Bivariate spline solution of time dependent nonlinear PDE for a population density over irregular domains (original) (raw)

Numerical and Analtyical Solutions of a Model of Population Dynamics

VFAST Transactions on Mathematics, 2015

The study presents numerical and approximate analytical approximations to a model of population dynamics with unbounded mortality function. The mathematical model involves a nonlocal boundary condion. A finite difference method is implemented for the numerical solution while the homotopy analysis method (HAM) is applied to obtain the approximate series solution. The HAM solution contains an auxiliary parameter which provides a convenient way of controlling the convergence regions of series solution. Results are presented for typical test problem provided in literature. Comparison of the results of both methods show validity and efficiency of the methods.

Advanced Nonlinear Dynamics of Population Biology and Epidemiology

Abstract and Applied Analysis, 2014

Modern biology and epidemiology have become more and more driven by the need of mathematical models and theory to elucidate general phenomena arising from the complexity of interactions on the numerous spatial, temporal, and hierarchical scales at which biological systems operate and diseases spread. Epidemic modeling and study of disease spread such as gonorrhea, HIV/AIDS, BSE, foot and mouth disease, measles, and rubella have had an impact on public health policy around the world which includes the United Kingdom, The Netherlands, Canada, and the United States. A wide variety of modeling approaches are involved in building up suitable models. Ordinary differential equation models, partial differential equation models, delay differential equation models, stochastic differential equation models, difference equation models, and nonautonomous models are examples of modeling approaches that are useful and capable of providing applicable strategies for the coexistence and conservation of endangered species, to prevent the overexploitation of natural resources, to control disease's outbreak, and to make optimal dosing polices for the drug administration, and so forth. This special issue is concerned with the nonlinear dynamic modeling and related analysis of interacting populations and important epidemic diseases. All papers submitted to this special issue went through a thorough peer-refereeing process. Based on the reviewer's reports, we collect 50 original research articles by more than 100 active international researchers on differential equations. In the following, we briefly review each of the papers by highlighting the significance of the key contributions.

A discrete spatially varying population model

We will consider a discrete population model on a one-dimensional spatial grid where the local dynamics are given by the standard logistic population model. We will show that the usual approach for partial differential equations will not work in the difference equation approach. Therefore, for the PdE (partial difference equation) model we find that we cannot have migration and reproduction occurring simultaneously. We introduce a new model for the biological problem. We will give a limited number of analytic results and several numerical examples for both Neumann and Dirichlet-like boundary conditions.

Numerical solutions of a linear age-structured population model

We study numerical solutions of an age-structured population model. The model is a linear partial differential equation (PDE) with nonlocal boundary condition. With this model, we aim to understand how the populations of individuals change in both time and age. We use finite difference operators to replace the partial derivatives and composite trapezoidal method to compute the integral at the boundary. By numerically solving the PDE in Matlab, we generate preliminary results about age-structured populations, and explore convergence of the numerical scheme.

Mathematical Models in Population Dynamics and Ecology

Biomathematics, 2006

We introduce the most common quantitative approaches to population dynamics and ecology, emphasizing the different theoretical foundations and assumptions. These populations can be aggregates of cells, simple unicellular organisms, plants or animals. The basic types of biological interactions are analysed: consumer-resource, prey-predation, competition and mutualism. Some of the modern developments associated with the concepts of chaos, quasi-periodicity, and structural stability are discussed. To describe short-and long-range population dispersal, the integral equation approach is derived, and some of its consequences are analysed. We derive the standard McKendrick age-structured density dependent model, and a particular solution of the McKendrick equation is obtained by elementary methods. The existence of demography growth cycles is discussed, and the differences between mitotic and sexual reproduction types are analysed.

MATHEMATICAL METHODS IN POPULATION DYNAMICS (Atena Editora)

MATHEMATICAL METHODS IN POPULATION DYNAMICS (Atena Editora), 2023

Without a doubt, the study in mathematical modeling is current, it is of vital importance in ecological studies, naturally some arise in population dynamics, highlighting the role of mathematical analysis that allows optimizing decision making in a certain behavior. The main objective is the study of the logistic model, to give way to the classic model of the Lotka-Volterra type without diffusion. At the end, an analysis of the model case with diffusion is made, and much attention is paid to linear and nonlinear models. Algorithmic results make sense to explain natural phenomena, growth that can be: a cell, an organ, a human being, a plant or a population, better if it is a fundamental problem of biology.

Age-structured population models and their numerical solution

Ecological Modelling, 2005

This paper considers the state of the art of the numerical solution of age-structured population models. The different numerical approaches to this kind of problems and the stability and convergence results for them are reviewed. Both characteristic curves methods and finite difference methods are compared with regards to accuracy, efficiency and their qualitative behaviour depending on the compatibility conditions between initial and boundary data of the problems. The paper is the first of a series of two considering the numerical solution of general structured population models.

Numerical approximations for population growth models

Applied Mathematics and Computation, 2005

This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF) collocation approach to solve the Volterra's model for population growth of a species within a closed system. This model is a nonlinear integro-differential equation where the integral term represents the effect of toxin. This approach is based on orthogonal functions which will be defined. The collocation method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare these methods with some other numerical results and show that the present approach is applicable for solving nonlinear integro-differential equations.