Extending and Strengthening Extending and Strengthening

the Pipeline the Pipeline

in Computer Sciencein Computer Science

MODELING AND VISUALIZATION OF THE

MARSHES VEGETATION

Alexey L. Sadovski1, Paul A. Montagna2, Scott King3 1Department of Mathematics and Statistics

2Department of Physical and Environmental Sciences 3School of Engineering and Computing Sciences

Texas A&M University - Corpus Christi

6300 Ocean Dr. Corpus Christi, Texas 78412 USA

UNESCO 2015, Paris, France

Introduction • Coastal marshes are

important, yet 50% of the marshes nationwide have disappeared since the founding of the United States.

• The Nueces Delta marsh has degraded because of high salinities and reduced flows in the Nueces River below Calallen tidal barrier.

• The Rincon Bayou Overflow channel was built to divert fresh water into Nueces Delta and improve marsh condition

Nueces River entering Nueces Bay

and surrounding marsh lands.

Modeling Roadmap • Mathematical model (equations

and relations)

• Simulation and visualization

• Model fit to existing data

(validation)

• Restoration plan based on

optimal control of water levels

Nueces delta and marsh.

Mathematical Modeling • Create a mathematical model

for marsh vegetation in wetlands at the Nueces Delta.

• Population model driven by spatial and temporal changes in water levels due to climate

• Variations and/or human impact.

• Significance: Such mathematical models could be modified as needed and applied to marshes in other regions of the country that are susceptible to similar environmental impacts from construction and water resource development.

Vegetation Pattern for Borrichia frutescens and Salicornia virginica in the Nueces Delta

Source: Figure 5.12 from Rasser, M.K. (2009) The role of biotic and abiotic processes in the zonation of salt marsh plants in the Nueces River Delta, Texas. Ph.D. dissertation, University

of Texas.

Nueces Marsh areas: blue = water, light green = lower marsh, medium green = middle marsh, dark green = high marsh, black – non marsh areas,

mainly open water (such as ponds and Nueces Bay).

Mathematical Model • Suppose we have an area of marshlands

within a specific boundary (Ω).

• There are N different plant species in which growth and spreading depends on the hydroperiod of the water supply, which regulates water levels on (under) the surface of marshes.

• We will consider population dynamics in the form of density of given species over the surface of marshlands.

Mathematical Model (contd.)

The system of N equations has the following form:

• ui (x,y,t) is the density of the i-th species at (x,y) at time t

• ri is the rate of reproduction

• Li(x, y) is the maximum possible density (carrying capacity of the i-th species at the point (x, y)

• ε i is the diffusion (or dispersion) coefficient of i-th species (usually quite small).

2

1

( , , )(1 )

( , ) ( , )

Ni k i

i i i

k k k

u x y t u uru

t L x y L x y

Some Remarks • In reality all L(x,y) mostly depend on a

difference between marsh elevation at the point (x,y) and a mean water level at the same point as well as salinity (which in turn may depend on water levels).

• In the similar way we can write system of difference equations (discrete model) in space and time by replacing partial derivatives with respect to x, y, and t by Δx, Δy, and Δt.

• We have made (using MatLab) an example (toy-model) of this model to investigate behavior of the system: o three different species o Over 1500 squares of marshland o arbitrary initial conditions and arbitrary

carrying capacity for each area and species.

Mathematical Model (contd.)

By introducing we can rewrite system

in the following form:

( , )

ii

i

uv

L x y

2

1

( , , )( , ) ( , ) (1 )

Ni

i i i i k i i

k

v x y tL x y L x y rv v v

t

Stability of the Model

Under some conditions (such as smoothness) on functions L(x,y) and parameters r and ε the following statements take place:

• Proposition 1. Any solution of the above boundary value problem of parabolic equations is bounded.

• Proposition 2. Multi-species system of equations has a solution which is either asymptotically stable or a limit cycle.

Simulation and Visualization

• Simulation is needed for understanding of the

stability of the system of equations and this model

under different initial conditions. Namely, it helps us

to evaluate system behavior (where system goes

with time) for different regions of the phase space.

• Visualization will help us to better understand

behavior of the system and may give us some input

for improvement of mathematical model.

Next two slides are an illustration of visualization over

the rectangle region for 3 different species.

Simulation and Visualization

Simulation and Visualization

Color Scale for Simulations with Two

Species

Modeled marsh coverage over 10-year run during wet conditions for two species. Clonal dominant (CD) species is red, and clonal stress tolerant (CST) species is green. At full red, that species is at 100% coverage, mixed coverage is combinations of red and green

colors (yellow when equal).

Moderate Conditions

Dry Conditions

Restoration • Formulate goal functions (criteria) such

as to maximize biomass of vegetation and minimize cost of restoration of marshes. Problem with more than one goal could be solved.

• Define controls, in this case water levels, and constraints.

• Apply theory of optimal control (analytical solution is not possible due to nonlinearity and complexity of the system of equations) to find satisfactory solution to the restoration problem.

Remarks on Restoration (contd.)

• Use of the local Geoid to estimate water levels for determination of carrying capacities for different species.

• Estimations to evaluate cost and impact of intervention to improve environmental condition in the marshlands.

• For example, evaluate the cost to increase water levels by 1 centimeter and compare it to the ecological benefits of the change in diversity and/or biomass.

Bibliography • [1] Ward, G.H., M.J. Irlbeck, and P.A. Montagna, Experimental river diversion for marsh enhancement,

Estuaries Vol. 25, No. 6, 2002, pp. 1416-1425.

• [2] Montagna, P.A., M. Alber, P. Doering, and M.S. Connor, Freshwater inflow: Science, policy, management, Estuaries, Vol. 25, No. 6, 2002, pp. 1243-1245.

• [3] Alexander, H.D. and K.H. Dunton, Freshwater inundation effects on emergent vegetation of a hypersaline salt marsh, Estuaries, Vol. 25, No. 6, 2002, pp. 1424-1435.

• [4] Rasser, M.K., The Role of Biotic and Abiotic Processes in the Zonation of Salt Marsh Plants in the Nueces River Delta, Texas, Ph.D. Dissertation, Department of Marine Science, University of Texas at Austin, 2009, 153 p.

• [4] Montagna, P.A., R.D. Kalke, and C. Ritter, Effect of restored freshwater inflow on macrofauna and meiofauna in upper Rincon Bayou, Texas, USA, Estuaries, Vol. 25, No. 6, 2002, pp. 1436-1447.

• [5] Anderson, J.R. and K. Deng, Global existence for nonlinear diffusion equations, J. Math. Anal. Appl., Vol. 196, 1995, pp. 479–501.

• [6] Bandle C. and H. A. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc., Vol. 316, 1989, pp. 595–622.

• [7] Leiva, H. and I. Sequera, Existence and stability of bounded solutions for a system of parabolic equations, J. Math. Anal. Appl., Vol. 279, 2003, pp. 495-507

• [8] Ramsey, F.L. and D.W. Schafer, The Statistical Sleuth: A Course in Methods of Data Analysis, Duxbury-Thomson Learning, 2002, p. 741

• [9] Frank P.J.S. “NPZ models of plankton dynamics: their constructions, coupling to physics, and applications” Journal of Oceanography, 58,(2), 379-387, 2002.

• [10] Sadovski. A., Preference Ranking and Decisions Based on Fuzzy Expert Information, “Advances in Fuzzy Systems and Evolutionary Computation,” World Scientific Engineering Society Press, USA, 2001, pp. 44-48.

• [11] Alexey A Sadovski, Paul A. Montagna, Spatial-Temporal Model of Multi-Species Vegetation in Marshlands, “Selected Topics in Mathematical Methods and Computational Techniques in Electrical Engineering”, WSEAS Press, 2010, pp.161-164

Acknowledgements This work is supported and funded by the following entities:

• United States Army Corps of Engineers.

• NOAA CAMEO Program

• Texas Research Development Fund.

• Harte Research Institute at Texas A&M University-Corpus Christi.

• College of Science and Engineering at Texas A&M University-Corpus Christi.