MODEMAT project, funded by the Escuela Politécnica Nacional de Quito ($9.800). June 2012-May 2013.

Ecological-spatial population growth models describe the spread of a biological entity, in a territory over time. It is of practical interest to simulate and control these phenomena. For instance, it is possible to avoid or control a plague, to develop harvesting plans, etc.

A convenient way to model these phenomena is to use reaction-diffusion equations and formulate associated optimal control problems. However, the numerical resolution of these models requires a high computational effort, due to the several variables involved and the complexity of the models. This fact makes these problems hard to be solved, and, usually, they are implemented in high performance computers.

In this project, we focused on the optimal control problem of population dynamics described by the Fisher equation.

$$ min_{(u,y)} J(u,y) := \frac{1}{2} \|y(T,.) - y_\Omega \|_{L^2(\Omega)}^2 dx + \frac{\lambda}{2}\|u\|_{L^2(\Omega)}^2 dt $$ subject to: $$ y_t - \gamma \Delta y + u(t,x)y = ry\left( 1 - \frac{y}{\kappa} \right) \text{ in }Q\\ y(x,t) = 0, \text{ in } \Sigma\\ y(x,0) = y_0 \text{ in } \Omega $$

We proposed a model reduction approach to this problem, by using the Proper Orthogonal Decomposition (POD) method. This technique allowed us to analyse an equivalent problem involving less variables. Thus, the numerical resolution of this new problem was feasible in computers with limited resources.

References

• [CC] R. Cantrell and C. Cosner. Spatial Ecology via Reaction-Difussion Equations. West Sussex : John Wiley & Sons, 2003.
• [VC] V. Arnautu and V. Capasso An Introduction to Optimal Control Problems in Life Sciences and Economics. Birkhauser, 2011.
• [TV] F. Tröltzsch and S. Volkwein. POD a-posteriori error estimates for linearquadratic optimal control problems. Computational Optimization and Applications, 44 (2009), 83-115.
• [T] F. Trölzsch. Optimal Conrol of Partial Differential Equations. American Mathematical Society, 2010.
• [KV] K. Kunisch and S. Volkwein. Galerkin proper orthogonal decomposition for parabolic problems. Numerische Mathematik, 90(2001), 117-148
• [GRS] C. Grossmann, H. Roos and M. Stynes. Numerical Treatment of Partial Differential Equations. Springer, 2007.
• [KV] M. Kahlbacher and S. Volkwein. POD a-posteriori error based inexact SQP method for bilinear elliptic optimal control problems. ESAIM:M2AN, 46 (2012) 491-511.