Gabriel Dimitriu

An optimal control problem for a two-prey and one-predator model with diffusion

Abstract An optimal control problem is studied for a reaction–diffusion system that models an ecosystem composed by one predator and two prey populations. One proposed to maximize the total density of the three populations. To do this, one proves the existence of an optimal solution and one establishes first and second order optimality conditions. Several numerical simulations performed in both one-dimensional and two-dimensional isolated environments and using different Holling type functional responses support the theoretical results.

On a prey-predator reaction-diffusion system with Holling type III functional response

In this paper we study a prey-predator model defined by an initial-boundary value problem whose dynamics is described by a Holling type III functional response. We establish global existence and uniqueness of the strong solution. We prove that if the initial data are positive and satisfy a certain regularity condition, the solution of the problem is positive and bounded on the domain Q=(0,T)x@W and then we deduce the continuous dependence on the initial data. A numerical approximation of the system is carried out with a spectral method coupled with the fourth-order Runge-Kutta time solver. The biological relevance of the comparative numerical results is also presented.

Comparative numerical analysis using reduced-order modeling strategies for nonlinear large-scale systems

We perform a comparative analysis using three reduced-order strategies-Missing Point Estimation (MPE) method, Gappy POD method, and Discrete Empirical Interpolation Method (DEIM)-applied to a biological model describing the spatio-temporal dynamics of a predator-prey community. The comparative study is focused on the efficiency of the reduced-order approximations and the complexity reduction of the nonlinear terms. Different variants are discussed related to the projection-based model reduction framework combined with selective spatial sampling to efficiently perform the online computations. Numerical results are presented.

Application of POD-DEIM Approach for Dimension Reduction of a Diffusive Predator-Prey System with Allee Effect

In this work we carry out an application of DEIM combined with POD to provide dimension reduction of a system of two nonlinear partial differential equations describing the spatio-temporal dynamics of a predator-prey community, where the prey per capita growth rate is damped by the Allee effect. DEIM improves the efficiency of the POD approximation reducing the computational complexity of the nonlinear term and regains the full model reduction expected from the POD model. Numerical results show that the dynamics of the predator-prey model in the full-order system of dimension 2048 can be captured accurately by the POD-DEIM reduced system with the computational time reduced by a factor of \(\mathcal{O}(10^4)\).

Optimal Control for Lotka-Volterra Systems with a Hunter Population

Of concern is an ecosystem consisting of a herbivorous species and a carnivorous one. A hunter population is introduced in the ecosystem. Suppose that it acts only on the carnivorous species and that the number of the hunted individuals is proportional to the number of the existing individuals in the carnivorous population. We find the optimal control in order to maximize the total number of individuals (prey and predators) at the end of a given time interval. Some numerical experiments are also presented.

Numerical simulations with data assimilation using an adaptive POD procedure

In this study the proper orthogonal decomposition (POD) methodology to model reduction is applied to construct a reduced-order control space for simple advection-diffusion equations Several 4D-Var data assimilation experiments associated with these models are carried out in the reduced control space Emphasis is laid on the performance evaluation of an adaptive POD procedure, with respect to the solution obtained with the classical 4D-Var (full model), and POD 4D-Var data assimilation Despite some perturbation factors characterizing the model dynamics, the adaptive POD scheme presents better numerical robustness compared to the other methods, and provides accurate results.

Numerical Approximation of a Free Boundary Problem for a Predator-Prey Model

This paper is concerned with the numerical approximation of a free boundary problem associated with a predator-prey ecological model. Taking into account the local dynamic of the system, a stable finite difference scheme is used, and numerical results are presented.

Comparative Study with Data Assimilation Experiments Using Proper Orthogonal Decomposition Method

In this work the POD approach to model reduction is used to construct a reduced-order control space for the simple one-dimensional transport equations. Several data assimilation experiments associated with these transport models are performed in the reduced control space. A numerical comparative study with data assimilation experiments in the full model space indicates that with an appropriate selection of the basis functions the optimization in the POD space is able to provide accurate results at a reduced computational cost.

Sensitivity study for a SEIT epidemic model

In this paper we perform a local sensitivity analysis with respect to parameters and initial conditions for a SEIT (susceptible, exposed in the latent period, infectious, and treated/recovery) epidemic model. Applying QR decomposition with column pivoting to the relative sensitivity matrix, we evaluate the relative identifiability and sensitivity of the parameters in the inverse problem, and order the parameters with respect to their identifiability.

Convergence Rate for a Convection Parameter Identified Using Tikhonov Regularization

In this paper we establish a convergence rate result for a parameter identification problem. We show that the convergence rate of a convection parameter in an elliptic equation with Dirichlet boundary conditions is O(??), where ? is a norm bound for the noise in the data.