Abstract
Approaches to modeling population dynamics using discrete-time models are described in this two-part review. The development of the scientific ideas of discrete time models, from the Malthus model to modern population models that take into account many factors affecting the structure and dynamics, is presented. The most important and interesting results of recurrent equation application to biological system analysis obtained by the authors are given. In the first part of this review, the population dynamic effects that result from density-dependent regulation of population, the age and sex structures, and the influence of external factors are considered.
Highlights
The “golden age” of mathematical biology began in the first half of the 20th century, with an enchanting “splash” of works that determined for a long time the development of theoretical ecology and mathematical population genetics and provided the foundations of the synthetic theory of evolution (Haldane, 1924; Lotka, 1925; Fisher, 1930; Volterra, 1931; Wright, 1931; Kostitzin, 1937; and others)
If for mathematicians recurrent equations serve as an aid or a method to study the dynamics of complex continuous-time models, for mathematical biologists and systemic ecologists they are an object of independent study, convenient for population dynamics simulations
A colossal amount of time series of empirical data indicating that biological systems have a complex dynamics has been accumulated
Summary
The “golden age” of mathematical biology began in the first half of the 20th century, with an enchanting “splash” of works that determined for a long time the development of theoretical ecology and mathematical population genetics and provided the foundations of the synthetic theory of evolution (Haldane, 1924; Lotka, 1925; Fisher, 1930; Volterra, 1931; Wright, 1931; Kostitzin, 1937; and others). At relatively large values of a2 or ρ (ρ ≥ 1) (i.e., in the case when the older age class primarily contributes to reproduction, and the birth rate is limited primarily by the younger age class), the loss of stability, as in the original one-dimensional Ricker model, is accompanied by the appearance of two-year fluctuations and a subsequent period-doubling cascade. Further study of the model with two age classes taking into account the influence of climatic factors on reproductive processes showed that the population abundance constantly moves from one basin to another and only in certain years is into the range of parameter values at which similar dynamics are observed (Neverova et al, 2019). The two-year fluctuations in abundance in a periodically changing environment can be both synchronous and asynchronous to the fluctuations in the environment
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