Spatio-temporal pattern formation in coupled models of plankton dynamics and sh school motion

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1 Nonlinear Analysis: Real World Applications 1 (2000) Spatio-temporal pattern formation in coupled models of plankton dynamics and sh school motion Horst Malchow a;, Birgit Radtke a, Malaak Kallache a, Alexander B. Medvinsky b, Dmitry A. Tikhonov b, Sergei V. Petrovskii c a Department of Mathematics and Computer Science, Institute of Environmental Systems Research, University of Osnabruck, Artilleries tr 34, D Osnabruck, Germany b Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Moscow Region, Russia c Shirshov Institute of Oceanology, Russian Academy of Sciences Nakhimovsky Prospect 36, Moscow , Russia Received 11 October 1999 Keywords: Plankton dynamics; Predator prey model; Bistability; Reaction diusion system; Rule-based sh school motion; Spiral waves; Chaos 1. Introduction Many mechanism of the spatio-temporal variability of natural plankton populatinos are still nuclear. Pronounced physical patterns like thermoclines, upwelling, fronts and eddies often set the frame for the biological processes [28]. However, under conditions of relative physical uniformity, the temporal and spatio-temporal variability can be a consequence of the coupled nonlinear biological and chemical dynamics [19,66]. Hence, the formation and stabilization of dissipative temporal, spatial and spatio-temporal structures by nonlinear systems dynamics is of continuous interest in theoretical biology and ecology. Conceptual minimal models are an appropriate tool for searching and understanding the basic mechanisms of this pattern formation. Several examples of their usefulness in studies of plankton dynamics like patchiness and phytoplankton blooms are known, cf. [5,10,58,65 70,80,86]. Corresponding author /00/$ - see front matter? 2000 Elsevier Science Ltd. All rights reserved. PII: S X(99)

2 54 H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) In 1952, Turing [81] showed that the reaction and diusion of at least two agents with considerably dierent diusivities can generate spatial structure of an uniform distribution. In 1972, Segel and Jackson [61] were the rst to apply Turing s idea to a problem in population dynamics: The dissipative instability in the prey predator interaction of phytoplankton and herbivorous copepods. Levin and Segel [19] suggested this scenario of spatial pattern formation for a possible origin of planktonic patchiness. Recently, local bistability, predator prey limit-cycle oscillations, plankton front propagation and the generation and drift of planktonic Turing patches were found in a minimal phytoplankton zooplankton interaction model [22 25,27] that was originally formulated by Scheer [56], accounting for the eects of nutrients and planktivorous sh on alternative local equilibria of the plankton community. Routes to chaos through seasonal 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 models and data of systems with three or more interacting plankton species have been studied as well [1,57]. The emergence of diusion-induced spatio-temporal chaos along a linear nutrient gradient has been found by Pascual [41,42] in Scheer s model without sh predation. Plankton-generated chaos in a sh population has been reported by Horwood [14]. Conditions for the emergence of three-dimensional spatial and spatio-temporal patterns after dierential-ow-induced instabilities (DIFICI [54]) of spatially uniform populations were derived by Malchow [24 26] and illustrated by patterns in Scheer s model. Instabilities of the spatially uniform distribution can appear if phytoplankton and zooplankton move with dierent velocities but regardless of which one is faster. This mechanism of generating patchy patterns is much more general than the Turing mechanism which depends on strong conditions on the diusion coecients. One can imagine a wide range of applications in population dynamics. The eect of external hydrodynamical forcing on the appearance and stability of nonequilibrium spatio-temporal patterns in Scheer s model was also studied by Malchow and Shigesada [24]. A channel under tidal forcing served as a hydrodynamical model system with a relatively high detention time of matter. Examples were provided on dierent time scales: The simple physical transport and deformation of a spatially non-uniform initial plankton distribution as well as the biologically determined formation of a localized spatial maximum of phytoplankton biomass. Other processes of spatial pattern formation after instability of spatially homogeneous 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 eects of nonuniform environmental potentials [21,64]. In this paper we focus on the inuence of sh on the spatio-temporal pattern formation of interacting plankton populations in uniform and non-uniform environments. A general class of planktonic prey predator systems is introduced, however, the model by Scheer [56] and Pascual [41] is used as an example. The sh will be considered as localized in schools, cruising and feeding according to dened rules. The process of aggregation of individual shes and the persistence of schools under environmental 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.

3 H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) Fig. 1. Sketch of simplied marine food chain. 2. The Plankton interaction diusion model The considered basic marine food chain from nutrients to planktivorous sh is sketched 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 = {x 1 ;x 2 ;x 3 }, driven by nutrients N, sh F as well as diusivities D P and D Z as external control = N H N + N P cp2 HP n + Z + D PP; (1) Pn P n P n = e HP n + Z Pn Zq HZ k + F + D ZZ: (2) Zk 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], Scheer [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 coecient of phytoplankton, e is the prey assimilation eciency of zooplankton, is the mortality of zooplankton and is the feeding rate of sh on zooplankton. H P, H Z and H N are the half-saturation constants of functional responses and nutrient limitation. Time and length x i ;i=1; 2; 3, are measured in days [d] and meters [m], respectively. P, Z, F, H P and H Z are usually measured in milligrams of dry weight per litre [mg dw=l]; N and H N are given in relative units; e is a dimensionless parameter; the dimension of, and is [d 1 ], is measured in [(mg dw=l) 1 q d 1 ], whereas c is expressed in [(mg dw=l) 1 d 1 ]. The diusion coecients D P and D Z are measured in [m 2 d 1 ]. is the Laplace operator. The exponents n and k describe dierent types of functional response of zooplankton and sh, respectively, e.g. n = k = 2 the eect of predator switching, whereas q = 1 and 2 stand for simple density-dependent mortality and intraspecic competition of zooplankton, cf. the work by Ludwig [20], Matsuda et al. [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 standard logistic form, introducing the carrying capacity K and the intrinsic logistic growth rate of phytoplankton, i.e., K = N ; = Kc: (3) c(h N + N)

4 56 H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) Possible spatio-temporal variations of growth, mortality and predation will be modelled later by certain functions of space R and time. Eqs. (1) and (2) read = P 1 P ) P n K HP n + Z + D Pn P P; (4) P n = e HP n + Z Pn Zq HZ k + F + D Zk Z Z: (5) Now, dimensionless quantities of densities, time and space are introduced: X 1 = P K ; X 2 = Z ek ; F f= ek ; t =; {x; y; z} = s L {x 1;x 2 ;x 3 }; (6) where L is the total length in all three space dimensions, s is an integer scale factor which models the scale of the expected patchy patterns and is the spatial mean of in the considered area volume V = L 3 : = 1 dv: (7) V V One nds the dimensionless 1 = rx 1 (1 X 1 ) an X1 1+b n X1 n X 2 + d 1 X 1 ; = an X1 n 1+b n X1 n X 2 m q X q 2 g k X2 k 1+g k X2 k f + d 2 X 2 (9) with r = ; b= K ( e ) ( ) 1=n 1=q ; a= b ; m= ek ; H P ek g = ek ; d 1 = s2 D P H Z L 2 ; d 2 = s2 D Z L 2 : (10) This model is the basis for the following investigation of the local processes and their sensitivity to external forcing by sh predation Local behaviour of the model The local properties, i.e., the emergence and stability of spatially uniform stationary states in the absence of diusion, can be analyzed by means of the zero-isocline representations: X 2 (X 1 )=r(1 X 1 ) 1+bn X1 n a n X n 1 ; (11) 1 [ ] 1=n (X 2 ) X 1 (X 2 )= a n b n ; (X 2 )=m q X q g k X k 1 2 (X 2 ) 1+g k X2 k f: (12) Dierent combinations of the exponents n =1; 2, q =1; 2 and k =1; 2 may create a dierent dynamics like bistability, tristability, oscillations and excitability. However, as mentioned before, the considerations will be restricted to the Scheer Pascual model, i.e., Holling-type II functional response and simple density-dependent mortality of

5 H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) Fig. 2. Nullclines X 2 (X 1 ) and X 1 (X 2 ) of the local predator prey plankton system (13,14) with r =1; a = b =5; m =0:5; g = 10 for f 0 = 0 (planktonic limit cycle), f 1 =0:035; f 2 =0:065 (bistability) and f 3 =0:095 (phytoplankton dominance). zooplankton (n = q = 1) as well as type III functional response of sh (k = = rx 1 (1 X 1 ) ax 1 1+bX 1 X 2 ; = ax 1 1+bX 1 X 2 mx 2 g2 X g 2 X 2 2 f: (14) The corresponding zero-isoclines are drawn in Fig. 2 for a certain xed parameter set and increasing sh predation pressure. Without sh (f 0 ) and for low values (f 1 ), the model exhibits phytoplankton zooplankton prey predator oscillations around the unstable single stationary state. For intermediate values of sh (f 2 ), the system is bistable, whereas it returns to a monostable but phytoplankton-dominated state for high sh predation pressure (f 3 ). The possible sequences of bifurcations [82,83] have been explored in detail [56,72,73], also for 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 all parameters [60] have been investigated and rather small windows of deterministic chaos have been obtained for certain intervals of the amplitude of the forcing oscillation.

6 58 H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) 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 model that 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 stable non-oscillating phytoplankton-dominated state, cf. f 0 and f 3 in Fig. 2. The dynamics will be investigated in the horizontal (x; y)-plane with no-ux boundary conditions north and south (x =0 and 1) and periodic boundary conditions west and east (y =0 and 1). The turbulent diusion coecients d 1 =d 2 =d are chosen according to Okubo s oceanic diusion diagrams [39] and correspond to a real system length L of about 10 6 m. The equations are solved on a quadratic spatial grid of points with a time step of 0.01, i.e., the real distance between the grid points is about 10 4 m and 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 phytoplankton patch is at its carrying capacity whereas the smaller zooplankton patch of arbitrary density is concentric inside the phytoplankton patch. The diusion will force the propagation of these initial population fronts. A 1D sketch of this initial situation is drawn in Fig Rules of sh school motion Fish are considered as localized in a number of schools with specic characteristics, i.e., these schools are treated as super-individuals [59]. They feed on zooplankton and move on the numerical grid for the integration of the plankton-dynamical reaction diusion equations, according to the following rules: 1. The sh schools feed on zooplankton down to its protective minimal density and then move. Fig. 3. Localized initial conditions for plankton.

7 H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) Fig. 4. Rules of sh school motion. 2. The sh schools might even have to move before reaching the minimal food density because of a maximum residence time which can be due to the protection against higher 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 and right of the previous direction with some decreasing weight. 4. At the reecting northern and southern boundaries the sh schools obey some mixed physical and biological laws of reection. 5. The sh schools act independently of each other. They do not change their specic characteristics of size, speed and maximum residence time. The rules of motion posed are as simple but also as realistic as possible, following related reports, cf. [8,11,34]. They are sketched in Fig Impacts on local dynamics Now, the impact of ve food-searching, feeding and cruising sh schools on the developing spatio-temporal plankton patterns will be investigated. The special choice of the initial condition implies the emergence of propagating non-periodic but still concentric circular diusive waves, i.e., an irregular target pattern behind the zooplankton front which is much slower than the phytoplankton front. Corresponding results for the 1D 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., m 2. Their rst cruising direction is randomly chosen. If l=1; 2;:::;5 is the number of the sh school,

8 60 H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) 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 to0:6 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 in Fig. 5. It has been shown in a previous paper [33] that such traces can be regarded as fractional Brownian motion. The interplay of sh school motion and plankton dynamics leads to a remarkable perturbation and deformation of the spatio-temporal planktonic structures. The top view of a sequence of developing patterns in the considered area is shown in Fig. 6. The darker the colour the less phytoplankton is present. The zooplankton 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 huge double spiral. It has been checked that this spiral is stable for numerical runs of 10 6 iterations which are equivalent more than 50 real time years. The corresponding trajectory of the spatially averaged plankton densities compared with the prey predator limit cycle of the local system (13,14) is shown in Fig. 7. The contraction of the attractor which is typical for spiral wave organization [33] is readily seen. A selected part 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 in Fig. 9.

9 H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) Fig. 6. Development of a giant double spiral from the localized initial condition. Scale factor s=1. Parameters as in Fig. 2, except d 1 = d 2 =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.

10 62 H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) Fig. 8. Irregular spatially averaged phytoplankton and zooplankton oscillations. Fig. 9. Next-maximum map of spatially averaged phytoplankton oscillations in Fig. 8.

11 H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) Fig. 10. Development of chaotic travelling waves along a linear north south gradient of phytoplankton growth from the localized initial condition. Scale factor s = 1. Parameters as in Fig. 6, except r south =2; r north =0:6. The irregularity of the spatially averaged oscillations and the resulting cloud of next-maximum map points is not only due to the plankton-sh dynamics. There are three additional reasons: The rst reason is the choice of the spatially non-uniform initial conditions. However, this is not an articial but a rather natural assumption. Spatially uniform initial conditions do not favor the formation of such a highly organized structure, at least not at this large scale s = 1 [16,33]. The second reason is the not so realistic assumption of a long-term uniform environment. Any supercritical variability of parameters would certainly destroy such a sensitive pattern. An example for the latter situation is the pattern dynamics with nutrient gradient, cf. the work of Pascual [41,42] for the 1D case without sh. For the same initial and boundary conditions, almost the same dynamics develop in the rst row of Fig. 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 which is destroyed, however, with ongoing time. Finally, the expected plane chaotic travelling 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, the spatial 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 spatially uniform plankton cycle. The feeding places of the sh schools get out of phase and a net of diusively coupled nonlinear oscillators in space appears. This is known to create

12 64 H. Malchow et al. / Nonlinear Analysis: Real World Applications 1 (2000) numerous forms of spatio-temporal dissipative structures. It has been demonstrated that the structures do not strongly depend on the sh school rules, given in Section Concluding remarks A conceptual coupled biomass- and rule-based model of phytoplankton zooplankton- sh dynamics has been investigated for conditions of temporal, spatial and spatiotemporal dissipative pattern formation. Growth, interaction and transport of plankton have been modelled by reaction diusion equations, i.e., continuous in space and time. The sh have been assumed to be localized in a number of schools, obeying certain dened behavioral rules of feeding and moving which essentially depend on the local zooplankton density and the specic maximum residence time. The schools themselves have been treated as static super-individuals, i.e., they do not have any inner dynamics like 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 generation of highly organized spatio-temporal plankton population patterns through the impact of feeding and cruising sh schools. Supercritical variations of the environment suppress the formation of such structures and create their own patterns in space and time. Here the 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 the upper discrete rule-based model layer have been of minor relevance for the generation and 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 , by NATO Linkage Grant no. OUTRG.LG971248, by DFG Grant no. 436 RUS 113=447 and by RFBR Grant no References [1] F.A. Ascioti, E. Beltrami, T.O. Carroll, C. Wirick, Is there chaos in plankton dynamics? J. Plankton Res. 15 (1993) [2] E. Beltrami, A mathematical model of the brown tide, Estuaries 12 (1989) [3] E. Beltrami, Unusual algal blooms as excitable systems: the case of brown-tides, Environ. Modeling Assessment 1 (1996) [4] R.W. Blake, Fish Locomotion, Cambridge University Press, Cambridge, [5] J.S. Collie, P.D. Spencer, Modeling predator prey dynamics in a uctuating environment, Canad. J. Fish. Aquat. Sci. 51 (1994) [6] D.H. Cushing, Marine Ecology and Fisheries, Cambridge University Press, Cambridge, 1975.

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