Bifurcations of equilibria of a non-linear age structured model

M. E. Gurtin and R. C. MacCamy investigated a non-linear age-structured population dynamical model, which served as one of the basic non-linear population dynamical models in the last three decades. They described a characteristic equation but they did not use it to discuss stability of equilibria of the system in certain special cases. In a recent paper, M. Farkas deduced a characteristic equation in another form. This characteristic equation enabled us to prove results about the stability of stationary age distributions of the system. In the present paper we are going to investigate how equilibria arise and change their stability as a basic parameter of the system varies

Net reproduction functions for nonlinear structured population models

The goal of this note is to present a general approach to define the net reproduction function for a large class of nonlinear physiologically structured population models. In particular, we are going to show that this can be achieved in a natural way by reformulating a nonlinear problem as a family of linear ones; each of the linear problems describing the evolution of the population in a different, but constant environment. The reformulation of a nonlinear population model as a family of linear ones is a new approach, and provides an elegant way to study qualitative questions, for example the existence of positive steady states. To define the net reproduction number for any fixed (constant) environment, i.e. for the linear models, we use a fairly recent spectral theoretic result, which characterizes the connection between the spectral bound of an unbounded operator and the spectral radius of a corresponding bounded operator. For nonlinear models, varying the environment naturally leads to a net reproduction function

Structured populations: The stabilizing effect of the inflow of newborns from an external source and the net growth rate

We investigate the effect of a positive population inflow of individuals from an external source on the dynamical behaviour of certain physisologically structured population models. We treat a size-structured model with constant inflow and nonlinear birth rate and an age-structured model with nonlinear (density dependent) inflow and linear birth rate. Analogously to the inherent net reproduction rate we introduce a net growth rate and discuss how this net growth rate can be related to our stability/instability conditions

Stability conditions for a non-linear size-structured model

In this paper we consider a general non-linear size-structured population dynamical model with size- and density-dependent fertility and mortality rates and with size-dependent growth rate. Based on M. Farkas (Appl. Math. Comput. 131 (1) (2002) 107-123) we are able to deduce a characteristic function for a stationary solution of the system in a similar way. Then we establish results about the stability (resp. instability) of the stationary solutions of the system

Asymptotic behavior of size-structured populations via juvenile-adult interaction

In this work a size structured juvenile-adult population model is considered. The linearized dynamical behavior of stationary solutions is analyzed using semigroup and spectral methods. The regularity of the governing linear semigroup allows to derive biologically meaningful conditions for the linear stability of stationary solutions. The main emphasis in this work is on juvenile-adult interaction and resulting consequences for the dynamics of the system. In addition, we investigate numerically the effect of a non-zero population inflow, due to an external source of newborns, on the dynamical behavior of the system in a special case of model ingredients

On a strain-structured epidemic model

We introduce and investigate an SIS-type model for the spread of an infectious disease, where the infected population is structured with respect to the different strain of the virus/bacteria they are carrying. Our aim is to capture the interesting scenario when individuals infected with different strains cause secondary (new) infections at different rates. Therefore, we consider a nonlinear infection process, which generalises the bilinear process arising from the classic mass-action assumption. Our main motivation is to study competition between different strains of a virus/bacteria. From the mathematical point of view, we are interested whether the nonlinear infection process leads to a well-posed model. We use a semilinear formulation to show global existence and positivity of solutions up to a critical value of the exponent in the nonlinearity. Furthermore, we establish the existence of the endemic steady state for particular classes of nonlinearities

Mathematical Analysis of a Clonal Evolution Model of Tumour Cell Proliferation

We investigate a partial differential equation model of a cancer cell population, which is structured with respect to age and telomere length of cells. We assume a continuous telomere length structure, which is applicable to the clonal evolution model of cancer cell growth. This model has a non-standard non-local boundary condition. We establish global existence of solutions and study their qualitative behaviour. We study the effect of telomere restoration on cancer cell dynamics. Our results indicate that without telomere restoration, the cell population extinguishes. With telomere restoration, exponential growth occurs in the linear model. We further characterise the specific growth behaviour of the cell population for special cases. We also study the effects of crowding induced mortality on the qualitative behaviour, and the existence and stability of steady states of a nonlinear model incorporating crowding effect. We present examples and extensive numerical simulations, which illustrate the rich dynamic behaviour of the linear and nonlinear models

The interaction of the Ifrs 9 expected loss approach with supervisory rules and implications for financial stability

This paper examines the interaction of the International Financial Reporting Standard (IFRS) 9 expected credit loss (ECL) model with supervisory rules and discusses potential implications for financial stability in the European Union. Compared to the incurred loss approach of IAS 39, the IFRS 9 ECL model incorporates earlier and larger impairment allowances and is more closely aligned with regulatory expected loss. The earlier recognition of credit losses will reduce the build-up of loss overhangs and the overstatement of regulatory capital. In addition, extended disclosure requirements are likely to contribute to more effective market discipline. Through these channels IFRS 9 might enhance financial stability. However, due to the reliance on point-in-time estimates of the main input parameters (probability of default and loss given default) IFRS 9 ECLs will increase the volatility of regulatory capital for some banks. Furthermore, the ECL model provides significant room for managerial discretion. Bank supervisors might play an important role in the implementation of IFRS 9, but too much supervisory intervention bears the risk of introducing a prudential bias into loan loss accounting that compromises the integrity of financial reporting. Overall, the potential benefits of the standard will crucially depend on its proper and consistent application across jurisdictions