Carrying Capacity of Spatially Distributed Metapopulations

Abstract

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 twopatch 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. A Brief History of Carrying Capacitya Fundamental but Confusing Concept Carrying capacity (commonly defined as the upper limit on the size of the population), has been one of the most important concepts in ecology for the last century. As such, it has been broadly used, from cell populations up to that of ecological communities at landscape and ecosystem levels [1,2]. Wildlife biologists introduced the term in the early 20th century as a tool in wildlife management. Aldo Leopold viewed carrying capacity as the population density reached at a particular site, determined by both the resources available and intraspecific competition [3]. Although, in Leopold's view, the realized carrying capacity was usually less than the maximum population density reached under optimum conditions. He called this the saturation pointthe maximum density that could be achieved by careful habitat manipulation. Leopold's definition was by no means the only one held among ecologists. For instance, Paul Errington viewed carrying capacity as the maximum size that a population could reach if there was refuge from predation available. Dhondt [4] documented the use of both Leopold's and Errington's definitions by other wildlife biologists and ecologists, noting, for example, that Dasmann [5] carried distinctions further by introducing four different definitions related to carrying capacity: subsistence density, optimum density, security density, and tolerance density. Dhondt [4] reviewed the multiplicity of views of carrying capacity and called it confusing, concluding that, at least for wildlife biology, the term should be avoided. However, carrying capacity had already entered the mainstream of ecology. Odum [6] took the first step of giving carrying capacity a formal mathematical meaning. He defined it as the constant K in the Pearl-Verhulst form of the logistic population equation: dN dt ¼ r 1− N K N ½1 where N is population size and r is the intrinsic population growth rate. This equation defines the carrying capacity as the equilibrium point that a population would always approach from lesser or

Risks to Fish Populations in Dryland Rivers From the Combined Threats of Drought and Instream Barriers

“…Local populations are extirpated by drought as waterholes dry or become inhospitable, and connectivity during flow events, permitting dispersal, allows local population recolonisation, so maintaining metapopulation viability. However, as metapopulations can survive only if the recolonisation rate of local populations is at least equal to their extinction rate (Datry et al, 2017), dispersal plays a fundamental role in maintaining populations in patchy environments (Zhang et al, 2021) such as fish in dryland rivers. This implies that if the extirpation of local fish populations increases, as predicted under the influences of drought, climate change and other stressors to waterhole refuge persistence and function outlined above, metapopulations would remain viable only if there is an equivalent rate of recolonisation maintained.…”

Individual heterogeneity and its importance for metapopulation dynamics

“…Disrupting spatial phenotypic heterogeneity through randomization decreases in consequence both population sizes and their variability across the connectedness gradient. Or, put alternatively, the maintenance of this phenotypic organization leads to more (asynchronous) fluctuating metapopulations of larger size, conditions known to rescue metapopulations at intermediate levels (Wang and Loreau 2014; Wang, Haegeman, and Loreau 2015), but that can also potentially lead the metapopulations to be more vulnerable towards local extinctions if said fluctuations become too pronounced or erratic (Abbott 2011; Zhang, DeAngelis, and Ni 2021).…”

“…Connectedness impacts local and regional fluctuations in population sizes, as well as the synchronicity of these changes within the entire metapopulation (De Roissart, Wang, and Bonte 2015; Wang, Haegeman, and Loreau 2015). Theory predicts that lower effective dispersal between disconnected patches raises both local and regional temporal variabilities in population size thereby increasing local patch extinctions while decreasing rescue from immigration (Abbott 2011; Wang, Haegeman, and Loreau 2015; Zhang, DeAngelis, and Ni 2021). On the other hand, a more connected network will also show a higher level of spatial synchronicity (Gouhier, Guichard, and Gonzalez 2010), which makes the whole system also more vulnerable to global extinction because of declines in rescuing potential when all local populations decrease at the same time (Lloyd and May 1999; Sabelis et al 2005).…”

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

“…Hence, investigating system response and tolerance to external stressors at a higher scale requires a solid mechanistic understanding of the diversity of responses to environmental change among species that contribute to the same ecosystem function (Elmqvist et al, 2003;Walker et al, 2004) and the consumer-resource model can be an appropriate tool to scale up individual resilience to the population, community levels. More importantly, most species in nature disperse across the landscape, which is a critical process of response diversity (Elmqvist et al, 2003) that scales up population dynamics to the metapopulation level (Wang et al, 2015;Zhang et al, 2020b). This consumer-resource model is also an appropriate model system to project the role of dispersal, especially in a heterogeneous environment.…”