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• W3096463338 abstract "The logistic equation, with carrying capacity, K, and growth rate, r, has traditionally been used to describe dynamics of ecological populations.Experiments confirmed the prediction that dispersal could increase metapopulation abundance in heterogeneous environments, whereas they rejected the prediction that heterogeneous environments support a larger metapopulation abundance than homogeneous environments with the same sum over K values.Consumer-resource models, which explicitly consider the resource inputs and time scales of feedbacks between organisms and their resource, agree consistently with experimental results, suggesting they are more appropriate for describing populations in space.The theoretical results have important management implications on wildlife, such as the important role of dispersal, or habitat connectivity, in influencing population abundance in patchy environments. 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. 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. 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.Chapman E.J. Byron C. The flexible application of carrying capacity in ecology.Glob. Ecol. Consev. 2018; 13e00365Google Scholar,2.Del Monte-Luna P. et al.The carrying capacity of ecosystems.Glob. Ecol. Biogeogr. 2004; 13: 485-495Crossref Scopus (121) Google Scholar]. 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.Sayre N.F. The genesis, history, and limits of carrying capacity.Ann. Assoc. Am. Geogr. 2008; 98: 120-134Crossref Scopus (148) Google Scholar]. 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 point – the 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.Dhondt A.A. Carrying-capacity - a confusing concept.Acta. Oecol-Oec. Gen. 1988; 9: 337-346Google Scholar] documented the use of both Leopold’s and Errington’s definitions by other wildlife biologists and ecologists, noting, for example, that Dasmann [5.Dasmann R. Wildlife Management. John Wiley & Sons, 1964Google Scholar] carried distinctions further by introducing four different definitions related to carrying capacity: subsistence density, optimum density, security density, and tolerance density. Dhondt [4.Dhondt A.A. Carrying-capacity - a confusing concept.Acta. Oecol-Oec. Gen. 1988; 9: 337-346Google Scholar] 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.Odum E. Fundamentals of Ecology. W. B. Saunders, 1953Google Scholar] 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:dNdt=r1−NKN[1] 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 greater values, and hence is regulated around K. Odum [6.Odum E. Fundamentals of Ecology. W. B. Saunders, 1953Google Scholar] assumed that the value of K depends on the limiting resource of the population. It appears in the logistic equation as a constant value, although Odum acknowledged that it could vary as the environment changes. Although the use of the logistic equation with carrying capacity K to describe ecological populations might seem to have ended the confusion over its meaning, both the logistic and the carrying capacity concept it formalizes have continued to be criticized from the empirical side. While logistic population growth is often observed in laboratory studies of microbial populations, Botkin [7.Botkin D.B. Discordant Harmonies: a New Ecology for the Twenty-First Century. Oxford University Press, 1990Google Scholar] noted that it has never been observed in nature, and many ecologists have embraced more qualitative concepts of population regulation, such as density-vague regulation [8.Strong D.R. Density-vague population change.Trends Ecol. Evol. 1986; 1: 39-42Abstract Full Text PDF PubMed Scopus (116) Google Scholar]. Nevertheless, the Pearl–Verhulst form of the logistic equation, with carrying capacity K being an equilibrium determined by resources, has been standard in ecology textbooks since the 1970s [4.Dhondt A.A. Carrying-capacity - a confusing concept.Acta. Oecol-Oec. Gen. 1988; 9: 337-346Google Scholar], and has a central place in theoretical ecology. For that reason, we focus on carrying capacity based on Equation 1 and show that there are serious complexities related to this model when extended to heterogeneous space. Predicting population dynamics at a landscape or regional level is a paramount problem in ecology, especially under the changing environment and human disturbance [9.Wu J. et al.Multiscale analysis of landscape heterogeneity: scale variance and pattern metrics.Cartogr. Geogr. Inf. Sci. 2000; 6: 6-19Google Scholar,10.van de Koppel J. et al.Spatial heterogeneity and irreversible vegetation change in semiarid grazing systems.Am. Nat. 2002; 159: 209-218Crossref PubMed Scopus (141) Google Scholar]. It has been said that ‘the emerging discipline of landscape ecology must serve as the foundation for effective biodiversity conservation programming’ [11.Harris L.D. et al.Landscape processes and their significance to biodiversity conservation.Population dynamics in ecological space and time. 1. 1996: 319-347Google Scholar]. For example, forest fragmentation is creating barriers that will hinder, or respectively, slow, dispersal capacity [12.Hansen M.C. et al.High-resolution global maps of 21st-century forest cover change.Science. 2013; 342: 850-853Crossref PubMed Scopus (5806) Google Scholar, 13.Taubert F. et al.Global patterns of tropical forest fragmentation.Nature. 2018; 554: 519-522Crossref PubMed Scopus (253) Google Scholar, 14.Arroyo-Rodríguez V. et al.Designing optimal human-modified landscapes for forest biodiversity conservation.Ecol. Lett. 2020; 23: 1404-1420Crossref PubMed Scopus (132) Google Scholar]. As dispersal rates decline with habitat fragmentation, understanding the combined effect of heterogeneity and dispersal on the attainable population size will be essential for fostering population persistence of desired species [15.Robertson E.P. et al.Isolating the roles of movement and reproduction on effective connectivity alters conservation priorities for an endangered bird.Proc. Natl. Acad. Sci. 2018; 115: 8591-8596Crossref PubMed Scopus (24) Google Scholar, 16.Fobert E.K. et al.Dispersal and population connectivity are phenotype dependent in a marine metapopulation.Proc. R. Soc. B. 2019; 286: 20191104Crossref PubMed Scopus (12) Google Scholar, 17.Cornelius C. et al.Habitat fragmentation drives inter-population variation in dispersal behavior in a neotropical rainforest bird.Perspect. Ecol. Conserv. 2017; 15: 3-9Google Scholar] or limiting invasive species in these environments [18.Milt A.W. et al.Minimizing opportunity costs to aquatic connectivity restoration while controlling an invasive species.Conserv. Biol. 2018; 32: 894-904Crossref PubMed Scopus (31) Google Scholar]. Additionally, general patterns of species range shifts (to higher elevations and latitudes) are anticipated with warming temperatures [19.Lenoir J. Svenning J.C. Climate-related range shifts–a global multidimensional synthesis and new research directions.Ecography. 2015; 38: 15-28Crossref Scopus (485) Google Scholar, 20.Parmesan C. Yohe G. A globally coherent fingerprint of climate change impacts across natural systems.Nature. 2003; 421: 37Crossref PubMed Scopus (7151) Google Scholar, 21.Zhang X. et al.Ecological contingency in species shifts: downslope shifts of woody species under warming climate and land-use change.Environ. Res. Lett. 2019; 14: 114033Crossref Scopus (10) Google Scholar], so projecting shifting population dynamics along environmental gradients is essential. Much previous work has focused on understanding the forces that determine the size of natural populations, using the assumption of a spatially homogeneous (environmentally uniform) system [22.Steudel B. et al.Biodiversity effects on ecosystem functioning change along environmental stress gradients.Ecol. Lett. 2012; 15: 1397-1405Crossref PubMed Scopus (118) Google Scholar, 23.Sibly R.M. et al.How environmental stress affects density dependence and carrying capacity in a marine copepod.J. Appl. Ecol. 2000; 37: 388-397Crossref Scopus (64) Google Scholar, 24.Liu F. et al.Effects of six selected antibiotics on plant growth and soil microbial and enzymatic activities.Environ. Pollut. 2009; 157: 1636-1642Crossref PubMed Scopus (344) Google Scholar]. With that assumption, the population dynamics can be described by the logistic Equation 1 in which r and K have constant values across the spatial environment [25.Groffman P.M. et al.Earthworms increase soil microbial biomass carrying capacity and nitrogen retention in northern hardwood forests.Soil Biol. Biochem. 2015; 87: 51-58Crossref Scopus (56) Google Scholar]. However, almost all environments are heterogeneous, with habitat quality varying either continuously or occurring as discrete, disjointed patches [26.Andrewartha H.G. Birch L.C. The distribution and abundance of animals. University of Chicago press, 1954Google Scholar]. Habitat heterogeneity has increased due to fragmentation by human activity and the conversion of natural ecosystems into agricultural or urban areas [27.Fahrig L. Habitat fragmentation: A long and tangled tale.Glob. Ecol. Biogeogr. 2019; 28: 33-41Crossref Scopus (62) Google Scholar,28.Cote J. et al.Evolution of dispersal strategies and dispersal syndromes in fragmented landscapes.Ecography. 2017; 40: 56-73Crossref Scopus (110) Google Scholar], and the effect of environmental change [29.Kling M.M. et al.Multiple axes of ecological vulnerability to climate change.Glob. Chang. Biol. 2020; 26: 2798-2813Crossref PubMed Scopus (19) Google Scholar,30.Kays R. et al.Terrestrial animal tracking as an eye on life and planet.Science. 2015; 348aaa2478Crossref PubMed Scopus (721) Google Scholar]. Such heterogeneous environments consist of a variety of local population growth rates and carrying capacities; hence, varying r and K. Importantly, a growing body of research has highlighted that spatial heterogeneity may be as important as the spatial mean for assessing environmental impacts on populations [31.Rudgers J.A. et al.Climate sensitivity functions and net primary production: a framework for incorporating climate mean and variability.Ecology. 2018; 99: 576-582Crossref PubMed Scopus (57) Google Scholar]. Including spatial heterogeneity, in combination with population dispersal, can alter and even reverse certain population-level predictions based only on the mean [32.Zimmermann N.E. et al.Climatic extremes improve predictions of spatial patterns of tree species.Proc. Natl. Acad. Sci. 2009; 106: 19723-19728Crossref PubMed Scopus (274) Google Scholar, 33.Kroeker K.J. et al.Meta-analysis reveals negative yet variable effects of ocean acidification on marine organisms.Ecol. Lett. 2010; 13: 1419-1434Crossref PubMed Scopus (1082) Google Scholar, 34.Uriarte M. Menge D. Variation between individuals fosters regional species coexistence.Ecol. Lett. 2018; 21: 1496-1504Crossref PubMed Scopus (22) Google Scholar]. As a result, understanding the role of environmental heterogeneity in affecting natural population dynamics is crucial [35.Anderson K.E. et al.Scaling population responses to spatial environmental variability in advection-dominated systems.Ecol. Lett. 2005; 8: 933-943Crossref Scopus (40) Google Scholar]. Dispersal, as a fundamental ecological process, plays an important role in shaping population dynamics [36.Jønsson K.A. et al.Tracking animal dispersal: from individual movement to community assembly and global range dynamics.Trends Ecol. Evol. 2016; 31: 204-214Abstract Full Text Full Text PDF PubMed Scopus (40) Google Scholar,37.Jacob S. et al.Variability in dispersal syndromes is a key driver of metapopulation dynamics in experimental microcosms.Am. Nat. 2019; 194: 613-626Crossref PubMed Scopus (15) Google Scholar], community diversity and composition [38.Albright M.B. Martiny J.B. Dispersal alters bacterial diversity and composition in a natural community.ISME J. 2018; 12: 296-299Crossref PubMed Scopus (41) Google Scholar], and ecosystem functioning [39.Little C.J. et al.Dispersal syndromes can impact ecosystem functioning in spatially structured freshwater populations.Biol. Lett. 2019; 15: 20180865Crossref PubMed Scopus (9) Google Scholar]. All organisms, including microbes, disperse within their ranges and dispersal can occur in different ways. Two classic dispersal patterns are the ideal free distribution (IFD) and random dispersal. According to the IFD, if other factors besides carrying capacity of a habitat patch, such as predator density, can be ignored, and animals are free to move, they will continue to move until further movement cannot improve their fitness. That is the point when the maximum population abundance attained at equilibrium is equal to the sum of the carrying capacities on the individual patches, or is the integral of continuously varying carrying capacity over the whole area. In the alternative to the IFD, random movement, if there is no directional bias, is called symmetric dispersal, of which diffusion is a special case where the dispersal proceeds by local steps. In the absence of knowledge of how individuals of a population actually move, symmetric dispersal is assumed in the preponderance of ecological models with dispersal [40.Levin S.A. et al.Theories of simplification and scaling of spatially distributed processes. Princeton University Press, Princeton, NJ1997: 271-295Google Scholar], such as reaction–diffusion models [41.Okubo A. Diffusion and Ecological Problems: Mathematical Models. Springer, 1980Google Scholar]. Therefore, we use this assumption with the logistic model (Equation 1) to show how random movement complicates the total size that a population can reach in a heterogeneous environment. Note, however, that the results here can be extended to asymmetric dispersal (Box 1).Box 1Mathematical Explanation of Prediction 1Extending the logistic model to heterogeneous space can be done by including logistic growth as the reaction term in a reaction-diffusion model, or by modeling space as a collection of discrete patches, among which populations can disperse. To demonstrate the effects of population movement, consider the latter approach, simplified to two patches and described by an equation for each patch, in which population dispersal rates are the same in both directions and carrying capacities and intrinsic growth rates differ on the two patches;dN1dt=r11−N1K1N1−DN1+DN2;[Ia] dN2dt=r21−N2K2N2−DN2+DN1.[Ib] where D is the symmetric dispersal between the two patches. As found by Freedman and Waltman [87.Freedman H.I. Waltman P. Mathematical-models of population interactions with dispersal .1. Stability of 2 habitats with and without a predator.SIAM J. Appl. Math. 1977; 32: 631-648Crossref Google Scholar] and Holt [88.Holt R.D. Population dynamics in two-patch environments: some anomalous consequences of an optimal habitat distribution.Theor. Popul. Biol. 1985; 28: 181-208Crossref Scopus (552) Google Scholar] and stated in corrected form by Arditi [89.Arditi R. et al.Asymmetric dispersal in the multi-patch logistic equation.Theor. Popul. Biol. 2018; 120: 11-15Crossref PubMed Scopus (26) Google Scholar], in the limit of large dispersal rate (D → ∞), which corresponds to diffusion occurring on a much faster time scale than population change, the equilibrium abundance that can be reached by the population in this system isTotal population=K1+K2+K1−K2r1K2−r2K1r1K2+r2K1,[II] This has the implication that the intrinsic growth rates on the two patches, r1 and r2, have an effect on the total equilibrium population when the population disperses symmetrically between the two patches. If K1 > K2 and r1/K1 > r2/K2, the population in a heterogeneous two-patch system could reach a total equilibrium population size greater than the sum of the carrying capacities of the two patches (equivalent to the population not diffusing) (Prediction 1). Conversely, if r1/K1 < r2/K2, the total equilibrium population is less than K1 + K2. An analogous result was found in continuous space by Lou [90.Lou Y. On the effects of migration and spatial heterogeneity on single and multiple species.J. Differ. Equations. 2006; 223: 400-426Crossref Scopus (171) Google Scholar], using a partial differential reaction diffusion equation. Arditi et al. [68.Arditi R. et al.Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation.Theor. Popul. Biol. 2015; 106: 45-59Crossref PubMed Scopus (48) Google Scholar] showed that making the dispersal asymmetric does not change the results qualitatively. Although these results are for D → ∞, simulations show that the total population deviates from K1 + K2 for smaller values of D as well, though approaching K1 + K2 as D → ∞. Therefore, dispersal rate, along with the intrinsic growth rate, influence the equilibrium total population size in a heterogeneous environment.Thus, the total size of a diffusing population can differ from that of the sum of the local carrying capacities, if the intrinsic growth rates also differ. Prediction 1 was confirmed experimentally [47.Zhang B. et al.Carrying capacity in a heterogeneous environment with habitat connectivity.Ecol. Lett. 2017; 20: 1118-1128Crossref PubMed Scopus (52) Google Scholar]. Extending the logistic model to heterogeneous space can be done by including logistic growth as the reaction term in a reaction-diffusion model, or by modeling space as a collection of discrete patches, among which populations can disperse. To demonstrate the effects of population movement, consider the latter approach, simplified to two patches and described by an equation for each patch, in which population dispersal rates are the same in both directions and carrying capacities and intrinsic growth rates differ on the two patches;dN1dt=r11−N1K1N1−DN1+DN2;[Ia] dN2dt=r21−N2K2N2−DN2+DN1.[Ib] where D is the symmetric dispersal between the two patches. As found by Freedman and Waltman [87.Freedman H.I. Waltman P. Mathematical-models of population interactions with dispersal .1. Stability of 2 habitats with and without a predator.SIAM J. Appl. Math. 1977; 32: 631-648Crossref Google Scholar] and Holt [88.Holt R.D. Population dynamics in two-patch environments: some anomalous consequences of an optimal habitat distribution.Theor. Popul. Biol. 1985; 28: 181-208Crossref Scopus (552) Google Scholar] and stated in corrected form by Arditi [89.Arditi R. et al.Asymmetric dispersal in the multi-patch logistic equation.Theor. Popul. Biol. 2018; 120: 11-15Crossref PubMed Scopus (26) Google Scholar], in the limit of large dispersal rate (D → ∞), which corresponds to diffusion occurring on a much faster time scale than population change, the equilibrium abundance that can be reached by the population in this system isTotal population=K1+K2+K1−K2r1K2−r2K1r1K2+r2K1,[II] This has the implication that the intrinsic growth rates on the two patches, r1 and r2, have an effect on the total equilibrium population when the population disperses symmetrically between the two patches. If K1 > K2 and r1/K1 > r2/K2, the population in a heterogeneous two-patch system could reach a total equilibrium population size greater than the sum of the carrying capacities of the two patches (equivalent to the population not diffusing) (Prediction 1). Conversely, if r1/K1 < r2/K2, the total equilibrium population is less than K1 + K2. An analogous result was found in continuous space by Lou [90.Lou Y. On the effects of migration and spatial heterogeneity on single and multiple species.J. Differ. Equations. 2006; 223: 400-426Crossref Scopus (171) Google Scholar], using a partial differential reaction diffusion equation. Arditi et al. [68.Arditi R. et al.Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation.Theor. Popul. Biol. 2015; 106: 45-59Crossref PubMed Scopus (48) Google Scholar] showed that making the dispersal asymmetric does not change the results qualitatively. Although these results are for D → ∞, simulations show that the total population deviates from K1 + K2 for smaller values of D as well, though approaching K1 + K2 as D → ∞. Therefore, dispersal rate, along with the intrinsic growth rate, influence the equilibrium total population size in a heterogeneous environment. Thus, the total size of a diffusing population can differ from that of the sum of the local carrying capacities, if the intrinsic growth rates also differ. Prediction 1 was confirmed experimentally [47.Zhang B. et al.Carrying capacity in a heterogeneous environment with habitat connectivity.Ecol. Lett. 2017; 20: 1118-1128Crossref PubMed Scopus (52) Google Scholar]. It might seem that determining the total carrying capacity of a randomly dispersing population in heterogeneous environments would be a straightforward summation of each carrying capacity, Ki, over all the local habitats. However, when mathematical ecologists analyzed this seemingly simple extension of the logistic equation beyond its nonspatial form to a population dispersing randomly in heterogeneous space, they found the following surprising mathematical results (Box 1, Box 2), which we will refer to as predictions to be tested empirically.Box 2Mathematical Explanation of Prediction 2Prediction 2 follows as a special case of Equation II in Box 1. It states that a diffusing population can reach a greater equilibrium population size in the heterogeneous case than in the homogeneous case when the two carrying capacities differ but sum to the same in the two cases. That is, in the homogeneous case, K1 = K2 = K, while in the heterogeneous case K1 = K + ∆, K2 = K − ∆, where is a deviation in K. From Equation II in Box 1 it can be found thatTotal population=2K+2∆Kr1−r2+∆r1+r2Kr1+r2+∆r1−r2,[I] which exceeds 2K when r1 > r2. Prediction 2 is mathematically correct, but it does not agree with the experimental results. The reason is that the experiment differed in an important way from the model based on coupled logistic equations. In the experiment, input of a limiting resource/nutrient, was provided to the yeast populations. The input was regulated such that the sum of the inputs to the two patches was same in the homogeneous and heterogeneous cases, although the amounts going to each patch differed in the heterogeneous case. It can be shown that Equation I for r1 > r2 violates the equality of resource inputs in the heterogeneous and homogeneous cases.To show this, note that in terms of the coupled logistic equations, the rate of input of resource at equilibrium is r1N1∗ + r2N2∗ = r1K1 + r2K2, which can be seen by adding Equations [Ia], [Ib] in Box 1 at equilibrium. Now, comparing the heterogeneous case, r1(K + ∆) + r2(K − ∆) with the homogeneous case, r1K + r2K, it can be seen that the total input cannot be the same for the two cases unless r1 = r2. Therefore, it is impossible for the total population to exceed K1 + K2 while keeping total input constant. It can further be shown that, if total resource input is kept the same in the heterogeneous and homogeneous cases, the population abundance will always be the same or larger in the homogeneous than the heterogeneous case, as found in the experiment. The opposite can occur if r1 and r2 are different. This is a special case of a more general result (Theorem 11 in [91.Guo Q. et al.On the effects of carrying capacity and intrinsic growth rate on single and multiple species in spatially heterogeneous environments.J. Math. Biol. 2020; 81: 403-433Crossref PubMed Scopus (7) Google Scholar]; see also [53.Wang Y.S. DeAngelis D.L. Energetic constraints and the paradox of a diffusing population in a heterogeneous environment.Theor. Popul. Biol. 2019; 125: 30-37Crossref PubMed Scopus (19) Google Scholar]). If a consumer–resource model is instead of the logistic model, the equality of input in the heterogeneous and homogeneous cases is satisfied in a more straightforward way (Box 3). Prediction 2 follows as a special case of Equation II in Box 1. It states that a diffusing population can reach a greater equilibrium population size in the heterogeneous case than in the homogeneous case when the two carrying capacities differ but sum to the same in the two cases. That is, in the homogeneous case, K1 = K2 = K, while in the heterogeneous case K1 = K + ∆, K2 = K − ∆, where is a deviation in K. From Equation II in Box 1 it can be found thatTotal population=2K+2∆Kr1−r2+∆r1+r2Kr1+r2+∆r1−r2,[I] which exceeds 2K when r1 > r2. Prediction 2 is mathematically correct, but it does not agree with the experimental results. The reason is that the experiment differed in an important way from the model based on coupled logistic equations. In the experiment, input of a limiting resource/nutrient, was provided to the yeast populations. The input was regulated such that the sum of the inputs to the two patches was same in the homogeneous and heterogeneous cases, although the amounts going to each patch differed in the heterogeneous case. It can be shown that Equation I for r1 > r2 violates the equality of resource inputs in the heterogeneous and homogeneous cases. To show this, note that in terms of the coupled logistic equations, the rate of input of resource at equilibrium is r1N1∗ + r2N2∗ = r1K1 + r2K2, which can be seen by adding Equations [Ia], [Ib] in Box 1 at equilibrium. Now, comparing the heterogeneous case, r1(K + ∆) + r2(K − ∆) with the homogeneous case, r1K + r2K, it can be seen that the total input cannot be the same for the two cases unless r1 = r2. Therefore, it is impossible for the total population to exceed K1 + K2 while keeping total input constant. It can further be shown that, if total resource input is kept the same in the heterogeneous and homogeneous cases, the population abundance will always be the same or larger in the homogeneous than the heterogeneous case, as found in the experiment. The opposite can occur if r1 and r2 are different. This is a special case of a more general result (Theorem 11 in [91.Guo Q. et al.On the effects of carrying capacity and intrinsic growth rate on single and multiple species in spatially heterogeneous environments.J. Math. Biol. 2020; 81: 403-433Crossref PubMed Scopus (7) Google Scholar]; see also [53.Wang Y.S. DeAngelis D.L. Energetic constraints and the paradox of a diffusing population in a heterogeneous environment.Theor. Popul. Biol. 2019; 125: 30-37Crossref PubMed Scopus (19) Google Scholar]). If a consumer–resource model is instead of the logistic model, the equality of input in the heterogeneous and homogeneous cases is satisfied in a more straightforward way (Box 3). Prediction 1: in a heterogeneous system, where both growth rate and carrying capacity vary spatially, total population abundance of a dispersing population may exceed total population abundance of a nondispersing population. Prediction 2: the total population abundance of a dispersing population in a heterogeneous environment can be higher than in the homogeneous environment, even if the sum of all local carrying capacities is the same for both cases. This is a special case of Equation I in Box 1, which is the" @default.
• W3096463338 created "2020-11-09" @default.
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• W3096463338 date "2021-02-01" @default.
• W3096463338 modified "2023-10-09" @default.
• W3096463338 title "Carrying Capacity of Spatially Distributed Metapopulations" @default.
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• W3096463338 doi "https://doi.org/10.1016/j.tree.2020.10.007" @default.
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