Biological turbulence as a generic mechanism of plankton patchiness

Biological turbulence as a generic mechanism of plankton patchiness Sergei Petrovskii Department of Mathematics, University of Leicester, UK... Lorentz Center, Leiden, December 8-12, 214

Plan of the talk Introduction Baseline model: diffusion-reaction equations Patch dynamics Advanced model: nonlocal diffusion Conclusions

Plankton patchiness at different 3.2. Local scales (single-patch) dynamics. The spatial structures resulting from modeling the features with the remote-sensing images of chloro (Figure 4).. Figure 4. Mesoscale spatial. variability of remot Small scale ( 1m 1m) California Large (composed scale ( 3km 3km) SeaWiFS image on Marc modeling of patches formation as a result of pr (21). The next natural step is to consider the dynamics o or habitat, created by relatively stable temperature

Introduction & motivation Spatiotemporal patchiness, usually irregular, is a common property of plankton systems A number of explanations have been suggested such as effect of the turbulent flows environmental heterogeneity

Plankton patchiness at different scales (contd.) (Mackas, 1977) (Weber et al., 1986)

There are physical and biological ranges Small and large patches behave differently Physical range Biological range Patch size (Levin, 1995)

Generic model The balance equation: ( Rate of change in population density ) = ( Local population dynamics ) + Dispersal. Hence the simplest (baseline) model is u(r, t) t = F(u) + D 2 u(r, t), where u is the population density at position r and time t. KISS model: the existence of the critical patch size, the patch will only survive if L > L cr.

Prediction from the single-species model

Interspecific interactions are important (Sketch) 28 Feb 22 fish detritus zooplankton phytoplankton bacteria nutrients

Mathematical model of the plankton dynamics u(r, t) t v(r, t) t = D T 2 u(r, t) + f (u)u r(u)v, = D T 2 v(r, t) + κr(u)v Mv where u, v are the phyto- and zooplankton densities. In dimensionless variables: u(r, t) t v(r, t) t = 2 u(r, t) + γu(u β)(1 u) uv 1 + Λu, = 2 v(r, t) + uv 1 + Λu mv (assuming the strong Allee effect for phytoplankton) Note that, for Λ =, these equations coincide with the SI model of a host-pathogen system with a strong infection.

2D case, simulations The initial conditions: u(x, y, ) = u for x 11 < x < x 12, y 11 < y < y 12, u(x, y, ) = otherwise, v(x, y, ) = v for x 21 < x < x 22, y 21 < y < y 22, v(x, y, ) = otherwise.

For the infection (predation) not very strong 3 3 t=6 t=2 2 2 1 1 1 2 3 1 2 3

When infection becomes stronger... 3 t=8 3 t=235 2 2 1 1 1 2 3 1 2 3

Space, y 1 2 3 4 5 Larger space, longer time 1 (a) (c) 2 3 4 5 (b) (d) 5 5 5 5 11 4 4 1.2 1 4 4 1.8.8 3 3.8.8 3 3.6.6 2 2.4.4 Space, y 1 1.2.2 1 2 2 3 3 4 4 5 5 1.6.6 2 2.4.4 1 1.2.2 1 2 2 3 3 4 4 5 5 1 Space, x 5 4 Initial(c)growth Biological(d) turbulence in the large-time limit 5 1 4.8 Biological turbulence is a common 3 property of multi-species 3 diffusion-reaction systems.6 2 2 1.2 1.8.6

Very strong infection (nonspatial) Transition of the infection type from lysogenic to lytic??.8.8.6.6 Predator / Infected.4 3 1 Predator / Infected.4.2.2 2 1 2.2.4.6.8 1 Prey / Susceptible.2.4.6.8 1 Prey / Susceptible The nonspatial system is not viable

Very strong infection (lytic), spatial 3 (a) 3 t=9 (b) t=145 2 2 1 1 1 2 3 (c) t=19 3 1 2 3 (d) t=455 3 2 2 1 1 1 2 3 1 2 3 The spatial system persists through the patchy spread

Larger space, longer time, stronger infection... 6 5 4 3 2 1 6 t=1 t=7 5 4 3 2 1 2 4 6 2 4 6

Advanced model Application of the Fickian diffusion to turbulence may be disputable as the turbulent transport is known to be nonlocal. Hence we consider an alternative framework: u t+1 (r) = v t+1 (r) = Ω Ω k ( r r ) f ( u t ( r ), v t ( r )) dr, k ( r r ) g ( u t ( r ), v t ( r )) dr, where, for the turbulent transport, a fat-tailed kernel seems to be more relevant.

Examples of possible kernels Reference case: k F ( r r ) = 1 2πα 2 i exp ( r r 2 2α 2 i ). Cauchy kernels, type I: and type II: k CI (r, r ) = β 2 i π(β i + r r ) 3, k CII (r, r ) = γ i. 2π γ 2i + ( r r 2 ) 3

Simulations, kernel type I 1 Prey Distribution 1 Prey Distribution 1 1 1 1 1 1 (t=2) (t=15)

Simulations, kernel type II 1 Prey Distribution 1 Prey Distribution 1 1 1 1 1 1 (t=2) (t=95)

Conclusions Interplay between the turbulent diffusion and infection or predation can result in a patchy spatiotemporal dynamics Stronger infection (e.g. transition from lysogenic to lytic) makes the patchiness more prominent The patchy dynamics is observed both in the diffusionreaction model and in the nonlocal integral-difference model

Acknowledgements Horst Malchow (Osnabrueck, Germany) Andrew Morozov (Leicester, UK) Diomar Mistro (Santa Maria, Brazil) Luiz Rodrigues (Santa Maria, Brazil)

Selected references (milestones) Medvinsky, A.B., Petrovskii, S.V., Tikhonova, I.A., Malchow, H., and Li, B.-L. (22). Spatiotemporal complexity of plankton and fish dynamics. SIAM Review 44, 311-37. Malchow, H., Petrovskii, S.V., and Venturino, E. (28) Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, Simulations, Chapman & Hall / CRC Press, 443 p. Malchow, H., Hilker, F.M., Petrovskii, S.V., and Brauer, K. (24) Oscillations and waves in a virally infected plankton system, I. The lysogenic stage. Ecological Complexity 1, 211-223. Petrovskii, S.V., Morozov, A.Y., and Venturino, E. (22) Allee effect makes possible patchy invasion in a predator-prey system. Ecology Letters 5, 345-352. Petrovskii, S.V., Malchow, H., Hilker, F.M., and Venturino, E. (25) Patterns of patchy spread in deterministic and stochastic models of biological invasion and biological control. Biological Invasions 7, 771-793. Rodrigues, L.A.D., Mistro, D.C., Cara, E.R., Petrovskaya, N., and Petrovskii, S.V. Patchy invasion of stage-structured alien species with short-distance and long-distance dispersal, submitted.

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