Inter-specific Variability in Demographic Processes Affects Abundance-Occupancy Relationships

Species with large local abundances tend to occupy more sites. One of the mechanisms proposed to explain this widely reported inter-specific relationship is a cross-scale hypothesis based on dynamics at the population level. Called the vital rates mechanism, it uses within-population demographic processes of population growth and density dependence to explain how positive inter-specific abundance-occupancy relationships can arise. Even though the vital rates mechanism is mathematically simple, it has never been tested directly because of the difficulty in estimating the demographic parameters involved. Here, using a recently introduced mark-recapture analysis method on 17 bird species, we show that inter-specific variability in density dependence strength can weaken both abundance-occupancy relationships and the expected corollaries of the vital rates mechanism. We demonstrate that one of the key assumptions of vital rates mechanism, that density dependence strength should be similar among species, is not met for these 17 species. Additionally, the mathematical structure of vital rates mechanism that relate population-level abundance and intrinsic growth rate is only weakly observed in our data. We argue that this mismatch of mathematical structure and data together with the violation of density dependence assumption weakens the expected positive abundance-occupancy association. Vital rates mechanism also predicts conditions under which positive abundance-occupancy association is weakened or even reversed; our results are consistent with these predictions. More generally, our findings support a cross-scale mechanism of macroecological abundance-occupancy relationship emerging from density-dependent dynamics at the population level.


Introduction 34
Exploring ecological processes and interactions across scales of biological organization 35 allows a deeper understanding of patterns of biodiversity. One well-documented macro-36 ecological pattern is the abundance-occupancy relationship: across phylogenetically similar 37 species, local average abundance and proportion of sites occupied are positively correlated 38 (Hanski, 1982;Bock & Ricklefs, 1983;Brown, 1984Brown, , 1995Gaston & Blackburn, 2000). This 39 pattern has been observed in a variety of taxa, such as mammals , birds 40 (Lacy & Bock, 1986;Blackburn et al., 1997), butterflies (Conrad et al., 2001), mollusks (Russell 41 & Lindberg, 1988), and plants (Guo et al., 2000). A meta-analysis on abundance-occupancy 42 relationships found strong positive effect size but also reported high variability across species 43 realms (Blackburn et al., 2006). Some species groups, however, show zero or negative 44 correlation between abundance and occupancy (Novosolov et al., 2017), which may be because 45 of an ecological mechanism (Reif et al., 2006;Symonds & Johnson, 2006;Ferenc et al., 2016;46 Freeman, 2019), or a result of the metrics used to represent abundance and occupancy (Wilson,47 density dependence, and the similarity of response to environmental gradients, among species. 126 Here, we mainly focus on density dependence strength and how it, together with other 127 demographic parameters, can affect macroecological patterns. 128 2) More than 14 adults, and more than 14 juveniles re-captured. 139

Materials and Methods
These filters reduced the dataset to 62 species. We applied CJS-pop to only 42 species 140 which had reasonable data size (e.g. maximum model convergence time was one week). For each 141 species, we treated separate MAPS locations (a cluster of mist-netting and banding stations) to 142 be independent breeding populations. We only included data from populations which have been 143 monitored for at least 5 years. 144

CJS-pop 145
CJS-pop uses robust design mark-recapture data as a single data source for parameterizing 146 demographic models in a Bayesian hierarchical framework (Şen & Akçakaya, 2020; also see (survival and fecundity) as well as density-dependence and process variance in these vital rates 149 (temporal or spatial variability). Here, we model survival and fecundity as functions of 150 environmental covariates in addition to density dependence. An example model structure, here 151 illustrated for , , , the survival probability of a stage invidiual, at population , and at year , 152 with two environmental covariates is given as: 153 logit( , , ) = − 1 ⋅ , + 2 ⋅ 1, , + 3 ⋅ 2, , + . 2 154 where is the survival probability of a stage individual in logit scale at mean population size 156 and at mean of environmental covariates 1 and 2 ; 1 is the change in survival in logit scale 157 with one unit change in population density index; is the population density index at population 158 , and at year (See appendix S2 for density index calculation); 2 and 3 are the change in 159 survival in logit scale with one unit change in 1 and 2 , respectively; is the temporal 160 random effect at year , and 2 is the temporal process variance of survival at logit scale; = 161 1,2, … , ; = 1,2,3. . . , ; = 1,2,3. . . , . We denote = 1 as juveniles and, = 2 as adults. 162 Similarly, fecundity , at population and year can be modelled as: 163 log( , ) = − 1 ⋅ ,( −1) + 2 ⋅ 1, , + 3 ⋅ 2, , + . 3 164 where is the average fecundity in log scale at mean population size and at mean of 166 environmental covariates 1 and 2 ; 1 is the change in fecundity in log scale with one unit 167 change in population density index; 2 and 3 are the changes in fecundity in log scale with one 168 unit change in 1 and 2 , respectively; is the temporal random effect at year , and 2 is the 169 temporal process variance of fecundity in log scale. where, is a vector with population level intrinsic growth rates. 216 We only used species with a negative density-dependence effect on growth rate (growth 217 rate at density index 2 was lower than growth rate at density index 0). Intrinsic growth rate is not 218 defined for circumstances in which increasing density also increases the population growth; in 219 such a case intrinsic growth rate loses its meaning as a theoretical maximum. Additionally, 220 positive density dependence effects are biologically meaningful only when Allee effects are 221 considered. Because we are not modelling Allee effects with CJS-pop, we removed species that 222 showed a positive density dependence effect on growth rate from further analysis.  (Table 1). Among these 17 species, population size showed a right skewed 273 distribution with many small and few large populations (Appednix S1 : Fig. S1a). The majority of 274 these populations have > 0 with unimodal distribution that is centered on 0.14 (Appendix S1: 275  Fig. 1c). 281

Is there a positive correlation between and among populations of each species? 282
There was a positive relationship between and for 13 out of 17 species. The strength 283 and uncertainty of these relationships are highly variable among species (Fig. 2). Seven species 284 have ( > 0) > 0.9, indicating strong evidence for the inferred positive relationship between 285 intrinsic growth rate and population size at the population level. Downy Woodpecker has the 286 highest correlation coefficient with the lowest uncertainty ( = 0.42 (0.31 − 0.53)). Wrentit is 287 the only species with a strong negative relationship between and (Fig. 2).  Figs. 1d and 1f). 297 ( ) and , and strong positive correlation between and , which indicates that species with 300 small average population size and large intrinsic growth rate are associated with stronger density 301 dependence in survival (Fig.s 4a and 4c). Relationship between density dependence in fecundity 302 ( ) and and shows the opposite trend to their survival counter parts, albeit with weaker 303 estimates (Figs. 4b and 4d). 304

Discussion 305
The four questions we asked follow the fundamental logic behind Holt et al., (1997)'s 306 vital rates mechanism. First, if there is a positive relationship between abundance and occupancy 307 across species and if that relationship is driven by vital rates mechanism, population-level 308 abundance and intrinsic growth rate should be correlated within a species (Eq. 1). We detected 309 weaker relationships between abundance and different metrics of occupancy (Figs. 1a, 1b, Webb et al., 2007). Additionally, only seven 312 species showed a high probability of positive correlation between and among their 313 populations, and the strength of these correlations tend to be weak as well (Fig. 2). 314 Secondly, as a corollary of vital rates mechanism and its assumption that density 315 dependence strength among species should be similar, species-level growth rate and abundance 316 should also be positively correlated. However, we observe the exactly opposite: a strongly 317 negative relationship between and (Fig. 3). We argue that the main reason we don't see a 318 positive relationship between and is the violation of the assumption that the strength of 319 density dependence is similar among species. Variability in the strength of density dependence, together with the weak relationship between and among populations of each species, 321 prevents a positive abundance occupancy association and results in weak relationships of both 322 and with occupancy metrics in Fig. 1. If all species have similar density dependence (e.g., in 323 terms of the mechanism, but especially in terms of strength), then species with higher average 324 growth rate will also have higher average abundance and higher occupancy; thus, all three 325 variables would be correlated among species. However, if the strength and other properties of 326 density dependence varies among species, such a pattern would not exist. Holt et al., (1997) 327 predicted that variable density dependence strength among species might break the positive 328 association between abundance and occupancy, writing that "[a] species with small but weak 329 density dependence can obtain enormous local abundances compared to another species with 330 large but intense density dependence". We hypothesize that this is exactly the pattern we report 331 here. 332 Holt et al., (1997) predicted when the relationship between intrinsic growth rate and 333 density dependence should break down positive occupancy-abundance relationships. The 334 relationship between density dependence strength and growth rate (Fig. 4c)  to the relationship in Fig. 4c. It seems likely that this variability is responsible for the weak 338 relationship detected between , and occupancy metrics (Fig. 1). Although we do not know the 339 exact ecological process that lead to these relationships (Fig. 4), there seems to be a pattern to the 340 trade-off among growth rate, average abundance, and density dependence across species. Species 341 with lower growth rate have higher abundance and are strongly regulated by density dependent 342 processes, especially acting on survival rates. Species with higher growth rate, on average, have lower abundance and are more weakly regulated by density dependent processes especially 344 acting on fecundity. We are not aware of any other study exploring these patterns, so a better 345 understanding of the relationship of density dependence with other demographic parameters and 346 the macroecological patterns they might cause requires further research of these patterns in other 347 taxonomic groups. 348 The correlation between and among populations of a species is the mathematical 349 foundation of the vital rates mechanism. The structure of this version of logistic equation implies 350 perfect correlation (assuming that density dependence strength is a species-specific trait and 351 therefore constant across populations of the species). To explore whether such strong 352 correlations exist among populations, a model should allow and to vary independently. CJS-353 pop is one such modelling framework in which any kind of relationship between and can 354 naturally arise. The majority of the 17 bird species considered show positive correlation between 355 and among their populations, but these correlations tend to be weak (Fig. 2). We argue there 356 is more nuance to this relationship between and , and their correlation should not be assumed 357 by default. These weak correlations likely play a role in weakening and sometimes reversing the 358 positive abundance-occupancy relationships observed in Fig. 1

. A weak correlation between 359
and can also indicate a spatially varying density dependence strength among the populations 360 of a species. This would mean that both resource availability and the response of a population to 361 available resources varies across populations for a given species. This mechanism is also 362 hypothesized to prevent positive abundance-occupancy relationships (Holt et al., 1997). While 363 CJS-pop can be modified to fit a population specific density dependence strength, none of the 364 species have enough data to provide such information. We believe this is an exciting future 365 research topic. negative abundance-occupancy relationships. Below we explore if other ecological processes can 368 lead to the patterns we report here. For a discussion on how statistical artefacts can affect the 369 patterns we report here see Appendix S2. 370

Alternative Processes to Density Dependence 371
An alternative mechanism that may cause a weak relationship between abundance and 372 occupancy is local adaptation in range-restricted species. For example, adaptation of 373 mountainous bird species to their local environments, especially in the afro-tropical region, has 374 been hypothesized to break down the correlation between range size and abundance (Reif et al., 375 2006;Symonds & Johnson, 2006;Ferenc et al., 2016;Freeman, 2019). These mountainous 376 species are highly adapted to high elevation areas and can reach abundance levels similar to 377 lower elevation species while still occurring across a narrower range. Although this mechanism 378 can explain the lack of a positive abundance occupancy relationship of specialist species with a 379 narrow range, it is not applicable to our study because majority of the17 bird species we modeled 380 are wide-ranging across US. 381 Another important mechanism that can affect the abundance-occupancy relationships is 382 dispersal and colonization-extinction dynamics. Individuals of the species we have modeled are 383 rarely captured in multiple populations (MAPS locations), so we cannot estimate dispersal rate 384 from the data we are using and at the scale we are applying CJS-pop. A future goal is to use 385 multiple data sets, one for species movement and another for species demography, to explore 386 how their interaction can affect abundance-occupancy relationships. 387 In addition to similarity of density-dependence strength across species, one of the most 388 important assumptions of the vital rates mechanism is that species should show a similar responses can explain the patterns we observed. Here, this response is quantified in the slopes 391 2) The response to an environmental gradient that is described by Holt et al., (1997) is 398 more akin to the concept of fundamental niche. CJS-pop, or any other statistical 399 method, may not be able to estimate the "true" response to environmental variability 400 for the simple fact that species may not occur in every suitable site because of 401 competitive exclusion. Inter-species biotic interactions are effectively missing from 402 the vital rates mechanism. 403 Patterns of distribution of life on Earth are interesting in themselves, but even more so 404 when they are associated with, and explained by, mechanisms at different levels of biological 405 organization. Holt et al., (1997) presented one such mechanism for the widely reported positive 406 abundance-occupancy relationships. It is, however, likely that the abundance-occupancy patterns 407 are simultaneously determined by dispersal, inter-specific and intra-specific interactions, species 408 demography and response to environmental gradients, as well as the sampling schemes used to 409 explore these relationships. Here, we only demonstrated the effect of within-population 410 processes. Ideally, macroecological patterns would be explored in frameworks that include 411 dynamics at all relevant scales, including processes that are at the population, metapopulation, frameworks would help us get a clearer picture of the conditions under which abundance-414 occupancy and other macroecological relationships emerge. 415 Table 1. Names of the17 selected species to be used in testing the vital rates mechanism.