Generally, the temporary discretization of the Navier-Stokes equations despise the convective term, and consider a boundary ∂Ω = . This paper introduces a Galerkin scheme designed to solve by means of the finite element method for the Oseen problem in three dimensions written in terms of speed, vorticity and pressure, in a viscous and incompressible fluid that flows through a porous medium, this problem is obtained from a temporary discretization of the Navier-Stokes equations and we consider a partitioned boundary ∂Ω = Γ1 ∪ Γ2 and disjoint, that is, the velocity is of the homogeneous Dirichlet type on Γ1, while the tangential velocity and pressure They are of the non-homogeneous Dirichlet type on Γ2. In the variational formulation the speed is completely decoupled, which allows you to approximate the vorticity and pressure independently. The speed is recovered from the vorticity and pressure. Galerkin’s scheme is based on Nédélec finite elements and continuous polynomials to pieces of the same order, for vorticity and pressure, respectively. Likewise, convergence rates are obtained for vorticity, speed and pressure in natural norms. Finally, a numerical example is provided that illustrates the behavior of the model.

Fast-growing forest plantations play an important role in reducing global warming and have great potential for carbon capture. In this study, we aimed to model the dynamics of carbon capture in fast-growing plantations. A mathematical model is proposed consisting of a tridimensional nonlinear system. The variables involved are the amount of living biomass, the intrinsic growth of biomass, and the burned area by forestry fire. The environmental humidity is also considered, assumed as a parameter by simplicity. The solutions of the model are approximated numerically by the Runge-Kutta fourth-order method. Once the equilibria of the model have been obtained and its local stability determined, the analysis of the model reveals that the living biomass, as well as the stored carbon, decreases in each harvest cycle as a consequence of the negative effects of fire on soil properties. Furthermore, the model shows that the maximum area burned is attained always after the maximum volume of biomass is obtained. Numerical simulations show that the model solutions are reasonable for the growth dynamics of a plantation, from a theoretical perspective. The mathematical results suggest that a suitable optimal management strategy to avoid biomass losses in the successive regeneration cycles of the plantation is the prevention of fires together with soil fertilization, applied to fast-growing plantations.

Plantations with fast-growing species play a crucial role in reducing global warming and have great carbon capture potential. Therefore, determining optimal management strategies is a challenge in the management of forest plantations to achieve the maximum carbon capture rate. The objective of this work is to determine optimal rotation strategies that maximize carbon capture in forest plantations. By evaluating an ecological optimal control problem, this work presents a method that manages forest plantations by planning activities such as reforestation, felling, thinning, and fire prevention. The mathematical model is governed by three ordinary differential equations: live biomass, intrinsic growth, and burned area. The characterization of the optimal control problem using Pontryagin’s maximum principle is analyzed. The model solutions are approximated numerically by the fourth-order Runge–Kutta method. To verify the efficiency of the model, parameters for three scenarios were considered: a realistic one that represents current forestry activities based on previous studies for the exotic species Pinus radiata D. Don, another pessimistic, which considers significant losses in forest productivity; and a more optimistic scenario which assumes the creation of new forest areas that contribute with carbon capture to prevent the increase in global temperature. The model predicts a higher volume of biomass for the optimistic scenario, with the consequent higher carbon capture than in the other two scenarios. The optimal solution for the felling strategy suggests that, to increase carbon capture, the rotation age should be prolonged and the felling rate decreased. The model also confirms that reforestation should be carried out immediately after felling, applying maximum reforestation effort in the optimistic and pessimistic scenarios. On the other hand, the model indicates that the maximum prevention effort should be applied during the life cycle of the plantation, which should be proportional to the biomass volume. Finally, the optimal solution for the thinning strategy indicates that in all three scenarios, the maximum thinning effort should be applied until the time when the fire prevention strategy begins.

Drought is one of the main environmental factors that limit plant growth. For this reason, it is necessary to apply nursery cultural practices to produce quality seedlings for successful reforestation in drought- prone sites. In this study, the extreme learning machines and multilayer are applied to predict survival in 5-month-old Pinus radiataseedlings belonging to 98 families of a genetic improvement program and subjected to a period of water restriction in the nursery. After applying the water restriction, survival was registered in each seedling as a categorical variable (1 = alive seedling, 0 = dead seedling). Additionally, the following morphological attributes of each seedling were also measured: total height, root collar diameter, slenderness index, dry weight of needles, stems and roots, total dry weight, and the root to shoot ratio. The extreme learning machines predicted with a better rate the survival of the “alive” class compared to the “dead” class. On the other hand, the multilayer-extreme learning machines improved the precision of survival concerning the class of “dead” seedlings. According to the results of the model, an overall precision of 74% was obtained. This may be due to the great genetic variability presented by each of the Pinus radiatafamily used in the database. However, this technique allowed predicting the survival of a group of seedlings grown in the nursery, which can be a tool to support the selection process of high quality planting stock.

Physics continues to be an inexhaustible source of problems that have inspired the theoretical development of mathematics, particularly in the development of the theory of dynamical systems that have their origin in the physics of the 15th century, with the beginning of differential calculus by Newton and Leibniz; differential equations are an appropriate tool for research in different areas of knowledge. Our objective is to show an example of how we can use dynamical systems in a problem framed in forestry and agricultural systems, ecology, and epidemiology. This work is inspired by one of the essential primary crops in the energy intake of the human diet, wheat; as it turns out, the productivity of this food can be affected by several biotic stresses, including viral diseases and their associated vectors. Among the viral infections is the barley yellow dwarf virus, which causes 11% to 33% yield losses in wheat fields; the virus moves through the agricultural landscape via different aphid species (vectors). The main problem is that aphids prefer colonizing plants with the virus, causing rapid growth of infected plants and substantially decreasing crop productivity; therefore, we propose to study the dynamics of aphid-plant interaction, in which the impact of aphid preference for infected plants is determined, and we obtained as a result that a key parameter is the quantity and quality of food consumed by aphids.

The partial differential equations for fluid flow dynamics based on the Brinkman equations, written in terms of velocity-vorticity and pressure in three dimensions, are essential for predicting climate, ocean currents, water flow in a pipe, the study of blood flow and any phenomenon involving incompressible fluids through porous media; having a significant impact in areas such as oceanographic engineering and biomedical sciences. This paper aims to study the Brinkman equations with homogeneous Dirichlet boundary are studied, the existence and uniqueness of solution at a continuous level through equivalence of problems is presented. It is discretized to approximate the solution using Nédélec finite elements and piecewise continuous polynomials to approximate vorticity and pressure. The velocity field is recovered, obtaining its a priori error estimation and order of convergence. As a result, ensuring a single prediction of the flow behavior of an incompressible fluid through porous media. Finally, a numerical example in 2D with the standard L2 is presented, confirming the theoretical analysis.

Physical, biological, and economic factors remain inexhaustible sources when determining silvicultural management strategies that maximize the net benefit of timber volume. The objective of this paper is to formulate an optimal control problem to maximize the net benefit of timber volume and to demonstrate the existence of optimal solutions to this problem. The optimal control theory will be used, through the formulation of a bioeconomic optimal control problem subjected to a system of three ordinary differential equations that describe the interactions between the living biomass, the intrinsic growth of the biomass, and the burned area. Four control variables are associated with these differential equations, which are reforestation, felling, thinning, and fire prevention. Assuming adequate conditions on the control variables, the existence of optimal solutions to the optimal control problem is obtained using Filippov’s theorem. In conclusion, the optimal control problem is well formulated and solutions exist. This will allow us to define forest management strategies that optimize the economic gain of the process, taking into consideration the physical, biological, and economic phenomena involved in the dynamics of forest plantations.

Determining the response to the forces applied to an elastic solid containing an ideal fluid with constant density is essential in the engineering and biomedical fields. Therefore this paper aims to present and analyze a mixed finite element method for an interaction problem solid-fluid that contributes to understanding these areas. It is assumed transmission conditions are maintained at the fluid boundary and are given by the balance of forces and the equality of normal displacements. The mixed variational formulation that avoids the locking phenomenon, for the coupled problem is in terms of displacement, stress tensor, and rotation in the solid and by pressure and scalar potential in the fluid, the main contribution of this work. The first transmission condition is imposed in the definition of the space and the rest of the conditions appear naturally, which means Lagrange multipliers are not needed at the coupling border. The unknowns for the fluid and the solid are approximated by finite element subspaces of Lagrange and Arnold-Falk-Winther of order 1, which lead to a Galerkin scheme for the continuous problem. Also, the resulting Galerkin scheme is convergent and derives optimal convergence rates. Finally, the model is illustrated using a numerical example.