Delay differential systems. A generalization of the Lotka-Volterra model

Abstract

The first mathematical laws used in the theory of survival were introduced by Th. Malthus (1798-Principle of population; 1803-The theory of population), B. Gompertz (1860), and W. Makeham (1874). They proposed different exponential laws to describe the intensity of death, which then formed the basis for the biometric functions of survival. In the twentieth century, the development of demographic research, as well as the study of various epidemics affecting populations, led to the emergence and evolution of new methods for solving problems in survival theory. Lotka-Volterra (in their Predator-Prey model) described the differential equations of population growth (in humans, microorganisms, and other species). These models were later improved by E. M. Wright and P. J. Wangersky (1978), who more plausibly assumed that the phenomena of immigration-emigration could represent the biological reaction of self-regulation of a population, and that this factor acts with a certain delay. The use of random processes and, later, stochastic differential equations contributed to the evolution and diversification of mathematical models in biological systems. In this article, we have developed a variational functional framework for the description of problems associated with delayed differential systems. We have thus made a generalization of the Lotka-Volterra model.