Interconnections accelerate collapse in a socio-ecological metapopulation

Resource over-exploitation can have profound effects on both ecosystems and the human populations residing in them. Models of population growth based on a depletable resources have been studied previously, but relatively few consider metapopulation effects. Here we analyze a socio-ecological metapopulation model where resources grow logistically on each patch. Each population harvests resources on its own patch to support population growth, but can also harvest resources from other patches when their own patch resources become scarce. We find that allowing populations to harvest from other patches significantly accelerates collapse and also increases the parameter regime for which collapse occurs, compared to a model where populations are not able to harvest resources from other patches. As the number of patches in the metapopulation increases, collapse is more sudden, more severe, and occurs sooner. These effects also persist under scenarios of asymmetry and inequality between patches. We conclude that metapopulation effects in socio-ecological systems can be both helpful and harmful and therefore require urgent study.


Background
sustainably on the planet, and finds that three of these boundaries have already been transgressed [9].
More recently, researchers have introduced frameworks to describe the process by which over-25 consumption occurs. For instance, the red/green sustainability framework describes how populations 26 become increasingly disconnected from their impacts as they urbanize [10]. In a 'green' phase, popula-27 tions are highly dependent on their local environment for their subsistence, and therefore feedback from 28 environmental implications of human activity are quick to down-regulate the human activity. However, as 29 populations develop and draw their resources from a global resource pool, their economic activities cause 30 environmental impacts that are no longer felt by them but rather by geographically distant populations, 31 weakening the short-term coupling between humans and their environment. This process is captured 32 by, for instance, the linkages between local deforestation and high pressure for international agricultural 33 exports [11], and the large dependence seafood markets in Japan, the United States, and the European 34 Union on foreign sources [12]. 35 Although logistic growth models and its variations are most widely used in ecology, the application of 36 population growth models to resource-limited human populations has also received attention, perhaps on 37 account of our growing awareness of the possible ramifications of resource over-exploitation, especially in 38 the face of environmental change [13]. Mathematical models have been used to study phenomena such as 39 human population collapse in a model with resource dynamics [14] and conflict among metapopulations 40 arising over common resources [15,16]. Models have also been used to study historical human population 41 collapses such as in the people of Easter Island [17,18,19,20,21,22,23,24,25,26,27], the Kayenta that coupling human populations together through exchange of resources, migration and technology can stabilize the entire metapopulation [27]. 53 In this paper we build on a previous single population model [18] to create a metapopulation model 54 of resource-limited growth that captures mechanisms similar to the red/green sustainability framework.

55
Populations grow logistically by exploiting a depletable resource that obeys a resource dynamic. Local 56 populations can take resources not only from their own patch but also from other patches, when resources 57 in their own patch become scarce. Our research objective was to determine whether the metapopulation 58 collapses faster or more often when patches are allowed to harvest resources from other patches. We 59 describe the model in the next section, followed by Results and a Discussion section.

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We build on a previous single-population model analyzed by Basener and Ross [18] who formulated a 62 model whereby the population grows logistically to a carrying capacity that is proportional to the resource 63 level. A second equation describes the logistic growth of resources to a separate carrying capacity, minus 64 harvesting. We develop both two-patch and ten-patch versions of our model. 66 In the two-patch model, patch 1 has population size P 1 and resource level R 1 , and patch 2 has population 67 size P 2 and resource level R 2 :

Two-patch model
of resources that patch 1 takes from patch 2, and similarly for b 2 . In this model, the carrying capacity of 72 the human populations is determined by how much resource is available to support them, either from their 73 own patch or taken from the other patch. When b 1 = b 2 = 0 we recover the original model by Basener 74 and Ross [18]. 75 We set b 1 = b 1 (R 1 , P 1 ) and assume that patch 1 will attempt to harvest more resources from patch 76 2 when the resources from patch 1 are not enough to support the patch 1 population. Similarly, b 2 = 77 b 2 (R 2 , P 2 ). These functions take the form These are sigmoidal functions where the rate at which patch 1 harvests from patch 2 is higher when P 1 /R 1 79 is higher, and vice versa, where β 1 > 0 controls the location of the mid-point of the sigmoid, and where 80 γ 1 > 0 controls how steep the curve is. Parameters β 2 and γ 2 are similarly defined.

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Ten-patch model 82 We also analyzed a version of the model where ten patches are interconnected and can take resources from 83 one another. The dynamics of patch i in the ten-patch model are given by where parameter definitions are the same as in the two-patch case.

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Baseline parameter values 86 The baseline values of our parameters appear in Harvesting efficiency of population 1, 2 0.008/year calibrated K 1,2 Carrying capacity of resources in patch 1, 2 1,000,000 calibrated β 1,2 Controls location of the mid-point of the sigmoid for population 1, 2 3.5 calibrated γ 1,2 Controls steepness of the sigmoid for population 1, 2 5 calibrated for several centuries regardless of their rates of resource use, and so their harvesting efficiency was high 91 enough that there were consequences to over-exploitation but not high enough to make resource use 92 incredibly costly. At these parameter values, the population size of a single patch grows to 650, 000 and 93 then declines somewhat to an equilibrium population size of 480, 000 over a timescale of several hundred 94 years.

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The parameters controlling the midpoint and steepness of the sigmoid function (β and γ) were obtained 96 through calibration by analyzing the effect they had on how much and when the populations would take 97 from neighboring populations. To calibrate β , our intention was that the populations did not take much 98 from neighbors when they were not in need. In contrast, they would take more when their resources began 99 to dwindle and neighbour's resources were needed to survive. To calibrate γ we choose a value such 100 that the switch between these two described states was relatively gradual. In particular, we required β i 101 and γ i to satisfy the property that if P i /R i = 1/2 and thus resources were abundant, then b i was roughly  136 We also studied how the time to collapse depends on parameter values for the isolated and interconnected demonstrating that the interconnection of the two populations is detrimental to the stability of the system 145 ( Fig. 2, 3). The isolated case is more resilient to collapse, as we see that the model often survives  (Table 1).

Time to collapse
also robust under these parameter variations (Fig. 2). As the number of populations increases from 2 to 10, metapopulation may stave off collapse. values, we observe that sustainability is a more frequent outcome than in the symmetric case, but occurs 185 less frequently than in the isolated case (electronic supplementary material, Figure S1).

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To observe the effect of inequality on system dynamics, we created an additional scenario involving two 188 unequal populations. Population 1 has a higher starting population size, population growth rate, resource 189 growth rate and harvesting efficiency, but a lower carrying capacity than population 2, which has more 190 resources but a lower starting population size and growth rate. Population 1 is also more prone to take 191 resources from population 2 than vice versa. The inequality scenario was simulated with and without 192 interconnections. Parameter values can be found in electronic supplementary material, Table S1 and the 193 initial conditions were P 1 (0)=50,000, P 2 (0)=25,000, R 1 (0) = 250,000, R 2 (0)=1,000,000.

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In the interconnected case (electronic supplementary material, Figure S2), population 1 grows relatively 195 quickly (Fig. S2a), reaching their maximum population size nearly 100 years before population 2. In the 196 process, they exhaust all of their resources early in the simulation (Fig. S2b). However, this causes very 197 little disturbance to population 1 since there is only a small, nearly non-existent, decrease in population 198 size at the time of resource depletion. This is due to their early dependence on population 2's resources 199 (Fig. S2g) dampening the effect that over-exploitation has on their own population. After this point, both populations continue to consume population 2's resources (Fig. S2d) until the inevitable depletion, causing 201 both populations to collapse.

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In the corresponding isolated but unequal case (electronic supplementary material, Figure S3), the 203 outcomes are very different. Population 2 begins a similar population increase as in the interconnected 204 case, but the population avoids complete collapse and instead recovers to a stable state ( Figure S3c).

205
However, population 1 grows unsustainably, over-depletes their resource, and collapses (Fig. S3a,b). 206 Hence, for these parameter values, we observe that the dichotomy between outcomes in the isolated and 207 interconnected scenario persists when the two populations are unequal.  This effect was robust under a wide range of parameter variation. We also found that asymmetry in 223 parameter values between the two patches does not change the qualitative results, but does tend to stave 224 off collapse. We speculate that models with greater heterogeneity (such that each patch has a unique set of 225 parameter values) might replicate this feature, but we leave this for future work. We furthermore found 226 that collapse can occur in a scenario of inequality between the two patches, although we did not test the 227 13/19 robustness of this finding to parameter variation.
In some respects, our model embodies some of the ideas of "red and green loop" dynamics as