Conservation Biology and SLOSS , Part II: Parallel Processing and Network Topology

Consider the following two system conditions and theoretical assumptions. First, a large nature reserve has a smaller periphery than several small ones with the same total area. Thus, all other factors equal, a smaller periphery pr. unit area of a large reserve implies a higher implicit survival rate for its embedded species. Fewer animals are “lost in space” due to accidentally diffusing out of their reserve. Consequently, a larger reserve is expected to have a proportionally larger species abundance of animals than a fragmented mosaic of smaller reserves, right? Second, again considering all other factors equal, distant reserves are functionally less connected. In other words, the population dynamics of two reserves in close proximity are assumed to be more in sync or anti-sync from intra-population dynamics than more distant ones, right? The conventional answers are obviously “yes”, but…

In my previous post I referred to empirical results on snail kite Rostrhamus sociabilis plumbeus in Florida (Reichert et al. 2016; Valle et al. 2017) that challenged the intuitive and unconditional “yes” answer to these two questions. Here I elaborate on this paradox of potential “no” answers, which may be resolved – as Reichert et al. (2016) and Valle et al. (2017) contributed to – by considering the network topology of the respective populations’ respective individuals.

Lost in space or an example of an  extremely long exploratory move? For a period of a few days in May 2017 this red-rumped swallow Cecropis daurica foraged together with a flock of local barn swallows in Ulsteinvik, Norway – a distance of about 2,500 km from the species’ ordinary habitat in the southern parts of Europe. Photo: AOG.

Species with a cognitive capacity for spatial memory may require a novel modelling approach at all level of system abstraction; individual, population and community. The “lost in space” assumption becomes less valid, because individuals may return non-randomly even after long-lasting and distant exploratory moves. Returning individuals to a specific local site – steered to some extent by strategic spatial behaviour at the landscape scale – may represent a small number of all immigration events to this site. However, the accumulated effect on local population dynamics of such deterministic and occasionally long distance returns, when summed over the larger temporal scale of a reproductive cycle, should be expected to be profound. Individuals may take advantage of the extended knowledge of both local and distant conditions. Consequently, the SLOSS concept under conservation biology may require a theoretical rethinking.

Distance in space may to a smaller or larger degree be de-coupled from distance in network-topological terms. Supported by the Florida snail kite example, the difference in local population turnover within virtual sections of a given area may be expected to be less spatially correlated, and the functional effect from a variable distance between specific reserves becomes blurred. For example, if the red-rumped swallow this particular week in May (image above) just performed an occasional sally – albeit an extremely long one – followed by a memory-driven return to its original habitat, the two localities would have been drawn closer together in terms of network topological distance.

As a complementary view of this network theoretical aspect, the “anti-dispersal effect” from intraspecific cohesion (a tendency to counteract free dispersal and effectively gluing a population together, as modelled by my Zoomer model) cannot be mimicked within a traditional, Markovian/mechanistic framework. Without spatial memory at the individual level, there is no or little glue to keep the population together in the long run. Individuals are easily lost in space, whether they – for example – happen to disperse into the surrounding matrix environment from large or small reserves. On the other hand, a capacity for targeted, memory-driven returns obviously improves survival and thus fitness, whether the context is some large or several small reserves. 

Recall from the Zoomer model for complex population dynamics that the influx of individuals to a given locality during a given interval may on average be equally distributed between short- and long distance immigrants (Gautestad 2015). This model property may sound counter-intuitive and paradoxical under the the traditional model architectures, unless one considers the premise of scale-free population flow during “zooming”. Statistically, a given individual in a distant location has a small probability of turning up at the given local site*. However, when summed over all individuals within this coarser spatial scale the migration rate becomes equal in magnitude to the probability from influx from more closely located individuals. Since number of individuals within a larger periphery are more numerous, their summed contribution to immigration to a finer-scale locality becomes equal to the immigration that originated from individuals more close-by. That is, given that individuals have shown equal weight to fine- and coarse-scale movement and space use on average (the default condition of both the MRW and the Zoomer models).

Targeted, goal-driven moves by individuals in the Zoomer model design drive the emergence of complex network topology of inter-population migration. Closely linked nodes in the network may be less dependent on physical distance between nodes (like reserves in a SLOSS context). This aspect, which has now found empirical support by studying multi-scaled individual movement’s effect on population redistribution, was illustrated by the snail kite papers that were reviewed in my previous post.

In other words, individuals, populations and communities in distant patches may be functionally connected in a manner that is inexplicable in classic metapopulation terms. A multi-scale kind of system analysis is necessary to understand both population dynamics and community ecology. Without it, paradoxes in local changes of population abundance may prevail and the predictive power of individual- and population level models in conservation ecology will remain low.

NOTE

*) Under classic models such a probability for visitors from a distant site is practically zero, due to the premise of a diffusion compliant migration; i.e., an exponentially declining distribution of individual displacement lengths. On the other hand, under the Zoomer model this exponential equation is replaced by a power law tail, which is thin but very long (up to the cut-off scale).

REFERENCES

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

Reichert, B. E., R. J. Fletcher, C. E. Cattau, and W. M. Kitchens. 2016. Consistent scaling of population structure across landscapes despite intraspecific variation in movement and connectivity. Journal of Animal Ecology 85:1563-1573.

Valle, D., S. Cvetojevic, E. P. Robertson, B. E. Reichert, H. H. Hochmair and R. J. Fletcher. 2017. Individual Movement Strategies Revealed through Novel Clustering
of Emergent Movement Patterns. Scientific Reports 7 (44052):1-12.

Conservation Biology and SLOSS , Part I: Time to Challenge System Assumptions

The contrasting ideas of a single large or several small (SLOSS) habitat reserves ignited a heated debate in conservation biology (Diamond 1975; Simberloff and Abele 1982). The recent development in movement ecology – in particular the theoretical aspects of spatial memory and scale-free space use of individuals – makes time ripe to initiate a study of the SLOSS concept under this contemporary perspective. In order to produce realistic predictions community, population and individual processes need to be understood from a coherent system theory involving all levels of system abstraction. Under this premise the original SLOSS concept seems to fall apart.

A single large reserve was argued to be preferable to several smaller reserves whose total areas were equal to the larger (Diamond 1975). On the other hand, if the smaller reserves had unshared species it was possible that two smaller reserves it sum could have more species than a single large reserve of the same total area (Simberloff and Abele). Opposing the latter view, it was argued that habitat fragmentation is probably the major threat to the loss of global biological diversity (Wilcox and Murphy 1985).

However, the SLOSS concept – originating in the field of conservation biology – was understood from the perspective of dispersion of species under various landscape configurations. In my view that approach drove the debate into a dune of sand, due to over-simplification or ignorance of a community system’s lower-level dynamics.

In a traditional metapopulation system, immigration to a given local population (green area) can be concptualized as a fixed rate, representing the average number of emigrant individuals from other populations that by chance happen to reach the present population pr. unit time. In the alternative Zoomer model; the population level formulation of the individual MRW model, immigrants represent a mixture of individuals from other populations that perform exploratory moves over a wider range of scales than assumed by the classic model; i.e., scale-free movement, in addition to individuals that return in a spatially memory-dependent manner (red arrows). Such goal-oriented spatial behaviour leads to the emergence of complex population kinetics.

The original SLOSS debate was considering relative presence and absence of species in various settings of reserve designs. From this classic community system perspective the theory tended to ignore population dispersal effects explicitly. However, to the extent that dispersal was considered [primarily in the context of metapopulation theory; see, for example Robert (2009)] it was assumed that emigration was a deterministic process at the spatial scale of local populations while immigration was a stochastic process at this scale (random influx). This direction-dependent toggle between deterministic and stochastic population flow follows logically from the traditional premise of a diffusion-advection framework for metapopulation systems. I criticized this core assumption of metapopulation theory in this post, and in the Figure to the right I summarize the argument. My critique pinpointed the theoretical consequences of allowing for complex population kinetics (the Zoomer model). The basic metapopulation principle of slow rate of population mixing between subpopulations may in this system variant be additionally influenced by some degree of returning individuals; i.e., a deterministic component of population flow even of the immigration rate. This potential is facilitated by spatial memory and temporally multi-scaled displacements (parallel processing). In other words, system complexity.

Such complexity plays havoc with a classic metapopulation system. After occasional sallies to other habitat reserves/sub-populations, individuals may perform directed and potentially long distance returns to a previous reserve. And such returns could be a function of the respective individuals’ current conditions both at source and target reserves.

Consequently, individuals are – according to this paradigm-opposing assumption – occasionally able to transcend the matrix of intermediate environment between refuges in an energetically effective manner and with sufficiently low travel risk to make such coarse-scale moves statistically worthwhile and positive in fitness terms (see this post). This property of the Multi-scaled random walk (MRW) theory is now gaining additional anecdotal support, for example from studies on Fowler’s toads Anaxyrus fowleri (Marchand et al. 2017) and free-ranging bison Bison bison.

To be more precise, let’s first assume that we are considering a “several small” reserve system where the spatial scale (size) of these local reserves is reflecting – say – 50-60% of the median displacement distance of the respective population constituents at the time resolution of a reproductive season. If we consider day-to-day movement rather that the coarser time scale of a season, long displacements beyond this defined 50-60% limit accumulate a small part of the area under the displacement distribution (the scale-free dispersal kernel becomes more apparent at fine temporal scales). In this manner, due to the “thin” long-tale part of displacements under the premise of scale-free space use, the system complies with the basic metapopulation property of a low inter-season migration rate between subpopulations relative to the higher intra-population mixing rate at finer scales*. Also consider that the long-tail part of the displacement distribution extends to the scale of this  system’s spatial extent. Hence, individuals may occasionally displace themselves over the entire arena (exploratory sallies of various length and duration), and occasionally perform directed returns to previous locations within the same scale range.

This assumed property of scale-free and memory-influenced returns at the individual level violates the traditional metapopulation theory at the population level and – of course – the traditional SLOSS theory at the community level. In addition to the relatively high frequency mixing of individuals at the local scale a less frequent and partly memory-driven mixing takes place at coarser scales. The latter contributes to the emergence of an inter-connected network of complex migration.

The very interesting studies on snail kite Rostrhamus sociabilis plumbeus in Florida (Reichert et al. 2016; Valle et al. 2017) illustrates the profound potential this kind of system complexity may generate, with consequence also for community ecology in general and the SLOSS debate in particular. The results indicate that individuals of this dietary specialist show a surprising capacity to rapidly adapting to changing conditions over a large range of spatial scales from localized home ranges to state-wide network of snail-rich wetland patches. I cite from my discussion of Valle et al.’s paper:

For example; under multi-scaled space use, if distant patches show improvement with respect to key resources, a functional response driven by spatial memory and parallel processing may represent a net pull effect; i.e., expressed as a net directed emigration rate relative to the local habitat with more constant conditions.

Consequently, the actual “force” driving long-distance pull in a population could be explained as the coarse-scale experience that emerges from a low frequency of “occasional sallies” by an individual outside its normal day of life of habitat explorations. […] In my view it is not the distance as such that is that main point here (the snail kite can easily traverse long distances in s short period of time), but the fact that the natural experiment provided by the exotic snail showed how some distant patches occasionally showed stronger modular connectivity than intermediate patches. This property of space use is in direct violation of key assumptions of – for example – metapopulation theory (one of the branches of the Paradigm), where spatially close subpopulations cannot be more weakly connected than more distant subpopulations that are separated by intermediate ones.

In particular, observe that even long distance returns could be a function of the respective individuals’ current conditions both at source and target reserves. Referring to the illustration above, the relative number of directed returns (deterministic behaviour, red arrows) may suddenly – at a finer temporal scale than the inter-season population change –  increase or decrease substantially. The cause of such events cannot be understood simply from local conditions, as under traditional mechanistic modelling. Coarser scale conditions both in time and space need to be studied in parallel. A single, more or less fixed immigration rate as applied in traditional metapopulation modelling does not suffice. Citing from above, “…some distant patches occasionally showed stronger modular connectivity than intermediate patches“. This is system complexity in a nutshell, whether we are considering single- or multi-species scenaria. Much work is needed to achieve a better understanding of multi-scaled behaviour under influence of spatial memory.

NOTE

*) As explained in other posts (for example, here), a scale-free displacement curve for individuals is expected to become more truncated if the time scale is coarsened, for example from short term displacements to longer sampling intervals. The reason is the increased influence from intermediate return events as observation intervals slide from the fine-grained time scale of population kinetics (observing fast mixing from individual movement) towards the coarser-grained dynamics of changing dispersion for example at the scale of a season (Gautestad 2012).

REFERENCES

Diamond, J.M. 1975. The Island Dilemma: Lessons of Modern Biogeographic Studies for the Design of Natural Reserves. Biological Conservation 7:129–146.

Gautestad, A. O. 2012. Brownian motion or Lévy walk? Stepping towards an extended statistical mechanics for animal locomotion. Journal of the Royal Society Interface 9:2332-2340.

Reichert, B. E., R. J. Fletcher, C. E. Cattau, and W. M. Kitchens. 2016. Consistent scaling of population structure across landscapes despite intraspecific variation in movement and connectivity. Journal of Animal Ecology 85:1563-1573.

Robert, A. 2009. The effects of spatially correlated perturbations and habitatconfiguration on metapopulation persistence. Oikos 118:1590-1600.

Simberloff, D. S. and L. G. Abele. 1982. Refuge design and island biogeographic theory – effects of fragmentation. American Naturalist 120:41-56.

Valle, D., S. Cvetojevic, E. P. Robertson, B. E. Reichert, H. H. Hochmair and R. J. Fletcher. 2017. Individual Movement Strategies Revealed through Novel Clustering
of Emergent Movement Patterns. Scientific Reports 7 (44052):1-12.

Wilcox, B. A., and D. D. Murphy. 1985. Conservation strategy – effects of fragmentation on extinction. American Naturalist 125:879-887.

 

The Inner Working of Parallel Processing

The concept of scale-free animal space use becomes increasingly difficult to avoid in modeling and statistical analysis of data. The empirical support for power law distributions continue to pile up, whether the pattern appears in GPS fixes of black bear movement or in the spatial dispersion of a population of sycamore aphids. What is the general class of mechanism, if any? In my approach into this challenging and often frustrating field of research on complex systems, one particular conjecture – parallel processing (PP) – percolates the model architecture. PP requires a non-mechanistic kind of dynamics. Sounding like a contradiction in terms? To illustrate PP in a simple graph, let’s roll dice!

Please note: the following description represents novel details of the PP concept, still awaiting journal publication. Thus, if you are inspired by this extended theory of statistical mechanics to the extent that it percolates into your own work, please give credit by referring to this blog post (or my book). Thank you.

The basic challenge regards how to model a process that consists of a mixture of short term tactics and longer time (coarser scale) strategic goals. Consider that the concept of “now” for a tactical response regards a temporally finer-grained event than “now” at the time scale for executing a more strategic event, which consequently takes place within a more “stretched” time frame relative to the tactical scale.

Strategy is defined in a hierarchy theoretical manner; coarser scale strategy consequently invokes a constraint on finer scaled events (references in my book). For example, while an individual executes a strategic change of state like starting a relatively large-distance displacement (towards a goal), finer-scaled events during this execution – consider shorter time goals – are processed freely but within the top-down constraint that they should not hinder the execution of the coarser goals. Hence, the degrees of process freedom increases with the scale distance between a given fine-scaled goal and a coarser-scaled goal.

To illustrate such a PP-compliant scale range from tactics to strategy within an extended statistical-mechanical system, consider the two-dimensional graph to the right. The x-axis represents a sequence of unidirectional classic time and the y-axis represents a log2-scaled* expression of time’s orthogonal axis, “elacs” (ε) along this sequence.

The continuous x-y plane has been discretized for simpler conceptualization, and each (x,y) pair shows a die. This die represents a potential change of state of the given process at the given point in time and at the given temporal scale. An actual change of state at a given (t,ε) location is marked by a yellow die, while a white die describes an event still in process at this scale. The respective number of eyes on each die could represent a set of available states for a given system variable at this scale. To illustrate complex dynamics (over-)simplistically in terms of concepts from quantum mechanics, consider each magnitude of ε at the y-axis to represent a wave length in a kind of “complex system” wave function and each yellow die represents a “collapse” of this probability wave into a specific execution of the given event at a given point of unit time this time scale.

As the system is viewed towards coarser time scales (larger ε), the average frequency of change of state vanishes proportionally with 1/ε = 1/bz, where b is the logarithmic base and increasing z describes increasing scale level of bz. In other words, the larger the z, the more “strategic” a given event at this scale. In short, consider that each die on scale level 1 [log(b0)=1] is rolled at each time increment t=1, t=2, …, t=8; each die at level 2 [log(b1)=2] is on average rolled each second time increment, an so on.

In the illustrative example above, no events have taken place during the eight time increments at the two coarsest scales bz where z=7 (ε=128) and z=8 (ε=256). A substantial increase of the observation period would be needed to increase the probability of actually observing such coarse-scaled change of system state.

More strategic events are executed more rarely. Strategic events at a given scale bare initiated in a stochastic manner when observed from a finer time scale (smaller z), but increasingly deterministic when observed from coarser time scales. At finer scales such a strategic event may be inexplicable (thus appearing unexpectedly at a given point in time), while the causal relationship of the given process is established (visible) when the process is observed at the properly coarsened time scale. However, at each time scale there is an element of surprise factor, due influence from even coarser scale constraints and even lower frequency change of state of the system at these coarser scales. 

The unit time scale, log(b0)=1, captures the standard time axis, which is one-dimensional as long as the system can be described as non-complex. In other words, the y-axis’ dynamics do not occur, and – consequently – it makes no sense to talk about a parallel process in progress**. In this standard scale-specific framework, time is one-dimensional and describes scale-specific processes realistically. This includes the vast theories of low order Markovian processes (“mechanistic” modeling), the  mathematical theory of differential equations (calculus), and standard statistical mechanics.

For a deeper argument why a PP kind of fundamental system expansion seems necessary for a realistic description of system complexity, read my book and my previous blog posts. By the way, it should of course be considered pieces of a theoretical framework in progress.

The ε-concept was introduced in my book to allow for complex dynamics within a non-Markovian physical architecture. In other words, to allow for a proper description of parallel processing the concept of time as we know it in standard modeling in my view needs to be heuristically expanded to a two-dimensional description of dynamics.

The bottom line: it works! In particular, it seems to survive the acid tests when applied on empirical data, both with respect to individual space use and population dispersion.

Environment is hereby expanded with a two-dimensional representation of dynamical time. This implies that an individual’s environment not only consists of its three-dimensional surroundings at a given point in time but also its temporal “surroundings” due to the log compliant (scale-free) scale-stretching of time. In this manner an implementation of parallel processing turns the common Markovian, mechanistically modeled framework into a special case. According to the special case of standard mechanistic dynamics a given process may be realistically represented either by a scale-specific process at a given (unit) scale or a trivial linear superposition of such processes (e.g., a composite random walk toggling between different magnitudes of the diffusion parameter for each “layer”). On the other hand, complexity arises when such a description that is based on one-dimensional time is not sufficient to reproduce the system realistically.

Observe that in a PP-system several events (change of system state) may be executed in parallel! In the illustration above, see for example the situation for t=5 where events at three time scales by chance are initiated simultaneously but at different time scales as defined by ε. Such a kind of dynamics represents a paradox within the constraint of a Markovian (mechanistic) system.

An earlier illustration of the PP framework was given here. For other examples, search this blog for “parallel processing” or read my book.

Various aspects of scaling in animal space use; from power law scaling of displacement lengths (Lévy like distribution), fractal dispersion of GPS fixes (the home range ghost model) and scale free distribution of populations (Taylor’s power law and the Zoomer model) may be natural outcomes of systems that obey the PP conjecture.

NOTE

*) The base, b, of the logarithm does not matter. Any positive integer introduces scaling of the ε-axis.

**) in a standard, mechanistic process an event describes a change of system state at a given point in space at a given point it time. No “time stretching” takes place.

 

Positive and Negative Feedback Part II: Populations

Examples of positive feedback loops in population dynamics abound. Even if the majority of models are focusing on negative feedback, like the logistic growth function, non-equilibrium “boom and bust” kind of model designs have also been developed. In this post I elaborate on the particular kind of positive feedback loop that emerges from cross-scale dual-direction flow of individuals that is based on the parallel processing conjecture.

The image to the right illustrates – in simplistic terms – a spatially extended population model of standard kind (e.g., a coupled map lattice design) where each virtually demarcated local population j at spatial resolution i and at a given point in time t contains Nij individuals. No borders for local migration are assumed; i.e., the environment is open both internally and externally towards neighbouring sites.Typically, these individuals are set to be subject to a locally negative feedback loop in accordance to principles of density dependent regulation*. The larger the N the larger the probability of an increased death rate and/or and increased emigration rate from time t to t+1, eventually leading both the local and the over-all population to a steady state. This balancing** condition lasts until some change (external perturbation) is forcing the system into a renewed loop of negative feedback-driven dynamics. In a variant of this design, density regulation may be formulated to be absent until a critical local density is reached, leading to boom and bust (“catastrophic” death and emigration), which may be more or less perturbed by random immigration rate from asynchronous developments in respective surrounding Nij. More sophisticated variants abound, like inclusion of time lag responses, interactions with other trophic levels, and so on.

As previously explained in other posts, this kind of model framework depends on a premise of Markov-compliant processes at the individual level (mechanistic system), and thus also at the population level (local or global compliance with the mean field principle). In this framework intrinsic dynamics may be density dependent or not, but from the perspective of a given Nij, extrinsic influence – like immigration of individuals – is always stochastic and thus density independent with respect to Nij.  In other words, the net immigration rate during a given time increment is not influenced by the state of the population in this location (i,j). You can search my blog or read my book to find descriptions and details on all these concepts.

To implement cross-location and dual-direction deterministic dynamics, multi-scaled behaviour and spatial memory needs to be introduced. My parallel processing conjecture; which spins off various testable hypotheses, creates turmoil in this standard system design for population dynamics because it explicitly introduces such system complexity. For example, positive feedback loops may emerge. Positive feedback as described below may effectively also counteracting the paradoxical Allée effect, which all “standard” population models are confronted with at the border zone of a population in an open environment**.

The dynamic driver of the complexity is the introduction of spatial memory in combination with a scale-free kind of dynamics along both the spatial and the temporal dimensions. In statistical-mechanical terms, parallel processing is incompatible with a mechanistic system. Thus, a kind of extended statistical mechanics is needed. I refer to the post where I describe the scale-extended description of a metapopulation system.

For the most extensive individual-level test of the parallel processing conjecture until now (indirectly also verifying positive feedback of space use), see our paper on statistical analysis of space use by red deer Cervus elaphus (Gautestad et al. 2013; Gautestad and Mysterud 2013). In my blog I have also provided several anecdotal examples of third party research potentially supporting the parallel processing conjecture. For the sake of system coherence, if parallel processing is verified for individual space use of a given species and under given ecological conditions, this behaviour should also be reflected in the complementary population dynamical modelling of the given species and conditions.

Extending the standard population model. As explained in a range of blog posts, my Zoomer model represents a population level system design that is coherent with the individual-level space use process (in parsimonious terms), as formulated by the Multi-scaled random walk model. In my previous post I described the latter in the context of positive feedback from individual-level site fidelity. Below I illustrate positive feedback also at the population level, where site fidelity get boosted by conspecific attraction. In other words, conspecifics become part of the individuals’ resource mapping at coarser scales, as it is allowed for by spatial memory. Consequently, a potential for dual-direction deterministic flow of individuals is introduced (see above). Conspecific attraction is assumed to be gradually developed by individual experience of conspecifics’ whereabouts during exploratory moves.

In the Zoomer model , some percentage of the individuals are redistributing themselves over a scale range during each time increment. Emigration (“zooming out”) is marked by dotted arrows, and immigration (“zooming in”) is shown as continuous-line arrows. Numbers refer to scale level of the neighbourhood of a given locality. This neighbourhood scales logarithmically; i.e., in a scale-free manner, in compliance with exploratory moves in the individual-level Multi-scaled random walk model. Zooming in depends on spatial memory by the individuals, and introduces a potential for the emergence of positive feedback at the population level.

First, consider the zooming process, whereby a given rate, z, of individuals (for example, z=5% on average at a chosen time resolution Δt) at a “unit” reference scale (k=i) are redistributing themselves over a scale range beyond this unit scale***. During a given Δt consider that 100 individuals become zoomers from the specific location marked by the white circle. In parallel with the zooming out-process the model describes a zooming in-process with a similar strength. The latter redistributes the zoomers in accordance to scale-free immigration of individuals under conspecific attraction.Thus, number of individuals (N) at this location j at scale i, marked as Nij, will at the next time t+1 either embed N-100 individuals if they all leave location j and end up somewhere in the neighbourhood of j, or the new number will be N -100 + an influx of immigrants, where these immigrants come from the neighbourhood at scale i (those returning home again), scale i+1 (immigration from locations nearby), i+2 (from an even more distant neighbourhood), etc.

In the ideal model variant of zooming we are thus assuming a scale-free redistribution of individuals during zooming, with zooming to a neighbourhood at scale ki+x takes place with probability 1/ki+x (Gautestad and Mysterud 2005). Under this condition, zoomers to successively coarser scales become “diluted” over proportionally larger neighbourhood area, the maximum number of immigrants in this example is 100 + N’, where N’ is the average number of zoomers pr. location at unit scale k=i within the coarsest defined system scale k=i(max) for zooming surrounding location j at scale i.

As a consequence of this kind of scale-free emigration of zoomers, the population system demonstrates zooming with equal weight of individual redistribution from scale to scale over the defined scale range (Lévy-like in this respect, with scaling exponent β≈2; see Gautestad and Mysterud 2005). By studying the distribution of step lengths, this “equal weight” hypothesis may be tested, when combinded with othe rstatistical fingerprints (in particular, verifying memory-dependent site fidelity; see Gautestad and Mysterud 2013).

Putting this parsimonious Zoomer model with its system variables and parameters into a specific ecological context implies a huge and basically unexplored potential for ecological inference under condition of scale-free space use in combination with site fidelity.

Positive feedback in the Zoomer model. As shown in my series of simulations of the Zoomer model a few posts ago, a positive feedback loop emerges from locations with relatively high abundance of individuals having a relatively larger chance of received a net influx of zoomers during the next increment, and vice versa for locations with low abundance. The positive feedback emerges from the conspecific attraction process, linking the dynamics at different scales together in a parallel processing manner.

This positive feedback loop from conspecific attraction also counteracts extinction from a potential Allée effect (see this post and this post), which have traditionally been understood and formulated from the standard population paradigm. The Zoomer model represents an alternative description of a process that effectively counteracts this effect.

NOTES

*) The migration rates connects the local population to surrounding populations. Immigration is – by necessity from the standard model design – density independent with respect to the dynamics in Nij.

**) Since the process is assumed to obey a Markovian and the mean field principles (standard, mechanistic process), the arena and population system must either be assumed to be infinitely large or the total set of local populations has to be assumed to be demarcated by some kind of physical border. Otherwise, net emigration and increased death rate in the border zone will tend to drive N towards zero  in open environments (extinction from standard diffusion in combination with local N drifting below critical density where Allée kicks in). Individuals will “leak” from an open border zone to the surroundings where N is lower.

***) The unit temporal scale for a population system should be considered coarser than the unit scale at the individual level, since the actual scale range under scrutiny typically is larger for population systems. In particular, to find the temporal scale where for example 5% of the local population can be expected to be moving past the inter-cell borders of a given unit spatial grid resolution ki=1, one should be expected to find Δt substantially larger than Δt at the individual level.

Consider that the difference in Δt is a function of the difference of the area of short-range versus long range displacements under the step length curve for individual displacements, where the ∼5% long-step tail of this area represents the relative unit time in comparison to the rest of the distribution (thereby defined as intra-cell moves). Since this area is a fraction of the area for the remaining 95% of the displacements, the difference in Δt should scale accordingly.

REFERENCES

Gautestad, A. O., and I. Mysterud. 2005. Intrinsic scaling complexity in animal dispersion and abundance. The American Naturalist 165:44-55.

Gautestad, A. O., and A. Mysterud. 2013. The Lévy flight foraging hypothesis: forgetting about memory may lead to false verification of Brownian motion. Movement Ecology 1:1-18.

Gautestad, A. O., L. E. Loe, and A. Mysterud. 2013. Inferring spatial memory and spatiotemporal scaling from GPS data: comparing red deer Cervus elaphus movements with simulation models. Journal of Animal Ecology 82:572-586.

 

The Balance of Nature?

To understand populations’ space use one needs to understand the individual’s space use. To understand the individuals’ space use one needs to acknowledge the profound influence of spatio-temporal memory capacity combined with multi-scale landscape utilization, which continues to be empirically verified at a high pace in a surprisingly wide range of taxa. Complex space use has wide-ranging consequences for the traditional way of thinking when it comes to formulate these processes in models. In a nutshell, the old and hard-dying belief in the balance of nature needs a serious re-formulation, since complexity implies “strange” fluctuations of abundance over space, time and scale. A  fresh perspective is needed with respect to inter-species interactions (community ecology) and environmental challenges from habitat destruction, fragmentation and chemical attacks. We need to address the challenge by rethinking also the very basic level of how we perceive an ecosystem’s constituents: how we assume individuals, populations and communities to relate to their surroundings in terms of statistical mechanics.

Stuart L. Pimm summarizes this Grand Ecological Challenge well in his book The Balance of Nature? (1991). Here he illustrates the need to rethink old perceptions linked to the implicit balancing principle of carrying capacity*, and he stresses the importance of understanding limits to how far population properties like resilience and resistance may be stretched before cascading effects appear. In particular, he advocates the need to extend the perspective from short-series local-scale population dynamics to long-term and broad scale community dynamics. In this regard, his book is as timely today as it was 27 years ago. However, in my view the challenge goes even deeper than the need to extending spatio-temporal scales and the web of species interactions.

Balancing on a straw – an Eurasian wren Troglodytes troglodytes (photo: AOG).

My own approach towards the Grand Ecological Challenge started with similar thoughts and concerns as raised by Pimm**. However, as I gradually drifted from being a field ecologist towards actually attempting to model parsimonious population systems I found the theoretical toolbox to be void of key instruments to build realistic dynamics. In fact, the current methods were in many respects even seriously misleading, due to what I considered some key dissonant model assumptions.

In my book (Gautestad 2015), and here in my subsequent blog, I have summarized how – for example – individual-based modelling generally rests on a very unrealistic perception of site fidelity (March 23, 2017: “Why W. H. Burt is Now Hampering Progress in Modern Home Range Analysis“). I have also found it necessary to start from scratch when attempting to build what I consider a more realistic framework for population dynamics (November 8, 2017: “MRW and Ecology – Part IV: Metapopulations?“), for the time being culminating with my recent series of post on “Simulating Populations” (part I-X).

I guess the main take-home message from the present post is:

  • Without a realistic understanding; i.e., modelling power, of individual dispersion over space, time and scale it will be futile to build a theoretical framework with deep explanatory and predictive value with respect to population dynamics and population ecology. In other words, some basic aspects of system complexity at the “particle level” needs to be resolved.
  • Since we in this respect typically are considering either the accumulation of space use locations during a time interval (e.g., a series of GPS fixes) or a population’s dispersion over space and how it changes over time, we need a proper formulation of the statistical mechanics of these processes.In other words, when simplifying extremely complicated systems into a manageable set of smaller set of variables, parameters and key interactions, we have to invoke the hidden layer.
  • With a realistic set of basic assumptions in this respect, the modelling framework will in due course be ready to be applied on issues related to the Grand Ecological Challenge – as so excellently summarized by Pimm in 1991. In other words, before we can have any hope of a detailed prediction of a local or regional faith of a given species or community of species under a given set of circumstances, we need to build models that are void of the classical system assumptions that have cemented the belief in the so-called balance of nature.

NOTES

*) The need to rethink the concept of carrying capacity and accompanying “balance” (density dependent regulation) should be obvious from the simulations of the Zoomer model. Here a concept of carrying capacity (called CC) is introduced at a local scale only, where – logically – the crunch from overcrowding is felt by the individuals. By coarse-graining to a larger pixel than this finest system resolution we get a mosaic of local population densities where each pixel contains a heterogeneous collection of intra-pixel (local) CC-levels. If “standard” population dynamic principles applies, the population change when averaging the responses over a large number of pixels with similar density should be the same whether one considers the density at the coarser pixel or the average density of the embedded finer-grained sub-pixels. This mathematical simplification follows from the mean field principle. In other words, the sum equals the parts. On the other hand, if the principle of multi-scaled dynamics applies, two pixels at the coarser scale containing a similar average population density may respond differently during the next time increment due to inter-scale influence. At any given resolution the dynamics is as a function not only of the intra-pixel heterogeneity within the two pixels but also of their respective neighbourhood densities; i.e., the condition at an even coarser scale. The latter is obviously not compliant with the mean field principle, and thus requires a novel kind of population dynamical modelling.

**) In the early days I was particularly inspired by Strong et al. (1984), O’Neill et al. (1986) and L. R. Taylor; for example, Taylor (1986).

REFERENCES

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

O’Neill, R. V., D. L. DeAngelis, J. B. Wade, and T. F. H. Allen. 1986. A Hierarchical Concept of Ecosystems. Monographs in Population Biology. Princeton, Princeton University Press.

Pimm, S. L. 1991, The balance of nature? Ecological issues in the conservation of species and communities. Chicago, The University of Chicago Press.

Strong, D.E., Simberloff, D., Abele, L.G. & Thistle, A.B. (eds). 1984. Ecological Communities: Conceptual Issues and the Evidence. Princeton,Princeton University Press.

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

Simulating Populations X: Confronting Theory With Real Data

In Part IX a standard Coupled map lattice model was shown to be able to display a 1/f power spectrum, by careful tuning of one of the conditions for population mixing. On the other hand, I also showed that the Zoomer model under similar general conditions showed a higher resilience with respect to the 1/f property. In the present post I explore this statistical resilience of the Zoomer model further, by stressing the population system towards other corners of extreme conditions. However, I start by elaborating on this pressing question: Why this recurrent focus on 1/f noise? To stimulate your curiosity I also reproduce from my book two analyses of empirical data; the sycamore aphid Drepanosiphum platanoides and the leaf miner Leucoptera meyricki

First, a brief bird’s view of complex dynamics. I have repeatedly given partly answers to the question “why focusing on 1/f noise?”, but here I seek to give you a broader perspective, starting with a citation from Scholarpedia:

1/f fluctuations are widely found in nature. During 80 years since the first observation by Johnson (1925), long-memory processes with long-term correlations and 1/fα (with 0.5 ≲ α ≲ 1.5) behavior of power spectra at low frequencies f have been observed in physics, technology, biology, astrophysics, geophysics, economics, psychology, language and even music…
…1/f noise can not be obtained by the simple procedure of integration or of differentiation of such convenient signals. Moreover, there are no simple, even linear stochastic differential equations generating signals with 1/f noise. The widespread occurrence of signals exhibiting such behavior suggests that a generic mathematical explanation might exist. Except for some formal mathematical descriptions like fractional Brownian motion (half-integral of a white noise signal), however, no generally recognized physical explanation of 1/f noise has been proposed. Consequently, the ubiquity of 1/noise is one of the oldest puzzles of contemporary physics and science in general.
Ward and Greenwood (2007)

As scientists – whether we are ecologists or working in other fields – we are of course curious about exploring this hard nut to crack, in particular since natural populations (paradoxically, if judged by expectation from standard theory) tend to show 1/f noise in their spatial variation and temporal fluctuations. Here are two examples from my book:

Both series show close agreement with 1/f noise; and their respective derivatives trivially show similar compliance with f noise (power changing proportionally with frequency, shown as dashed plots). The first example is a spatial transect of sycamore aphids (my own data), and the second example shows my power spectrogram analysis based on a time series collected from Bigger and Tapley (1969) of the coffee leaf miner Leucoptera meyricki. Both data sets and methods are described in detail in my book.

A flock of ruffs Philomachus pugnax along the coast of Norway. Migrating birds do not follow diffusion laws and they do not follow biased random walk models. They relate to their environment over a wide range of spatio-temporal scales, and recent research verifies that they are influenced by a memory map. Example of complex migration: see this post. Photo: AOG.

The burning question then arises: what kind of spatially extended population model may reproduce 1/f noise over space, time and scale? Within the current paradigm the coupled map lattice (CML) approach fails, since it by design cannot reproduce long distance and far back influence on a present location’s dynamics*.The standard calculus approach (ordinary and partial differential equations) also fails, since models of this design rests on the mean field approximation. In short a novel approach is needed. As previously stated, for the time being and to my knowledge the Zoomer model is the only candidate standing the test when confronting simulations with real data properties with respect to statistical mechanics.

In a series of posts I have now in a step-by step manner presented the Zoomer model, and compared it with the paradigmatic approach in the field, the coupled map lattice design (CML). In Part IX the CML model was shown to be more sensitive to the population’s response to local overcrowding (the CC level). Only 100% redistribution of individuals where CC was temporally exceeded led to a 1/f pattern. The Zoomer model showed 1/f, whether 40% or 100% of individuals emigrated. In other words, strong statistical resilience in this regard. However, the best way to test a model’s intrinsic dynamics is to stress it to its limit. Thus, below I test the effect from changing the redistribution behaviour of the individuals that emigrate from the “CC overshoot” localities.

So far, these individuals have been set to redistribute themselves to other cells in “swarms” of X% each of all swarmers at the actual point in time. For example, if two cells have an overshoot event at this point in time with a total number of dispersers of – say – 10,000 individuals, these dispersers settle in 5% of the other cells (of 1024 cells available) in swarms of ≈ 195 individuals pr. receiving cell, chosen randomly.

Below I explore two variants of this behaviour; (1) the dispersers settle randomly and individually among the available cells, and (2) the dispersers settle in only 1% of the cells. The former implies a more smooth dispersal kernel, and the latter a more collective kind of large swarms settling in just a few cells.

 

 

In short, the power spectrum of the time series of the smooth dispersal condition shows white noise (1/f0), while the extremely clumped dispersal shows more like red noise (1/f1.8).**

To conclude, the 1/f property in the Zoomer model seems to be critically sensitive to the behaviour of the dispersing individuals during local crowding (boom/bust) events. Whether the bust leads to local meltdown (0% remaining) or not (60% remaining), the 1/f pattern was robust. However, how the emigrating individual re-settled themselves (independently or in a contagious manner) determined 1/f compliance. An intermediate level of contagion was in best compliance with 1/f, and thereby with the empirical results shown above. Intermediate contagion implies intermediate level of population disturbance frequency***.

This conclusion also fits well with the general model property of zooming, which reflects conspecific attraction in over-all terms. The “balanced” and intermediate kind of zooming follows prom the generic condition that the strength of zooming is distributed evenly over that actual range of spatial scales (kb zoomers to kb larger grid area, with b=1). However, while zoomers re-distribute themselves contagiously in a spatially explicit manner from a process that depends on the individuals’ spatial memory of past experiences, the swarms of emigrants in the current Zoomer design settle randomly. “Stressed” individuals (emigrants from local crunch events) are assumed to redistribute themselves in a tactical manner, temporarily following the collective behaviour of a swarm during the actual time increment.

 

Interestingly, the spatial pattern log(M,V) for the white noise condition shows – as expected – a similarly rattled pattern, with slope b ≈ 1 and log(a) >> 0. However, in the contageous dispersal scenario, the spatially self-similar property b ≈ 2 and log(a) ≈ 0 is maintained. Hence, the spatial pattern is still fractal-compliant with similar properties as for other levels of contagion during re-distribution events, despite the more classic “random walk-like” time series of fluctuations.

To conclude, by studying a population’s variation of abundance over space, time and scale it should be possible to analyze a wide range of key ecological signatures from the data series’ statistical properties; for example to what extent the population under the given environmental conditions adheres to a multi-scaled and scale-free kind of intrinsically driven population dynamics/kinetics, and how the population responds to local crowding events. Hopefully, this statistical-mechanical approach may lead to more realistic theory and thus better predictive power. Bringing population dynamical modelling and some basic empirical properties of real populations in closer agreement is long overdue in this important field of ecology.

Crucially, these basic analyses should also be able to cast light on to what extent the dynamics are compliant with expectations from a classical modeling regime; i.e., a Markov- and mean field compliant kind of statistical mechanics, or the alternative framework based on the parallel processing conjecture for individual space use (the MRW model and its population level version, the Zoomer model). This challenge is of course the first obstacle that has to be passed. Hopefully the two insect examples above will inspire others to perform broader and deeper tests – for example, based on my recent series of blog posts.

REFERENCES

Bigger, M., and R. G. Tapley. 1969. Prediction of outbreaks of coffee leaf-miners on
Kilimanjaro. Bulletin of Entomological Research 58:601-618.

Ward, L. M. and P. E. Greenwood. 2007. 1/f noise. Scholarpedia 2(12):1537.

NOTES

*) The 1/f pattern from a CML example in Part IX was a 1/f look-alike due to some tweaking of conditions, but it failed when other statistical aspects were scrutinized.

**) As in previous scenaria, the time scale is set to be smaller than the population’s net growth rate of 0-1%, giving a focus of the higher rate population mixing processes; diffusion 1% for the CML examples, zooming 5% in the Zoomer examples (distributed with 1% pr. spatial scale), and a stochastic magnitude of intrinsic re-distribution of individuals from local CC-linked events under both platforms.

***) By intermediate frequency of disturbance from intrinsic reshuffling of individuals I mean on one side that a given locality’s frequency of disruptions by swarms of dispersing individuals from other localities depends on swarm size. Larger swarms mean fewer locations where they settle. On one hand, large swarms (stronger contagion among emigrants) appear more rarely at a given locality in statistical terms but will have stronger local impact when a swarm arrives. On the other hand, an additional disruptive event from such an influx may happen within the next time increment if the sum of existing and arriving individuals surpass the local CC level. One also has to consider that a higher local abundance due to arriving immigrants  may change a declining local abundance to an increasing one due to the zooming process of conspecific attraction.The abundance level as seen from a broader neighbourhood scale may increase sufficiently to toggle this neighbourhood (including the actual cell receiving the immigrants) from being a net supplier; i.e., subject to net local population decline, to becoming net receiver of individuals in the time ahead.

This perspective again underscores the importance of studying population dynamics not only over space and time, but also over respective scale axes and in a Parallel processing manner. Hence, a Markovian-compliant model framework is not sufficient to understand complex dynamics/kinetics, including the 1/f property.