A discrete spatially varying population model (original) (raw)

A numerical method for a model of population dynamics with spatial diffusion

Computers & Mathematics with Applications, 1990

We propose a procedure to approximate the solution of a degenerate parabolic equation describing the dynamics of a one-sex population with spatial diffusion. We use a finite difference method along the characteristic age-time direction combined with finite elements in the spatial variables. Error estimates are derived for this approximation.

Population Dynamics in Heterogeneous Environments: A Discrete Model

International Journal of Bifurcation and Chaos, 2000

We study the dynamics of a multidimensional coordinate-dependent mapping governing the time evolution of a population spread over a one-dimensional lattice. The nonlinearity is of mean-field type and the dependence on coordinates, given by the so-called fitness, allows to take into account the spatial heterogeneities of the habitat. A global picture of the dynamics is given in the case without diffusion and in the case with diffusion when the fitness is homogeneous and leads to a periodic orbit. Moreover it is shown that, periodic fitnesses close to homogeneous ones impose their periodicity on the asymptotic dynamics when the latter is time-periodic.

Spatial effects in discrete generation population models

Journal of Mathematical Biology, 2004

A framework is developed for constructing a large class of discrete generation, continuous space models of evolving single species populations and finding their bifurcating patterned spatial distributions. Our models involve, in separate stages, the spatial redistribution (through movement laws) and local regulation of the population; and the fundamental properties of these events in a homogeneous environment are found. Emphasis is placed on the interaction of migrating individuals with the existing population through conspecific attraction (or repulsion), as well as on random dispersion. The nature of the competition of these two effects in a linearized scenario is clarified. The bifurcation of stationary spatially patterned population distributions is studied, with special attention given to the role played by that competition.

Dynamics of populations with individual variation in dispersal on bounded domains

Journal of biological dynamics, 2018

Most classical models for the movement of organisms assume that all individuals have the same patterns and rates of movement (for example, diffusion with a fixed diffusion coefficient) but there is empirical evidence that movement rates and patterns may vary among different individuals. A simple way to capture variation in dispersal that has been suggested in the ecological literature is to allow individuals to switch between two distinct dispersal modes. We study models for populations whose members can switch between two different nonzero rates of diffusion and whose local population dynamics are subject to density dependence of logistic type. The resulting models are reaction-diffusion systems that can be cooperative at some population densities and competitive at others. We assume that the focal population inhabits a bounded region and study how its overall dynamics depend on the parameters describing switching rates and local population dynamics. (Traveling waves and spread rat...

Population growth in space and time: spatial logistic equations

2003

How great an effect does self-generated spatial structure have on logistic population growth? Results are described from an individual-based model (IBM) with spatially localized dispersal and competition, and from a deterministic approximation to the IBM describing the dynamics of the first and second spatial moments. The dynamical system incorporates a novel closure that gives a close approximation to the IBM in the presence of strong spatial structure.

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.

The simplest model of spatially distributed population with reasonable migration of organisms

Eprint Arxiv Q Bio 0510004, 2005

The simplest model of a smart spatial redistribution of individuals is proposed. A single-species population is considered, to be composed of two discrete subpopulations inhabiting two stations; migration is a transfer between them. The migration is not random and yields the maximization of net reproduction, with respect to the transaction costs. The organisms are supposed to be globally informed. Discrete time model is studied, since it shows all the features of a smart migrations, while the continuous time case brings no new knowledge but the technical problems. Some properties of the model are studied and discussed.

Mathematical population dynamics models in deterministic and stochastic environments

2014

In this thesis, we consider mathematical population dynamics models in deterministic and stochastic environments. For deterministic environments, we study three models: an intraguild model with the e↵ects of spatial heterogeneous environment and fast migration of individuals; a fishery model with Marine Protected Area where fishing is prohibited and an area where the fish population is harvested; a predator-prey model which has one prey and two predators with Beddington-DeAngelis functional responses. For stochastic environments, we study SIRS epidemic model and predator-prey models under telegraph noise. We try to present the dynamical behavior of these models and show out the existence or vanishing of species in the models.

Spatiotemporal dynamics of reaction–diffusion models of interacting populations

Applied Mathematical Modelling, 2014

The present investigation deals with the necessary conditions for Turing instability with zero-flux boundary conditions that arise in a ratio-dependent predator-prey model involving the influence of logistic population growth in prey and intra-specific competition among predators described by a system of non-linear partial differential equations. The prime objective is to investigate the parametric space for which Turing spatial structure takes place and to perform extensive numerical simulation from both the mathematical and the biological points of view in order to examine the role of diffusion coefficients in Turing instability. Various spatiotemporal distributions of interacting species through Turing instability in two dimensional spatial domain are portrayed and analyzed at length in order to substantiate the applicability of the present model.