Abstract
The logistic equation with delay or Hutchinson’s equation is one of the fundamental equations of population dynamics and is widely used in problems of mathematical ecology. We consider a family of mappings built for this equation based on central separated differences. Such difference schemes are usually used in the numerical simulation of this problem. The presented mappings themselves can serve as models of population dynamics; therefore, their study is of considerable interest. We compare the properties of the trajectories of these mappings and the original equation with delay. It is shown that the behavior of the solutions of the mappings constructed on the basis of the central separated differences does not preserve, even with a sufficiently small value of the time step, the basic dynamic properties of the logistic equation with delay. In particular, this map does not have a stable invariant curve bifurcating under the oscillatory loss of stability of a nonzero equilibrium state. This curve corresponds in such mappings to the stable limit cycle of the original continuous equation. Thus, it is shown that such a difference scheme cannot be used for numerical modeling of the logistic equation with delay.
Highlights
E presented mappings themselves can serve as models of population dynamics; their study is of considerable interest
We compare the properties of the trajectories of these mappings and the original equation with delay
It is shown that the behavior of the solutions of the mappings constructed on the basis of the central separated di erences does not preserve, even with a su ciently small value of the time step, the basic dynamic properties of the logistic equation with delay
Summary
Простейшие свойства логистического уравнения с запаздыванием Среди фундаментальных моделей популяционной динамики особое место занимает логистическое уравнение с запаздыванием. В котором неотрицательная функция ( ) моделирует нормированную плотность численности популяции, а положительный параметр характеризует скорость ее роста. Для изложения полученных нами результатов понадобятся простейшие свойства его решений, полученные в этих работах. Через [−1, 0] ниже обозначается пространство непрерывных на отрезке [−1, 0] функций со стандартной нормой. 1. Для уравнения (1) выполняется теорема существования и единственности решений, т.е. Для каждого значения 0 и каждой начальной функции ( ) ∈ [−1, 0], при всех > 0, существует и единственное решение ( , ) уравнения (1), для которого ( 0 + , ) = ( ). Существует такое 0 > 0, что при всех ∈ (0, 0] уравнение (1) имеет орбитально асимптотически устойчивый цикл 0( , ), для которого имеет место асимптотическое представление
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