^{1}and Erik M. Bollt

^{1}

### Abstract

Given multiple images that describe chaotic reaction-diffusion dynamics, parameters of a partial differential equation (PDE) model are estimated using autosynchronization, where parameters are controlled by synchronization of the model to the observed data. A two-component system of predator-prey reaction-diffusion PDEs is used with spatially dependent parameters to benchmark the methods described. Applications to modeling the ecological habitat of marine plankton blooms by nonlinear data assimilation through remote sensing are discussed.

When attempting to model many physical processes, a frequent roadblock is the inability to observe certain variables and parameters. For example, hyperspectral satellite imagery provides a means to model phytoplanktonecology in the ocean. However, a realistic model includes predator-prey interaction with zooplankton, the primary controller of phytoplankton blooms, and unobservable by satellite imagery. Data assimilation, wherein measured data are incorporated into model output, is crucial to obtain a model that fits real-time observation. We address the problem of data assimilation and parameter estimation for a system of reaction-diffusion partial differential equations(PDEs) wherein only one species is observable such that modeling oceanecology by means of satellite imagery is plausible. This article extends former results in ordinary differential equations by so-called autosynchronization to the PDE case; the results permit an important generalization, estimating parameters that are spatially heterogeneous.

This work was supported by the Office of Naval Research under Grant No. N00014-09-1-0647. The authors would also like to thank the anonymous referees for helpful comments and suggestions on the manuscript.

I. INTRODUCTION

II. THE PARAMETER ESTIMATION METHOD

III. RESULTS AND SIMULATIONS OF AUTOSYNCHRONIZATION PARAMETER ESTIMATION

IV. SYNCHRONIZATION BY SAMPLING ONLY ONE SPECIES

V. ON INCOMPLETE OBSERVATION DATA

VI. CONCLUSION

### Key Topics

- Partial differential equations
- 33.0
- Phytoplankton
- 21.0
- Zooplankton
- 17.0
- Spatial analysis
- 15.0
- Synchronization
- 12.0

## Figures

Three sets of spatially dependent parameters used in simulations. Figures 1(a) and 1(b) are described by Eq. (12) , with on the left and on the right. Below, with the same ordering, are the parameters described by Eq. (13) . Finally, the swirly parameters are shown in Figures 1(e) and 1(f) .

Three sets of spatially dependent parameters used in simulations. Figures 1(a) and 1(b) are described by Eq. (12) , with on the left and on the right. Below, with the same ordering, are the parameters described by Eq. (13) . Finally, the swirly parameters are shown in Figures 1(e) and 1(f) .

Autosynchronization of species in Eqs. (8) and (9) . Each figure shows drive (top) and response (bottom) pairs. P(x, y, 0) and in (a), P(x, y, 1000) and in (c), and P(x, y, 4788) and in (e). Z(x, y, 0) and in (b), Z(x, y, 1000) and in (d), and Z(x, y, 4788) and in (f). Model parameters are and .

Autosynchronization of species in Eqs. (8) and (9) . Each figure shows drive (top) and response (bottom) pairs. P(x, y, 0) and in (a), P(x, y, 1000) and in (c), and P(x, y, 4788) and in (e). Z(x, y, 0) and in (b), Z(x, y, 1000) and in (d), and Z(x, y, 4788) and in (f). Model parameters are and .

Autosynchronization of response parameters in Eqs. (8) and (9) . Each figure shows drive (top) and response (bottom) pairs. and in (a), and in (c), and and in (e). and in (b), and in (d), and and in (f).

Autosynchronization of response parameters in Eqs. (8) and (9) . Each figure shows drive (top) and response (bottom) pairs. and in (a), and in (c), and and in (e). and in (b), and in (d), and and in (f).

Autosynchronization of species in Eqs. (8) and (9) . Each figure shows drive (top) and response (bottom) pairs. P(x, y, 0) and in (a), P(x, y, 1000) and in (c), and P(x, y, 10 660) and in (e). Z(x, y, 0) and in (b), Z(x, y, 1000) and in (d), and Z(x, y, 10 660) and in (f). Model parameters are and .

Autosynchronization of species in Eqs. (8) and (9) . Each figure shows drive (top) and response (bottom) pairs. P(x, y, 0) and in (a), P(x, y, 1000) and in (c), and P(x, y, 10 660) and in (e). Z(x, y, 0) and in (b), Z(x, y, 1000) and in (d), and Z(x, y, 10 660) and in (f). Model parameters are and .

Autosynchronization of response parameters in Eqs. (8) and (9) . Each figure shows drive (top) and response (bottom) pairs. and in (a), and in (c), and and in (e). and in (b), and in (d), and and in (f).

Globally averaged relative synchronization error between drive and response PDE components and parameters on a log scale. Figures 6(a) and 6(b) correspond to parameters built by Eq. (12) and simulation displayed in Figures 2 and 3 , respectively. Figures 6(c) and 6(d) show globally averaged relative synchronization error for species and parameters built by Eq. (13) , corresponding to simulations in Figures 4 and 5 , respectively.

Globally averaged relative synchronization error between drive and response PDE components and parameters on a log scale. Figures 6(a) and 6(b) correspond to parameters built by Eq. (12) and simulation displayed in Figures 2 and 3 , respectively. Figures 6(c) and 6(d) show globally averaged relative synchronization error for species and parameters built by Eq. (13) , corresponding to simulations in Figures 4 and 5 , respectively.

Autosynchronization of species in Eqs. (14) and (15) . Each figure shows drive (top) and response (bottom) pairs. P(x, y, 0) and in (a), P(x, y, 1000) and in (c), and P(x, y, 9360) and in (e). Z(x, y, 0) and in (b), Z(x, y, 1000) and in (d), and Z(x, y, 9360) and in (f). Model parameters are those shown in Figures 1(e) and 1(f) .

Autosynchronization of species in Eqs. (14) and (15) . Each figure shows drive (top) and response (bottom) pairs. P(x, y, 0) and in (a), P(x, y, 1000) and in (c), and P(x, y, 9360) and in (e). Z(x, y, 0) and in (b), Z(x, y, 1000) and in (d), and Z(x, y, 9360) and in (f). Model parameters are those shown in Figures 1(e) and 1(f) .

Autosynchronization of parameters in Eqs. (14) and (15) . Each figure shows drive (top) and response (bottom) pairs. and in (a), and in (c), and and in (e). and in (b), and in (d), and and in (f). Model parameters are those shown in Figures 1(e) and 1(f) .

Autosynchronization of parameters in Eqs. (14) and (15) . Each figure shows drive (top) and response (bottom) pairs. and in (a), and in (c), and and in (e). and in (b), and in (d), and and in (f). Model parameters are those shown in Figures 1(e) and 1(f) .

Globally averaged relative synchronization error between drive and response PDE components and parameters on a log scale, estimating perhaps more realistic spiral parameters. Figures (a) and (b) correspond to parameters shown in Figures 1(e) and 1(f) and simulation displayed in Figures 7 and 8 , respectively.

Globally averaged relative synchronization error between drive and response PDE components and parameters on a log scale, estimating perhaps more realistic spiral parameters. Figures (a) and (b) correspond to parameters shown in Figures 1(e) and 1(f) and simulation displayed in Figures 7 and 8 , respectively.

Locally averaged patches over which drive system is sampled shown in black. Sampled on subset of 3 × 3 grid points with a distance of 3grid points between patches.

Locally averaged patches over which drive system is sampled shown in black. Sampled on subset of 3 × 3 grid points with a distance of 3grid points between patches.

Comparison of three different sampling schemes. Shown are relative synchronization errors between drive and response systems for sampling over 3 × 3 grid points (blue) with a distance of 3 grid points between subsequent patches, 2 × 2 grid points (red) with a distance of 2 grid points between subsequent patches, and 1 × 1 grid points (black) with a distance of 1 grid points between subsequent patches. Phytoplankton synchronization errors on left and zooplankton synchronization errors shown on right.

Comparison of three different sampling schemes. Shown are relative synchronization errors between drive and response systems for sampling over 3 × 3 grid points (blue) with a distance of 3 grid points between subsequent patches, 2 × 2 grid points (red) with a distance of 2 grid points between subsequent patches, and 1 × 1 grid points (black) with a distance of 1 grid points between subsequent patches. Phytoplankton synchronization errors on left and zooplankton synchronization errors shown on right.

Autosynchronization results shown at t = 2000. Both species and both parameters shown compared with drive species and true parameters. Effect of adding diffusion to parameter equations is clearly visible in estimated parameters.

Autosynchronization results shown at t = 2000. Both species and both parameters shown compared with drive species and true parameters. Effect of adding diffusion to parameter equations is clearly visible in estimated parameters.

Globally averaged relative synchronization errors shown for species and parameters. Local sampling destroys stability of the identical synchronization manifold, however, spatial characteristics of parameters are still observed.

Globally averaged relative synchronization errors shown for species and parameters. Local sampling destroys stability of the identical synchronization manifold, however, spatial characteristics of parameters are still observed.

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