Optimal Rotation Age in Fast Growing Plantations: A Dynamical Optimization Problem.

Forest plantations are economically and environmentally relevant, as they play a key role in timber production and carbon capture. It is expected that the future climate change scenario affects forest growth and modify the rotation age for timber production. However, mathematical models on the effect of climate change on the rotation age for timber production remain still limited. We aim to determine the optimal rotation age that maximizes the net economic benefit of timber volume in a negative scenario from the climatic point of view. For this purpose, a bioeconomic optimal control problem was formulated from a system of Ordinary Differential Equations (ODEs) governed by the state variables live biomass volume, intrinsic growth rate, and area affected by fire. Then, four control variables were associated to the system, representing forest management activities, which are felling, thinning, reforestation, and fire prevention. The existence of optimal control solutions was demonstrated, and the solutions of the optimal control problem were also characterized using Pontryagin's Maximum Principle. The solutions of the model were approximated numerically by the Forward-Backward Sweep method. To validate the model, two scenarios were considered: a realistic scenario that represents current forestry activities for the exotic species Pinus radiata D. Don, and a pessimistic scenario, which considers environmental conditions conducive to a higher occurrence of forest fires. The optimal solution that maximizes the net benefit of timber volume consists of a strategy that considers all four control variables simultaneously. For felling and thinning, regardless of the scenario considered, the optimal strategy is to spend on both activities depending on the amount of biomass in the field. Similarly, for reforestation, the optimal strategy is to spend as the forest is harvested. In the case of fire prevention, in the realistic scenario, the optimal strategy consists of reducing the expenses in fire prevention because the incidence of fires is lower, whereas in the pessimistic scenario, the opposite is true. It is concluded that the optimal rotation age that maximizes the net economic benefit of timber volume in P. radiata plantations is 24 and 19 years for the realistic and pessimistic scenarios, respectively. This corroborates that the presence of fires influences the determination of the optimal rotation age, and as a consequence, the net economic benefit.

A soil water indicator for a dynamic model of crop and soil water interaction.

Water scarcity is a critical issue in agriculture, and the development of reliable methods for determining soil water content is crucial for effective water management. This study proposes a novel, theoretical, non-physiological indicator of soil water content obtained by applying the next-generation matrix method, which reflects the water-soil-crop dynamics and identifies the minimum viable value of soil water content for crop growth. The development of this indicator is based on a two-dimensional, nonlinear dynamic that considers two different irrigation scenarios: the first scenario involves constant irrigation, and the second scenario irrigates in regular periods by assuming each irrigation as an impulse in the system. The analysis considers the study of the local stability of the system by incorporating parameters involved in the water-soil-crop dynamics. We established a criterion for identifying the minimum viable value of soil water content for crop growth over time. Finally, the model was calibrated and validated using data from an independent field study on apple orchards and a tomato crop obtained from a previous field study. Our results suggest the advantages of using this theoretical approach in modeling the plants' conditions under water scarcity as the first step before an empirical model. The proposed indicator has some limitations, suggesting the need for future studies that consider other factors that affect soil water content.

Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist predators.

In the ecological literature, mutual interference (self-interference) or competition among predators (CAP) to effect the harvesting of their prey has been modeled through different mathematical formulations. In this work, the dynamical properties of a Leslie-Gower type predation model is analyzed, incorporating one of these forms, which is described by the function $g\left(y\right) =y^{\beta }$, with $0<\beta <1$. This function $g$ is not differentiable for $y=0$, and neither the Jacobian matrix of the system is not defined in the equilibrium points over the horizontal axis ($x-axis$). To determine the nature of these points, we had to use a non-standard methodology. Previously, we have shown the fundamental properties of the Leslie-Gower type model with generalist predators, to carry out an adequate comparative analysis with the model where the competition among predators (CAP) is incorporated. The main obtained outcomes in both systems are: (i) The unique positive equilibrium point, when exists, is globally asymptotically stable (g.a.s), which is proven using a suitable Lyapunov function. (ii) There not exist periodic orbits, which was proved constructing an adequate Dulac function.

Dynamics of a modified Leslie-Gower predation model considering a generalist predator and the hyperbolic functional response.

In the ecological literature,many models for the predator-prey interactions have been well formulated but partially analyzed.Assuming this analysis to be true and complete,some authors use that results to study a more complex relationship among species (food webs).Others employ more sophisticated mathematical tools for the analysis,without further questioning.The aim of this paper is to extend,complement and enhance the results established in an earlier article referred to a modified Leslie-Gower model.In that work,the authors proved only the boundedness of solutions,the existence of an attracting set,and the global stability of a single equilibrium point at the interior of the first quadrant.In this paper,new results for the same model are proven,establishing conditions in the parameter space for which up two positive equilibria exist.Assuming there exists a unique positive equilibrium point,we have proved,the existence of:i) a separatrix curve Σ,dividing the trajectories in the phase plane,which can have different ω-limit,ii) a subset of the parameter space in which two concentric limit cycles exist,the innermost unstable and the outermost stable.Then,there exists the phenomenon of tri-stability,because simultaneously,it has:a local stable positive equilibrium point, a stable limit cycle,and an attractor equilibrium point over the vertical axis.Therefore,we warn the model studied have more rich and interesting properties that those shown that earlier papers.Numerical simulations and a bifurcation diagram are given to endorse the analytical results.

Bifurcations and multistability on the May-Holling-Tanner predation model considering alternative food for the predators.

In this paper a modified May-Holling-Tanner predator-prey model is analyzed, considering an alternative food for predators, when the quantity of prey i scarce. Our obtained results not only extend but also complement existing ones for this model, achieved in previous articles. The model presents rich dynamics for different sets of the parameter values; it is possible to prove the existence of: (i) a separatrix curve on the phase plane dividing the behavior of the trajectories, which can have different ω-limit; this implies that solutions nearest to that separatrix are highly sensitive to initial conditions, (ii) a homoclinic curve generated by the stable and unstable manifolds of a saddle point in the interior of the first quadrant, whose break generates a non-infinitesimal limit cycle, (iii) different kinds of bifurcations, such as: saddle-node, Hopf, Bogdanov-Takens, homoclinic and multiple Hopf bifurcations. (iv) up to two limit cycles surrounding a positive equilibrium point, which is locally asymptotically stable. Thus, the phenomenon of tri-stability can exist, since simultaneously can coexist a stable limit cycle, joint with two locally asymptotically stable equilibrium points, one of them over the y-axis and the other positive singularity. Numerical simulations supporting the main mathematical outcomes are shown and some of their ecological meanings are discussed.

Multiple limit cycles in a Gause type predator-prey model with Holling type III functional response and Allee effect on prey.

This work aims to examine the global behavior of a Gause type predator-prey model considering two aspects: (i) the functional response is Holling type III and, (ii) the prey growth is affected by the Allee effect. We prove the origin of the system is an attractor equilibrium point for all parameter values. It has also been shown that it is the ω-limit of a wide set of trajectories of the system, due to the existence of a separatrix curve determined by the stable manifold of the equilibrium point (m,0), which is associated to the Allee effect on prey. When a weak Allee effect on the prey is assumed, an important result is obtained, involving the existence of two limit cycles surrounding a unique positive equilibrium point: the innermost cycle is unstable and the outermost stable. This property, not yet reported in models considering a sigmoid functional response, is an important aspect for ecologists to acknowledge as regards the kind of tristability shown here: (1) the origin; (2) an interior equilibrium; and (3) a limit cycle of large amplitude. These models have undoubtedly been rather sensitive to disturbances and require careful management in applied conservation and renewable resource contexts.