Pursuit-evasion predator-prey waves in two spatial dimensionsV. N. Biktashev, J. Brindley, A. V. Holden, and M. A. Tsyganov Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 14, 988 (2004); doi: 10.1063/1.1793751 View online: http://dx.doi.org/10.1063/1.1793751 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/14/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A plasma source driven predator-prey like mechanism as a potential cause of spiraling intermittencies in linearplasma devices Phys. Plasmas 21, 032302 (2014); 10.1063/1.4867492 Spatial dynamics in a predator-prey model with herd behavior Chaos 23, 033102 (2013); 10.1063/1.4812724 Spatially dependent parameter estimation and nonlinear data assimilation by autosynchronization of a system ofpartial differential equations Chaos 23, 033101 (2013); 10.1063/1.4812722 SpatioTemporal Oscillations in PredatorPrey Systems AIP Conf. Proc. 795, 217 (2005); 10.1063/1.2128368 A Short Guide to Predator-Prey Lattice Models Comput. Sci. Eng. 6, 62 (2004); 10.1109/MCISE.2004.1255822

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.209.6.50 On: Fri, 19 Dec 2014 11:06:39

Pursuit-evasion predator-prey waves in two spatial dimensionsV. N. Biktasheva)

Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, United Kingdom

J. BrindleyDepartment of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom

A. V. HoldenSchool of Biomedical Sciences, University of Leeds, Leeds LS2 9JT, United Kingdom

M. A. TsyganovInstitute of Theoretical and Experimental Biophysics, Pushchino, Moscow Region 142290, Russia

(Received 11 March 2004; accepted 28 July 2004; published online 21 October 2004)

We consider a spatially distributed population dynamics model with excitable predator-prey kinet-ics, where species propagate in space due to their taxis with respect to each other’s gradient inaddition to, or instead of, their diffusive spread. Earlier, we have described new phenomena in thismodel in one spatial dimension, not found in analogous systems without taxis: reflecting andself-splitting waves. Here we identify new phenomena in two spatial dimensions: unusual patternsof meander of spirals, partial reflection of waves, swelling wave tips, attachment of free wave endsto wave backs, and as a result, a novel mechanism of self-supporting complicated spatiotemporalactivity, unknown in reaction-diffusion population models. ©2004 American Institute of Physics.[DOI: 10.1063/1.1793751]

To describe spatiotemporal dynamics of populations oneneeds, apart from local interaction of populations, to in-voke some mechanism of spatial interaction, e.g., spreadof individuals in space. Often this is done in terms ofdiffusion, which is a macroscopic way of describing ran-dom, nondirected movement of individuals. This makes apopulation dynamics model a reaction-diffusion system,and pattern formation in reaction-diffusion systems iswell researched. However, living organisms do not neces-sarily move randomly, and often show some directedmovement in response to exogeneous factors, a behaviorusually described as taxis. Macroscopic representation oftaxis yields equations different from reaction-diffusionsystems, and pattern formation in such models has beenstudied much less. Our aim is to elucidate some new pat-tern formation mechanisms specific to this type of model.

I. INTRODUCTION

In our previous work,1,2 we have described new phenom-ena observed numerically in a reaction-diffusion-taxismodel, with excitable predator-prey local kinetics.3 In addi-tion to or instead of diffusion, we introduced taxis of eachspecies on the other’s gradient: predators pursuing prey andprey escaping the predators. We chose excitable rather thanmore traditional limit-cycle predator-prey dynamics, as it ismethodologically easier to deal with solitary populationwaves than with wave trains, and identified some new fea-tures that are typical in our excitable pursuit-evasion model,but unknown or very rare in reaction-diffusion models ofsimilar systems. These include ability of propagating waves

to penetrate through each other or reflect from impermeableboundaries(rather than simply annihilate, as is typical forreaction-diffusion waves), split and emit backward waves.

This was done in models with one spatial dimension. Inmost ecological applications, however, two spatial dimen-sions are more realistic. In reaction-diffusion systems it iswell known that two-dimensional models can demonstrate amuch wider variety of nontrivial solutions than one-dimensional. Thus, in the present paper, we describe phe-nomenology observed in two-dimensional simulations of ourpursuit-evasion excitable predator-prey model. As far as weare aware, this is the first study of this sort, so rather thangiving an exhaustive description, we have identified phe-nomena that are qualitatively different from what is observedin two-dimensional reaction-diffusion excitable systems.

II. THE MODEL

We consider a two-dimensional version of the model,studied earlier in Refs. 1 and 2:

]P

]t= fsP,Zd + D¹2P + h− ¹ sP ¹ Zd,

s1d]Z

]t= gsP,Zd + D¹2Z − h+ ¹ sZ ¹ Pd,

where Psx,y,td and Zsx,y,td are biomass densities of theprey and predator populations,h− is the coefficient determin-ing the taxis of prey down the gradient of predators(eva-sion), h+ is the coefficient determining the taxis of predatorsup the gradient of prey(pursuit), D is their diffusion coeffi-cient, assumed equal for simplicity, andf andg are “kinetic”terms, describing local interaction of the species with eacha)Electronic mail: vnb@liv.ac.uk

CHAOS VOLUME 14, NUMBER 4 DECEMBER 2004

1054-1500/2004/14(4)/988/7/$22.00 © 2004 American Institute of Physics988

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.209.6.50 On: Fri, 19 Dec 2014 11:06:39

other. We assume that this local interaction takes theTruscott–Brindley3 form

fsP,Zd = bPs1 − Pd − ZP2/sP2 + n2d,

s2dgsP,Zd = gZP2/sP2 + n2d − wZ,

which describes excitable kinetics. Equations(1) and(2) aresolved numerically with the following parameter values:n=0.07,b=1, w=0.004. Parametersg andD have been usedmostly in two fixed combinations,(A) g=0.01,D=0.04, and(B) g=0.016,D=0. Parametersh−, h+ varied between simu-lations. From one-dimensional simulations, described in Ref.2, we know that in(A), purely reaction diffusion waves with-out taxis termssh−=h+=0d are possible, where as in(B),such waves are not possible.

All two-dimensional simulations were performed in arectangular domainsx,ydP f0,Lxg3 f0,Lyg, with no-fluxboundary conditions

U ]sP,Zd]x

Ux=0,Lx

= U ]sP,Zd]y

Uy=0,Ly

= 0.

The calculations have been performed using finite differencediscretization with explicit approximation of the time deriva-tive and the upwind approximation of the spatial derivativesin the taxis terms. A detailed description of the upwind ap-proximation in one spatial dimension can be found in Ref. 2.In two dimensions, we used similar schemes to approximateboth components ofh−¹ sP¹Zd=s] /]xdfPs]Z/]xdg+s] /]yd3fPs]Z/]ydg and similarly forh+¹ sZ¹Pd. In the majorityof calculations, we used discretization stepsdx=0.5 anddt=0.005. The accuracy of the scheme was assessed by com-paring the solutions at successively smaller discretizationsteps. In that way, the earlier steps showed an accuracy ofapproximately 1%, estimated by the value of the establishedspeed of a solitary wave in one spatial dimension, comparedto the one at much smaller steps,dx=0.01,dt=4310−6. Wealso repeated parts of selected two-dimensional simulationsat smaller steps, down todx=0.2, dt=10−3, to check that allimportant qualitative two-dimensional phenomena are repro-duced correctly.

In figures representing the results of the simulations, wedesignate the domain size asLx3Ly, choice ofg ,D combi-

nation as (A) or (B), the values of taxis coefficients assh−,h+d, and where the panels are presented with a regulartime interval, it is given asDt.

III. PURSUIT-EVASION WAVES IN ONE DIMENSION

A. Mechanism of propagation

It is useful to first recapitulate the one-dimensional re-sults, and we start with propagation of solitary waves. Figure1 shows typical profiles of steadily propagating waves in areaction-diffusion system and in a reaction-taxis system, andschematically illustrates various mechanism involved. In thereaction-diffusion wave[panel(a)], the density of predatorschanges only slightly through the front. The propagation ofthe wave front is mostly due to interaction between diffusionand nonlinear local dynamics of the prey: influx of prey dueto the diffusion triggers the prey-escape mechanism, whenthe prey multiply faster than the predators and thereforegrow unchecked until reaching the carrying capacity of thehabitat. Between the front and the back of the wave the preyand predators are in quasiequilibrium, the predators slowlyconsuming the prey and growing themselves. On the back,the concentration of the predators is so large that the highconcentration of prey is no longer sustainable, and their dy-namics are such that the transition to their low state is sud-den, and is therefore also influenced by their diffusion, as atthe front. The sequence of events in the reaction-taxis waveis entirely different[panel(b)]. The advancing front of preyattracts the predators in the retrograde direction. This leavesspace relatively free of predators, which triggers the prey-escape mechanism. Another mechanism, driving the preyforward, is evasion, i.e., their taxis away from the predators.This feedback loop of prey evading predators and predatorspursuing prey leads to spatial oscillations of both(similar tothe way the feedback loop of prey benefiting predators andpredators harming prey tends to produce their temporal os-cillations). The similarity with reaction-diffusion waves isthat the front peaks when the prey reach their high quasiequi-librium. The difference, apart from oscillatory character ofthe taxis front, is that the density of predators does not re-main nearly constant during the front, but instead shows amarked tip followed by a rise, due to predators first movingback towards advancing prey, and then stopping that move-

FIG. 1. A stationary propagating wave in(a) reaction diffusion systemsD=0.04,h±=0d, (b) reaction-taxis systemsD=0,h−=2,h+=1d, andg=0.01 and otherparameters standard in both cases. Arrows illustrate the main factors affecting the dynamics of the front and the back of the wave. These include: preydynamics(growth and decrease due to grazing pressure) in both (a) and(b), diffusion of prey in(a), and taxis(pursuit by predators and evasion by prey) in(b).

Chaos, Vol. 14, No. 4, 2004 Pursuit-evasion waves 989

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.209.6.50 On: Fri, 19 Dec 2014 11:06:39

ment when the gradient of prey is over. The events on thewave back are also quite different. There we observe an ad-vancing wave of predators pursuing prey, and the recedingwave of prey escaping predators. This recession of prey isenhanced also by the fact that they are consumed by pursu-ing predators. Thus the position of the back is not directlyrelated to the time taken by predators to grow to a certainconcentration or to consume a certain numbers of prey, butrather, by time and space needed for the pursuit-evasion dy-namics to form the back structure. As a result, the duration ofthe taxis wave is much shorter than that of the diffusionwave. So, for the waves shown on Fig. 1, the duration of thediffusion wave, measured at the levelP=0.5, is about 106time units, and the duration of the similarly measuredpursuit-evasion waveat the same kinetic parametersis onlyabout 32 time units.

B. Mechanism of reflection

Figure 2 illustrates the process of reflection of the taxiswave from an impermeable boundary. When the front of thewave arrives at the boundary, the system there switches tothe high-prey state, close to the carrying capacity of the habi-tat, i.e., the maximal stationary density of prey if predatorsare absent, which in our model isP=1 st= t0+200d. This issimilar to what happens in a reaction-diffusion wave, and inboth cases is mainly due to the nonlinear dynamics of theprey population. The difference comes when the back of theexcitation wave approaches the boundaryst= t0+200. . .t0+400d, because in the taxis wave, unlike the diffusion wave,the back of the wave profile is formed by prey evading thepeak of predators following them, rather than by consump-tion of prey by predators. As a result, in the taxis wave, thedensity of predators at the back of the wave is insufficient toreduce prey density to a low level during the time that thetail actually lasts. Thus, when the evading prey are stoppedby the impermeable boundary, the predators catching up withthem do not consume them all. Instead, a substantial densityof prey survivesst= t0+400d. Those prey that survive after

the maximum of the predator density has passed them escapein the retrograde directionst= t0+600d, and if their number islarge enough, initiate a new backward propagating wavest= t0+800d.

Thus we see that the mechanism of reflection of taxiswaves is entirely different from known mechanisms of re-flection of reaction-diffusion waves, e.g., positive overlap ofoscillatory tails of the colliding waves discussed in Ref. 4,which required a very close proximity of the local dynamicsto a Hopf bifurcation. In that case the reflection property wasrestricted to a narrow region in the parameter space. With thepresent mechanism the reflection is rather robust, both interms of local dynamics, and of taxis and diffusioncoefficients.2

IV. SPIRAL WAVES

Among the most ubiquitous, dramatic and intensivelystudied phenomena observed in two-dimensional(2D)reaction-diffusion excitable systems are re-entrant excitationwaves, commonly known as spiral waves.5 They are ob-served in a variety of experimental systems, and in an evengreater variety of mathematical models, that includes spa-tially extended mathematical models of predator-prey inter-action of the reaction-diffusion type.6 Their significancestems from the fact that they can occur in an event of a wavebreak in a propagating excitation for whatever reason, andthus serve as a specific route to spatiotemporal chaos. Thuswe have started to study details of reaction-diffusion-taxiswaves in 2D through the effect that the taxis terms have onthe spiral waves. We have found that even very small taxiscan significantly destabilize spirals.

Figure 3 shows a typical rigidly rotating spiral wave so-lution, observed in the purely reaction-diffusion model. Fromthe behavior of one-dimensional(1D) solutions,1,2 qualita-tively new behavior in 2D based on 1D phenomenology, e.g.,reflection from the boundaries, is expected when the taxisterms are large enough in comparison with coefficientD,which is =0.04 in our simulations. The relevant values are ofthe order of 1, which corresponds to comparable fluxes gen-erated by the diffusion and the taxis terms, at the values ofthe variables at around the stable equilibrium. Comparison offluxes gives a reasonable way to compare the relative signifi-cance of taxis and diffusion terms, ash± are coefficients ofnonlinear terms and have different dimensionality from thatof D.

FIG. 2. Reflection of a taxis wave from an impermeable boundary. Param-eters areD=0, h−=2, h+=1, g=0.01, same as in Fig. 1(b). This combinationof parameters is specially selected for illustration purposes. Tics on thehorizontal axis are in 5 units, on the vertical axis in 1 unit.

FIG. 3. Spiral wave in a purely diffusive medium,(A) 2003200 (0,0), Dt=150. Here and on the next figures with spiral waves, the trajectory of thetip is superimposed. The tip is defined as a point whereP=0.49 andZ=0.21; normally, there is only one such point per tip of a spiral wave.

990 Chaos, Vol. 14, No. 4, 2004 Biktashev et al.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.209.6.50 On: Fri, 19 Dec 2014 11:06:39

Figure 4 shows the behavior of the reaction-diffusion-taxis spiral wave at values of the taxis coefficients muchsmaller than those at which reflection happens. Already thesesmall taxis terms change the behavior of the spiral very sig-nificantly, as, with the same local kinetics as the stationaryspiral of Fig. 3, it now meanders wildly. This kind of non-stationary behavior of spiral waves is not unknown in excit-able media, where the transition from steady to meanderingrotation is usually associated with change of the parametersof the reaction; here we see such a transition for the samereaction parameters but with the spatial terms altered.

As the taxis coefficients increase, their effect on the spi-ral wave dynamics increases further. Figure 5 shows that thespiral wave meander becomes so extended that the spiralsoon drifts out of the medium and annihilates at the bound-ary. This happens at taxis coefficients that are still muchsmaller than those needed for quasisoliton behavior.

V. ALTERNATIVE BEHAVIOR OF THE WAVE TIPS

In reaction-diffusion excitable systems, if an excitationwave is broken, its tip will either grow(“germinate”), pro-trude and curl up into a spiral wave, if the medium is“strongly excitable,” or it may shrink and retract, if it is“weakly excitable.” Figure 6 shows an example of areaction-taxis excitable system, where neither takes place.The tip of a broken wave protrudes, but does not curl up intoa spiral; instead new characteristic phenomena are observed.

A. Wave tip swelling

See Fig. 6,t=40. . .60. The consequence of the wave-break protruding may be seen as “swelling” of the tip, as itgoes through a “blobbing” stage, when an initially smalloval-shape region of medium is excited, which subsequentlygrows and reforms into circular expanding wave stage. A

phenomenological interpretation of this swelling phenom-enon follows from the empirical observation that it usuallyhappens when the evasion coefficient is large compared tothe pursuit coefficient. Thus the prey near the wave breakhave the capacity to escape from the predators sideways. Asubpopulation of prey then finds itself in a region relativelyfree from predators. In this predator-free zone prey start mul-tiplying intensively and form a circular expanding region. Asthere is no chasing wave of predators inside this circularregion, the high-prey state ends only when the predatorsmultiply to large densities, taking much longer than in astandard reaction-taxis propagating wave where the end ofthe prey wave is determined by the chasing wave of preda-tors. So the swollen tip can grow to form a large patch ofprey, before turning into an annular shape wave, when thebackstructure of the taxis wave is formed. Subsequently, thepredator wave catches up with the prey and the circular ex-panding region develops into a annulus-shaped expandingwave.

B. Self-attachment

See Fig. 6,t=60. . .240. After the tip swelling phase, nowavebreak can be identified based on the distribution of theprey population. Nevertheless, a wavebreak, or tip, can beformally defined, e.g., as an intersection of suitable isolinesof the prey and predator populations, as illustrated on Fig. 7.For topological reasons, such a tip cannot disappear, but hasto move continuously until reaching the medium boundary orannihilating with another tip of the opposite chirality. Such a“tip” is found near where the circular wave, newly born fromthe swollen tip, touches the back of the mother wave. Nowthe whole old+new wave can be seen as a single wave, afree end of which is attached somewhere to its own back.This could not happen in a reaction-diffusion medium, be-cause of the refractory period behind an excitation wave. In

FIG. 5. Unstable spiral at larger taxis coefficients,(A) 2003200 (0.1,0.1),Dt=225.

FIG. 4. Meandering spiral in a medium with small taxis coefficients,(A)2003200 (0.05,0.05), Dt=400.

FIG. 6. Wave break that does not lead to spiral wave initiation,(B) 2503250(5,1). At t=60: predators have multiplied in the center of the circular prey patch,ending the “swollen tip” phase. Att=100: dashed rectangle is shown in detail later in Fig. 7. Att=240: a piece of the wave has reflected from the easternboundary.

Chaos, Vol. 14, No. 4, 2004 Pursuit-evasion waves 991

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.209.6.50 On: Fri, 19 Dec 2014 11:06:39

terms of the present population dynamics system, the refrac-tory period is characterized by such high density of predatorsthat triggering a prey-escape mechanism is not feasible. Thedifference with taxis excitation waves is that they are notcharacterized by significant difference in the predators’ den-sity before and after the wave. To put it briefly, the preywave finishes not just when all prey are consumed by preda-tors, but rather when those that are not consumed have es-caped forward. Similarly, the predators’ wave finishes notwhen all predators die out because of starvation, but ratherwhen all predators have moved forward in the pursuit ofprey. Thus the equilibrium concentration of predators andprey behind the wave establishes rather quickly and due to amechanism quite different from reaction-diffusion waves. Asa result, there is little or no “refractory tail” behind a taxiswave, and this makes the self-attachment of a wave’s tip toits own back possible.

The structure of the attachment site is illustrated in moredetail on Fig. 7. The high prey regions of the mother waveand the circular wave are connected, whereas their highpredator regions are disjoint, or, more precisely, connectedby an isthmus of much smaller concentration of predatorsthan elsewhere, because the predators at the junction havebeen attracted backwards by the circular wave. Note that theattached wave tip has to move considerably faster than aplane solitary wave, as it participates simultaneously in themovement of the mother wave, and the circular wave, whichmoves at an angle to the mother wave. One more reason forsuch faster movement is the unusual nonlinear dispersionrelationship for taxis waves, where speed increases as perioddecreases. This property is illustrated on Fig. 8, which showsvelocity of waves propagating on a one-dimensional intervalwith periodic boundary conditions. Initially a wave wasstarted on an interval of a large lengthL. This length thenwas decreased in small steps, after each of which sufficienttime was allowed for the circulating wave to approach itsstationary velocityvsLd, which was then recorded, and thenext decrease ofL was made. To circumvent the interpreta-tion difficulty related to the difference of spatial scales indiffusion and taxis terms, we plot not the absolute velocity,but the velocity relative to the velocity of a solitary wave,vs`d. For the same reason, the horizontal coordinate in Fig. 8is not the spatial period of wavesL, but their temporal periodT calculated asL /vsLd.

Another way to view the mechanism of the attachmentof the free end is through kinetics of the prey, which multiplymore efficiently because of the gap in the predators’ popula-

tion at the site of the junction; the wave grows in the gaprather than propagates into it(in fact, of course, prey growthis an essential component of the wave propagation, so bothexplanations are valid).

VI. PARTIAL REFLECTION

As we have already mentioned, at appropriate taxis co-efficients, one-dimensional waves can reflect from theboundaries and penetrate through each other. In two dimen-sions, there are new aspects characterizing impact of thewaves with boundaries and each other, not available in onedimension: the curvature of the waves at the moment of im-pact, and angle of incidence. In reaction-diffusion systems,the wave propagation velocity is known to depend on thecurvature of the wave. We have found that in our reaction-taxis model, the curvature, or angle of incidence, or both,affect the result of the impact, i.e., whether the wave will bereflected from the medium boundary or another wave, or willannihilate at it. A representative example is shown in Fig. 9.A circular wave was initiated at an asymmetrically locatedsite within a rectangular region, so its distances to all fourboundaries were different. The result is that the wave hasmostly annihilated at the northern and western boundaries(except small parts of the wave at the further ends of thoseboundaries), which the wave reached first while having ahigher curvature. Then the wave has mostly reflected fromthe southern and eastern boundaries, which it reached laterwhile having a lower curvature. The waves reflected from theeastern and southern boundaries partly annihilate on impact,namely, where their collision is more head-on. Meanwhile,where their collision is more slantwise, the waves penetrate

FIG. 7. Structure of the self-attachment site,(B) 2503250 (5,1). (a) Distri-bution of P, (b) distribution of Z, (c) isolines P=const (solid) and Z=const(dashed).

FIG. 8. Dispersion curves of periodic waves, as normalized velocityvsLd /vs`d vs time periodT=L /vsLd, at different parameter values as indi-cated by the legend. While reaction-diffusion waves go slower if frequent,taxis tends to speed frequent waves up.

FIG. 9. Circular wave: some parts are reflected from the boundaries, someare not.(B) 2003160 (5,1).

992 Chaos, Vol. 14, No. 4, 2004 Biktashev et al.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.209.6.50 On: Fri, 19 Dec 2014 11:06:39

each other. One more factor that can affect the outcome ofthe collision, is the “age” of the propagating wave. This isrelated to delicate asymptotic properties of the taxis waveswhich will be considered in more detail elsewhere.7 For thepresent paper, the important observation is that a collidingwave may either annihilate or penetrate/reflect in the samemedium, depending on the local details of the collision.Therefore, the outcome of collision may be different for dif-ferent parts of the same colliding wave, which means that thewave will be broken.

VII. SELF-SUPPORTING ACTIVITY

We have identified two specifically two-dimensionalphenomena that can happen to pursuit-evasion waves: partialreflection/penetration and tip swelling. These two phenom-ena can work together to generate a self-supporting spa-tiotemporal activity. An example of such self-supporting ac-tivity is shown in Fig. 10. This is the continuation of Fig. 6,where a broken wave has failed to initiate a spiral. The se-quence of events on this figure is typical for a self-supprtingactivity and is as follows. Unlike waves in reaction-diffusion

systems, the free ends do not curl up into spirals, but swelland produce circular waves with the free end eventually at-tached to the back of its mother wave. These waves interactwith each other and with medium boundaries. Partialreflection/penetration of the waves in such interaction createsmore broken waves with free ends, and so this sequence ofevents repeats itself. We have, as a result, a spatiotemporal“chaos.” The mechanism described is different from thatknown before in reaction-diffusion excitable systems, or in-deed in any other nonlinear spatially distributed systems. Wehave found similar activity in simulations with different ini-tial conditions, such as localized overthreshold lumps of preyconcentration, generating initially circular waves, and withparameters, including both kinetics A and B with variouscombinations ofsh−,h+d, and with and without the diffusionterms. So, this is a robust phenomenon, and not an exotic onerestricted to some carefully chosen experimental setup. Theself-supporting activity can last long, but not necessarily forever, as there is always a possibility that all waves annihilateon collisions so that no activity survives. Obviously theprobability of that happening is higher in a smaller system.This conclusion has also been confirmed by our simulations.

FIG. 10. Self-supporting activity. Continuation of Fig. 6,(B) 2503250 (5,1).

Chaos, Vol. 14, No. 4, 2004 Pursuit-evasion waves 993

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.209.6.50 On: Fri, 19 Dec 2014 11:06:39

VIII. CONCLUSIONS

We have demonstrated that two-dimensional excitablereaction-diffusion-taxis systems can exhibit new properties,unavailable in one-dimensional systems of the same kind,and different from two-dimensional purely reaction-diffusionsystems with the same kinetics. Even relatively small taxisterms can make spiral waves very unstable in conditionswhere they would be stable without such terms. If the taxis(pursuit/evasion) terms are large enough, compared to diffu-sion, to cause quasi-soliton behavior in 1D, then the 2D be-havior is entirely different and is no longer dominated byprocesses of spiral waves generation and births, as is typicalfor reaction-diffusion systems. Instead, a new type of self-supporting activity takes place, which is mediated by threequalitatively new types of events:

• partial reflection of waves from boundaries, or their partialpenetration through each other, which produces brokenwaves;

• “swollen tips,” i.e., circular wave sources, produced byfree ends of broken waves; and

• attachment of free ends of broken waves to the wavebacks.

The first of these phenomena is due to sensitive dependenceof quasisoliton behavior on the circumstances of collision.The second is due to the increased possibility of the sidewardescape of prey from the head of the propagating taxis wavenear the wave break, when their mobility is higher than thatof the predators. The third is due to short supply of predatorsat the site of attachment, which allows prey to multiplyquicker to seal the gap. All three phenomena are related tothe peculiar feature of the reaction-taxis waves, the virtualabsence of the refractory tail, which is a prominent feature ofreaction-diffusion excitation waves.

As we noted earlier, quasisoliton behavior can be ob-served in some reaction-diffusion excitable systems if its ki-netics are close to a Hopf bifurcation.4 In two spatial dimen-sions, such systems also demonstrate unusual types ofbehavior, such as converging, concave spirals,8–10 but againin very limited regions of parameter space.

The value of this study is twofold. First, this illustratesthe importance of taking into account of directed movementof species in modeling spatially distributed interacting popu-lations, as this can produce completely different phenomenafrom those that occur in reaction-diffusion systems. Second,this is interesting in a broader nonlinear science context, asan example of a new type of nonlinear dissipative waves,with some properties similar to those known before, andsome completely new.

This study is, of course, far from exhaustive, and furtherinvestigation is required, both to identify and classify thewhole range and variety of characteristic behavior in thisclass of models, and to lay the foundations for theoreticalunderstanding of the phenomenology. There is always a pos-sibility of new regimes in the same type of systems at differ-

ent parameters or initial conditions. Besides, beyond thescope of Refs. 1, 2, and 7 and the present paper remains thequestion of the behavior of systems with similar spatial in-teraction, but with oscillatory, rather than excitable, local ki-netics. In such systems, solitary wave solutions are clearlyimpossible, so it is not obvious, what would be the analogue,if any, of the quasisoliton wave interaction in such systems,let alone the possible richness of two-dimensional behaviorsthere. Of particular interest would be to study the effects ofpursuit-evasion behavior in more realistic population dynam-ics models, for example.6 Evidence of quasisoliton behaviorin a real two-dimensional population dynamics system, agrowing bacterial population, has been presented in Ref. 11(see also Ref. 2). That experiment can be viewed as a limitcase of a predator-prey system with bacteria as predators andnutrient as prey, with negligible vital dynamics of prey. Out-puts from such studies could suggest ways of analyzing andinterpreting actual field data to establish the importance ofpursuit-evasion strategies in ecological contexts. Clearly, fur-ther investigations are required to bring the theoretical un-derstanding and the practical applicability of this class ofmodels anywhere near to those of reaction-diffusion systems.

ACKNOWLEDGMENTS

This study was supported in part by EPSRC Grant No.GR/S08664/01(UK) and RFBR Grant No. 03-01-00673(Russia).

1M. A. Tsyganov, J. Brindley, A. V. Holden, and V. N. Biktashev, “Quasi-soliton interaction of pursuit-evasion waves in a predator-prey system,”Phys. Rev. Lett.91, 218102(2003).

2M. A. Tsyganov, J. Brindley, A. V. Holden, and V. N. Biktashev, “Prop-erties of population taxis waves in a predator-prey system with pursuit andevasion, 2004,” Physica D197, 18–33(2004).

3J. E. Truscott and J. Brindley, “Equilibria, stability and excitability ingeneral class of plankton population models,” Philos. Trans. R. Soc. Lon-don, Ser. A 347, 703–718(1994).

4O. A. Mornev, O. V. Aslanidi, R. R. Aliev, and L. M. Chaylakhyan,“Soliton regimes in the FitzHugh–Nagumo model: Reflection of collidingpulses of excitation,” Dokl. Biophys.346–348, 21–23(1996).

5A. T. Winfree, “Spiral waves of chemical activity,” Science175, 634–636(1972).

6J. A. Sherratt, X. Lambin, C. J. Thomas, and T. N. Sherratt, “Generation ofperiodic waves by landscape features in cyclic predator-prey systems,”Proc. R. Soc. London, Ser. B269, 327–334(2002).

7M. A. Tsyganov and V. N. Biktashev, “Half-soliton interaction of popula-tion taxis waves in predator-prey systems with pursuit and evasion,” Phys.Rev. E 70, 031901(2004).

8O. A. Mornev, O. V. Aslanidi, and L. M. Chailakhyan, “Solitonic mode inthe FitzHugh–Nagumo equations: Dynamics of a rotating spiral wave,”Dokl. Akad. Nauk 353, 682–686(1997).

9O. A. Mornev, O. V. Aslanidi, and I. M. Tsyganov, “Soliton-like regimes,echo, and concave spiral waves in mathematical models of biological ex-citable media,” Macromol. Symp.160, 115–122(2000).

10O. A. Mornev, I. M. Tsyganov, O. V. Aslanidi, and M. A. Tsyganov,“Beyond boundaries of the Kuramoto–Zeldovich theory: Stably rotatingconcave spiral waves and their link to the echo phenomenon,” JETP Lett.77, 319–325(2003).

11M. A. Tsyganov, I. B. Kresteva, A. B. Medvinsky, and G. R. Ivanitsky, “Anovel mode bacterial population wave interaction,” Dokl. Akad. Nauk333, 532–536(1993).

994 Chaos, Vol. 14, No. 4, 2004 Biktashev et al.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.209.6.50 On: Fri, 19 Dec 2014 11:06:39