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.

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.

Carrying Capacity of Spatially Distributed Metapopulations.

Carrying capacity is a key concept in ecology. A body of theory, based on the logistic equation, has extended predictions of carrying capacity to spatially distributed, dispersing populations. However, this theory has only recently been tested empirically. The experimental results disagree with some theoretical predictions of when they are extended to a population dispersing randomly in a two-patch system. However, they are consistent with a mechanistic model of consumption on an exploitable resource (consumer-resource model). We argue that carrying capacity, defined as the total equilibrium population, is not a fundamental property of ecological systems, at least in the context of spatial heterogeneity. Instead, it is an emergent property that depends on the population's intrinsic growth and dispersal rates.

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.

Effect of Stressors on the Carrying Capacity of Spatially Distributed Metapopulations.

Stressors such as antibiotics, herbicides, and pollutants are becoming increasingly common in the environment. The effects of stressors on populations are typically studied in homogeneous, nonspatial settings. However, most populations in nature are spatially distributed over environmentally heterogeneous landscapes with spatially restricted dispersal. Little is known about the effects of stressors in these more realistic settings. Here, we combine laboratory experiments with novel mathematical theory to rigorously investigate how a stressor's physiological effect and spatial distribution interact with dispersal to influence population dynamics. We prove mathematically that if a stressor increases the death rate and/or simultaneously decreases the population growth rate and yield, a homogeneous distribution of the stressor leads to a lower total population size than if the same amount of the stressor was heterogeneously distributed. We experimentally test this prediction on spatially distributed populations of budding yeast (Saccharomyces cerevisiae). We find that the antibiotic cycloheximide increases the yeast death rate but reduces the growth rate and yield. Consistent with our mathematical predictions, we observe that a homogeneous spatial distribution of cycloheximide minimizes the total equilibrium size of experimental metapopulations, with the magnitude of the effect depending predictably on the dispersal rate and the geographic pattern of antibiotic heterogeneity. Our study has implications for assessing the population risk posed by pollutants, antibiotics, and global change and for the rational design of strategies for employing toxins to control pathogens and pests.

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.

Carrying capacity in a heterogeneous environment with habitat connectivity.

A large body of theory predicts that populations diffusing in heterogeneous environments reach higher total size than if non-diffusing, and, paradoxically, higher size than in a corresponding homogeneous environment. However, this theory and its assumptions have not been rigorously tested. Here, we extended previous theory to include exploitable resources, proving qualitatively novel results, which we tested experimentally using spatially diffusing laboratory populations of yeast. Consistent with previous theory, we predicted and experimentally observed that spatial diffusion increased total equilibrium population abundance in heterogeneous environments, with the effect size depending on the relationship between r and K. Refuting previous theory, however, we discovered that homogeneously distributed resources support higher total carrying capacity than heterogeneously distributed resources, even with species diffusion. Our results provide rigorous experimental tests of new and old theory, demonstrating how the traditional notion of carrying capacity is ambiguous for populations diffusing in spatially heterogeneous environments.

Dispersal and spatial heterogeneity: single species.

A recent result for a reaction-diffusion equation is that a population diffusing at any rate in an environment in which resources vary spatially will reach a higher total equilibrium biomass than the population in an environment in which the same total resources are distributed homogeneously. This has so far been proven by Lou for the case in which the reaction term has only one parameter, m(x), varying with spatial location x, which serves as both the intrinsic growth rate coefficient and carrying capacity of the population. However, this striking result seems rather limited when applies to real populations. In order to make the model more relevant for ecologists, we consider a logistic reaction term, with two parameters, r (x) for intrinsic growth rate, and K(x) for carrying capacity. When r (x) and K(x) are proportional, the logistic equation takes a particularly simple form, and the earlier result still holds. In this paper we have established the result for the more general case of a positive correlation between r (x) and K(x) when dispersal rate is small. We review natural and laboratory systems to which these results are relevant and discuss the implications of the results to population theory and conservation ecology.

Effects of dispersal on total biomass in a patchy, heterogeneous system: Analysis and experiment.

An intriguing recent result from mathematics is that a population diffusing at an intermediate rate in an environment in which resources vary spatially will reach a higher total equilibrium biomass than the population in an environment in which the same total resources are distributed homogeneously. We extended the current mathematical theory to apply to logistic growth and also showed that the result applies to patchy systems with dispersal among patches, both for continuous and discrete time. This allowed us to make specific predictions, through simulations, concerning the biomass dynamics, which were verified by a laboratory experiment. The experiment was a study of biomass growth of duckweed (Lemna minor Linn.), where the resources (nutrients added to water) were distributed homogeneously among a discrete series of water-filled containers in one treatment, and distributed heterogeneously in another treatment. The experimental results showed that total biomass peaked at an intermediate, relatively low, diffusion rate, higher than the total carrying capacity of the system and agreeing with the simulation model. The implications of the experiment to dynamics of source, sink, and pseudo-sink dynamics are discussed.