strategy of adaptive threshold harvesting (ATH) to control fluctuations in population size. In this strategy, the population is harvested whenever population size has grown beyond a certain proportion in comparison to the previous generation. Taking such population increases into account, ATH intervenes also at smaller population sizes than the strategy of threshold harvesting. Moreover, ATH is the harvesting version of adaptive limiter control (ALC) that has recently been shown to stabilize population oscillations in both experiments and theoretical studies. We find that ATH has similar stabilization properties as ALC and thus offers itself as a harvesting alternative for the control of pests, exploitation of biological resources, or when restocking interventions required from ALC are unfeasible. We present numerical simulations of ATH to illustrate its performance in the presence of noise, lattice effect, and Allee effect. In addition, we propose an adjustment to both ATH and ALC that restricts interventions when control seems unnecessary, i.e. when population size is too small or too large, respectively. This adjustment cancels prolonged transients.

Analysis of dispersal effects in metapopulation models

The interplay between local dynamics and dispersal rates in discrete metapopulation models for homogeneous landscapes is studied. We introduce an approach based on scalar dynamics to study global attraction of equilibria and periodic orbits. This approach applies for any number of patches, dispersal rates, or landscape structure. The existence of chaos in metapopulation models is also discussed. We analyze issues such as sensitive dependence on the initial conditions or short/intermediate/long term behaviours of chaotic orbits.