# A road map for synthesizing the scaling patterns in ecology

Cang Hui  Centre for Invasion Biology, Department of Botany and Zoology,
Stellenbosch University, Matieland 7602, South Africa; E-mail<EMAIL_ADDRESS>
###### Abstract

Ecology studies biodiversity in its variety and complexity. It describes how
species distribute and perform in response to environmental changes.
Ecological processes and structures are highly complex and adaptive. In order
to quantify emerging ecological patterns and investigate their hidden
mechanisms, we need to rely on the simplicity of mathematical language. This
becomes especially apparent when dealing with scaling patterns in ecology.
Indeed, nearly all of ecological patterns are scale dependent. Such scale
dependence hampers our predictive power and creates problems in our inference.
This challenge calls for a clear and fundamental understanding of how and why
ecological patterns change across scales. As Simon Levin stated in his
MacArthur Award lecture, the problem of relating phenomena across scales is
the central problem in ecology and other natural sciences. It has become clear
that there is currently a drive in ecology and complexity science to develop
new quantitative approaches that are suitable for analysing and forecasting
patterns of ecological systems. Here I provide a road map for future works on
synthesizing the scaling patterns in ecology, aiming (i) to collect and sort a
diverse array of ecological patterns, (ii) to present the dominant parametric
forms of how these patterns change across spatial and temporal scales, (iii)
to detect the processes and mechanisms using mathematical models, and finally
(iv) to probe the physical meaning of these scaling patterns. This road map is
divided into three parts and covers three main concepts of scale in ecology:
heterogeneity, hierarchy and size. Using scale as a thread, this road map and
its following works weave the kaleidoscope of ecological scaling patterns into
a cohesive whole.

PACS numbers

87.23

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

Ecological patterns are emerging structures observed in populations,
communities and ecosystems. Elucidating drivers behind ecological patterns can
greatly improve our knowledge on how ecosystems assemble, function and respond
to change and perturbation. Due to the non-random nature, most, if not all,
ecological patterns change with measurement, characteristic and organization
scales and exhibit distinct scaling properties. Such scaling properties can be
broadly grouped into patterns related to heterogeneity, hierarchy and size.
The road map introduces the three groups of scaling patterns. The emphasis
here is not to provide solutions to these outlined research questions; rather,
by grouping relevant scaling patterns under unique banners, I attempt to
highlight the challenges and connect the emerging clues for building a unified
theory for scaling patterns in ecology in the near future.

##  I Heterogeneity

This section on the scaling pattern of heterogeneity aims to investigate how
aggregated structures of organisms, diversity and ecosystem service change
with measurement scales and which/why biological patterns resonate with
underlying processes at the same characteristic scales.

Aggregation. Species distributions are not uniform across space, reflecting
the interplay between habitat heterogeneity and the underlying nonlinear
biotic regulation ref1 . Such non-random, aggregated patterns not only can be
the indicative of non-equilibrium dynamics (e.g. during range expansion mihaja
; ecography ; cecile ) but also self-organized pattern emergence (e.g.
ploszhang ; ncer ; ncem ). When ecologists examine species distributions
across scales, the Modifiable Areal Unit Problem presents itself ref2 ; maup .
The problem can be described as the change in species distribution
characteristics as the unit of measurement changes, both in terms of size and
shape of the sample unit. Such scaling patterns of aggregation follow three
general parametric forms: logarithmic, power-law and lognormal shape ref1 .
Following on recent progress of using the Bayesian rule for cross-scale
extrapolation ref3 ; ref4 , further advancement in this field is to provide a
consistent description of aggregation when scaling up and down, and therefore
a universal basis of comparison for distributions in differing contexts. A
fully-functional model with predictive power for up- and down-scaling species
distribution is needed. Under certain conditions, this model should further
allow extrapolating fine-scale occupancy and population densities from coarse-
scale observations (e.g. ecoappl ; ecoscience ). Great potential exists to
apply such a predictive model in various cross-scale pattern analyses
espcially when detection is imperfect (e.g. springer ; jae ).

Space-for-time substitution. The directionality of community succession is an
important concept in conservation biology Odum ; pickett ; it is analogous to
the irreversibility of time in physics that has revolutionised the
understanding of complex adaptive systems jorg . By definition, succession is
an orderly process of community change after disturbance Odum . Knowing the
directionality of succession is necessary for (i) distinguishing new from
mature communities (i.e. defining the age of a community), (ii) understanding
how communities evolve and respond to disturbance (e.g. habitat loss and
climate change), and (iii) designing more efficient conservation and
restoration plans nuria . However, popularising the concept of directionality
in succession is challenging for two reasons. Firstly, acknowledging this
concept demands the acceptance of inherent bias in nature which contradicts
the null hypothesis of a random and isotropic world jorg . Secondly,
appropriate long-term data required for detecting the succession direction are
scarce, and indices and analytical methods for such computation are lacking.
To this end designing alternative tests (e.g. the space-for-time substitution;
pickett  that can capture the essence of directionality and irreversibility in
community development but which can be applied to available data becomes
crucial.

The spatial and temporal scales of ecological processes are intertwined.
Processes that account for the spatial distribution of species also underpin
its temporal dynamics. This means that we can potentially forecast the future
or rebuild the history based solely on current spatial distribution, without
resorting to long-term time series ref5 ; ref6 . In other words, the need to
wait years and decades to measure changes in distribution can be averted
through the ability to make sufficiently accurate predictions based only on
the spatial distribution of species at the current time. As the ability to
forecast the temporal trend of a focal species provides crucial information on
its performance and viability ref5 ; ref7 , the methodology of space-for-time
substitution is extremely appealing, especially because our ability to obtain
spatial records has been drastically improved. This area of research calls for
a model that can relate the scaling pattern of species’ current spatial
distribution to the near-future population trend and performance.

Scale resonance. Just as two tuning forks of the same characteristic frequency
resonate, so do ecological patterns and processes working at the same scale.
Species distributions are regulated by a variety of abiotic and biotic
processes working in concert but at different scales mcgill . Those processes
identified as key biotic drivers using methods such as multivariate statistics
often resonate with the scale of the study. That is, information being picked
up represents a product of the measurement method, rather than the intrinsic
cross-scale mechanism. Such a pattern of scale resonance has been observed
when synthesizing a series of collaborative works on identifying the factors
of the distribution of Argentine ants at local ref8 , regional ref9  and
global scales ref10 . This finding brings into question many regional
management planning practices that are based on the upscale extrapolation of
local-scale studies. Future research needs to explore the mechanism behind
scale resonance in ecology and to present a statistical remedy for cross-scale
inference.

Co-distribution. To exploit resources while mitigating conflict, species often
partition available habitats, forming co-distribution patterns of association
or dissociation. Null models based on permutation have been widely applied for
detecting signals of association or dissociation from co-distribution
patterns, from which the type of biotic interactions can be inferred. Future
research needs to present a model that incorporates biotic interactions and
also captures the transition from fine-scale dissociation to coarse-scale
association (e.g. ref4 ), explaining why this co-distribution pattern changes
across scales and how this scale dependence affects the pairwise measure of
species turnover. It should be reconcile the debate between the Rich-Get-
Richer phenomenon in invasion biology and the opposing Competitive-Exclusion-
Principle.

Biodiversity. Species diversity patterns, such as the species-area curve
ref11a , endemics-area relationship, distance decay of similarity and
occupancy frequency distribution ref11 ; ref12 , are just a few interrelated
patterns of scale dependence emerging from complex ecological systems. The
integration of patterns of species diversity patterns is central to
understanding the processes that drive species assembly 125 . Changing the
measurement scale will lead to a coordinated change in all diversity patterns.
I envisage a new diversity pattern – delta diversity – that connects all
commonly known diversity scaling patterns, using delta diversity as building
blocks. This model should be able to further explain Raunkiaer’s bimodal law
of frequency ref13  and resolves the debate on the ceiling of species richness
in a community.

Ecosystem function and service Ecosystem services are by-products from the
function of ecosystem processes that sustain the basic needs of humans and
their socioeconomic activities. According to the Millennium Ecosystem
Assessment, ecosystem services are generated from interactions ranging from
specialist taxa to all biodiversity, and the functional units of the variety
of services range from local populations to global biogeochemical cycles. At
the local scale, we benefit from services of pollination, pest control, soil
fertility and seed dispersal that are related to biodiversity. At regional
scale, we benefit from services of air and water purification, flood and
drought mitigation, and waste decomposition that are delivered by plants and
micro-organisms. At the global level, we benefit from services of climate
stability and UV protection from plants and biogeochemical cycles.
Understanding how different ecosystem services change with spatial scales and
potentially conceptualizing into a model for extrapolating the level of
service across scales warrants great attention. Recent studies showed the
scale dependence of a crude ecosystem service indicator – biocapacity and the
resultant sustainability index ref14 ; ref15 , and the robustness and
invasibility of recipient ecosystems to biological invasions ref16 .
Exploiting ecosystem services within their maximum sustainable level can
ensure a reliable service provision without triggering a regime shift.
Achieving this balance is a challenge for conservation management and
sustainable development.

##  II Hierarchy

The scaling pattern of hierarchy depicts how the structure and function of
asymmetrical ecological systems emerge and change with the system complexity.
Using ecological networks as the model system, this section aims to
investigate how cascade interactions affects the robustness and resilience of
networks, how network architectures, especially nestedness and
compartmentalization, emerge and function, and the role of network complexity
on the stability of ecological networks.

Cascade. Nodes and edges of a network are a good proxy of species and their
interactions in an ecosystem. Probing processes that can lead to the emergence
of large-scale network architectures, e.g. small-world networks and scale-free
networks, is a new wave in science. Scaling laws of food webs depict how a
biological network behaves as a function of its complexity. In reconciling
with May’s stability criterion of complex systems may , Cohen’s cascade model
cohen  and its later development provide a phenomenological explanation of
some of the scaling laws and scaling invariant patterns. The principle of
Maximum Entropy that identifies the unbiased estimates under constraints has
been widely used in ecology. Ecological networks are efficient energy
transporting, non-equilibrium systems that are adaptive to changes and
disturbances. This calls for the development of models based on the recently
developed principle of Maximum Entropy Production in non-equilibrium
thermodynamics dewar , in an attempt to explain the pattern emergence in
complex adaptive networks, in particular, the cascade interactions in food
webs. Such models will provide a physical understanding on the network
emergence and shed light on solving the complexity-stability debate.

Nestedness. Nested structure has been observed in many networks, in particular
bipartite mutualistic networks (e.g. pollination networks and seed-dispersal
networks bas ). To have models with quantitative accuracy and predictive
power, one needs to rely on process-based models. A key feature in ecological
networks is the adaptive and innovative nature of the edges and nodes. Species
are constantly optimizing which partners they should interact with for maximum
fitness gain, as according to the optimal foraging theory. This can be
achieved at two time scales: 1) At the ecological time scale, interaction
switching reflects seeking optimal fitness gain ref17 ; anim ; 2) At the
evolutionary time scale, interaction switching reflects coevolutionary
dynamics between interacting traits.

Compartmentalization. One important hierarchical structure in ecological
networks is compartmentalization, i.e. the formation of functional modules,
where interactions are most likely to occur within the same module, that is to
say “like is connected to like” in a network ref18 . Energy flows
directionally in resource-exploitation networks (e.g. host-parasite networks),
forming compartmentalized structures. This research area calls for a process-
based model that optimizes energy transport via adaptive interaction switching
partners for local fitness gain. The strong predictive power requires further
demonstration by comparing the level of compartmentalization observed versus
predicted for real networks savannah . This, together with the previous two
research areas, provides a physical understanding of pattern emergence in
complex adaptive networks.

##  III Size

This section on the scaling pattern of size investigates how form and function
change as organisms get larger or as their traits change. The section aims to
investigate how allometric laws of metabolism emerge and how phenotypes of
biological traits form through co-evolution with other traits, species and the
environment and how this affects the path of evolution and diversification.

Allometry. Allometric scaling is the most salient pattern of how biological
rates, especially metabolic rate, are regulated by organism size allo , known
as Kleiber’s law. A model, depicting how fractal-structured circulatory or
vascular systems transport energy and matter throughout the body of the
organism, has been proposed for explaining the emergence of allometric
scaling, which is further summarized succinctly as the Fourth Dimension of
Life west . Recently, a not fully developed concept suggests that natural
selection can increase the ontogenetic and population growth rates to a
ceiling, above which the population will crash due to intrinsic instability
lev , in principle similar to the origin of planetary rings that are only
distributed on a thin surface surrounding the planet, with all other
directions, although possible, have been eliminated through inter-particle
collisions. This progress calls for further development of a conceptual model
for allometric scaling based on the principle of Maximum Entropy Production
and self-organized criticality. This model should be able to simulate the
process of cell differentiation in multicellular organisms.

Trait. This research area focuses on addressing two questions. First, what are
the mechanisms determining phenotypic traits? Evolution via natural selection
relies on heritable phenotypic variation and has long been regarded as being
solely reliant on direct expression of gene variation. This assumption may be
an exaggeration, as factors other than genetic variation can also dictate the
outcomes of natural selection and thus evolution. At the forefront of these
alternative platforms is the rapidly expanding field of epigenetics – the
biochemical modification of DNA without changes in its sequence – that gives
rise to differential gene expression and thus different phenotypes epi . A
revised Price equation that incorporates epigenetic mechanism for explaining
phenotypic variation is needed. Second, how do traits affect biotic
interactions and thus the path of evolution? Biotic interactions, largely to
do with resource competition and exploitation, are realised by specific
phenotypic traits that are used for searching and handling resources, for
competing and defending resources, for memorizing and recognizing beneficial
resources, and for escaping from being exploited as resources by predators or
parasites. In this regard, Adaptive Dynamics is a power tool to explore how
trait-mediated interactions can lead to diverse evolutionary phenomena ulf ,
from the Red Queen dynamics to speciation ec ; ref19 .

##  IV Epilogue

Nature never fails to amaze us. It coordinates the interplay of numerous
organisms in their environments, forming a complex functional system that
sustains us and many other species via ecosystem service. As the ultimate goal
of natural sciences, quantifying emerging patterns in nature and understanding
hidden mechanisms are the pinnacle of science. An important phenomenon in
quantifying ecological patterns is that nearly all of them are scale
dependent. This scale dependence creates problems in our inference, yet also
simultaneously provides opportunities for us to pry into the core of how
nature assembles, organizes and functions. Once again, we need to relate these
research areas back to the overarching research philosophy in studying natural
systems: (i) what patterns exist in nature (i.e. using statistical methods to
measure and quantify ecological patterns); (ii) how such patterns emerge (i.e.
proposing mathematical models to unveil mechanisms driving the ecological
pattern formation; (iii) why nature organizes itself in such a way (i.e. using
physical laws to reveal the adaptive and/or optimal nature of ecological
systems). This has been eloquently presented as the Ouroboros of scales in
Just Six Numbers by Martin Rees. William Blake also expressed the role of
scales in his Auguries of Innocence in 1803, ‘To see a world in a grain of
sand, and a heaven in a wild flower; Hold infinity in the palm of your hand,
and eternity in an hour’.

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