Interpretation and Dynamics of the Lotka–Volterra Model in the Description of a Three-Level Laser

In this work, the Lotka–Volterra equations where applied to laser physics to describe population inversion and the number of emitted photons. Given that predation and stimulated emissions are analogous processes, two rate equations where obtained by finding suitable parameter transformations for a three-level laser. This resulted in a set of differential equations which are isomorphic to several laser models under accurate parameter identification. Furthermore, the steady state provided two critical points: one where light amplification stops and another where continuous-wave operation is achieved. Lyapunov’s first method of stability yielded the conditions for the convergence to the continuous-wave point, whereas a Lyapunov potential provided its stability regions. Finally, the Q-Switching technique was modeled by introducing a periodic variation of the quality Q of the cavity. This resulted in the transformation of the asymptotically stable fixed point into a limit cycle in the phase space.

Artificial selection of communities drives the emergence of structured interactions

Species-rich communities, such as the microbiota or environmental microbial assemblages, provide key functions for human health and ecological resilience. Increasing effort is being dedicated to design experimental protocols for selecting community-level functions of interest. These experiments typically involve selection acting on populations of communities, each of which is composed of multiple species. Numerical explorations allowed to link the evolutionary dynamics to the multiple parameters involved in this complex, multi-scale evolutionary process. However, a comprehensive theoretical understanding of artificial selection of communities is still lacking. Here, we propose a general model for the evolutionary dynamics of species-rich communities, each described by disordered generalized Lotka-Volterra equations, that we study analytically and by numerical simulations. Our results reveal that a generic response to selection for larger total community abundance is the emergence of an isolated eigenvalue of the interaction matrix that can be understood as an effective cross-feeding term. In this way, selection imprints a structure on the community, which results in a global increase of both the level of mutualism and the diversity of interactions. Our approach moreover allows to disentangle the role of intraspecific competition, interspecific interactions symmetry and number of selected communities in the evolutionary process, and can thus be used as a guidance in optimizing artificial selection protocols.

Inverse problems of the dynamics of observation interpretation systems

Based on the analysis and systematization of the inverse problems of dynamics, the study of the properties and features of the types of dynamic models under consideration, an approach is proposed for the development of appropriate methods of mathematical modeling based on the use and implementation of integral models in the form of Volterra equations of the I and II kind, their functional capabilities are determined in the study of various classes of problems, and also formulated the features that affect the choice of methods for their numerical solution. Methods for obtaining integral models are proposed, which are the basis for constructing algorithms for solving inverse problems of dynamics for a fairly wide class of dynamic objects. Integral methods for the identification of dynamic objects have been developed, which make it possible to obtain stable non-optimization algorithms for calculating the parameters of mathematical models. Recurrent methods of parametric identification of transfer functions of dynamic objects with an arbitrary input action are proposed (the obtained parameters of the transfer functions are also coefficients of the corresponding differential equations, which makes it possible to obtain equivalent mathematical models in the form of integral equations). The study of algorithms that implement the proposed identification methods allows us to conclude about their efficiency in terms of the amount of computation and ease of implementation, as well as the high accuracy of calculating the model parameters.

Stability of Nonlocal Stochastic Volterra Equations

In this paper, we study Ulam-Hyers-Rassias stability of solutions for nonlocal stochastic Volterra equations. Sufficient conditions for the existence and stability of solutions are derived using the Gronwall lemma. The advantage of our model equation is that it allows for additional measurements leading to better results compared to models with local initial conditions. Examples are solved to illustrate the applications of the results.

Lorenz and Lotka-Volterra Equations Using MATLAB Resolution

In Mathematics, a differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology.Mainly the study of differential equations consists of the study of their solutions, and of the properties of their solutions.Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly.In our report, we are interested by the resolution of two differential equations, the famous Lorenz dynamic system which was developped to understand the chaotic character of meteorology, and the Predator prey model.In the folowwing report, we are going to resolve these equations in order to understand their meanings using Matlab, but first of all, we should introduce each problem, then develop and explain both mathematical and numerical issues, our main goal is to resolve these systems with an adequate Matlab formulation, using the ode45 function and finally we should discuss the results.

Theoretical Clues for Agroecological Transitions: The Conuco Legacy and the Monoculture Trap

The multiple ecological crisis that we are facing forces us to ponder the transition toward sustainable agricultural systems. Two key uncertainties need to be unveiled in addressing this problem; first, we need to identify the general features of alternative models that make them sustainable, and second, we need to explore how to build them from the (flawed) existing systems. In this work we explore these two questions using an ethnoecological and theoretical approach. In the exploration of alternative models, we evaluate an ancestral farming system, the conuco, characterized by, (i) the use of the ecological succession to constantly renew its properties, (ii) the increase of its biodiversity over time (in the horizontal and vertical components), and (iii) the self-regulation of the associated populations. Next, we characterize the topology of ecological networks of agroecosystems along the transition from a monoculture to a conuco-like agroecological system. We use topologies obtained from field information of conventional and agroecological systems as starting and arrival points. To model the dynamics of the systems and numerically simulate the transitions, we use a model based on Generalized Lotka-Volterra equations, where all types of population interactions are represented, with outcomes based on a density-dependent conditionality. The results highlight the relevance of increasing the connectance and diminishing the degree centrality of the conventional systems networks to promote their sustainability. Finally, we propose that the transitions between the monoculture and the agroecological systems could be figuratively interpreted as a cusp catastrophe, where the two systems are understood as alternative stable states and the path from one to the other cannot be reverted by just reversing the values of the control parameter. That is, once a system is in either of these states there is a tendency to stay and a resistance to move away from it. This implies that in the process of transition from a monoculture to a multi-diverse system, it is prudent not to despair if there are no immediate improvements in the performance of the system because once a certain point is reached, the system may experience an abrupt improvement.