A pair-formation model with a discrete set of offsprings and child care

We present a discrete newborns set-based deterministic model for a two-sex population structured by age and marital status. The model includes the spatial migration, a weighted harmonic mean-type pair formation function, and strong parental care and neglects the separation of pairs. Each sex has pre-reproductive and reproductive age intervals. All adult (of reproductive age) individuals are divided into single males, single females, and permanent pairs. All pairs are of two types: pairs without offsprings under parental care at the given time and pairs taking care of their young offsprings. All individuals of pre-reproductive age are divided into young (under parental care) and juvenile (offsprings who can live without parental care but cannot produce offsprings) groups. It is assumed that births can only occur from couples and after the death of any of the pair partner all young offsprings of this pair die. The model consists of integro-partial differential equations subject to the conditions of integral type. The number of these equations depends on the biologically possible maximal newborns number of the same generation produced by a pair. A class of separable solutions is studied for this model. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 1, pp. 93–129, January–March, 2007.     

An Age-Structured Population Dynamics Model with Females' Pregnancy and Child Care

We present a deterministic model for an age-structured population dynamics taking into account females' pregnancy, maternal care of offsprings, and environmental pressure with or without spatial migration. The model is based on the age-density notion for a group formed by a mother and her offsprings under maternal care. A harmonic-mean-type mating function of sexes without formation of permanent pairs is used. It is assumed that each sex has the pre-reproductive, reproductive, and post-reproductive age intervals. All adult individuals are divided into males, single females, fertilized females, and females taking child care. Individuals of post-reproductive age belong to the group of singles. All individuals of pre-reproductive age are divided into the young and juvenile groups. Only young offsprings are assumed to be under maternal care. Juvenile individuals can live without maternal care. The model consists of integro-PDEs subject to the conditions of integral type. The existence and uniqueness theorem is proved in the case of unlimited population. Separable solutions and their long-time behavior are studied for the limited nondispersing population. In the case of random migration two types of separable solutions and their long-time behavior for the homogeneous Dirichlet and Neumann boundary conditions are studied. In the case of directed migration in one-dimensional domain with special initial and Dirichlet boundary conditions, the unlimited invasive population dynamics is studied. In particular, an explicit formula for the migration front is given.     

A population dynamics model with nonautonomous past

Abstract

In this paper we study a two-phase population model, which distinguishes the population by two different stages

By the standard technique of characteristics, this population equation is transformed as the ordinary differential equation with nonautonomous past

where 1 ≤ p < ∞ and I = [?r, 0] (finite delay) or I = (?∞, 0] (infinite delay), E a Banach space, Φ : W^1,p(I, E) → E a linear delay operator and B a nonlinear operator on E. The main result of this paper is the well-posedness of this delay equation by using the (right) multiplicative perturbation result of Desch and Schappacher in [8].     

A size-structured population dynamics model of Daphnia

The stability of a size-structured population dynamics model of Daphnia coupled with the dynamics of an unstructured algal food source is investigated for the case where there is also an inflow of newborns from an external source. We determine the steady states and study the stability of the nontrivial steady states. We also identify a demographic-algae parameter that determines a condition for the stability.     

Nursing care demand prediction based on a decomposed semi-markov population model

The semi-markovian population model introduced by Kao for the planning of progressive care hospitals is adapted to the prediction of nursing care demand at the level of a care unit in a general hospital. Assuming a feedback admission policy which refills the unit as soon as discharges occur, it is shown that the care unit can be decomposed into B independent subsystems corresponding to each of the B beds in the unit.For each bed the semi-Markov model permits the computation of the expected care demand and its variance for each of the seven forthcoming days. The model permits also the prediction of admissions of new patients. A prediction formula can thus be obtained where the expected care demand is expressed as a linear function of the expected number of admissions in the forthcoming days.Finally this methodology is illustrated on real data obtained in the gynaecology department of the Montreal Jewish General Hospital.     

A general model of size-dependent population dynamics with nonlinear growth rate

We study a general model of size-dependent population dynamics with nonlinear growth rate. The existence of a local solution is shown by using Schauder's fixed point theorem. Uniqueness and continuous dependence on initial data are also established under additional conditions.     

Complex dynamics of a diffusive epidemic model with strong Allee effect

In this paper, we investigate the dynamics of a diffusive epidemic model with strong Allee effect in the susceptible population. We show some properties of solutions of the model, the asymptotic stability of the equilibria. Especially, we show that there exists a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. Furthermore, we give the conditions of Turing instability and determine the Turing space in the parameters space. Based on these results, we perform a series of numerical simulations and find that the model exhibits complex pattern replication: spots, spot–stripe mixtures and stripes patterns.     

Non-autonomous dynamics of a semi-Kolmogorov population model with periodic forcing

In this paper we study a semi-Kolmogorov type of population model, arising from a predator–prey system with indirect effects. In particular we are interested in investigating the population dynamics when the indirect effects are time dependent and periodic. We first prove the existence of a global pullback attractor. We then estimate the fractal dimension of the attractor, which is done for a subclass by using Leonov’s theorem and constructing a proper Lyapunov function. To have more insights about the dynamical behavior of the system we also study the coexistence of the three species. Numerical examples are provided to illustrate all the theoretical results.     

A queueing model of activities in an intensive care unit

^** Email: griffiths{at}cardiff.ac.uk Activities in an intensive care unit (ICU) at a major teachinghospital are modelled by means of a queue-theoretic approach.Using data supplied by the ICU relating to the admissions process,the bed availability and the length of stay of patients, itwas possible to fit theoretical distributions to the observed‘arrival’ and ‘service’ distributions.Queueing equations relevant to a multi-channel system havingrandom arrivals and hyper-exponential service times for eachchannel are set up, and solved iteratively. Results obtainedmatch well with observations, and the model is then utilisedto investigate several ‘what if? ’ scenarios. Referenceis made to a simulation model developed in conjunction withthe queueing model.     

An improved model of age-structured population dynamics

The classical model of age-dependent population dynamics is improved. Instead of the traditional renewal equation, a new approach is developed to describe the reproduction process of the population. The composition of a population is redefined to contain the pre-birth individuals, and the disadvantages of the classical model avoided. Moreover, the improved model turns out to be an initial value problem, which is mathematically more convenient to deal with. Existence and uniqueness results for the nonlinear nonautonomous system of model equations are obtained. It is shown that the classical model and its time delay generalization are two degenerate cases of the improved model.     

Fast game dynamics coupled to slow population dynamics: A single population with hawk-dove strategies

We study a model of a population subdivided into two subpopulations corresponding to hawk and dove tactics. It is assumed that the hawk and dove individuals compete for a resource every Day, I.e., at a fast time scale. This fast part of the model is coupled to a slow part which describes the growth of the subpopulations and the long term effects of the encounters between the individuals which must fight to have an access to the resource. We aggregate the model into a single equation for the total population. It is shown that in the case of a constant game matrix, the total population grows according to a logistic curve whose τ and K parameters are related to the coefficients of the hawk-dove game matrix. Our result shows that high equilibrium density populations are mainly doves, whereas low equilibrium density populations are mainly hawks. We also study the case of a density dependent game matrix for which the gain is linearly decreasing with the total density.     

A finite element method for a two-sex model of population dynamics

A finite element scheme is described to approximate the solution of a nonlinear and non-local system of integro-differential equations that models the dynamics of a two-sex population. Crank-Nicolson time discretization is used and error estimates are derived for the appoximation.     

Galerkin methods for a model of population dynamics with nonlinear diffusion

A numerical method is proposed to approximate the solution of a nonlinear and nonlocal system of integro-differential equations describing age-dependent population dynamics with spatial diffusion. We use a finite difference method along the characteristic age-time direction combined with finite elements in the spatial variable. Optimal order error estimates are derived for this approximation. © 1996 John Wiley & Sons, Inc.     

A population model with time-dependent delay

We present analytical and computational results concerning the linear stability and instability of the uniform steady-state solution of a system of reaction-diffusion equations where a parameter in the kinetic terms is periodic in time. Under suitable assumptions the system is equivalent to a scalar equation with a periodically varying delay. Such a varying delay can model the seasonal fluctuations to the regeneration time of a resource. We study the effect such a varying delay can have on the stability of the spatially uniform steady-state. Analytical results reveal that instability can set in if the delays are large, while computational methods of analysing the stability equations reveal the precise shape of the instability boundary. The nonlinear stability of the uniform state is also examined using ladder methods.     

A population model with nonlinear diffusion

A model is presented for a single species population moving in a limited one-dimensional environment. The birth-death process is specialized by assuming a constant death modulus and a birth modulus which is an exponential in the age. The diffusion mechanism is nonlinear and results in a problem for the space population density which has a degenerate parabolic form and is similarly to the porous media equation. It is shown that the effect of the nonlinearity in the diffusion is to produce an approach to steady state even when the process is birth dominant. The interaction of the birth-death and diffusion processes is studied and is shown to yield a modified birth-death mechanism which is both time and space dependent.     

A planning and routing model for patient transportation in health care

In this paper, a problem concerning both the planning of health care services and the routing of vehicles, for patients transportation is addressed. An integrated approach, based on the column generation technique, is proposed to solve the planning and routing problem. Preliminary results on real data show the effectiveness of the proposed approach.     

A non-local PDE model for population dynamics with state-selective delay: Local theory and global attractors

We propose a non-local PDE model for the evolution of a single species population that involves delayed feedback, where the delay such as the maturation time in the delayed birth rate, is selective and the selection depends on the status of the system. This delay selection, in contrast with the usual state-dependent delay widely used in ordinary delay differential equation, ensures the Lipschitz continuity of the nonlinear functional in the classical phase space. We also develop the local theory, and the existence and upper semi-continuity of the global attractor with respect to parameters.     

On local stability of a population dynamics model with three development stages

We consider an age-dependent model of population dynamics, and obtain a sharp effective coefficient criterion of asymptotic stability for the non-trivial equilibrium point.