Carrying capacity in a heterogeneous environment with habitat connectivity

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.

Landscape heterogeneity provides co-benefits to predator and prey

Predator populations are imperiled globally, due in part to changing habitat and trophic interactions. Theoretical and laboratory studies suggest that heterogeneous landscapes containing prey refuges acting as source habitats can benefit both predator and prey populations, although the importance of heterogeneity in natural systems is uncertain. Here, we tested the hypothesis that landscape heterogeneity mediates predator-prey interactions between the California spotted owl (Strix occidentalis occidentalis)-a mature forest species-and one of its principal prey, the dusky-footed woodrat (Neotoma fuscipes)-a younger forest species-to the benefit of both. We did so by combining estimates of woodrat density and survival from live trapping and very high frequency tracking with direct observations of prey deliveries to dependent young by owls in both heterogeneous and homogeneous home ranges. Woodrat abundance was ~2.5 times higher in owl home ranges (14.12 km2 ) featuring greater heterogeneity in vegetation types (1805.0 ± 50.2 SE) compared to those dominated by mature forest (727.3 ± 51.9 SE), in large part because of high densities in young forests appearing to act as sources promoting woodrat densities in nearby mature forests. Woodrat mortality rates were low across vegetation types and did not differ between heterogeneous and homogeneous home ranges, yet all observed predation by owls occurred within mature forests, suggesting young forests may act as woodrat refuges. Owls exhibited a type 1 functional response, consuming ~2.5 times more woodrats in heterogeneous (31.1/month ± 5.2 SE) versus homogeneous (12.7/month ± 3.7 SE) home ranges. While consumption of smaller-bodied alternative prey partially compensated for lower woodrat consumption in homogeneous home ranges, owls nevertheless consumed 30% more biomass in heterogeneous home ranges-approximately equivalent to the energetic needs of producing one additional offspring. Thus, a mosaic of vegetation types including young forest patches increased woodrat abundance and availability that, in turn, provided energetic and potentially reproductive benefits to mature forest-associated spotted owls. More broadly, our findings provide strong empirical evidence that heterogeneous landscapes containing prey refuges can benefit both predator and prey populations. As anthropogenic activities continue to homogenize landscapes globally, promoting heterogeneous systems with prey refuges may benefit imperiled predators.

The comparison of dispersal rate between invasive and native species varied by plant life form and functional traits

A long dispersal distance is widely used to indicate high invasiveness, but it ignores the temporal dimensions of plant invasion. Faster dispersal rates (= distance/time) of invasive species than native ones have been widely used in modeling species invasion and planning control management. However, the comparison of dispersal rate between invasive and native plants, particularly for dispersal on a local or landscape scale, has not been tested with a comprehensive dataset. Moreover, both the effects of plant functional traits on the dispersal rate and variation in the functional-trait effects between invasive and native plants remain elusive. Compiling studies from 30 countries globally, we compared seed dispersal rates (km/year) on a local or landscape scale between 64 observations of invasive and 78 observations of native plants given effects of plant life forms, disturbance levels, and measurement methods. Furthermore, we compared the effects of functional traits on dispersal rate between invasive and native species. We found that: (1) Trait values were similar between the invasive and native plants except for the greater height of woody native plants than woody invasive ones; (2) Compared within the same plant life form, the faster dispersal rates of invasive species were found in herbaceous plants, not in woody plants, and disturbance level and measurement methods did not affect the rate comparison; (3) Plant height and seed length had significant effects on dispersal rates of both invasive and native plants, but the effect of leaf dry matter content (LDMC) was only significant on herbaceous invasive plants. The comparison of dispersal rate between invasive and native plants varied by plant life form. The convergent values but divergent dispersal effects of plant traits between invasive and native species suggest that the trait effects on invasiveness could be better understood by trait association with key factors in invasiveness, e.g., dispersal rate, than the direct trait comparison between invasive and native plants.

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.

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.

Dynamics of consumer-resource reaction-diffusion models: single and multiple consumer species

A consumer-resource reaction-diffusion model with a single consumer species was proposed and experimentally studied by Zhang et al.(Ecol Lett 20:1118-1128, 2017). Analytical study on its dynamics was further performed by He et al.(J Math Biol 78:1605-1636, 2019). In this work, we completely settle the conjecture proposed by He et al.(J Math Biol 78:1605-1636, 2019) about the global dynamics of the consumer-resource model for small yield rate. We then study a multi-species consumer-resource model where all the consumer species compete with each other through depression of the limited resources by consumption and there is no direct competition between them. We show that in this case, all consumer species persist uniformly, which implies that "competition exclusion" phenomenon will never happen. We also clarify its dynamics in both homogeneous and heterogeneous environments under various circumstances.

Resource allocation in a PDE ecosystem model

The effects of habitat heterogeneity on a diffusing population are investigated here. We formulate a reaction-diffusion system of partial differential equations to analyze the effect of resource allocation in an ecosystem with resource having its own dynamics in space and time. We show a priori estimates to prove the existence of state solutions given a control. We formulate an optimal control problem of our ecosystem model such that the abundance of a single species is maximized while minimizing the cost of inflow resource allocation. In addition, we show the existence and uniqueness of the optimal control as well as the optimal control characterization. We also establish the existence of an optimal intermediate diffusion rate. Moreover, we illustrate several numerical simulations with Dirichlet and Neumann boundary conditions with the space domain in 1D and 2D.

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.

Population dynamics with resource-dependent dispersal: single- and two-species models

In this paper, we consider the population models with resource-dependent dispersal for single-species and two-species with competition. For the single-species model, it is well-known that the total population supported by the environment is always greater than the environmental carrying capacity if the dispersal is simply random diffusion. However, we find that the total population supported can be equal or smaller than the environmental carrying capacity when the dispersal depends on the resource distribution. This analytical finding not only well agrees with the yeast experiment observation of Zhang et al. (Ecol Lett 20(9):1118-1128, 2017), but also indicates that resource-dependent dispersal may produce different outcomes compared to the random dispersal. For the two-species competition model, when two competing species use the same dispersal strategy up to a multiplicative constant (i.e. their dispersal strategies are proportional) or both diffusion coefficients are large, we give a classification of global dynamics. We also show, along with numerical simulations, that if the dispersal strategies are resource-dependent, even one species has slower diffusion, coexistence is possible though competitive exclusion may occur under different conditions. This is distinct from the prominent result that with random dispersal the slower diffuser always wipes out its fast competitor. Our analytical results, supported by the numerical simulations, show that the resource-dependent dispersal strategy has profound impact on the population dynamics and evolutionary processes.

Biotic Interactions in Soil are Underestimated Drivers of Microbial Carbon Use Efficiency

Microbial carbon use efficiency (CUE)-the balance between microbial growth and respiration-strongly impacts microbial mediated soil carbon storage and is sensitive to many well-studied abiotic environmental factors. However, surprisingly, little work has examined how biotic interactions in soil may impact CUE. Here, we review the theoretical and empirical lines of evidence exploring how biotic interactions affect CUE through the lens of life history strategies. Fundamentally, the CUE of a microbial population is constrained by population density and carrying capacity, which, when reached, causes species to grow more quickly and less efficiently. When microbes engage in interspecific competition, they accelerate growth rates to acquire limited resources and release secondary chemicals toxic to competitors. Such processes are not anabolic and thus constrain CUE. In turn, antagonists may activate one of a number of stress responses that also do not involve biomass production, potentially further reducing CUE. In contrast, facilitation can increase CUE by expanding species realized niches, mitigating environmental stress and reducing production costs of extracellular enzymes. Microbial interactions at higher trophic levels also influence CUE. For instance, predation on microbes can positively or negatively impact CUE by changing microbial density and the outcomes of interspecific competition. Finally, we discuss how plants select for more or less efficient microbes under different contexts. In short, this review demonstrates the potential for biotic interactions to be a strong regulator of microbial CUE and additionally provides a blueprint for future research to address key knowledge gaps of ecological and applied importance for carbon sequestration.

Evaluating water resource carrying capacity using the deep learning method: a case study of Yunnan, Southwest China

Water resource carrying capacity (WRCC) is an important index for measuring the relations between water resource systems and socio-economic-environmental development. In view of the difficulty in describing the complex and nonlinear relationships between the WRCC and indicators using traditional methods, this study introduces deep learning theory and proposes a novel deep neural network named WRCC-Net for WRCC assessment. Unlike typical network structures, we constructed a hierarchical structure that can indicate the index system in WRCC evaluation. Furthermore, we utilized a residual learning technique to increase the network depth for fitting the complex relationship between the WRCC state and indicators. The proposed deep learning method was applied to solve the real-world WRCC problem by taking the Yunnan province (Southwest China) as the case area. The WRCC was assessed from the following five dimensions: the water resources, social, economic, ecological environment, and coordination subsystems. Performance evaluation shows the advantages of the proposed WRCC-Net over the typical deep feed-forward network and shallow methods. Therefore, the proposed method provides a new way of evaluating the WRCC state and has potential for WRCC research. Overall, the WRCC evaluation using the WRCC-Net shows that the state of the WRCC in Yunnan constantly decreased from 2008 to 2018. These central-eastern areas in the Yunnan province, such as Kunming, Qujing, and Yuxi, are under an unfavorable capacity state. Measures, such as improving water resources management and increasing water utilization efficiency, should be considered in water resource planning in Yunnan province for the sustainable development of water resources.

Directed movement changes coexistence outcomes in heterogeneous environments

Understanding mechanisms of coexistence is a central topic in ecology. Mathematical analysis of models of competition between two identical species moving at different rates of symmetric diffusion in heterogeneous environments show that the slower mover excludes the faster one. The models have not been tested empirically and lack inclusions of a component of directed movement toward favourable areas. To address these gaps, we extended previous theory by explicitly including exploitable resource dynamics and directed movement. We tested the mathematical results experimentally using laboratory populations of the nematode worm, Caenorhabditis elegans. Our results not only support the previous theory that the species diffusing at a slower rate prevails in heterogeneous environments but also reveal that moderate levels of a directed movement component on top of the diffusive movement allow species to coexist. Our results broaden the theory of species coexistence in heterogeneous space and provide empirical confirmation of the mathematical predictions.

Foraging behaviours lead to spatiotemporal self-similar dynamics in grazing ecosystems

Biological behaviour‐driven self‐organized patterns have recently been confirmed to play a key role in ecosystem functioning. Here, we develop a theoretical phase‐separation model to describe spatiotemporal self‐similar dynamics, which is a consequence of behaviour‐driven trophic interactions in short‐time scales. Our framework integrates scale‐dependent feedback and density‐dependent movement into grazing ecosystems. This model derives six types of selective foraging behaviours that trigger pattern formation for top‐down grazing ecosystems, and one of which is consistent with existing foraging theories. Self‐organized patterns nucleate under moderate grazing intensity and are destroyed by overgrazing, which suggests ecosystem degradation. Theoretical results qualitatively agree with observed grazing ecosystems that display spatial heterogeneities under variable grazing intensity. Our findings potentially provide new insights into self‐organized patterns as an indicator of ecosystem transitions under a stressful environment.

Model of two competing populations in two habitats with migration: Application to optimal marine protected area size

The standard model of a single population fragmented into two patches connected by migration, was first introduced in the 1970s by Freedman and Waltman, since generating long-term research interest, though its full analysis for arbitrary values of migration rate has only been completed relatively recently. Here, we present a model of two competing species in a two-patch habitat with migrations between patches. We derive equilibrium solutions of this model for three cases of migration rate resulting in isolated, well-mixed and semi-isolated habitats. We evaluate the full range of effects of habitat, life-history and migration parameters on population sizes. Finally, we add harvesting mortality and define conditions under which introduction of a no-harvesting (protected) area may lead to increased maximum sustainable yield. The results have applications in mixed fishery management and the design of wildlife protection zones, including marine protected areas (MPAs).

Comprehensive evaluation of water resources carrying capacity and analysis of obstacle factors in Weifang City based on hierarchical cluster analysis-VIKOR method

This study uses hierarchical cluster analysis (HCA) to screen the evaluation indexes and establishes a comprehensive evaluation index system for water resources carrying capacity (WRCC), based on the VIKOR method and the obstacle degree model for the identification of the main factors affecting the WRCC of Weifang City. The results show that the WRCC of Weifang City has steadily increased from 2008 to 2018. The subsystems referred to society and water environment are currently the main obstacles affecting Weifang's WRCC, but there is still space for improvement in the future. The areas with low WRCC was Kuiwen District in 2018, which was in a seriously overloaded state, mostly affected by the water resources subsystem. The implementation of measures such as efficiently improving the level of water resources management and the development of water conservancy projects is prominent in water resource planning in Kuiwen District. This study analyzes the current situation of water resources management in order to consider it in strategic decision-making in promoting the improvement of WRCC, which in turn may ensure the realization of a green and sustainable development strategy in the future for Weifang City.

The optimal controlling strategy on a dispersing population in a two-patch system: Experimental and theoretical perspectives

Invasive species, disease vectors, and pathogens are significant threats to biodiversity, ecosystem function and services, and human health. Understanding the optimal management strategy, which maximizes the effectiveness is crucial. Despite an abundance of theoretical work has conducted on projecting the optimal allocation strategy, almost no empirical work has been performed to validate the theory. We first used a consumer-resource model to simulate a series of allocation fractions of controlling treatment to determine the optimal controlling strategy. Further, we conducted rigorous laboratory experiments using spatially diffusing laboratory populations of yeast to verify our mathematical results. We found consistent results that: (1) When population growth is limited by the local resource, the controlling priority should be given to the areas with higher concentration of resource; (2) When population growth is not limited by the resource concentration, the best strategy is to allocate equal amount of controlling efforts among the regions; (3) With restricted budget, it is more efficient to prioritize the controlling effects to the areas with high population abundance, otherwise, it is better to control equally among the regions. The new theory, which was tested by laboratory experiments, will reveal new opportunities for future field interventions, thereby informing subsequent biological decision-making.

Impact of State-Dependent Dispersal on Disease Prevalence

Based on a susceptible-infected-susceptible patch model, we study the influence of dispersal on the disease prevalence of an individual patch and all patches at the endemic equilibrium. Specifically, we estimate the disease prevalence of each patch and obtain a weak order-preserving result that correlated the patch reproduction number with the patch disease prevalence. Then we assume that dispersal rates of the susceptible and infected populations are proportional and derive the overall disease prevalence, or equivalently, the total infection size at no dispersal or infinite dispersal as well as the right derivative of the total infection size at no dispersal. Furthermore, for the two-patch submodel, two complete classifications of the model parameter space are given: one addressing when dispersal leads to higher or lower overall disease prevalence than no dispersal, and the other concerning how the overall disease prevalence varies with dispersal rate. Numerical simulations are performed to further investigate the effect of movement on disease prevalence.

Dynamics of Competitive Systems with Diffusion Between Source-Sink Patches

This paper considers two-species competitive systems with one-species' diffusion between patches. Each species can persist alone in the corresponding patch (a source), while the mobile species cannot survive in the other (a sink). Using the method of monotone dynamical systems, we give a rigorous analysis on persistence of the system, prove local/global stability of the equilibria and show new types of bi-stability. These results demonstrate that diffusion could lead to results reversing those without diffusion, which extend the principle of competitive exclusion: Diffusion could lead to persistence of the mobile competitor in the sink, make it reach total abundance larger than if non-diffusing and even exclude the opponent. The total abundance is shown to be a distorted function (surface) of diffusion rates, which extends both previous theory and experimental observations. A novel strategy of diffusion is deduced in which the mobile competitor could drive the opponent into extinction, and then approach the maximal abundance. Initial population density and diffusive asymmetry play a role in the competition. Our work has potential applications in biodiversity conservation and economic competition.

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.

On the Lotka-Volterra competition system with dynamical resources and density-dependent diffusion

In this paper, we consider the following Lotka-Volterra competition system with dynamical resources and density-dependent diffusion in a bounded smooth domain [Formula: see text] with homogeneous Neumann boundary conditions, where the parameters [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] ([Formula: see text]) are positive constants, m(x) is the prey's resource, and the dispersal rate function [Formula: see text] satisfies the the following hypothesis: [Formula: see text], [Formula: see text] on [Formula: see text] and [Formula: see text]. When m(x) is constant, we show that the system (*) with has a unique global classical solution when the initial datum is in functional space [Formula: see text] with [Formula: see text]. By constructing appropriate Lyapunov functionals and using LaSalle's invariant principle, we further prove that the solution of (*) converges to the co-existence steady state exponentially or competitive exclusion steady state algebraically as time tends to infinity in different parameter regimes. Our results reveal that once the resource w has temporal dynamics, two competitors may coexist in the case of weak competition regardless of their dispersal rates and initial values no matter whether there is explicit dependence in dispersal or not. When the prey's resource is spatially heterogeneous (i.e. m(x) is non-constant), we use numerical simulations to demonstrate that the striking phenomenon "slower diffuser always prevails" (cf. Dockery et al. in J Math Biol 37(1):61-83, 1998; Lou in J Differ Equ 223(2):400-426, 2006) fails to appear if the non-random dispersal strategy is employed by competing species (i.e. either [Formula: see text] or [Formula: see text] is non-constant) while it still holds true if both d(w) and [Formula: see text] are constant.

Toward a Universal Theoretical Framework to Understand Robustness and Resilience: From Cells to Systems

Research across a range of biological subdisciplines and scales, ranging from molecular to ecosystemic, provides ample evidence that living systems generally exhibit both a degree of resistance to disruption and an ability to recover following disturbance. Not only do mechanisms of robustness and resilience exist across and between systems, but those mechanisms exhibit ubiquitous and scalable commonalities in pattern and function. Mechanisms such as redundancy, plasticity, interconnectivity, and coordination of subunits appear to be crucial internal players in the determination of stability. Similarly, factors external to the system such as the amplitude, frequency, and predictability of disruptors, or the prevalence of key limiting resources, may constrain pathways of response. In the face of a rapidly changing environment, there is a pressing need to develop a common framework for describing, assessing, and predicting robustness and resilience within and across living systems.

OPTIMAL RESOURCE ALLOCATION FOR A DIFFUSIVE POPULATION MODEL

The spatial distribution of resources for diffusive populations can have a strong effect on population abundance. We investigate the optimal allocation of resources for a diffusive population. Population dynamics are represented by a parabolic partial differential equation with density-dependent growth and resources are represented through their space- and time-varying influence on the growth function. We consider both local and integral constraints on resource allocation. The goal is to maximize the abundance of the population while minimizing the cost of resource allocation. After characterizing the optimal control in terms of the population solution and the adjoint functions, we illustrate several scenarios numerically. The effects of initial and boundary conditions are important for the optimal allocation of resources.

On the effects of carrying capacity and intrinsic growth rate on single and multiple species in spatially heterogeneous environments

We first consider a diffusive logistic model of a single species in a heterogeneous environment, with two parameters, r(x) for intrinsic growth rate and K(x) for carrying capacity. When r(x) and K(x) are proportional, i.e., [Formula: see text], it is proved by Lou (J Differ Equ 223(2):400-426, 2006) that a population diffusing at any rate will reach a higher total equilibrium biomass than the population in an environment in which the same total resources are distributed homogeneously. This paper studies another case when r(x) is a constant, i.e., independent of K(x). In such case, a striking result is that for any dispersal rate, the logistic equation with spatially heterogeneous resources will always support a total population strictly smaller than the total carrying capacity at equilibrium, which is just opposite to the case [Formula: see text]. These two cases of single species models also lead to two different forms of Lotka-Volterra competition-diffusion systems. We then examine the consequences of the aforementioned difference on the two forms of competition systems. We find that the outcome of the competition in terms of the dispersal rates and spatial distributions of resources for the two forms of competition systems are again quite different. Our results indicate that in heterogeneous environments, the correlation between r(x) and K(x) has more profound impacts in population ecology than we had previously expected, at least from a mathematical point of view.

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.

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.

Carrying Capacity of a Population Diffusing in a Heterogeneous Environment

The carrying capacity of the environment for a population is one of the key concepts in ecology and it is incorporated in the growth term of reaction-diffusion equations describing populations in space. Analysis of reaction-diffusion models of populations in heterogeneous space have shown that, when the maximum growth rate and carrying capacity in a logistic growth function vary in space, conditions exist for which the total population size at equilibrium (i) exceeds the total population that which would occur in the absence of diffusion and (ii) exceeds that which would occur if the system were homogeneous and the total carrying capacity, computed as the integral over the local carrying capacities, was the same in the heterogeneous and homogeneous cases. We review here work over the past few years that has explained these apparently counter-intuitive results in terms of the way input of energy or another limiting resource (e.g., a nutrient) varies across the system. We report on both mathematical analysis and laboratory experiments confirming that total population size in a heterogeneous system with diffusion can exceed that in the system without diffusion. We further report, however, that when the resource of the population in question is explicitly modeled as a coupled variable, as in a reaction-diffusion chemostat model rather than a model with logistic growth, the total population in the heterogeneous system with diffusion cannot exceed the total population size in the corresponding homogeneous system in which the total carrying capacities are the same.

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.

Population abundance of a two-patch chemostat system with asymmetric diffusion

This paper considers a two-patch chemostat system with asymmetric diffusion, which characterizes laboratory experiments and includes exploitable nutrients. Using dynamical system theory, we demonstrate global stability of the one-patch model, and show uniform persistence of the two-patch system, which leads to existence of a stable positive equilibrium. Analysis on the equilibrium demonstrates mechanisms by which varying the asymmetric diffusion can make the total population abundance in heterogeneous environments larger than that without diffusion, even larger than that in the corresponding homogeneous environments with or without diffusion. The mechanisms are shown to be effective even in source-sink populations. A novel finding of this work is that the asymmetry combined with high diffusion intensity can reverse the predictions of symmetric diffusion in previous studies, while intermediate asymmetry is shown to be favorable but extremely large or extremely small asymmetry is unfavorable. Our results are consistent with experimental observations and provide new insights. Numerical simulations confirm and extend the results.

Dynamics of a consumer-resource reaction-diffusion model : Homogeneous versus heterogeneous environments

We study the dynamics of a consumer-resource reaction-diffusion model, proposed recently by Zhang et al. (Ecol Lett 20(9):1118-1128, 2017), in both homogeneous and heterogeneous environments. For homogeneous environments we establish the global stability of constant steady states. For heterogeneous environments we study the existence and stability of positive steady states and the persistence of time-dependent solutions. Our results illustrate that for heterogeneous environments there are some parameter regions in which the resources are only partially limited in space, a unique feature which does not occur in homogeneous environments. Such difference between homogeneous and heterogeneous environments seems to be closely connected with a recent finding by Zhang et al. (2017), which says that in consumer-resource models, homogeneously distributed resources could support higher population abundance than heterogeneously distributed resources. This is opposite to the prediction by Lou (J Differ Equ 223(2):400-426, 2006. https://doi.org/10.1016/j.jde.2005.05.010 ) for logistic-type models. For both small and high yield rates, we also show that when a consumer exists in a region with a heterogeneously distributed input of exploitable renewed limiting resources, the total population abundance at equilibrium can reach a greater abundance when it diffuses than when it does not. In contrast, such phenomenon may fail for intermediate yield rates.

The importance and adaptive value of life-history evolution for metapopulation dynamics

The spatial configuration and size of patches influence metapopulation dynamics by altering colonisation-extinction dynamics and local density dependency. This spatial forcing as determined by the metapopulation typology then imposes strong selection pressures on life-history traits, which will in turn feed back on the ecological metapopulation dynamics. Given the relevance of metapopulation persistence for biological conservation, and the potential rescuing role of evolution, a firm understanding of the relevance of these eco-evolutionary processes is essential. We here follow a systems' modelling approach to quantify the importance of spatial forcing and experimentally observed life-history evolution for metapopulation demography as quantified by (meta)population size and variability. We therefore developed an individual-based model matching an earlier experimental evolution with spider mites to perform virtual translocation and invasion experiments that would have been otherwise impossible to conduct. We show that (a) metapopulation demography is more affected by spatial forcing than by life-history evolution, but that life-history evolution contributes substantially to changes in local- and especially metapopulation-level population sizes, (b) extinction rates are minimised by evolution in classical metapopulations, and (c) evolution is optimising individual performance in metapopulations when considering the importance of more cryptic stress resistance evolution. Ecological systems' modelling opens up a promising avenue to quantify the importance of eco-evolutionary feedbacks in spatially structured populations. Metapopulation sizes are especially impacted by evolution, but its variability is mainly determined by the spatial forcing. Eco-evolutionary dynamics can increase the persistence of classical metapopulations. Conservation of genetic variation and, hence, adaptive potential is thus not only essential in the face of environmental change; it also generates putative rescuing feedbacks that impact metapopulation persistence.

Carrying capacity of a spatially-structured population: Disentangling the effects of dispersal, growth parameters, habitat heterogeneity and habitat clustering

Carrying capacity, K, is a fundamental quantity in theoretical and applied ecology. When populations are distributed over space, carrying capacity becomes a complicated function of local, global and nearby environments, dispersal rate, and the relationship between population growth parameters, e.g., r and K. Expressions for the total carrying capacity, K_total, in an n-patch model that explicitly disentangle all of these factors are currently lacking. Therefore, here we derive K_total for a linear spatial array of n habitat patches with logistic growth and strong or weak random dispersal of individuals between adjacent patches. With strong dispersal, K_total depends on the mean r and K over all patches (〈r〉 and 〈K〉), the among-patch variance in K, and the linear regression coefficient of r on K, β_r,K. Strong dispersal increases K_total only if β_{r, K} > 〈r〉/〈K〉, which requires a positive convex or negative concave association between r and K, and decreases K_total if β_{r, K} < 〈r〉/〈K〉. Alternatively, weak dispersal increases K_total only if the within-patch covariance of r and K, cov(r, K) is greater than the spatial covariance between r and K, cov(r, K_m), defined as the average covariance between r in a focal patch and K in neighboring patches. Unlike the strong dispersal limit, this condition depends not only on the magnitude of environmental heterogeneity, but explicitly on the spatial distribution of heterogeneity (i.e., habitat clustering). This work clarifies how the interaction between dispersal, habitat heterogeneity, and population growth parameters shape carrying capacity in spatial populations, with implications for species management, conservation and evolution.

Dominant Fish and Macroinvertebrate Response to Flow Changes of the Geum River in Korea

This study presents the impact of natural flow patterns on downstream aquatic species habitats in a reach of the Geum River, Korea. The study reach is a 13.4 km long, located downstream of the Yongdam Dam. To assess such an impact, this study performed physical habitat simulations. The River2D model was used for the computation of the flow field and morphology, and the Habitat Suitability Index (HSI) model for the habitat simulation. Three habitat variables—flow depth, velocity, and substrate were used. The Zacco platypus and Baetis fuscatus were selected as the target fish and benthic macro-invertebrate, respectively. Using the building block approach (BBA), the scenarios for modifying dam operations were constructed in the study reach. Scenario 1, scenario 2, and scenario 3 were proposed by using the magnitude–duration concept, base flow allocation concept, and seasonally adjusted minimum flow allocation concept, respectively. Simulation results indicated that the scenarios’ effects significantly increased by about 14.3% for the weighted usable area (WUA). In addition, the morphology change with the restoration of flood events was investigated. It was revealed that the morphology change in the physical habitat simulations further increased by about 13% for the WUA. The change of dam operations through natural flow patterns is more advantageous to aquatic species.

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.