Advent of Extreme Events in Predator Populations
Sudhanshu Shekhar Chaurasia, Umesh Kumar Verma, Sudeshna Sinha
AAdvent of Extreme Events in Predator Populations
Sudhanshu Shekhar Chaurasia , Umesh Kumar Verma , and Sudeshna Sinha Indian Institute of Science Education and Research Mohali,Knowledge City, Sector 81, Manauli PO 140 306, India Indian Institute of Science Education and Research Tirupati, Tirupati, 517507, India (Dated: November 19, 2019)We study the dynamics of a ring of patches with vegetation-prey-predator populations, coupledthrough interactions of the Lotka-Volterra type. We find that the system yields aperiodic, recurrentand rare explosive bursts of predator density in a few isolated spatial patches from time to time.Further, the collective predator biomass also exhibits sudden uncorrelated occurrences of largedeviations as the coupled system evolves. The maximum value of the predator population in apatch, as well as the maximum value of the predator biomass, increases with coupling strength.These trends are further corroborated by fits to Generalized Extreme Value distributions, where thelocation and scale factor of the distribution increases markedly with coupling strength, indicating thecrucial role of coupling interactions in the generation of extreme events. These results indicate howoccurrences of extremely large predator populations can emerge in coupled population dynamics,and in a more general context they suggest a generic class of deterministic nonlinear systems thatcan naturally exhibit extreme events.
Introduction
Due to their huge impact in phenomena that rangefrom traffic jams to weather disturbances, the existenceof extreme events has triggered much research interest[1]. An extreme event can be considered as one wherea state variable (or variables) in an engineered or natu-ral system exhibits large deviations from the average, i.e.the system is interrupted by sudden excursions to val-ues that are significantly different from the mean value,with such deviations being aperiodic, recurrent and rare.Typically an extreme event can be said to have occurredif a variable is several standard deviations away from themean, and such unusually large values signal occurrencesof catastrophic significance. Examples of such extremeevents are found in weather patterns [2], ocean waves [3],financial crashes [4], black-outs in power grid networks[5, 6] and optical systems [7].The search for generic mechanisms that naturally yieldsuch extreme events is an issue of vital importance forthe basic understanding of complex systems, as well asreal world applications [8]. Efforts to obtain extremeevents typically involve stochastic models [9, 10], such asa recent random walk model of transport on networks[11]. In the arena of deterministic dynamical systems,there have also been a few recent studies on extremeevent generation in coupled systems. Such systems havetypically been composed of diffusively coupled individ-ual units that are excitable systems, which are capableof self-generating large deviations [12–14]. It is impor-tant however to find broad coupling classes that can pro-vide mechanisms to induce extreme events in dynami-cal systems that are not capable of generating such ex-treme events in isolation. Unearthing such determinis-tic systems would offer non-trivial examples of extremeevents arising from interactions [8], rather than intrin-sic or noise-driven large deviations in the states of theconstituent systems. Here we explore the emergence of extreme events ina ring of patches with vegetation-prey-predator popula-tions, coupled through interactions of the Lotka-Volterratype. Unlike many earlier models yielding extremeevents, our model has no stochastic environmental in-fluences or sources of random fluctuations, in either thestate variables or the parameters determining the dynam-ics of isolated sites. Rather, we present a new scenariofor the emergence of extreme events in both space andtime, in a system of populations coupled through genericLotka-Volterra type interactions, suggesting a genericcoupling class that can naturally yield extreme eventsin interactive deterministic nonlinear systems.
Coupled Population Model
In this work we consider a model for the populationfluctuation in snowshoe hare and the Canadian lynx, thatfits observed data well [15]. The dynamical equations de-scribing the time evolution of the coupled system of veg-etation (denoted by u ), prey (denoted by v ) and preda-tor (denoted by w ) populations is given by the followingfunctions f ( u, v, w ), g ( u, v, w ) and h ( u, v, w ):˙ u = f ( u, v, w ) = au − α f ( u, v )˙ v = g ( u, v, w ) = − bv + α f ( u, v ) − α f ( v, w )˙ w = h ( u, v, w ) = − c ( w − w (cid:63) ) + α f ( v, w ) (1)where the functions f ( u, v ) and f ( v, w ) describe twotypes of coupling. The functional form of f ( u, v ) is uv/ (1 + ku ), i.e. it is a Holling type II term representingthe coupling of vegetation ( u ) and prey ( v ) with cou-pling strength α . The second function f ( v, w ) = vw ,is the Lotka-Volterra term representing coupling of prey( v ) and predator ( w ) with coupling strength α . In thiswork we chose the parameters to be a = 1 . b = 1 . c = 10 . w (cid:63) = 0 . α = 0 . α = 1 . k = 0 . a r X i v : . [ n li n . C D ] N ov Now we expand the scope of the model above to mimica collection of such population patches, consisting of veg-etation, preys and predators. The populations in a patchinteract with other nearby patches in such a way thatthe predator of one patch can attack prey in neighbour-ing patches. Fig 1 shows a schematic diagram of theinteraction of a patch with nearby patches. Specifically,we consider a ring of such patches, with each patch in-dexed spatially by i , i = 1 , . . . N , where N is the totalnumber of patches in the ring. The populations of vege-tation, prey and predator in patch i is represented by u i , v i and w i respectively. FIG. 1: Schematic diagram of the interaction of a patch withnearest neighbour patches, in a ring of population patches.
The form of the predator-prey interaction betweenneighbouring patches is of the Lotka-Volterra type andis given by the following set of dynamical equations:˙ u i = f ( u i , v i , w i )˙ v i = g ( u i , v i , w i ) − C { v i w i − + v i w i +1 } (2)˙ w i = h ( u i , v i , w i ) + C { w i v i − + w i v i +1 } The coupling constant C reflects the strength of interac-tion among patches, and in this work we focus on thiscrucial parameter. Spatial distribution and temporal evolution ofthe population densities in the network
The first quantity of relevance in this system is the local population density in the patches, and their tempo-ral fluctuations. We will focus on the deviation of thelocal population densities from the mean value, i.e., wewill look for the emergence of extreme events in the localpatches, as evident in explosive population densities atspecific spatial locations.Fig. 2 shows the time evolution of the vegetation, preyand predator populations of one representative patchfrom the network at high coupling strength C = 1 .
0. Weobserve that, while the population densities are mostlyconfined to low values, the evolution is punctuated bysudden boosts to very high values. For instance, localpredator population densities can shoot up more than10 standard deviations away from the mean value. Thisis evident in the lower right panel of Fig. 2, where one u v t w u v t w FIG. 2: Time evolution of the vegetation u , prey v and preda-tor w populations of one representative patch, for the follow-ing cases: (left) uncoupled patches and (right) patches cou-pled to neighbouring patches, with coupling strength C = 1 . µ ) and ten times the standard deviation σ fromthe mean (i.e. µ + 10 σ ) respectively. can see instances where w exceeds the 10 σ threshold.The instants at which these large fluctuations occur arerelatively rare and completely uncorrelated in time andspace. These large deviations are also clearly evidentthrough the space-time plot of the evolution of predatorpopulations in different patches in the network, shown inFig. 3, where the extreme values of predator populationsare visible as sparse bright randomly located dots in thefigure.Note that if consider a lower threshold (e.g. µ + 4 σ ),the vegetation, prey and predator all exhibit events thatexceed the threshold, with prey and predator popula-tions yielding above-threshold events even when uncou-pled. However significantly, such occurrences in uncou-pled patches are strictly periodic , and stem from the in-trinsic pulse-like solutions of the constituent patches.
20 40 60 80 100 site ( i ) . . . . . . . . . t FIG. 3: Space-time density plot of the predator population w i ( t ) at site i = 1 , . . . C = 1 . In Fig. 4 we display the spatial distributions of thepredator populations w i for the patches i = 1 , . . . N ata representative instant of time, for the case of uncou-pled patches and coupled patches. It is clearly seen thatthere are a few patches in the coupled case that growexplosively and have a predator population much largerthan the mean value. These results qualitatively suggestthat the coupling of population patches give rise to ex-treme predator populations at a small number of spatiallocations, analogous to an extreme event in space. Suchcatastrophic events are rare in space, and occurs only ata couple of sites, but signal significant damage as theymay entail serious control costs. i w i w FIG. 4: Spatial distribution of the predator population w i ( i = 1 , . . . C =1 . µ ) and ten times the standard deviation σ from themean (i.e. µ + 10 σ ) respectively. Note that the predator pop-ulation w i in the most patches, as well as the spatial averageof w at an instant of time, is so low that it is barely visible inthe figures. Global maximum of predator populations
Now we quantitatively estimate the maximum vegeta-tion, prey and predator population densities in a patch,denoted by u max , v max and w max respectively, attainedin the course of the network dynamics [16, 17]. We esti-mate this by finding the global maximum of u i , v i and w i , for i = 1 , . . . N (where N is sufficiently large), sam-pled over a time interval T (where T is much longer thanthe intrinsic oscillation period). This will help us gaugethe magnitude of the extreme event and its relation tocoupling strengths. Here we present results with T = 50,with no loss of generality.Fig. 5 shows the maximum u max , v max and w max ,for a wide range of coupling strengths of the populationpatches. In the figure we depict the scaled values of themaxima, where u max , v max and w max are divided by theirvalues in the uncoupled case. These scaled quantitieshelp us assess the increase in the maxima in coupled net-works, compared to that obtained in uncoupled patches,allowing us to specifically gauge the coupling-induced ef-fects on the emergent maximum population densities. It C S c a l e d m a x i m u m p o pu l a t i o n FIG. 5: Dependence of the global maximum of the vegeta-tion population u max (red), the prey population v max (green)and the predator population w max (blue), occurring in thepatches in an interval of time T = 50, on coupling strength,for N = 100. Here we depict the scaled values of the maxi-mum, where u max , v max and w max are divided by their valuesin the uncoupled (i.e. C = 0) case. Note that larger systemsyield larger u max , v max and w max , with the values saturatingto a characteristic value in the limit of large system size. is evident from the simulation results that the magnitudeof the maximum vegetation and prey populations (i.e. u max and v max denoted by the red and green curves) donot increase much as coupling strength increases. How-ever, the maximum predator population increases verysignificantly with coupling strength, with w max at C = 1exceeding over six-fold the value obtained for uncoupledpatches. Temporal evolution of Biomass
The next quantity of interest is the total biomass of thevegetation, prey and predators, denoted by b u , b v and b w respectively. This represents a collective dynamicalquantity , and is given at an instant of time t as follows: b u ( t ) = N (cid:88) i =1 u i ( t ) , b v ( t ) = N (cid:88) i =1 v i ( t ) , b w ( t ) = N (cid:88) i =1 w i ( t )(3)where N is the system size. Again, we will examine thepresence of large excursions from mean-values, as suchexplosive growth indicate extreme events in time of acollective quantity.Fig. 6 shows the biomass of the vegetation ( b u ), prey( b v ) and predators ( b w ). The uncoupled case is shownalongside as a reference. It is clear that when the patchesare uncoupled, b u , b v and b w do not experience any largefluctuations. Further, the biomass of vegetation and preyfor the coupled case also stays bounded within the 4 σ threshold. However, interestingly, the predator biomassin coupled patches occasionally builds up to extreme val-ues, crossing the 4 σ threshold. This is also corrobo-rated through a comparison of the maximum values ofthe biomass of the vegetation, prey and predator in thecoupled network vis-a-vis the uncoupled patches. The b u b v t b w b u b v t b w FIG. 6: Time evolution of the biomass b u ( t ), b v ( t ) and b w ( t ) of the vegetation, prey and predator populations re-spectively, for the following cases: (left) uncoupled patchesand (right) patches coupled to neighbouring patches, withcoupling strength C = 1 . µ ) and four times the standard deviation σ from the mean (i.e. µ + 4 σ ) respectively. maximum prey biomass is almost unchanged on coupling,and the maximum vegetation biomass in a coupled net-work exhibits less than 20% change from uncoupled val-ues. On the other hand, the maximum predator biomassin a coupled network exhibits a three-fold increase com-pared to uncoupled patches (cf. Fig. 7). So coupling hasa very significant effect on the predator biomass, and onefinds clear evidence of the emergence of extreme eventsin time for this collective quantity. C S c a l e d m a x i m u m b i o m a ss FIG. 7: Dependence of the maximum of vegetation biomass b u (red), the prey biomass b v (green) and the predator biomass b w (blue), in an interval of time T = 50, on coupling strength,for N = 100. Here we depict the values of the maximumscaled by their values in the uncoupled case. In order to ascertain that the extreme values are uncor-related and aperiodic we examine the time intervals be-tween successive extreme events in the predator biomass evolution. Fig. 8 shows the return map of the intervalsbetween extreme events and it is clearly shows no regu-larity. The probability distribution of the intervals is alsoPoisson distributed and so the extreme predator popula-tion buildups are uncorrelated aperiodic events. ∆ t i ∆ t i + ∆ t P ( ∆ t ) FIG. 8: (Left) Return Map of ∆ t i +1 vs ∆ t i , and (right) Prob-ability distribution of ∆ t i fitted with exponentially decayingfunction, where ∆ t i is the i th interval between successive ex-treme events, where an extreme event is defined at the instantwhen biomass crosses the µ + 4 σ line (cf. Fig 6). Here thesystem size N = 100 and coupling strength C = 1 . Generalized Extreme Value distribution
Lastly we analyse the distribution of the maximum sizeof the predator populations, as well as the maximum sizeof the collective predator biomass. In order to obtaina quantitative measure of the extreme events generatedfor different coupling strengths, we fit these probabilitydistributions to the probability density function of thegeneralized extreme value distribution, given by: f ( y ; ζ ) = exp( − (1 − ζy ) /ζ )(1 − ζy ) (1 /ζ ) − , ζ > ,y ≤ /ζ exp( − exp( − y )) exp( − y ) , ζ = 0with y ( x, µ, σ ) = ( x − µ ) /σ Here ζ is the shape parameter, µ is the location pa-rameter and σ is the scale parameter. Note that thescale parameter is the most relevant parameter in thiscase as it determines the spread of the distribution.Fig. 10 shows the location and scale parameters ob-tained by best-fit to the Generalized Extreme Value dis-tribution of the maximum predator population in a patch w max , and the maximum predator biomass b w . Clearlythe location and scale parameters increase monotonicallywith coupling strength, for coupling strengths higherthan ∼ .
4, with the rise being approximately linear athigh C . Increasing location parameters indicate that theaverage predator population in a patch and the averagepredator biomass increase almost linearly with couplingstrength. Increasing scale parameters suggest that thedistribution becomes increasingly spread out, and the tail w max p ( w m a x )
20 40 60 80 100 b w p ( b w ) FIG. 9: Probability distribution of the (top) maximum preda-tor population in a patch w max , and (bottom) maximumpredator biomass, obtained from sampling the predator pop-ulations in a time interval T = 50 (with no loss of generality).The fit of the data from numerical simulations to the Gen-eralized Extreme Value distribution (cf. Eqn. 4) is shown bysolid lines, for coupling strength C = 0 . . . . of the distribution extends to larger values. So more ex-treme predator populations can be expected to occur inthe patches, from time to time, when the coupling be-tween the patches is stronger. C µ C σ C µ C σ FIG. 10: Location (left) and scale (right) parameters obtainedby best-fit to the Generalized Extreme Value distribution of(top) maximum predator population in a patch w max , and(bottom) maximum predator biomass b w , obtained from sam-pling a time interval of T = 50 (cf. Fig 9), for different cou-pling strengths C . Conclusions
In summary, we have studied the population dynamicsof a ring of patches with vegetation, preys and preda-tors. The population dynamics in the patches is givenby a model for the snowshoe hare and the Canadian lynxthat fits observed data well, and the patches are coupledthrough interactions of the Lotka-Volterra type. We findthat this system yields extreme events in the predatorpopulation in the patches, with bursts of explosive preda-tor population growth in a few isolated patches from timeto time. Further, the collective predator biomass alsoyields extreme values as the coupled system evolves. Themaximum value of the predator population in a patch, aswell as the maximum value of the predator biomass, in-creases with coupling strength, indicating the importanceof coupling in the generation of such extreme events. Fitsof the data from numerical simulations to GeneralizedExtreme Value distributions also quantitatively corrobo-rate these trends.Our results then are important, both from the view-point of general models of coupled nonlinear systems,as well as for the more specific implications it maypotentially hold for population dynamics. So first wehave demonstrated how a deterministic system, withgeneric Lotka-Volterra type of interactions, can give riseto extreme events in space and time . Such examples areuncommon, and so they are significant in the contextof general complex systems. Secondly, in the specificcontext of population dynamics, our model systemsuggests how predator population densities can growexplosively in certain patches. Though relatively rare,the magnitude of these extreme bursts of predator pop-ulation density is so huge, that the damage or ensuingcost to control the event, is considerable. Further thebiomass of predators can also grow extremely large atcertain points in time, and this is of significance due tothe catastrophic effects large predator populations canhave on the ecosystem as a whole. Lastly, interestingly,the Lotka–Volterra class of interactions has also beenused extensively in economic theory [18], and so ourresults may have some bearing on extreme events in thefinancial context. [1]
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