Could sampling make hares eat lynxes?
Brenno Caetano Troca Cabella, Fernando Meloni, Alexandre Souto Martinez
aa r X i v : . [ q - b i o . Q M ] J u l Could sampling make hares eat lynxes?
Brenno Caetano Troca Cabella ∗ , Fernando Meloni † , and AlexandreSouto Martinez ‡ Sapra Assessoria , Cid Silva César, 600, 13562-400 - São Carlos - SãoPaulo, Brazil Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP)Universidade of São Paulo (USP) Av. Bandeirantes 3900, 14040-901Ribeirão Preto, SP, Brazil Instituto Nacional de Ciência e Tecnologia em Sistemas Complexos(INCTSC/CNPq) Av. Bandeirantes 3900, 14040-901 Ribeirão Preto, SP,Brazil
Keywords ecological modeling, sampling, aliasing effect, cyclic dynamics, Lotka-Volterra model,hare-lynx paradox. ∗ [email protected] † [email protected] ‡ [email protected] bstract Cycles in population dynamics are widely found in nature. These cycles are un-derstood as emerging from the interaction between two or more coupled species.Here, we argue that data regarding population dynamics are prone to misinterpre-tation when sampling is conducted at a slow rate compared to the population cy-cle period. This effect, known as aliasing, is well described in other areas, such assignal processing and computer graphics. However, to the best of our knowledge,aliasing has never been addressed in the population dynamics context or in cou-pled oscillatory systems. To illustrate aliasing, the Lotka-Volterra model oscillatoryregime is numerically sampled, creating prey-predator cycles. Inadequate samplingperiods produce inversions in the cause-effect relationship and an increase in cy-cle period, as reported in the well-known hare-lynx paradox. More generally, slowacquisition rates may distort data, producing deceptive patterns and eventuallyleading to data misinterpretation. ntroduction Quantitative sampling provides the most important information source for ecologicalmodeling. Because the validation of sampling methods is a difficult issue, the results oftheoretical models are rarely achieved in in situ experiments. For instance, predictionsconcerning scaling and survival/extinction transition in population dynamics [4, 5, 22]have not been experimentally tested. Consequently, many ecological data may lack con-cordance, and/or the real systems may be profoundly misinterpreted.An important example, and not yet fully understood, lies in the periodic species abun-dance cycles in population dynamics. These cycles may appear in coupled systems, inwhich two or more species interact due to a cause-effect relationship. Using the historicdata series from Hudson’s Bay Company, MacLulich [17] and Elton and Nicholson [7]found regular cycles in the population of Snowshoe Hares (
Lepus americanus ) and Cana-dian Lynx (
Lynx canadensis ). Data of both of these species were matched and indi-cated an overlap with a small delay between the species abundance. The system wasinterpreted from the perspective of trophic interactions, as a regular predator-prey sys-tem, which was first labeled the Lotka-Volterra model (LVM) [19]. Some years later, themodel became more robust, considering finite limits in the oscillatory predation rate [20].Although predator-prey models are intuitively coherent and produce qualitative patternsfound in nature, such models provide poor adjustment to the field data, so their empiri-cism is still controversial [18]. In "Do hares eat lynx?" [8], the author fitted real datausing different coupled models, and beyond the poor fit, data displayed a cause-effectrelationship inversion. Hares seemed to negatively affect the lynx population. The pro-posed solution to this paradox was the human influence on data collection, a plausiblebut untestable hypothesis [18, 8]. Further, field data infer that hares and lynx presentregular population cycles with approximately ten years between the respective peaks, in-stead of the expected one-year period. However, neither several years of field researchnor theoretical approaches could identify which factors influence this period increase [14].Experimental evidence has shown that both the predator and prey densities affect thedynamics of hares. Conversely, space, food sources, diseases and parasites are variablesthat are neglected by the models and have been experimentally discarded as modulatorsof cycles in hare populations [14, 24, 26, 15, 16]. Therefore, extensive research only rein-forces the empiric value of cyclic models as the descriptor of the predator-prey dynamics,while the poor fit is the main argument against the use of such models.We argue that the cyclic dynamics are particularly influenced by the sampling rate. Ap-parent inversion of cycle direction or an increase in the cycle period, among other be-haviors, may be artifacts due to poor sampling. Here, we conjecture that the aliasingeffect could be a plausible explanation for the lack of concordance between oscillatorytheoretical models and field data in Ecology. Further, aliasing shows a predictive charac-ter, which allows one to avoid possible misinterpretations when sampling is the basis formodeling. 3 aterial and methods
Before describing the methods and numerical experiment, we first introduce the nec-essary theoretical background. We present the temporal aliasing effect and the Lotka-Volterra model. Next, we numerically solve the model and sample it with different rates.
Aliasing Effect in the Lotka-Volterra Model
A temporal aliasing effect occurs when the sampling rate is not fast enough compared tothe system’s natural cycle period. For example, in movies, the spiked wheels on horse-drawn wagons sometimes appear to turn backwards, the "wagon-wheel effect", which isdepicted in Fig. 1. A wheel indeed turns clockwise, but due to the slow sampling by thecamera (number of frames per second), a filmed wheel appears to turn counter-clockwise.This effect can be avoided considering the Nyquist-Shannon sampling theorem, whichstates that given a time series with minimum period τ e , the equally spaced intervals be-tween samples T s must be smaller than half the minimum period, i.e. , T s < ( τ e / .Figure 1: Example of an aliasing effect in the clockwise rotation of a wheel. The visual-ized behavior on film is a counter-clockwise rotation, known as the wagon-wheel effect.The long time interval between samples explains this curiosity.In Ecology, cycles are extensively found in systems in which species interact with eachother and with the environment [18]. To illustrate the manner in which the aliasingeffect may disturb the interpretation of population abundance cycles, consider a sim-ple prey-predator interaction described by the LVM: dx/dt = x ( a − by ) and dy/dt = y ( cx − d ) where x ( t ) and y ( t ) are the prey and predator population densities, respectively,at time t , a is the prey growth rate in the absence of predators, d is the predator deathrate in the absence of prey, and b and c are related to the interaction strength betweenboth of the species. The LVM equations have two fixed points: the mutual extinction, E ( x ∗ , y ∗ ) = (0 , , and the neutral center, E ( x ∗ , y ∗ ) = ( c/d, a/b ) . Solutions around thesingular point E are cycles with period τ e = 2 π/ √ ad . Although the LVM is not com-pletely adequate to quantitatively describe real-world community dynamics, it is suitable4ere because it implies a cause-effect relation, where the number of predators increasesafter the prey abundance increases. Numerical simulation and sampling rates
To demonstrate how sampling rates can shift the patterns in predator-prey systems, wehave numerically calculated a cyclic dynamic pattern using the LVM. The Lotka-Volterradifferential equations have been implemented in the MatLab R (cid:13) language, and their so-lutions have been obtained using the Dormand-Prince method. Fig. 2 a shows the prey(full line) and predator (dashed line) population cycles. The model parameters havebeen set to produce a unitary oscillation period τ e = 1 .Next, the prey (circles) and predators (triangles) were sampled within fixed time inter-vals, T s . We repeated the procedure, reducing the sampling rate from τ e / until τ e . Foreach sampling rate, we interpolated the points to build the respective time series to inferthe original series. Based on the peaks of the time series, we inferred the oscillation pe-riod and the dephasing of predator and prey abundances. In all the cases, we consideredall the individuals from both of the populations. Therefore, we avoided any influence ofspace or sampling deviation on population densities to only address the effect of sam-pling rate on population dynamics. Results
In the following, we present the results of the sampling of two coupled oscillating sys-tems. The main result is that the sampling rate influenced the retrieval of the originaltime series. For T s < τ e / / , the system real cycle period is correctly retrieved(Nyquist-Shannon theorem), as displayed in Fig. 2b, with T s = τ e / . The fraction τ e / means that there were 5 sampling periods T s within τ e . As T s increases, the signal re-trieval is increasingly biased. In Fig. 2c, T s = τ e / is the limiting period from which theoriginal signal can be properly retrieved. However, because there are only 2 sampling pe-riods in τ e , the interpolation between the periods produces a straight line. Therefore, therelative delay between prey and predator dynamics cannot be correctly retrieved, andthe populations seem to overlap.For a slightly greater value, T s = 51 τ e / , different patterns can be seen in the sametime series. Fig. 2d shows interspersed periods of synchronicity and desynchronicity. For T s = q / τ e , an irrational number, the time series depicted in Fig. 2e seems to be er-ratic, with no identified pattern because no integer sampling periods can fit in τ e .A further increase in T s causes an inversion of the prey-predator cycles and an enhance-ment of the population cycle period. In Fig. 2f, T s = 9 τ e / , the predator abundanceincreases before the prey abundance, and when the prey abundance increases, the num-ber of predators diminishes.The inverted cycle oscillations persist for even greater values of T s as the oscillation pe-riod increases to T s → τ e . When T s = τ e , there are no oscillations, as depicted in Fig. 2g.5ll the time series presented from Fig. 2b to Fig. 2g repeat for kτ e < T s < (2 k + 1) τ e / where k = 0 , , , ... . Discussion
There is a scientific consensus about field experiments that better samples lead to betterinterpretation, or inference, about the real pattern. However, the effects caused by inap-propriate sampling are not trivial to analyze. This difficulty has already been addressedin the spatial influence on population dynamics or by the numerical insufficiency of sam-ples [11, 21]. A very simple and controlled oscillatory behavior, such the one LVM simu-lates, may produce different time series due only to inappropriate sampling rates, Fig. 2bto Fig. 2g. In Fig. 2f, the data series suggests that prey animals are eating predators andtheir cycle period is almost ten times the real one. In the hare/lynx paradox, assumingthat hunting occurs approximately every 12 months ( T s ) and that the hare/lynx periodcycle ( τ e ) fluctuates around this value, the aliasing effect would occur because T s ≈ τ e .Therefore, it is possible that the inversion and enhancement of the prey-predator cyclesmay be due to sampling artifacts. In real world systems, this difficulty is amplified be-cause the populations’ periodicity is not necessarily constant and/or many species inter-actions may tangle the dynamics even more. These results indicates that coupled sys-tems (such as ecological systems) seem to be very sensitive to temporal aliasing.Regarding the practical implications of sampling rates, a paradox emerges from fieldstudies. The appropriated sampling rate always depends on ad hoc information aboutthe real period of a species cycle. However, this knowledge is generally obtained by sam-pling the species, creating a redundant uncertainty. This problem could be the case ofthe hare-lynx system, for which almost all of the studies have used few data sources, thatoften were acquired from circumstantial sampling, without an adequate experimentalplanningThe aliasing effect is not restricted to biological experiments; it is a statistical phenomenon,and therefore, we highlight the large scope of our finding. A search in the scientific lit-erature demonstrates that the aliasing effect is poorly explored, and its considerationmay have deep implications. For instance, delays in coupled systems are ordinarily inter-preted as competition effects [28], but here, we have demonstrated that these delays canalso emerge from inappropriate sampling. Benicà and collaborators [2, 3] have studied along time series of plankton communities, applying regular samples to measure severalspecies. The authors have found that the cause-effect relationship suggests a chaotic foodweb. Although aliasing could provide an alternative explanation to the plankton commu-nity food web, this hypothesis was not tested.Aliasing should be better evaluated in many other circumstances, such as the coupledaerosol-cloud-rain system, because the LVM is applied to modeling [13]. The influence ofclimate anomalies has been investigated as a driver of periods in population dynamics, asin the hare-lynx system [27, 30]. In this case, the poor fit explanation could be related toaliasing, but again, this hypothesis has not been tested yet. In applicable areas, species6bundance rates are the basis for evaluation of biological control success in crops, and insuch cases, aliasing can have great financial consequences [25]. Sampling effects also haveimplications for biological conservation and species management, as in marine ecosys-tems, where population levels are used as a criterion to regulate fishing [9]. Further,some theoretical approaches about the trade-offs in Ecology and Evolution also concernpredator-prey systems, trophic interactions or population cycles, so aliasing should beaddressed [1, 29, 12, 6, 23, 10].To conclude, we have stressed the importance of the aliasing effect in retrieving the be-havior of oscillatory dynamics, for instance, in a coupled system. We have numericallydemonstrated that slow sampling rates of this oscillatory regime, compared to the realcycle period, may lead to data misinterpretation, even when other influences are avoided.We have qualitatively compared our results with the hares/lynxes paradox and presenteda new approach to this classic problem. We highlight the wide scope of the aliasing ef-fect on oscillatory coupled systems and its influence on the interpretation of real-worldpatterns. The temporal aliasing hypothesis shows a predictive character and can providenew insights to old problems in Ecology and Biology. This effect should be considered infuture experimental designs involving population dynamics in time series. Acknowledgments
BCTC thanks the CNPq (127151/2012-5), FM thanks FAPESP (2013/06196-4), and ASMthanks the CNPq (305738/2010-0 and 485155-2013-3). We would like to acknowledge the or-ganizers of the Summer Course on Mathematical Methods in Population Biology, Roberto An-dré Kraenkel and Paulo Inácio de Knegt López de Prado, where the hare-lynx paradox was firstpresented to us. Special thanks to C. A. S. Terçariol for calling our attention to the irrationalsampling period effect. Thanks to Cristiano R. F. Granzotti, Olavo H. Menin and Tiago J. Ar-ruda for fruitful comments on the manuscript.
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(f ) T s = 9 / , an inversion and an extension of cycle period may beinterpreted as preys eating predators. (g) As T s → τ ee