On the Hutchinson Population Model for Differential Equations with Maxima
Keywords:
Hutchinson population model, functional–differential equations, existence and uniqueness, numerical methods, product of two nonlinear functionAbstract
This paper investigates a generalized Hutchinson population model formulated as
a nonlinear functional-differential equation involving a maximum operator. The proposed model
extends the classical delayed logistic equation by incorporating the maximum population density
attained over a recent time interval, thereby accounting for memory effects and delayed biological
responses. Such formulations arise naturally in population dynamics, ecology, economics, and
other systems characterized by hereditary behavior and feedback mechanisms.
The analytical properties of the model are examined using the method of steps, which reduces
the original nonlinear functional-differential equation to a sequence of ordinary differential equations on successive intervals. Explicit analytical solutions are derived for the initial stages of
the construction, while numerical approximations are obtained using the Euler, Runge–Kutta,
and Runge–Kutta–Merson methods. A comparison of the numerical schemes demonstrates the
superior accuracy and stability of higher-order Runge–Kutta techniques.
In addition, a more general nonlinear boundary value problem containing a maximum functional
and a product of nonlinear perturbation terms is considered. By transforming the problem into
an equivalent nonlinear Fredholm integral equation and applying the method of successive approximations together with Banach’s fixed-point theorem, sufficient conditions for the existence and
uniqueness of solutions are established. The obtained results contribute to the qualitative theory
of functional-differential equations with maxima and provide effective analytical and numerical
tools for studying population models with memory.
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