Intraspecific Cohesion From Conspecific Attraction, Part III: A Simple Test

In the real world, animals typically tend to congregate at the landscape scale, even when they show repellence at finer scales (territorial behaviour). In contrast, one of the basic assumptions of the vast majority of mathematical population models is independent space use by individuals. In other words, it is assumed that the individuals do not show conspecific attraction but adhere to “full mixing” from independent space use. A strange assumption, indeed! Using the part of the Parallel processing theory (PP) that was summarized in Part I and Part II it is a simple statistical exercise to test if a given population’s dispersion obeys the full mixing assumption (the Paradigm); or alternatively, indicating a positive feedback tendency from PP-compliant space use (the Zoomer model). 

The concept of conspecific attraction is verified among many species and taxa of animals. For example, lizards prefer to settle near conspecifics, even when unoccupied habitat is available nearby (Stamps 1987; 1988; 1991). Stamps (1991) searched the literature on territorial vertebrates but was unable to find any study in which a negative effect of the density of residents on the settlement of newcomers was demonstrated; i.e. the general assumption of habitat selection studies that prospective territory owners prefer relatively empty habitats proved to lack empirical evidence.

In contrast, in theoretical population models it is typically assumed that individuals both move and settle inter-independently. This “full mixing” assumption in models is a mathematical convenience. It is in fact a requirement, under the premise of the mean field framework, on which almost all population dynamical models adhere to. For example, to model population dynamics realistically with ordinary differential models one has to assume both full spatial mixing of individuals at the temporal scale of the analysis, and a closed system. If the system is open, one should apply partial differential equations, since this allows for an assumption of full mixing locally instead of system-wide (allowing for spatio-temporal “shifting mosaic” of local abundance) (Potts and Lewis 2019)*. The former is called a spatially implicit model, the latter is called a spatially explicit model.

The “full mixing” requirement in classic and contemporary theory of space use by populations is one of the main reasons why theoreticians and field ecologists often tend to drift apart. Logically, it does not make sense, even before considering behavioual fitness arguments! As I argued in Part I and Part II, independent movement and settlement in an open environment will over time drift a Markov-compliant population towards extinction from diffusion and Allée effects.

Empirical results show that (a) single-species populations tend to show scale-free spatial congregation, compliant with a power law (Taylor 1986), (b) empirical results continue to support Non-Markovian, spatial memory utilization by individuals, and (c) populations in general seem to adhere to the principle of conspecific attraction. References in by book and throughout this blog.

(a) Upper part: a superposition of five MRW series with stronger utilization distribution overlap than in Fig. 4 (inset). Lower part: a superposition of five series based on classic RW with homing tendency, with less spatial overlap in spatial utilization. (b) A log-transformed frequency histogram of local cell densities for MRWsuperpostions (filled circles) and RW-superpositions (open circles) shows that neither of the dispersion pattern at the population level in this case satisfy a power law.

Scientifically, the primary question is: how to perform the initial task (prior to making follow-up ecological inference) to test for inter-dependent or independent space use? In other words, how to test if local dispersion is influenced by conspecific attraction?

Consider the PP-based model to represent a specific alternative hypothesis, which should be tested against the null hypothesis given by the Paradigm (for example, a reaction-diffusion model for spatially explicit population dynamics).

The protocol can be quite simple, at least in the first-level approach of analysis of empirical data.

Again, consider results from simulations of PP, i addition to a Paradigm compliant population dispersion. In the Figure to the right you find situation both from scale-specific (Markovian compliant random walk, the Paradigm) and scale-free individual space use with extensive spatial overlap (the PP model), and where both conditions are void of intraspecific cohesion to make the scearia comparable in this respect. This condition contrasts with the scenario in Part I, where I showed how a population with PP compliant space use generated a scale-free dispersion under influence of conspecific attraction. In the present two scenaria the conspecific attraction factor is absent. Individuals use space inter-independently, and below you learn how to test statistically for this lack of conspecific attraction at the so-called “landscape scale” under two qualitatively distinct model frameworks.

The result of inter-independency is seen in the lower part of the Figure. Both situations result in a lack of fractal spatial dispersion of abundance of the pooled set of locations. The reason is that the distribution complies with a negative exponential function (semi-log linear, not shown) rather than power law compliant one (log-log linear).

When these two distinct system conditions are viewed from the population perspective, they both lead to mean field-like system properties with respect to the (M, F) regression at the population level. In other words, even the scenario where individual space use was PP compliant, the lack of conspecific attraction masked the fractal PP property of individuals when analyzed at the population level. Under log-log transformation of frequency of cells in respective bins of grid cell abundance the regression lines were not linear; i.e., not power law compliant. Basically, to test PP compliant space use under the additional property of intraspecific cohesion from conspecific attraction one needs to verify a scale-free (log-log linear) frequency distribution of local density of individuals.

This result illustrates the interesting system property where space use at the individual level may adhere to scale-free dispersion of locations of respective individuals (analyzed separately) while space use at the population level (local abundance of the pooled sets of locations) apparently shows mean field compliance: local fluctuations of abundance are negative exponential compliant! Conclusion: the full mixing premise of the Paradigm – the mean field compliance – can easily be tested on real data.

I’d like to finish this post with an additional study:

In sum, we experimentally tested in breeding mallards two alternative and mutually exclusive hypotheses of habitat selection rules, and found more support for the conspecific attraction rule. However, taking into account the pattern of habitat distribution of breeding mallards (see references in Introduction; this study) pairs certainly use other ways of habitat assessment than mere presence of conspecifics. Some lakes had relatively stable pair numbers while others remained empty independently of experimental treatment.
Pöysä et al. 1998, p287

As I stated in Part II, “A given number of individuals cannot be everywhere all the time”. Thus, some lakes should – as a logical consequence under the premise of conspecific attraction – always be expected to be void of breeding pairs…

NOTE

*) Using the toolbox of partial differential equations and some alternatives it is shown how “diffusion-taxis” equations may show system-intrinsically driven heterogeneity of local population abundance between populations (Potts and Lewis 2019). However, this phenomenon regards interspecific cohesion (or repellence) between separate populations, not intraspecific cohesion. It is assumed that full mixing of individuals is satisfied for each of the populations at the temporal scale of analysis.

REFERENCES

Potts, J. R. and M. A. Lewis. 2019. Spatial memory and taxis-driven pattern formation in model ecosystems. arXiv:1903.05381v05382.

Pöysä, H., J. Elmberg, K. Sjöberg, and P. Nummi. 1998. Habitat selection rules in breeding mallards (Anas platyrhynchos): a test of two competing hypotheses. Oecologia 114:283-287.

Stamps, J. A. 1987. Conspecifics as cues to territory quality: a preference of juvenile lizards (Anolis aeneus) for previously used territories. The American Naturalist 129:629-642.

Stamps, J. A. 1988. Conspecific attraction and aggregation in territorial species. The American Naturalist 131:329-347.

Stamps, J. A. 1991. The effects of conspecifics on habitat selection in territorial species. Behavioral Ecology and Sociobiology 28:29-36.

Taylor, L. R. 1986. Synoptic dynamics, migration and the Rothamsted insect survey. J. Anim. Ecol. 55:1-38.

Intraspecific Cohesion from Conspecific Attraction, Part II: Paradox Resolved

I briefly mentioned in Part I that the combination of spatial memory utilization and scale-free space use under the Parallel processing conjecture (PP) may lead to a fractal compliant population dispersion of intrinsic origin, given the additional condition of conspecific attraction. Below I elaborate on heterogeneous population dispersion as expected under the Paradigmatic framework (Markovian process, mean field compliance) and the contrasting PP kind of space use. In particular, one may find that two locations with different population density under the PP condition may reveal similar intensity of space use! Under the Paradigm such a result will appear paradoxical. Under the PP framework (the MRW and the Zoomer model) the paradox is resolved.

Before switching to empirical results, consider the following crucial question for ecological theory of space use. What is the driving force behind the typical pattern of a shifting mosaic of population abundance over a range of spatial scales, apparently even before the landscape structure (habitat heterogenity) is considered as a complicating factor? In other words, a population is heterogeneously scattered over space. This widespread phenomenon, widely documented in insects, was intensively debated during the last half of the 20th century. In particular, because the dispersion typically adhere to a power law of fluctuation of abundance, i.e., a scale-free phenomenon (Taylor 1986). However, despite many attempts over the years to model such complex patterns they still appear quite paradoxical, with little degree of consensus. My own attempt to drill into this phenomenon led to the idea and development of the PP concept already in the early 1990’s. I cite from my post “Simulating Populations VI: the Unrealism of Standard Models“, dated 10 March, 2018:

First, consider the condition where the individuals are living in an open environment, which is the general condition. No species is abundant everywhere, meaning that a given population is surrounded by unoccupied space […] In some cases this fringe zone may be easily understood from the perspective of unfriendly neighbourhood in habitat terms, but often the surroundings appear quite similar to the conditions inside the distribution range. For example, the over-all population may be spatially fragmented with respect to abundance; particularly along the core areas’ perimeters, with no apparent reason why small and large chunks of the intermediate areas should not be habitable. Pick any well-studied species, and ask an expert on its distributional range to explain population absence in some regions with apparently similar ecological conditions to the population’s present range. In an in my view unsatisfactory manner the way out of this dilemma (the “shoot from the hip” answer) is normally to point to some environmental factor still not revealed, or some kind of statistical chance effect. In my Zoomer model such apparently inexplicable “clumping” of a population is explained as an emergent property of conspecific attraction and scale free redistribution of some individuals. A given number of individuals cannot be everywhere all the time.

The concept of trivial and non-trivial congregation of individuals in a population below carrying capacity is illustrated in a hypothetical spatial arena, where the local carrying capacity (cc) is defined by circle size at grain scale 1/25 of arena size, and population density relative to local cc is defined by the black part of the circles. The two situations in the upper row show homogeneous arena conditions, while the lower parts show heterogeneous conditions. A trivial dispersion pattern (Paradigm-compliant) is illustrated to the left in both rows, where Poisson-variability in abundance is simplified as uniform mean density expectancy at similar local cc levels, and the expected abundance changing linearly (proportionally) with cc. Abundance below cc can – for example – be due to a population slowly recovering from a recent resource bottleneck event. A population growth rate that is slower than the rate of inter-population mixing is traditionally assumed in population dynamics models, due to their assumption of valid mean field approximations. Thus, individuals are expected to be distributed in linear proportion to local resource levels even below carrying capacity levels. A non-trivial abundance pattern (PP compliant), to the right in both rows, is defined as density levels that does not correlate proportionally with cc fluctuations, whether cc is uniformly dispersed or not.

To illustrate concept of system complexity from intraspecific cohesion, consider the sketch to the right (Gautestad and Mysterud 2006). As shown in the upper row, even in a hypothetical homogeneous habitat the process of conspecific attraction predicts heterogeneous space use (upper right). As stated above, “A given number of individuals cannot be everywhere all the time”. Individuals are congregating spatially within the limits that are set by local environmental constraints (size of the circles; upper row, right). In a heterogeneous environment (lower row, right), where these local constraints vary, this intrinsic force is mingling with the effect from local conditions*.

The sketch illustrates population dispersion at the given spatial scale of the virtual grid. Crucially, under PP a similar kind of heterogeneous dispersion with similar parameter values except for a trivial rescaling operation is expected if the observational scale is changed to another grid resolution. In other words, the dispersion is statistically scale-free, or fractal-compliant.

Consequently, both at the individual and the population level, to study PP-compliant space use where this property is verified (scale-free and memory-driven patch attraction or conspecific attraction, respectively) it is required to use PP-derived methods to infer intensity of local space use. This contrasts with the simpler expectation from standard mean field compliant modelling (the Paradigm), where a relatively straightforward positive correlation between local resource level and local abundance is expected (upper and lower left images in the Figure). As repeatedly shown elsewhere, under the Paradigm the intensity of space use is basically proportional with population density. Under the PP model, the intensity of use is less straightforward, and requires different quantification.

As an example of the statistical strength of applying PP-derived methods to study habitat selection where the animals had scale-free and memory-influenced space use, follow this link (PDF).

To summarize, one should first test whether a given space use dispersion complies with the Paradigm or the PP framework. Next, one should apply a method from respective statistical toolboxes to infer ecology, like quantifying local intensity of space use as a function of – for example – local resource level.

By applying a PP model, one may find that two locations with similar local density of individuals have different intensity of space use (indicating different strength of habitat selection). One may also find that two locations with different density may reveal similar intensity of space use! Under the Paradigm such results appear paradoxical. Under the PP framework (the MRW and the Zoomer model) the paradox is resolved.

NOTE

At the individual level and using the MRW model, a similar self-organized “clumping” emerges due to self-reinforcing patch use from targeted returns.

REFERENCES

Taylor, L. R. 1986. Synoptic dynamics, migration and the Rothamsted insect survey. J. Anim. Ecol. 55:1-38.

Intraspecific Cohesion From Conspecific Attraction, Part I: Overview

Animal survival requires some kind of intraspecific cohesion (“population glue”); an intrinsically driven tendency to counteract the diffusion effect from free dispersal. Populations of most species live in an open environment. Thus, without some kind of behavioural capacity to continuously or periodically seek and maintain contact with conspecifics only the most extreme and bizarre kind of environmental constraint would be required for the species’ long term survival. Despite the general agreement on this basic requirement, ecological models for individual movement and population dynamics have for a century maintained a different paradigm, a stubborn assumption that the animals follow the basic principles of mechanistic (Markovian) movement at the individual level and diffusion-advection laws at the population level. Fueled by empirical results and – in my view – common sense the Paradigm has from some researchers come under attack for many years, but mostly in vain. Ecological models and statistical methods are still deeply dependent on the Paradigm.

Visual, auditory and olfactory cues are often not satisfactory mechanisms for individuals to effectively return home when a sally brings them too far away to maintain sufficient population contact. Hence, under the premise of Paradigm compliant behaviour*, the result is an extensive “lost in space” risk, in particular at the fringe zones of a population. However, since field data typically reveals a capacity for targeted returns beyond sensory guidance, this property should be included in space use model designs where applicable.

What does “a capacity for targeted returns beyond sensory guidance” mean? In both the book, in papers and throughout this blog I have presented two complementary and parsimonious models for the individual and the population level respectively, the Multi-scaled random walk (MRW) and the Zoomer model. These inter-coherent approaches are based on two Paradigm-opposing assumptions; (a) individuals have a capacity for spatial memory, and (b) individuals are processing this spatio-temporal information in compliance with the Parallel processing conjecture (PP; providing a potential for scale-free habitat utilization along three axes; space, time and scale). Both PP aspects are hypothesized to be key to mimic real population dynamics realistically. For example, they open for “a capacity for targeted returns beyond sensory guidance”.

How to mimic conspecific attraction at the system resolution of individuals? The series show the accumulated spatial dispersion of relocations from seven simulations of MRW with partly inter-dependent series, by letting a uniformly sampled subset of locations, a ‘‘seed set’’, from one series become a starting point for another series. This seed set kernel, the common ‘‘return base’’, then represents a space use template, which the new series will tend to emphasize for the rest of the series. This can be interpreted as an animal that is utilizing its habitat similarly but not identically to another individual with respect to patch preferences over a range of scales. In particular, the individuals also share some common meeting points. The smaller the seed set, the more independent space use by the individuals. From Gautestad and Mysterud (2006).

Most crucially, I have presented several statistical methods that have the power to test PP-compliant space use against the Paradigm, including the challenging task to differentiate between PP and Markov-compliant execution of spatial memory. I have also presented several such tests on empirical data. On one hand, the spatial dispersion of an accumulated set of individual relocations (fixes) and on the other hand a snapshot of population dispersion of individuals produce clear statistical fingerprints using proper analyses, which can distinguish between a Paradigm- and a PP-compliant kind of behaviour.

If the PP conjecture continues to gain support, it is a short step to prolong this parsimonious modelling approach to explicitly explore the mathematically challenging aspect of intraspecific cohesion from conspecific attraction – outside the comfort zone of standard mechanistic and statistical-mechanistic (Markovian) designs. The illustration to the right shows an example using the MRW design. Alternatively, the Zoomer design could be explored if one prefers a population level abstraction of space use (an “ensemble” rather than a “particle” approach in statistical-mechanical terms).

Basically, both approaches require just one extra assumption to study intraspecific cohesion. This assumption states that contact with conspecifics represent a resource for the animals, in a sense in line with other resources like food, shelter, refuges from predators, and so on (Gautestad and Mysterud 2006). However, inclusion of conspecific attraction in compliance with the PP conjecture predicts a qualitatively different kind of dynamics than you find under the Paradigm, resulting in complex space use and population dispersion. In particular, space use dispersion of individuals and local density fluctuation of the population develop an emergent property of a statistical fractal (aggregations within aggregation within…).

A log–log transformed histogram of frequency of local grid cell densities of locations from the merged set of the seven series of relocations confirm a power law distribution with slope close to the expected -1 (superimposed, best fit). The result supports a statistical fractal of space use for the population of individuals. From Gautestad and Mysterud (2006).

By applying the PP-developed methods, both the MRW and the Zoomer model find support in data covering a wide range of species and taxa. Long-lasting but generally under-communicated paradoxes that appear when the Paradigm is confronted with real data find solution under the complementary PP models.

As always in ecology, exceptions to the general rule should be expected, but so far the PP compliance seem to be very comprehensive across the animal kingdom of vertebrates, and apparently also among many invertebrates (references are given in my book and here in my blog). Again, I challenge readers to test their own animal space use use data; whether they regard GPS fixes of mammal movement or multi-scale analysis of population dispersion of insects.

Unfortunately, my book (Gautestad 2015) is still the only extensive theoretical wrap-up of the Paradigm, why it needs to be replaced as the default framework of space use ecology, and how an  alternative approach may be developed, explored and empirically tested. A broader scrutiny – involving follow-up from expertise covering a wide range of fields from wildlife management to theoretical modelling – is needed.

NOTE

*) Model examples of Paradigm compliant population dynamics are spatially implicit differential equations, spatially explicit partial differential equations and difference equations (including metapopulation models).

REFERENCES

Gautestad, A. O. 2015. Animal Space Use: Memory Effects, Scaling Complexity, and Biophysical Model Coherence Dog Ear Publishing, Indianapolis.

Gautestad, A. O. and I. Mysterud. 2006. Complex animal distribution and abundance from memory-dependent kinetics. Ecological Complexity 3:44-55.