Post-bifurcational dynamics of spatiotemporal patterns in advection-reaction-diffusion systems

Transcription

1 Post-bifurcational dynamics of spatiotemporal patterns in advection-reaction-diffusion systems ANTONIOS ZAGARIS UNIVERSITY OF TWENTE APPLIED MATHEMATICS

2 dynamics of pistachio trees

3 dynamics of pistachio trees periodical cicadas

4 dynamics of pistachio trees guano subsidies periodical cicadas

5 dynamics of pistachio trees guano subsidies periodical cicadas Gaussian termite mounds

6 dynamics of pistachio trees guano subsidies periodical cicadas Gaussian termite mounds biological turbulence

7 dynamics of pistachio trees guano subsidies periodical cicadas Gaussian termite mounds biological turbulence stochastic rain

8 Colonial adventures in stereo ANTONIOS ZAGARIS UNIVERSITY OF TWENTE APPLIED MATHEMATICS

9 Temporal Variability in a Water Column Hawaii Ocean Time-Series Huisman, Pham Thi, Karl, Sommeijer Nature 2006

10 Light Limitation J. Math. Biology (1981) 12: dournaiof Mathematical 131oiog9 ~) by Springer-Verlag 1981 J. Math. Biology (1982) 16:1-24 Journal of Mathematical Biology 9 Springer-Verlag 1982 Analysis of the Self-Shading Effect on Algal Vertical Distribution in Natural Waters* Global Stability of Stationary Solutions to a Nonlinear Diffusion Equation in Phytoplankton Dynamics Nanako Shigesada** and Akira Okubo Marine Sciences Research Center, State University of New York, Stony Brook, N.Y , USA Abstract. Self-shading of light by algae growing in a column of water plays an important:role in the dynamics of algal blooms. Thus without self-shading the algal concentration would increase more rapidly, making the nutrient limitation too strong. Apart from the practical importance of self-shading, its inherent nonlinearity in the growth dynamics leads to an interesting mathematical problem, which warrants detailed analytical investigation. Our mathematical model for the self-shading effect includes vertical diffusion; algal settling, gross production, and collective losses of algae. Steady-state solutions of the model equation are investigated in detail by the phase plane method, and their stability examined. Finally we discuss the vertical profile of algal concentration. Key words: Self-shading-Nonlinear diffusion and reaction-algal vertical distribution Hitoshi Ishii I and Izumi Takagi 2 Department of Mathematics, Faculty of Science and Engineering, Chuo University, , Kasuga, Bunkyo-ku, Tokyo 112, Japan 2 Tokyo Metropolitan College of Aeronautical Engineering, , Minami-Senju, Arakawa-ku, Tokyo 116, Japan Abstract. We consider a nonlinear diffusion equation proposed by Shigesada and Okubo which describes phytoplankton growth dynamics with a selfshading effect. We show that the following alternative holds: Either (i) the trivial stationary solution which vanishes everywhere is a unique stationary solution and is globally stable, or (ii) the trivial solution is unstable and there exists a unique positive stationary solution which is globally stable. A criterion for the existence of positive stationary solutions is stated in terms of three parameters included in the equation. Key words: Global stability - Nonlinear diffusion equation - Self-shading

11 Light Limitation p ~ Io,: o,,0,f d s 1--r=~.,ir,j v i '\.\,r 0 qe qm _-q 0 qc qm =q

12 Light and Nutrient Co-limitation Limnol. Oceanogr., 46(8), 2001, q 2001, by the American Society of Limnology and Oceanography, Inc. Algal games: The vertical distribution of phytoplankton in poorly mixed water columns Christopher A. Klausmeier 1 and Elena Litchman 2 EAWAG, Seestrasse 79, CH-6047 Kastanienbaum, Switzerland Abstract Phytoplankton often face the dilemma of living in contrasting gradients of two essential resources: light that is supplied from above and nutrients that are often supplied from below. In poorly mixed water columns, algae can be heterogeneously distributed, with thin layers of biomass found on the surface, at depth, or on the sediment surface. Here, we show that these patterns can result from intraspecific competition for light and nutrients. First, we present numerical solutions of a reaction-diffusion-taxis model of phytoplankton, nutrients, and light. We argue that motile phytoplankton can form a thin layer under poorly mixed conditions. We then analyze a related game theoretical model that treats the depth of a thin layer of phytoplankton as the strategy. The evolutionarily stable strategy is the depth at which the phytoplankton are equally limited by both resources, as long as the layer is restricted to the water column. The layer becomes shallower with an increase in the nutrient supply and deeper with an increase in the light supply. For low nutrient levels, low background attenuation, and shallow water columns, a benthic layer occurs; for intermediate nutrient levels in deep water columns, a deep chlorophyll maximum occurs; and for high nutrient levels, a surface scum occurs. These general patterns are in agreement with field observations. Thus, this model can explain many patterns of algal distribution found in poorly mixed aquatic ecosystems. SIAM J. MATH. ANAL. Vol. 40, No. 4, pp c 2008 Society for Industrial and Applied Mathematics SIAM J. MATH. ANAL. Vol. 40, No. 4, pp c 2008 Society for Industrial and Applied Mathematics CONCENTRATION PHENOMENA IN A NONLOCAL QUASI-LINEAR PROBLEM MODELLING PHYTOPLANKTON I: EXISTENCE YIHONG DU AND SZE-BI HSU Abstract. We study the positive steady state of a quasi-linear reaction-diffusion system in one space dimension introduced by Klausmeier and Litchman for the modelling of the distributions of phytoplankton biomass and its nutrient. The system has nonlocal dependence on the biomass function, and it has a biomass-dependent drifting term describing the active movement of the biomass towards the location of the optimal growth condition. We obtain complete descriptions of the profile of the solutions when the coefficient of the drifting term is large, rigorously proving the numerically observed phenomenon of concentration of biomass for this model. Our theoretical results reveal four critical numbers for the model not observed before and offer several further insights into the problem being modelled. This is Part I of a two-part series, where we obtain nearly optimal existence and nonexistence results. The asymptotic profile of the solutions is studied in the separate Part II. Key words. quasi-linear, nonlocal dependence, phytoplankton, concentration phenomenon, reaction-diffusion equation CONCENTRATION PHENOMENA IN A NONLOCAL QUASI-LINEAR PROBLEM MODELLING PHYTOPLANKTON II: LIMITING PROFILE YIHONG DU AND SZE-BI HSU Abstract. This is Part II of our study on the positive steady state of a quasi-linear reactiondiffusion system in one space dimension introduced by Klausmeier and Litchman for the modelling of the distributions of phytoplankton biomass and its nutrient. In Part I, we proved nearly optimal existence and nonexistence results. In Part II, we obtain complete descriptions of the profile of the solutions when the coefficient of the drifting term is large, rigorously proving the numerically observed phenomenon of concentration of biomass for this model. Moreover, we reveal four critical numbers for the model and provide further insights to the problem being modelled. Key words. quasi-linear, nonlocal dependence, phytoplankton, concentration phenomenon, reaction-diffusion equation

13 Light and Nutrient Co-limitation Limnol. Oceanogr., 46(8), 2001, q 2001, by the American Society of Limnology and Oceanography, Inc. Algal games: The vertical distribution of phytoplankton in poorly mixed water columns Christopher A. Klausmeier 1 and Elena Litchman 2 EAWAG, Seestrasse 79, CH-6047 Kastanienbaum, Switzerland Abstract Phytoplankton often face the dilemma of living in contrasting gradients of two essential resources: light that is supplied from above and nutrients that are often supplied from below. In poorly mixed water columns, algae can be heterogeneously distributed, with thin layers of biomass found on the surface, at depth, or on the sediment surface. Here, we show that these patterns can result from intraspecific competition for light and nutrients. First, we present numerical solutions of a reaction-diffusion-taxis model of phytoplankton, nutrients, and light. We argue that motile phytoplankton can form a thin layer under poorly mixed conditions. We then analyze a related game theoretical model that treats the depth of a thin layer of phytoplankton as the strategy. The evolutionarily stable strategy is the depth at which the phytoplankton are equally limited by both resources, as long as the layer is restricted to the water column. The layer becomes shallower with an increase in the nutrient supply and deeper with an increase in the light supply. For low nutrient levels, low background attenuation, and shallow water columns, a benthic layer occurs; for intermediate nutrient levels in deep water columns, a deep chlorophyll maximum occurs; and for high nutrient levels, a surface scum occurs. These general patterns are in agreement with field observations. Thus, this model can explain many patterns of algal distribution found in poorly mixed aquatic ecosystems. Steady-state profiles No dynamics No surprises Fig. 1. Equilibrium vertical distributions of phytoplankton, nutrients, and light for varying v max determined by (A E) numerical solution of Eq. 1 and (F) our game theoretical approach (z* m from solving Eq. 4). Other parameters are given in Table 1. Note the changes in the biomass scale between parts.

14 A Spatial Problem in a Water Column Sinking phytoplankton Co-limited growth diffusion, advection light, sedimentary nutrient P t N t ¼ growth 2 loss 2 sinking þ mixing ¼ mðn;iþp 2 mp 2 v P z þ k 2 P z 2 ¼ 2uptake þ recycling þ mixing ¼ 2amðN; IÞP þ 1amP þ k 2 N z 2 Huisman, Pham Thi, Karl, Sommeijer Nature 2006 reaction advection diffusion model

15 A Spatial Problem in a Water Column Sinking phytoplankton Co-limited growth diffusion, advection light, sedimentary nutrient Huisman, Pham Thi, Karl, Sommeijer Nature 2006 stable pattern à oscillatory pattern à chaos

16 A Spatial Problem in a Water Column Sinking phytoplankton Co-limited growth diffusion, advection light, sedimentary nutrient u v = t "@zz 2 p "v@ z + h(z) ` 0 u "` 1h(z) "@ zz v 1 "` 1 (h(z) µ(z,u,v))p p AZ, Doelman, Pham Thi, Sommeijer SIAM J App Math 2009 reaction advection diffusion model

17 Slight Abstraction L models linear processes εm models passive processes affecting nutrient εk models nutrient sensitivity transport, optimal growth transport u v L 0 u v f(z,u,v; ") uv = t "K "M "g(z,u,v; ") AZ, Doelman, Pham Thi, Sommeijer SIAM J App Math 2009 consumer resource / exploiter victim

18 Linear Analysis L models linear processes εm models passive processes affecting nutrient εk models nutrient sensitivity transport, optimal growth transport u v L 0 u v f(z,u,v; ") uv = t "K "M "g(z,u,v; ") AZ, Doelman, Pham Thi, Sommeijer SIAM J App Math 2009 consumer resource / exploiter victim

19 Linear Analysis u v L 0 = "K "M t u v f(z,u,v; ") uv "g(z,u,v; ") AZ, Doelman Nonlinearity 2011 transcritical + small spectrum for passive processes

20 Linear Analysis V BL V* Trivial State DCM (0,0) n** n* n 1 AZ, Doelman, Pham Thi, Sommeijer SIAM J App Math 2009

21 Linear Analysis V BL V* Trivial State DCM (0,0) n** n* AZ, Doelman, Pham Thi, Sommeijer SIAM J App Math 2009 increased death rate n 1

22 Linear Analysis V BL V* Trivial State DCM (0,0) n** n* n 1 AZ, Doelman, Pham Thi, Sommeijer SIAM J App Math 2009 increased turbidity

23 u v p Nonlinear Dynamics? x 0 u0 v 0 center manifold reduction ẋ 0 = 0 x 0 a(0) x 2 0 R stable center manifold ODE unstable stable λ 0 unstable but what happens further?

24 u v p Nonlinear Dynamics? x 0 u0 v 0 center manifold reduction ẋ 0 = 0 x 0 a(0) x 2 0 center manifold ODE R but what happens further?

25 Reduced System slaving relation L Xr x r = "x 0 ( )(y) evolution equations ẋ 0 = 0 x 0 x 0 (y) y = My + x 0 Ku 0 extended center manifold reduction

26 Emerging Colony (x 0,y )= 0 a(0) 1, M 1 Ku 0 emerging profile a(0) + 0 a( ) = 0 a( ) = f 0 u 0 (M + ) 1 Ku 0, û 0 stability of emerging profile R stability coefficient a( ) = 1` ZZ f 0 (r)û 0 (r)u 0 (r)g(r, s; )h(s)u 0 (s)dsdr [0,1] 2 Doelman, Sewalt, AZ 2014

27 ing the next order term, we find p p p p tanh( ) a( ) a(0) sinh (1 z ) p 1 Water = 1 + " p in the a( ; 0 ) = p Back a( ) v cosh a(0; 0DCM ) = (1 z ) p 3/2 cosh p (1 z ) = sinh (1 z ) and (III.9) becomes BL p p p tanh( ) " p = v 0, (V.3 which determines infinitely many other eigen values 2 C. We proceed in studying (V.3) in a fashio that p resembles Section 4.4 in Ref. 17. We se = µ = µr +iµi and restrict arg(µ) to lie i [0, /2], because eigenvalues come in comple conjugate pairs. The stability equation for becomes, oscillatory destabilization no destabilization r " 3Doelman, Sewalt, AZ p(µ) = µ µ tanh(µ) = 0 > 0. (V.4 v

28 Conclusions Local extension of center manifold ODE-PDE problem largely linear Dynamical systems theory tools applicable Richer stability problem Can study Λ 0 >>1 limit