Mathematical modeling and control of population systems: Applications in biological pest control

The aim of this paper is to apply methods from optimal control theory, and from the theory of dynamic systems to the mathematical modeling of biological pest control. The linear feedback control problem for nonlinear systems has been formulated in order to obtain the optimal pest control strategy only through the introduction of natural enemies. Asymptotic stability of the closed-loop nonlinear Kolmogorov system is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution of the Hamilton-Jacobi-Bellman equation, thus guaranteeing both stability and optimality. Numerical simulations for three possible scenarios of biological pest control based on the Lotka-Volterra models are provided to show the effectiveness of this method. (c) 2007 Elsevier B.V. All rights reserved.

Generalizations and extensions of the Fokker-Planck-Kolmogorov equations

The problem of determining probability density functions of general transformations of random processes is considered in this thesis. A method of solution is developed in which partial differential equations satisfied by the unknown density function are derived. These partial differential equations are interpreted as generalized forms of the classical Fokker-Planck-Kolmogorov equations and are shown to imply the classical equations for certain classes of Markov processes. Extensions of the generalized equations which overcome degeneracy occurring in the steady-state case are also obtained.

The equations of Darling and Siegert are derived as special cases of the generalized equations thereby providing unity to two previously existing theories. A technique for treating non-Markov processes by studying closely related Markov processes is proposed and is seen to yield the Darling and Siegert equations directly from the classical Fokker-Planck-Kolmogorov equations.

As illustrations of their applicability, the generalized Fokker-Planck-Kolmogorov equations are presented for certain joint probability density functions associated with the linear filter. These equations are solved for the density of the output of an arbitrary linear filter excited by Markov Gaussian noise and for the density of the output of an RC filter excited by the Poisson square wave. This latter density is also found by using the extensions of the generalized equations mentioned above. Finally, some new approaches for finding the output probability density function of an RC filter-limiter-RC filter system driven by white Gaussian noise are included. The results in this case exhibit the data required for complete solution and clearly illustrate some of the mathematical difficulties inherent to the use of the generalized equations.

Biological controls on resuspension and the relationship between particle size and the Kolmogorov length scale in a shallow coastal sea

Measurements of suspended particle matter (SPM) and turbulence have been obtained over five tidal surveys during spring and summer 2010 at station L4 (5025 degrees N 04.22 degrees W, depth 50 m), in the Western English Channel. The relationship between turbulence intensity and bed stress is explored, with an in-line holographic imaging system evaluating the extent to which material is resuspended. Image analysis allows for the identification of SPM above a size threshold of 200 pm, capturing particle variability across tidal cycles and the two seasons. Dissipation of turbulent kinetic energy, which exceeds 10(-5) W kg(-1), yields maximum values of bed stress of between 0.17 and 0.20 N m(-2), frequently resulting in the resuspension of material from the bed. Resuspension is shown to promote aggregation of SPM into flocs, where the size of such particles is theoretically determined by the Kolmogorov microscale, l(k). During the spring surveys, flocs of a size larger than lk were observed, though this was not repeated during summer. It is proposed that the presence of gelatinous, biological material in spring allows flocculated particles to exceed l(k). This suggests that under specific circumstances, the limiting factor on the growth of flocculated SPM is not only turbulence, as previously thought, but the presence or absence of certain types of biological particle.

Global dynamics of stationary solutions of the extended Fisher-Kolmogorov equation

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Towards the construction of a validated numerical system to study the mesoscale dynamics of the North East Atlantic (2003-2006)

Máster en Oceanografía

The existence of an inverse limit of inverse system of measure spaces - a purely measurable case

The existence of an inverse limit of an inverse system of (probability) measure spaces has been investigated since the very beginning of the birth of the modern probability theory. Results from Kolmogorov [10], Bochner [2], Choksi [5], Metivier [14], Bourbaki [3] among others have paved the way of the deep understanding of the problem under consideration. All the above results, however, call for some topological concepts, or at least ones which are closely related topological ones. In this paper we investigate purely measurable inverse systems of (probability) measure spaces, and give a sucient condition for the existence of a unique inverse limit. An example for the considered purely measurable inverse systems of (probability) measure spaces is also given.