Accepting spatial Memory: Some Alternative Ecological Methods

In this post I present a guideline that summarizes how a memory-based model with an increasing pile of empirical verification covering many species – the Multi-scaled Random Walk model (MRW) – may be applied in ecological research. The methods are in part based on published papers and in part based on some of the novel methods which you find scattered throughout this blog. 

In the following, let us assume for a given data set that we have verified MRW compliance (using the standard memory-less models or alternative memory-implementing models as null hypotheses) by performing the various tests that have already been proposed in my papers, blog posts and book. Typically, a standard procedure should be to verify (a) site fidelity; i.e., presence of a home range, (b) scale-free space use by studying the step length distribution from high frequency sampling, and (c) the fractal dimension D≈1 of the spatial scatter of relocations in the resolution range between the dilution effect (very small pixels) and the space fill effect (very large pixels).

The obvious next step is to explore specific ecological hypotheses using the MRW as the model for space use. Here follows a quick tutorial:

  1. Study the Home range ghost model, I(N) = cNz, to estimate c, the individual’s characteristic scale of space use (CSSU). Search my blog for methods how to optimize grid resolution, and in particular also consider the recent breakthrough 11 months ago where I show how to estimate CSSU from auto-correlated data. Variations in CSSU quantifies difference in intensity of space use, which logically illuminates aspects of habitat selection.
  2. After proper estimation of CSSU study the power exponent, z, of the Home range ghost model. If your data lands on z≈0.5 – the default condition – you have verified that the individual has not only utilized its environment in a scale-free manner but also has put “equal weight” into relating to its habitat across the spatial scale range within its home range. On the other hand, finding 0.2<z<0.3 indicates that a model for an alternative movement class, the Markov-compliant MemRW, may be more suitable for your data. 0 < z < 0.3 indicates that the individual has concentrated its space utilization primarily towards finer resolutions, like you would expect from a Markov-compliant kind of cognitive processing. More detailed procedures should be applied to select model framework, since MemRW and MRW describes qualitatively disparate classes. Observe that the traditional KDE method is not able to discriminate between these classes; hence, typically showing weak N-dependence on area demarcation, due to its implicit assumptions that successive revisits to a local patch are independent events and that the process should be memory-less (see link above).
  3. Is the individual’s space use stationary, or is the home range drifting over time? Spatial autocorrelation in your series of fixes typically has two causes; high-frequency sampling of fixes from space use relative to a slower return frequency (ρ>>1; see my previous post) or high- or medium-frequency fix sampling under the condition of a drifting home range. Split the data into several subsets of magnitude Ns where the number of fixes (N) in each set is constant. Then study the overlap pattern of incidence I(N) at spatial resolution of CSSU (see method here). Low degree of overlap between successive subsets implies a non-stationary kind of home range. By comparing non-adjacent subsets in time one may even quantify the degree of non-stationarity (the speed by which the space use is drifting). These results can then be interpreted ecologically.
  4. What about the fractal dimension of the total set of fixes, for example by applying the box counting method? By default one expects D≈1 when z=0.5. Deviations from D=1 over specific spatial resolutions can be interpreted ecologically. For example, 1.5<D<2 at the coarsest resolutions may indicate that space use is constrained by some kind of borders. However, it could also appear from missing outlier fixes in the set (Gautestad and Mysterud 2012) or a simple statistical artifact (the space fill effect). On the other end, 1<<D can be hypothesized to emerge where the animal has concentrated its space use among a set of fine-scale patches rather than scattering is optimization more smoothly (in a statistical sense) over a wider range of scales. In Gautestad (2011) I simulated central place foraging, where i found 0.7<D<1. More sophisticated but logically simple methods can contribute to various system properties and statistical artifacts that contribute to deviation from D≈1, for example by varying the sample size of fixes as illustrated in the Figure to the right (copied from the link above).

The MRW theory also offers several other methods to study ecological and biological aspects of space use. For example, the data may reveal whether the temporal memory horizon has been constrained or unlimited (infinite memory, or remembering previous visits only over a limited, trailing time window). Temporally constrained memory will be shown by example in my next post. For more theoretical or technical details of the methods above please search this blog for the actual term, or find references in the subject index of my book.

REFERENCES

Gautestad, A. O. 2011. Memory matters: Influence from a cognitive map on animal space use. Journal of Theoretical Biology 287:26-36.

Gautestad, A. O. and I. Mysterud. 2012. The Dilution Effect and the Space Fill Effect: Seeking to Offset Statistical Artifacts When Analyzing Animal Space Use from Telemetry Fixes. Ecological Complexity 9:33-42.

Temporally Constrained Space Use, Part III: Critique of Common Models

There is no doubt among field ecologists that animals from a broad range of taxa and over wide range of ecological conditions utilize their environment in a spatial memory-influenced manner. Spatial map utilization have now been verified also well beyond vertebrates, like dragonflies and some solitary wasps. To me at least it is thus a mystery why theoretical models that are void of influence from a memory map; for example ARS, Lévy walk and CTRW (see Part I, II), are still dominating ecological research with mostly no critical questions asked about their feasibility.

It is a fact that the memory-less mainstream models all have a premise that the data should not be influenced by map-dependent site fidelity. In other words, applying ARS, Lévy walk and CTRW models as stochastic representation of space use also implies accepting that the animal’s path is self-crossing by chance only, and not influenced by targeted returns. Such returns can be expected to seriously disrupt results on – for example – habitat selection, since self-reinforcing patch utilization (positive feedback) obviously becomes a serious issue for methods that are based on memory-less space use where revisits are statistically independent events.

Despite performing hypothesis tests on data that obviously contradicts this hidden assumption about lack of spatial memory influence, for example movement in a home range context (where the home range is an emergent property from such returns), memory-less models are applied by cultural instinct or a misconception that alternatives do not exist. “Everybody else is using these standard models, so why not me?”

This attitude obviously hinders space use-related ecological research on its path towards becoming hard science at the level we are used to find in physics, chemistry and geology; i.e., models with strong predictive power. The laid-back excuse that animal ecology is not only more complicated but also basically more complex does not hold anymore. Biophysical research, for example based on inspiration from – or developed in compliance with – my parsimonious MRW model (Song et al. 2010; Boyer et al. 2012; Boyer and Solis-Salas 2014; Mercado-Vásquez and Boyer 2018), show how even complex space use systems may now be treated analytically with success.

So far, there still exists only one book (Gautestad 2015) that is dedicated to criticizing the sloppy culture of model selection in ecological research. The statistical errors that follow from ignoring the frequently violated assumption about memory-less space use are percolating both my book and my blog*.

MRW implements a combination of scale-free space use with memory-dependent, occasional returns to previous sites in accordance to the parallel processing conjecture. The average return interval tret to a previously visited location relative to the sampling interval tobsρ = tret/tobs, will lead to different analytical results a a function of ρ.

This important ratio defines how the observed distribution of step lengths is a function of  memory-influenced movement that complies with the MRW formulation: a mixture of scale-free exploratory steps and occasional returns to a previous location. I cite from Part II:

If the animal in question is utilizing spatial memory a lot of confusion, paradoxes and controversy may thus appear if the same data are analyzed on the basis of erroneously applying memory-less models within different regimes of ρ!

For example, an decreasing tret for a given tobs implies stronger site fidelity. The variable observer effect that is expressed by tobs becomes apparent within a quite wide transition range around tobs ≈ tret. For example, a Brownian motion-like form of the step length distribution may erroneously be found if ρ << 1, and a power law form can be expected when ρ >> 1, with truncated power law to be observed in-between. However, power law compliance may arise both in scale-free but spatially memory-less behaviour (Lévy walk) and MRW when ρ >> 1. Recall that MRW implies a combination of spatially memory-influenced and Lévy walk-like kind of movement in statistical terms.

The step length distributions to the right (Gautestad and I. Mysterud 2005)  illustrates from MRW-simulated data the effect on changing the ratio ρ >> 1 towards ρ < 1 apparently makes the step length distribution shape-shifting from a power law (apparently Lévy) to a negative exponential (apparently Brownian). This paradoxical pattern appears simply from changing sampling frequency of a given series of successive relocations.  As observation frequency becomes larger than the return frequency the paradox appears from comparing the expectation from erroneous of model based on the memory-less space use assumption; i.e., Brownian motion vs. Lévy walk. 

The Figure to the right (Gautestad and A. Mysterud 2013) illustrates the same transition more graphically. The hump (blue colour)  that is observed for for ρ = 10 towards the extreme tail of the distribution, leading to a hump-like “hockey stick” pattern, becomes almost invisible at ρ = 100 Appendix 1 in Gautestad and A. Mysterud, 2013; see also Gautestad 2012). This gradual appearance/disappearance of the hockey stick as a function of ρ >> 1 illustrates the pseudo-LW aspect of MRW. By the way, such a “hump” on the tail part of a power law distribution has in fact been found and commented in several analyses of empirical data. Citing from Gautestad and A. Mysterud (2013):

It is interesting that one of the main issues raised in this  respect regards the “problematic” occasional over-representation of very long step lengths even relative to an ideal Lévy walk distribution, invoking the term “Lévy walk-like” search (Sims and Humphries 2012; Sims et al. 2012). This “hump” in the long tail part of the distribution has been hypothesized to emerge from some kind of environmental forcing (Sims and Humphries 2012). However, here we have shown (Figure 3) that a similar hump – called a hockey stick – is in fact expected by default if MRW-compliant data are analysed within a specific range of the ratio between return events and observation interval.
Gautestad and Mysterud 2013, p14.

The take-home message from these two examples is stressing the importance of testing for spatial memory before choosing which statistical model(s) to apply for a specific analysis.

NOTE

*) In my research I also criticize memory-implementing models where spatial utilization beyond the individual’s current perceptual field builds on a mechanistic (Markov-compliant) kind of information processing. See, for example, this post. Consequently, in the Scaling cube, these Markov models are located in the lower right corner (MemRW), in contrast to the “parallel processing”-based MRW, which you find in the upper right corner. In Gautestad et al. (2013) we tested these alternative model classes on red deer Cervus elaphus, and found strong support for the MRW framework. The red deer moved both in compliance with a scale-free space utilization, in parallel with site fidelity from targeted returns in a manner which supported parallel processing. Additional research has also given support to to MRW lately; for example see Merkle et al. (2014), who tested a set of contemporary hypotheses on memory-influenced movement in free-ranging bison Bison bison and found support for a central premise of MRW in the summer ranges of this species.

REFERENCES

Boyer, D., M. C. Crofoot, and P. D. Walsh. 2012. Non-random walks in monkeys and humans. Journal of the Royal Society Interface 9:842-847.

Boyer, D. and C. Solis-Salas. 2014. Random walks with preferential relocations to places visited in the past and their application to biology. arXiv 1403.6069v1:1-5.

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.

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 I. 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.

Mercado-Vásquez, G. and D. Boyer. 2018. Lotka-Volterra systems with stochastic resetting. arXiv:cond-mat.stat-mech:1809.03975v03971.

Merkle, J. A., D. Fortin, and J. M. Morales. 2014. A memory-based foraging tactic reveals an adaptive mechanism for restricted space use. Ecology Letters Doi: 10.1111/ele.12294.

Sims, D. W. and N. E. Humphries. 2012. Lévy flight search patterns of marine predators not questioned: a reply to Edwards et al. ArXiv 1210.2288: [q-bio.PE].

Sims D. W, N. E. Humphries, R. W. Bradford and B. D. Bruce. 2012. Lévy flight and Brownian search patterns of a free-ranging predator reflect different prey field characteristics. Journal of Animal Ecology 81:432-442.

Song, C., T. Koren, P. Wang, and A.-L. Barabási. 2010. Modelling the scaling properties of human mobility. Nature Physics 6:818-823.

Temporally Constrained Space Use, Part II: Approaching the Memory Challenge

In Part I three models for temporally constrained space use were summarized. Here in Part II I put them more explicitly into the context of ecology with focus on some key assumptions for the respective models. Area restricted search (ARS), Lévy walk (LW) and Continuous time random walk (CTRW) are statistical representations of disparate classes of temporally constrained space use without explicit consideration of spatial memory effects. Hence, below I reflect on a fourth model, Multi-scaled Random Walk (MRW), where site fidelity gets a different definition relative to its spatially memory-less counterparts.

A cattle egret Bubulcus ibis is foraging within a wide perimeter surrounding its breeding site. Spatial memory is utilized not only to be able to return to the nest but also to revisit favored foraging locations during a bout, based on a memory map of past experience. Photo AOG.

First, ARS is typically formulated as a composite random walk-like behaviour in statistical terms, which could be suitable for situations where a Markovian compliant (“mechanistic”) behaviour is either verified or can be reasonably assumed (memory-less  and scale-specific movement in both time and space). In this scenario the diffusion exponent can be estimated for movement bouts in different habitats and time intervals, and the result can be interpreted behavioural-ecologically. For example, the diffusion rate can be expected to be smaller i optimal patches than elsewhere. In other words, the local staying time increases due to a more jagged path.

Second, Lévy walk is a special kind of random walk. Most steps are relatively short but others may be extremely long. Sequences of short steps in-between the long ones make the overall space use appear locally constrained during these periods*). Lévy walk is characterized by a spatially memory-less statistical representation of scale-free (“hierarchical”) movement within a given spatial scale range. Beyond this range the distribution of step lengths will show increased compliance with a non-scaling, truncated Lévy walk; i.e., a composite model with exponential tail rather than a power law for the extreme part of the step length distribution. By analyzing the step length distribution within the scale-free (power law) regime using different sampling intervals one should be able to verify model compliance from stationary power exponent. A Lévy walk is statistically self-similar in space, and thus the power exponent is expected to be relatively unaffected by the sampling scheme; see Reynolds (2008). Calculating the difference in the median step length for a given sampling interval when studying subsets of the movement data under different environmental conditions brings the model into the realm of ecology [see a practical method in; for example, Gautestad (2012)].

Third, Continuous Time Random Walk (CTRW) is suitable where the animal is found to occasionally stop moving. The temporal distribution of the duration of such resting episodes can then be fitted to statistical models; for example, a power law, a negative exponential, or a mixture as in the distribution for truncated power law. The spatial distribution of step lengths is in CTRW fitted independently of the temporal distribution. Bartumeus et al. (2010) applied the CTRW framework to study “intensive versus extensive searching” (scale-free sub-diffusive versus super-diffusive search) in foraging of Balearic shearwaters Puffinus mauretanicus and Cory’s shearwaters Calonectris diomedea along the coast of Spain (Bartumeus et al. 2010). The authors  interpreted the results ecologically with weight on difference between presence and absence of local trawling activity. See Part I, where I gave a brief summary.

However, is CTRW a proper framework for these seabirds? At the end of each foraging bout they obviously utilized spatial memory to successfully return to their breeding location. CTRW assumes consistently random crossing of the movement path due to the model’s lack of spatial memory description. To me it seems illogical to assume that these birds should toggle between memory-dependent and goal-oriented returns at the end (and possibly at the start) of each trip and memory-less Brownian motion (ARS-like?) during foraging when moving in the proximity of trawlers. The same argument about conditional memory switch-off may be raised for scale-free (Lévy-like) search in the absence of trawlers.

In the context of memory-less statistical modelling of movement (the three models above), site fidelity is defined by the strength of “slow motion”, and how the distribution of local staying times is expected to vary with ecological conditions. Compare this with the alternative model Multi-scaled Random Walk, where site fidelity is defined as the strength (frequency) of targeted returns to a previous location on a path. This return frequency may be interpreted as a function of ecological conditions. Hence, MRW explicitly invokes both spatial memory and its relative strength:

Three time scales are defined: the implicit interval between successive displacements in simulations (t), the average return interval to a previous location (tret), and the observation interval on the movement path (tobs). The latter represents GPS locations in real data, and is applied to study the effect from varying ρ = tret/tobs (relative strength of site fidelity for a given tobs).
Gautestad and Mysterud (2013), p4

Note that an increasing tret for a given tobs implies weakened site fidelity, and the functional form of the step length distribution is influenced by the ρ = tret/tobs ratio. For example, a Brownian motion-like form may be found if ρ << 1, and a power law form can be expected when ρ >> 1, with truncated power law (Lévy-like) to be observed in-between. See Figure 3 in Gautestad and I. Mysterud (2005) and Figure 3 in Gautestad and A. Mysterud (2013).

If the animal in question is utilizing spatial memory a lot of confusion, paradoxes and controversy may thus appear if the same data are analyzed on the basis of erroneously applying memory-less models within different regimes of ρ!

The MRW model may thus offer interesting aspects with a potential for alternative interpretation of the results of space use analyses when put into the context of – for example – foraging shearwaters. Thanks to the three times scales for MRW as above – where the third variable, t, represents the unit (t≡1) spatiotemporal scales for exploratory moves – it should be possible to test for example Lévy walk or CTRW against MRW using real movement data. 

More on this in Part III.

NOTE

*) While temporally constrained space use in ARS regards difference in environmental forcing, the occurrence of short-step intervals of random occurrence in a Lévy walk is by default due to intrinsic behaviour.

REFERENCES

Bartumeus, F., L. Giuggioli, M. Louzao, V. Bretagnolle, D. Oro, and S. A. Levin. 2010. Fishery discards impact on seabird movement patterns at regional scales. Current Biology 20:215-222.

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.

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.

Reynolds, A. 2008. How many animals really do the Lévy walk? Comment. Ecology 89:2347-2351.

Temporally Constrained Space Use, Part I: Three Models

Temporally constrained space use is a key property of animal movement. With respect to vertebrates three main statistical representations are particularly popular among modelers, based on disparate theoretical foundations. Which one should one use for analysis of a particular data set? As always in ecological research, one needs some simple protocol to distinguish between alternative model assumptions.

Animals paths are neither straight lines nor a dense dot of positions from juggling back and forth at the same spot. Typically we see a complicated combination of these two extreme patterns; some quite straightforward moves occasionally abrupted by more jagged movement. In order to infer behavioural and ecological results from space use one needs to study the data in the context of a realistic theoretical framework.

Outside the realm of temporal site fidelity; e.g., a drifting home range (Doncaster and Macdonald 1991),  ecological textbooks typically explain the mixture of straight and convoluted movement bouts as Area restricted Search (ARS).

Area-restricted search. A foraging pattern in which a consumer responds to an intake of food by slowing down its movement and remaining longer in the vicinity of the most recently located food item. This behaviour causes consumers to remain longer in areas where the density of food items is high than in areas where it is low.
A Dictionary of Ecology. Encyclopedia.com. 29 Jul. 2018.

In terms of statistical models, this rather qualitative description of behaviour may be formulated in many ways. A popular one is to combine classic or correlated random walk with two distinct parameter values for intrinsic step length distribution (the λ value) in F(r) ∝ e-λr. In this manner, movement varies with the jaggedness of the path (number of turns pr. period of time) rather than the movement speed. A larger λ implies smaller step lengths on average, which tend to increase local staying time during intervals when this movement mode is active. By fine-tuning respective λ1 and λ2 and  in such a superposition of two “randomly toggling modes” this so-called composite random walk can even be made to mimic the second main approach to model complicated paths, the Lévy flight model (Benhamou 2007; but see Gautestad 2013):

Lévy flights are, by construction, Markov processes. For general distributions of the step-size, satisfying the power-like condition, the distance from the origin of the random walk tends, after a large number of steps, to a stable distribution due to the generalized central limit theorem, enabling many processes to be modeled using Lévy flights.
https://en.wikipedia.org/wiki/Lévy_flight.

Lévy flights (and walks) are typically thought of as producing “fat tailed” step length distributions. However, within an often observed parameter range of the distribution (Pr(U>u = O(u)-k with 1<k<2) in real animals, one should not forget that half of the displacements in the distribution are in fact relatively short! In fact, the dominating step length bin is ultra-short moves, leading to a path that is conceptually (albeit not statistical mechanically) similar to the slow-down effect of more jagged moves during composite random walk.

Such “knots” of ultra-short moves of a Lévy path brings us to the third class of movement models, Continuous time Random Walk (CTRW). In this case the movement may be arrested for a shorter or longer period:

The step length distribution and a waiting time distribution (“resting” between steps) describe mutually independent random variables. This independence between jump lengths and waiting time to perform the next step makes the difference between CTRW and ordinary random walk (including Brownian motion).
Page 91 in: Gautestad, A. O. 2015, Animal Space Use: Memory Effects, Scaling Complexity, and Biophysical Model Coherence Indianapolis, Dog Ear Publishing.

Again, by fine-tuning the intrinsic model formulations and parameter conditions of CTRW one may in fact produce a Lévy flight-like movement pattern (Reynolds 2010; Schulz et al. 2013). Towards another extreme, one may achieve a classic random walk look-alike pattern. Conceptually, CTRW thus belongs on the left wall of the Scaling cube (x=0; y and z varies in accordance to model variant and parameter settings).

CTWR (which is closely connected to the concept of fractional diffusion) is typically invoked to explain a kind of switching – “intensive versus extensive searching” (scale-free sub-diffusive versus super-diffusive search) – that is now being found in real animal data like foraging of Balearic shearwaters Puffinus mauretanicus and Cory’s shearwaters Calonectris diomedea along the coast of Spain (Bartumeus et al. 2010).

Since I lack good pictures of shearwaters I decorate this post with a snapshot of another seabird, the sandwich tern Thalasseus sandvicensis. At least the picture was taken in the same general area of the Mediterranean. Photo: AOG.

In particular, they found that the birds in the presence of fisheries were more “restless” when taking advantage of fishery discards than in absence of trawlers, implying a higher probability of leaving a localized area pr. unit time during trawling activity. Specifically, during period of fishery discard utilization tended to show a smaller temporal scaling exponent for staying time (the temporal aspect of CTRW; larger β in their site fidelity function S(t) ∝ tβ, implying fewer events with a particularly prolonged staying time). In the spatial aspect of flight length distributions, when fisheries discard was present the birds tended to show good compliance with a negative exponential function*.

On the other hand, in the absence of trawlers a compliance with Lévy walk (truncated power law) was found. Space and time brought together, they found that the birds tended towards sub-diffusive foraging in the presence of fisheries discard (despite – somewhat counter-intuitively – a smaller local staying time within a given patch of a given spatial resolution), and super-diffusive foraging under natural conditions.

Despite the mathematical and numerical attractiveness of composite random walk, Lévy walk and CTRW and their well-explored statistical properties, they unfortunately all lack what may be a crucial component of foraging behaviour: spatial memory; i.e., the condition x>0 of the scaling cube.

Without spatial memory, self-crossing of an individual’s path happens by chance only, not intentionally by returns to a previous location (site fidelity, whether we consider short term or long term time scales).

  • May a model that implements spatial memory offer an alternative interpretation of the results presented by Bartumeus et al. 2010?
  • May this alternative hypothesis even offer a logical explanation for the apparent paradox that the birds were more restless locally when the movement simultaneously was more spatially constrained in overall terms?

The memory aspect of temporally constrained space use will be explored in Part II.

NOTE

*) Somewhat confusingly relative to common practice Bartumeus et al. (2010) use the exponential formula variant F(r) ∝ e-r/λ, which makes average step length proportional with λ rather than 1/λ.

REFERENCES

Bartumeus, F., L. Giuggioli, M. Louzao, V. Bretagnolle, D. Oro, and S. A. Levin. 2010. Fishery discards impact on seabird movement patterns at regional scales. Current Biology 20:215-222.

Benhamou, S. 2007. How many animals really do the Lévy walk? Ecology 88:1962-1969.

Doncaster, C. P., and D. W. Macdonald. 1991. Drifting territoriality in the red fox Vulpes vulpes. Journal of Animal Ecology 60:423-439.

Gautestad, A. O. 2013. Animal space use: Distinguishing a two-level superposition of scale-specific walks from scale-free Lévy walk. Oikos 122:612-620.

Reynolds, A. 2010. Bridging the gulf between correlated random walks and Lévy walks: autocorrelation as a source of Lévy walk movement patterns. J. R. Soc. Interface 7: 1753–1758.

Schulz, J. H. P., A. V. Chechkin, and R. Metzler. 2013. Correlated continuous time random walks: combining scale-invariance with long-range memory for spatial and temporal dynamics. J. Phys. A: Math. Theor. 46:1-22.

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.

 

Positive and Negative Feedback Part I: Individual Space use

The standard theories on animal space use rest on some shaky behavioural assumptions, as elaborated on in my papers, in my book and here in my blog. One of these assumptions regards the assumed lack of influence of positive feedback, in particular the self-reinforcing effect that emerge when individuals are moving around with a cognitive capacity for both temporal and spatial memory utilization. The common ecological methods to study individual habitat use; like the utilization distribution (a kernel density distribution with isopleth demarcations), use/availability analysis, and so on, explicitly build on statistical theory that not only disregards such positive feedback, but in fact requires that this emergent property is not influencing the system under scrutiny.

Unfortunately, most memory-enhanced numerical models to simulate space use are rigged to comply with negative rather than positive feedback effects. For example, the model animal successively stores its local experience with habitat attributes while traversing the environment, and it uses this insight in the sequential calculation of how long to stay in the current location and when to seek to re-visit some particularly rewarding patches (Börger et al. 2008, van Moorter et al. 2009, Spencer 2012, Fronhofer et al. 2013; Nabe-Nielsen et al. 2013). In other words, the background melody is still to maintain compliance with the marginal value theorem (Charnov 1976) and the ideal free distribution dogma, which both are negative feedback processes and not a self-reinforcing process that tends to counteract such a tendency.

Negative feedback (or balancing feedback) occurs when some function of the output of a system, process, or mechanism is fed back in a manner that tends to reduce the fluctuations in the output, whether caused by changes in the input or by other disturbances. Whereas positive feedback tends to lead to instability via exponential growth, oscillation or chaotic behavior, negative feedback generally promotes stability. Negative feedback tends to promote a settling to equilibrium, and reduces the effects of perturbations. Negative feedback loops in which just the right amount of correction is applied with optimum timing can be very stable, accurate, and responsive.
https://en.wikipedia.org/wiki/Negative_feedback

A common curlew Numenius arquata foraging on a field within its summer home range. Anecdotally, one may observe that a specific individual tends to revisit not only specific fields while foraging, but also specific parts of these fields. If this site fidelity is influenced by a rising tendency to prefer familiar space on expense of revisiting potentially equally rewarding patches based on previous visits, a positive feedback (self-reinforcing space use) has emerged.  This effect then interferes with the traditional ecological factors, like selection based on habitat heterogeneity, in a complex manner. Photo: AOG.

The above definitions follow the usual path to explain negative feedback as “good”, and positive feedback as something scary (I will return to this misconception in a later post in this series). It echoes the prevailing “Balance of nature” philosophy of ecology, which I’ve criticized at several occasions (see, for example, this post).

In a previous post, “Home Range as an Emergent Property“, I described how memory map utilization under specific ecological conditions may lead to a self-reinforcing re-visitation of previously visited locations (Gautestad and Mysterud 2006, 2010); in other words, a positive feedback mechanism*. Contemporary research on animal movement covering a wide range of taxa, scales, and ecological conditions continues to verify site fidelity as a key property of animal space use.

I use a literature search to test an assumption of the ideal models that has become widespread in habitat selection theory: that animals behave without regard for site familiarity. I find little support for such “familiarity blindness” in vertebrates.
Piper 2011, p1329.

Obviously, in the context of spatial memory and site fidelity it should be an important theme for research to explore to what extent and under which conditions negative and positive feedback mechanisms are shaping animal space use.

Positive feedback from site fidelity will fundamentally influence analysis of space use. For example, two patches with a priori similar ecological properties may end up being utilized with a disproportionate frequency due to initial chance effects regarding which patch happened to gain familiarity first**. Further, if the animal is utilizing the habitat in a multi-scaled manner (which is easy to test using my MRW-based methods), this grand error factor in a standard use/availability analysis cannot be expected to be statistically hidden by just studying the habitat at a coarser spatial resolution within the home range.

Despite this theoretical-empirical insight, the large majority of wildlife ecologists still tend to use classic methods resting on the negative feedback paradigm to study various aspects of space use. The rationale can be described by two end-points on a continuum: either one ignores the effect from self-reinforcing space use (assuming/hoping that the effect does not significantly influence the result), or one use these classic methods while holding one’s nose.

The latter category is accepting the prevailing methods’ basic shortcomings – either based on field experience or inspired by reading about alternative theories and methods – but the strong force from conformity in the research community is hindering bold steps out of the comfort zone. Hence, the paradigm prevails. Again I can’t resist referring to a previous post, “Why W. H. Burt is Now Hampering Progress in Modern Home Range Analysis“.

Within the prevailing modelling paradigm, implementing spatial memory utilization in combination with positive feedback-compliant site fidelity is a mathematical and statistical nightmare – if at all possible. However, as a reader of this blog you are at least fully aware of the fact that some numeric models have been developed lately, ouside the prevailing paradigm. These approaches not only account for memory map utilization but also embed the process of positive feedback in a scale-free manner (I refer to our papers and to my book for model details; see also Boyer et al. 2012).

 

NOTES

* The paper explores space use under the premise of positive feedback during superabundance of resources, in combination with negative feedback during temporal and local over-exploitation.

** In Gautestad and Mysterud (2010) I described this aspect as the distance from centre-effect; i.e., the utilization distribution falls off at the periphery of ulilized patches independently of a similar degradation of preferred habitat.

 

REFERENCES

Börger, L., B. Dalziel, and J. Fryxell. 2008. Are there general mechanisms of animal home range behaviour? A review and prospects for future research. Ecology Letters 11:637-650.

Boyer, D., M. C. Crofoot, and P. D. Walsh. 2012. Non-random walks in monkeys and humans. Journal of the Royal Society Interface 9:842-847.

Charnov, E. L. 1976. Optimal foraging: the marginal value theorem. Theor. Popula. Biol. 9:129-136.

Fronhofer, E. A., T. Hovestadt, and H.-J. Poethke. 2013. From random walks to informed movement. Oikos 122:857-866.

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

Gautestad, A. O., and I. Mysterud. 2010. Spatial memory, habitat auto-facilitation and the emergence of fractal home range patterns. Ecological Modelling 221:2741-2750.

Nabe-Nielsen, J., J. Tougaard, J. Teilmann, K. Lucke, and M. C. Forckhammer. 2013. How a simple adaptive foraging strategy can lead to emergent home ranges and increased food intake. Oikos 122:1307-1316.

Piper, W. H. 2011. Making habitat selection more “familiar”: a review. Behav. Ecol. Sociobiol. 65:1329-1351.

Spencer, W. D. 2012. Home ranges and the value of spatial information. Journal of Mammalogy 93:929-947.

van Moorter, B., D. Visscher, S. Benhamou, L. Börger, M. S. Boyce, and J.-M. Gaillard. 2009. Memory keeps you at home: a mechanistic model for home range emergence. Oikos 118:641-652.