spatio-temporalpatternformationincoupled … · 2014. 9. 12. ·...

Nonlinear Analysis: Real World Applications 1 (2000) 53–67www.elsevier.nl/locate/na

Spatio-temporal pattern formation in coupledmodels of plankton dynamics and �sh school

motion

Horst Malchowa ; ∗, Birgit Radtkea, Malaak Kallachea,Alexander B. Medvinskyb, Dmitry A. Tikhonovb, Sergei V. Petrovskiic

aDepartment of Mathematics and Computer Science, Institute of Environmental Systems Research,University of Osnabr�uck, Artilleries tr 34, D-49069 Osnabr�uck, Germany

bInstitute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino,Moscow Region, 142292 Russia

cShirshov Institute of Oceanology, Russian Academy of Sciences Nakhimovsky Prospect 36,Moscow 117218, Russia

Received 11 October 1999

Keywords: Plankton dynamics; Predator–prey model; Bistability; Reaction–di�usion system;Rule-based �sh school motion; Spiral waves; Chaos

1. Introduction

Many mechanism of the spatio-temporal variability of natural plankton populatinosare still nuclear. Pronounced physical patterns like thermoclines, upwelling, fronts andeddies often set the frame for the biological processes [28]. However, under conditionsof relative physical uniformity, the temporal and spatio-temporal variability can be aconsequence of the coupled nonlinear biological and chemical dynamics [19,66]. Hence,the formation and stabilization of dissipative temporal, spatial and spatio-temporal struc-tures by nonlinear systems dynamics is of continuous interest in theoretical biology andecology. Conceptual minimal models are an appropriate tool for searching and under-standing the basic mechanisms of this pattern formation. Several examples of theirusefulness in studies of plankton dynamics like patchiness and phytoplankton bloomsare known, cf. [5,10,58,65–70,80,86].

∗ Corresponding author.

1468-1218/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved.PII: S0362 -546X(99)00393 -4

54 H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) 53–67

In 1952, Turing [81] showed that the reaction and di�usion of at least two agentswith considerably di�erent di�usivities can generate spatial structure of an uniformdistribution. In 1972, Segel and Jackson [61] were the �rst to apply Turing’s ideato a problem in population dynamics: The dissipative instability in the prey–predatorinteraction of phytoplankton and herbivorous copepods. Levin and Segel [19] suggestedthis scenario of spatial pattern formation for a possible origin of planktonic patchiness.Recently, local bistability, predator–prey limit-cycle oscillations, plankton front prop-

agation and the generation and drift of planktonic Turing patches were found in aminimal phytoplankton–zooplankton interaction model [22–25,27] that was originallyformulated by Sche�er [56], accounting for the e�ects of nutrients and planktivorous�sh on alternative local equilibria of the plankton community. Routes to chaos throughseasonal oscillations of parameters have been extensively studied with several models[7,17,46,50,51,55,58,60,71–73,78]. Deterministic chaos in uniform parameter modelsand data of systems with three or more interacting plankton species have been stud-ied as well [1,57]. The emergence of di�usion-induced spatio-temporal chaos along alinear nutrient gradient has been found by Pascual [41,42] in Sche�er’s model without�sh predation. Plankton-generated chaos in a �sh population has been reported byHorwood [14].Conditions for the emergence of three-dimensional spatial and spatio-temporal pat-

terns after di�erential- ow-induced instabilities (DIFICI [54]) of spatially uniformpopulations were derived by Malchow [24–26] and illustrated by patterns in Sche�er’smodel. Instabilities of the spatially uniform distribution can appear if phytoplanktonand zooplankton move with di�erent velocities but regardless of which one is faster.This mechanism of generating patchy patterns is much more general than the Turingmechanism which depends on strong conditions on the di�usion coe�cients. One canimagine a wide range of applications in population dynamics.The e�ect of external hydrodynamical forcing on the appearance and stability

of nonequilibrium spatio-temporal patterns in Sche�er’s model was also studied byMalchow and Shigesada [24]. A channel under tidal forcing served as a hydrody-namical model system with a relatively high detention time of matter. Examples wereprovided on di�erent time scales: The simple physical transport and deformation of aspatially non-uniform initial plankton distribution as well as the biologically determinedformation of a localized spatial maximum of phytoplankton biomass.Other processes of spatial pattern formation after instability of spatially homo-

geneous species distributions have been reported too, e.g. bioconvection and gyrotaxis[43,45,77,84], trapping of populations of swimming microorganisms in circulation cells[18,74], and e�ects of nonuniform environmental potentials [21,64].In this paper we focus on the in uence of �sh on the spatio-temporal pattern

formation of interacting plankton populations in uniform and non-uniform environ-ments. A general class of planktonic prey–predator systems is introduced, however, themodel by Sche�er [56] and Pascual [41] is used as an example. The �sh will beconsidered as localized in schools, cruising and feeding according to de�ned rules.The process of aggregation of individual �shes and the persistence of schools underenvironmental or social constraints has already been studied by many other authors[4,6,12,13,15,36,40,47,49,52,66] and will not be considered here.

H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) 53–67 55

Fig. 1. Sketch of simpli�ed marine food chain.

2. The Plankton interaction–di�usion model

The considered basic marine food chain from nutrients to planktivorous �sh issketched in Fig. 1.The corresponding simple prey–predator model of the interaction and dispersion of

dynamic phytoplankton P and zooplankton Z at time � and position R = {x1; x2; x3},driven by nutrients N , �sh F as well as di�usivities DP and DZ as external controlparameters, reads

@P@�= �

NHN + N

P − cP2 − Pn

HnP + PnZ + DP�P; (1)

@Z@�= e

Pn

HnP + PnZ − �Zq − � Zk

HkZ + ZkF + DZ�Z: (2)

The interaction terms in Eqs. (1) and (2) account for the work by several authors,e.g. Beltrami [2,3], Brindley et al. [9,30,79], Malchow [22–25,27], Pascual [41,42],Sche�er [56–60] and Steele et al. [65–70]. � is the growth rate of phytoplankton, is the grazing rate of zooplankton on phytoplankton, c is the competition coe�cient ofphytoplankton, e is the prey assimilation e�ciency of zooplankton, � is the mortalityof zooplankton and � is the feeding rate of �sh on zooplankton. HP , HZ and HN arethe half-saturation constants of functional responses and nutrient limitation. Time �and length xi; i = 1; 2; 3, are measured in days [d] and meters [m], respectively. P,Z , F , HP and HZ are usually measured in milligrams of dry weight per litre [mgdw=l]; N and HN are given in relative units; e is a dimensionless parameter; thedimension of �, and � is [d−1], � is measured in [(mg dw=l)1−q d−1], whereas c isexpressed in [(mg dw=l)−1 d−1]. The di�usion coe�cients DP and DZ are measured in[m2 d−1]. � is the Laplace operator. The exponents n and k describe di�erent types offunctional response of zooplankton and �sh, respectively, e.g. n = k = 2 the e�ect ofpredator switching, whereas q = 1 and 2 stand for simple density-dependent mortalityand intraspeci�c competition of zooplankton, cf. the work by Ludwig [20], Matsuda etal. [29], May [31,32], Murdoch and Oaten [35], Noy-Meir [37,38], Rosenzweig [53],Svirezhev and Logofet [75], Steele [68,69], Teramoto et al. [76] and Wissel [85].Because the general phytoplankton-growth limiting nutrient N is not a dynamic state

variable here, one can reformulate the phytoplankton growth in Eq. (1) in standardlogistic form, introducing the carrying capacity K and the intrinsic logistic growth rate� of phytoplankton, i.e.,

K =�N

c(HN + N ); �= Kc: (3)


Possible spatio-temporal variations of growth, mortality and predation will be modelledlater by certain functions of space R and time �. Eqs. (1) and (2) read now

@P@�= �P

(1− P

K

)− Pn

HnP + PnZ + DP�P; (4)

@Z@�= e

Pn

HnP + PnZ − �Zq − � Zk

HkZ + ZkF + DZ �Z: (5)

Now, dimensionless quantities of densities, time and space are introduced:

X1 =PK; X2 =

ZeK; f =

�FeK ��

; t = ��; {x; y; z}= sL{x1; x2; x3}; (6)

where L is the total length in all three space dimensions, s is an integer scale factorwhich models the scale of the expected patchy patterns and �� is the spatial mean of �in the considered area volume V = L3:

��=1V

∫V� dV: (7)

One �nds the dimensionless equations@X1@t

= rX1(1− X1)− anX n11 + bnX n1

X2 + d1 �X1; (8)

@X2@t

=anX n1

1 + bnX n1X2 − mqX q2 − gkX k2

1 + gkX k2f + d2 �X2 (9)

with

r =��; b=

KHP; a= b

(e ��

)1=n; m= eK

(�e K ��

)1=q;

g=eKHZ; d1 =

s2DPL2 ��

; d2 =s2DZL2 ��

: (10)

This model is the basis for the following investigation of the local processes and theirsensitivity to external forcing by �sh predation.

2.1. Local behaviour of the model

The local properties, i.e., the emergence and stability of spatially uniform station-ary states in the absence of di�usion, can be analyzed by means of the zero-isoclinerepresentations:

X2(X1) = r(1− X1)1 + bnX n1

anX n−11

; (11)

X1(X2) =[

�(X2)an − bn�(X2)

]1=n; �(X2) = mqX

q−12 +

gkX k−12

1 + gkX k2f: (12)

Di�erent combinations of the exponents n = 1; 2, q = 1; 2 and k = 1; 2 may create adi�erent dynamics like bistability, tristability, oscillations and excitability. However, asmentioned before, the considerations will be restricted to the Sche�er–Pascual model,i.e., Holling-type II functional response and simple density-dependent mortality of


Fig. 2. Nullclines X2(X1) and X1(X2) of the local predator–prey plankton system (13,14) withr = 1; a = b = 5; m = 0:5; g = 10 for f0 = 0 (planktonic limit cycle), f1 = 0:035; f2 = 0:065 (bista-bility) and f3 = 0:095 (phytoplankton dominance).

zooplankton (n= q= 1) as well as type III functional response of �sh (k = 2):

@X1@t

= rX1(1− X1)− aX11 + bX1

X2; (13)

@X2@t

=aX1

1 + bX1X2 − mX2 − g2X 22

1 + g2X 22f: (14)

The corresponding zero-isoclines are drawn in Fig. 2 for a certain �xed parameter setand increasing �sh predation pressure.Without �sh (f0) and for low values (f1), the model exhibits phytoplankton–

zooplankton prey–predator oscillations around the unstable single stationary state. Forintermediate values of �sh (f2), the system is bistable, whereas it returns to a mono-stable but phytoplankton-dominated state for high �sh predation pressure (f3). Thepossible sequences of bifurcations [82,83] have been explored in detail [56,72,73], alsofor the excitable system with n= 2 [71].The externally forced local model has been studied as well. A periodically varying

phytoplankton growth rate [73] or �sh predation rate [48] or periodic changes of allparameters [60] have been investigated and rather small windows of deterministic chaoshave been obtained for certain intervals of the amplitude of the forcing oscillation.


3. Spatial model with mobile �sh schools

3.1. Parameters, boundary and initial conditions

The spatial model will be investigated for a parameter set of the local modelthat admits only stable plankton oscillations in the absence of �sh. The presence of�sh will immediately switch the system from these stable oscillations to the stablenon-oscillating phytoplankton-dominated state, cf. f0 and f3 in Fig. 2.The dynamics will be investigated in the horizontal (x; y)-plane with no- ux bound-

ary conditions north and south (x=0 and 1) and periodic boundary conditions west andeast (y=0 and 1). The turbulent di�usion coe�cients d1=d2=d are chosen accordingto Okubo’s oceanic di�usion diagrams [39] and correspond to a real system length L ofabout 106 m. The equations are solved on a quadratic spatial grid of 100× 100 pointswith a time step of 0.01, i.e., the real distance between the grid points is about 104 mand the real time step about half an hour, assuming a mean phytoplankton growth rate�� of about 0:5 d−1.The initial population distribution is spatially non-uniform. The phytoplankton and

zooplankton populations are localized in the center of the model area. The phytoplank-ton patch is at its carrying capacity whereas the smaller zooplankton patch of arbitrarydensity is concentric inside the phytoplankton patch. The di�usion will force the prop-agation of these initial population fronts. A 1D sketch of this initial situation is drawnin Fig. 3.

3.2. Rules of �sh school motion

Fish are considered as localized in a number of schools with speci�c characteristics,i.e., these schools are treated as super-individuals [59]. They feed on zooplankton andmove on the numerical grid for the integration of the plankton-dynamical reaction–di�usion equations, according to the following rules:

1. The �sh schools feed on zooplankton down to its protective minimal density andthen move.

Fig. 3. Localized initial conditions for plankton.


Fig. 4. Rules of �sh school motion.

2. The �sh schools might even have to move before reaching the minimal food densitybecause of a maximum residence time which can be due to the protection againsthigher predation or the security of oxygen demand.

3. The �sh schools memorize and prefer the previous direction of motion. Therefore,the new direction is randomly chosen within an “angle of vision” of ±90◦ left andright of the previous direction with some decreasing weight.

4. At the re ecting northern and southern boundaries the �sh schools obey some mixedphysical and biological laws of re ection.

5. The �sh schools act independently of each other. They do not change their speci�ccharacteristics of size, speed and maximum residence time.

The rules of motion posed are as simple but also as realistic as possible, followingrelated reports, cf. [8,11,34]. They are sketched in Fig. 4.

3.3. Impacts on local dynamics

Now, the impact of �ve food-searching, feeding and cruising �sh schools on thedeveloping spatio-temporal plankton patterns will be investigated. The special choiceof the initial condition implies the emergence of propagating non-periodic but still con-centric circular di�usive waves, i.e., an irregular target pattern behind the zooplanktonfront which is much slower than the phytoplankton front. Corresponding results for the1D case have been published recently [44,62,63].For the �rst 500 iterations, the plankton waves can develop without perturbations.

Then, the �ve schools invade the area, starting from the periodic east–west boundary.All schools have the same size of four grid compartments, i.e., 4× 108 m2. Their �rstcruising direction is randomly chosen. If l=1; 2; : : : ; 5 is the number of the �sh school,


Fig. 5. Trace of one cruising �sh school.

their speed is [18− 2 (l− 1)] time steps for the distance between two grid points, i.e.,from about 0:3 to 0:6 m=s. Their maximum residence time is [20+2(l−1)] time steps,i.e., from about 9 to 14 h. A part of the trace of the second �sh schools is given inFig. 5.It has been shown in a previous paper [33] that such traces can be regarded as frac-

tional Brownian motion. The interplay of �sh school motion and plankton dynamicsleads to a remarkable perturbation and deformation of the spatio-temporal planktonicstructures. The top view of a sequence of developing patterns in the considered areais shown in Fig. 6. The darker the colour the less phytoplankton is present. The zoo-plankton patterns are complementary to the shown phytoplankton structures.The �rst two pictures in the upper row show the expected propagation of the circular

but irregular plankton waves. The invading �sh schools destroy the target pattern,however, out of the irregular wave propagation, the system self-organizes in a hugedouble spiral. It has been checked that this spiral is stable for numerical runs of106 iterations which are equivalent more than 50 real time years. The correspondingtrajectory of the spatially averaged plankton densities compared with the prey–predatorlimit cycle of the local system (13,14) is shown in Fig. 7. The contraction of theattractor which is typical for spiral wave organization [33] is readily seen. A selectedpart of a long-term time series after establishment of the spiral is drawn in Fig. 8.The corresponding next-maximum map of the long-term simulation is presented inFig. 9.


Fig. 6. Development of a giant double spiral from the localized initial condition. Scale factor s=1. Parametersas in Fig. 2, except d1 = d2 = 10−4, m = 0:6, f = 0:5.

Fig. 7. The nullclines for f = 0 and the corresponding homogeneous plankton limit cycle as well as the�sh-induced contracted spatially averaged plankton cycle in phase space.


Fig. 8. Irregular spatially averaged phytoplankton and zooplankton oscillations.

Fig. 9. Next-maximum map of spatially averaged phytoplankton oscillations in Fig. 8.


Fig. 10. Development of chaotic travelling waves along a linear north–south gradient of phytoplankton growthfrom the localized initial condition. Scale factor s=1. Parameters as in Fig. 6, except rsouth = 2; rnorth = 0:6.

The irregularity of the spatially averaged oscillations and the resulting cloud ofnext-maximum map points is not only due to the plankton-�sh dynamics. There arethree additional reasons:The �rst reason is the choice of the spatially non-uniform initial conditions. How-

ever, this is not an arti�cial but a rather natural assumption. Spatially uniform initialconditions do not favor the formation of such a highly organized structure, at least notat this large scale s= 1 [16,33].The second reason is the not so realistic assumption of a long-term uniform en-

vironment. Any supercritical variability of parameters would certainly destroy such asensitive pattern. An example for the latter situation is the pattern dynamics with nutri-ent gradient, cf. the work of Pascual [41,42] for the 1D case without �sh. For the sameinitial and boundary conditions, almost the same dynamics develop in the �rst row ofFig. 10 except for some asymmetry caused by the gradient in phytoplankton growth.The �rst pattern in the second row shows the suggestion of a double spiral whichis destroyed, however, with ongoing time. Finally, the expected plane chaotic travel-ling waves along the gradient remain, slightly disturbed by the �sh schools [16]. Any“higher-order” complexity like the spiral formation is suppressed, at least at this scale.The third reason is, of course, the impact of the top predation by the cruising �sh

schools. It has been checked that at this spatial scale and without �sh schools, thespatial system would relax to the local prey–predator limit cycle shown in Fig. 7,now uniform in space. The �sh predation leads to local perturbations of the spatiallyuniform plankton cycle. The feeding places of the �sh schools get out of phase and anet of di�usively coupled nonlinear oscillators in space appears. This is known to create


numerous forms of spatio-temporal dissipative structures. It has been demonstrated thatthe structures do not strongly depend on the �sh school rules, given in Section 3.2.

4. Concluding remarks

A conceptual coupled biomass- and rule-based model of phytoplankton–zooplankton-�sh dynamics has been investigated for conditions of temporal, spatial and spatio-temporal dissipative pattern formation. Growth, interaction and transport of planktonhave been modelled by reaction–di�usion equations, i.e., continuous in space and time.The �sh have been assumed to be localized in a number of schools, obeying certainde�ned behavioral rules of feeding and moving which essentially depend on the localzooplankton density and the speci�c maximum residence time. The schools themselveshave been treated as static super-individuals, i.e., they do not have any inner dynamicslike age or size structure.It turned out that certain spatially non-uniform initial distribution of the plankton

populations in an uniform environment are an important precondition for the generationof highly organized spatio-temporal plankton population patterns through the impact offeeding and cruising �sh schools. Supercritical variations of the environment suppressthe formation of such structures and create their own patterns in space and time. Herethe formation as well as the suppression of a spiral pattern have been demonstrated.The formation and decay of structures do strongly depend on the spatial scale; only

the full-scale case has been described here. The details of the behavioural rules for theupper discrete rule-based model layer have been of minor relevance for the generationand stabilization of plankton population structures.

Acknowledgements

The authors acknowledge helpful discussions with J. Brindley (Leeds), M. Kirkilionis(Heidelberg) and E. Kriksunov (Moscow).This work is partially supported by INTAS Grant no. 96-2033, by NATO Linkage

Grant no. OUTRG.LG971248, by DFG Grant no. 436 RUS 113=447 and by RFBRGrant no. 98-04-04065.

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