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Register a new userSize-structured populations: immigration, (bi)stability and the net growth rate
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Authors
Jozsef Z. Farkas
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We consider a class of physiologically structured population models, a first order nonlinear partial differential equation equipped with a nonlocal boundary condition, with a constant external inflow of individuals. We prove that the linearised system is governed by a quasicontraction semigroup. We also establish that linear stability of equilibrium solutions is governed by a generalized net reproduction function. In a special case of the model ingredients we discuss the nonlinear dynamics of the system when the spectral bound of the linearised operator equals zero, i.e. when linearisation does not decide stability. This allows us to demonstrate, through a concrete example, how immigration might be beneficial to the population. In particular, we show that from a nonlinearly unstable positive equilibrium a linearly stable and unstable pair of equilibria bifurcates. In fact, the linearised system exhibits bistability, for a certain range of values of the external inflow, induced potentially by All{e}e-effect.
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Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task. In the present paper, we develop a numerical framework for computing approximations to stationary solutions of general evolution equations, which can also be used to produce existence and stability regions for steady states. In particular, we use the Trotter-Kato Theorem to approximate the infinitesimal generator of an evolution equation on a finite dimensional space, which in turn reduces the evolution equation into a system of ordinary differential equations. Consequently, we approximate and study the asymptotic behavior of stationary solutions. We illustrate the convergence of our numerical framework by applying it to a linear Sinko-Streifer structured population model for which the exact form of the steady state is known. To further illustrate the utility of our approach, we apply our framework to nonlinear population balance equation, which is an extension of well-known Smoluchowksi coagulation-fragmentation model to biological populations. We also demonstrate that our numerical framework can be used to gain insight about the theoretical stability of the stationary solutions of the evolution equations. Furthermore, the open source Python program that we have developed for our numerical simulations is freely available from our Github repository (github.com/MathBioCU).
In the present paper we analyze the linear stability of a hierarchical size-structured population model where the vital rates (mortality, fertility and growth rate) depend both on size and a general functional of the population density (environment). We derive regularity properties of the governing linear semigroup, implying that linear stability is governed by a dominant real eigenvalue of the semigroup generator, which arises as a zero of an associated characteristic function. In the special case where neither the growth rate nor the mortality depend on the environment, we explicitly calculate the characteristic function and use it to formulate simple conditions for the linear stability of population equilibria. In the general case we derive a dissipativity condition for the linear semigroup, thereby characterizing exponential stability of the steady state.
We investigate steady states of a quasilinear first order hyperbolic partial integro-differential equation. The model describes the evolution of a hierarchical structured population with distributed states at birth. Hierarchical size-structured models describe the dynamics of populations when individuals experience size-specific environment. This is the case for example in a population where individuals exhibit cannibalistic behavior and the chance to become prey (or to attack) depends on the individuals size. The other distinctive feature of the model is that individuals are recruited into the population at arbitrary size. This amounts to an infinite rank integral operator describing the recruitment process. First we establish conditions for the existence of a positive steady state of the model. Our method uses a fixed point result of nonlinear maps in conical shells of Banach spaces. Then we study stability properties of steady states for the special case of a separable growth rate using results from the theory of positive operators on Banach lattices.
Molecular circadian clocks, that are found in all nucleated cells of mammals, are known to dictate rhythms of approximately 24 hours (circa diem) to many physiological processes. This includes metabolism (e.g., temperature, hormonal blood levels) and cell proliferation. It has been observed in tumor-bearing laboratory rodents that a severe disruption of these physiological rhythms results in accelerated tumor growth. The question of accurately representing the control exerted by circadian clocks on healthy and tumour tissue proliferation to explain this phenomenon has given rise to mathematical developments, which we review. The main goal of these previous works was to examine the influence of a periodic control on the cell division cycle in physiologically structured cell populations, comparing the effects of periodic control with no control, and of different periodic controls between them. We state here a general convexity result that may give a theoretical justification to the concept of cancer chronotherapeutics. Our result also leads us to hypothesize that the above mentioned effect of disruption of circadian rhythms on tumor growth enhancement is indirect, that, is this enhancement is likely to result from the weakening of healthy tissue that are at work fighting tumor growth.
We consider a nonlinear structured population model with a distributed recruitment term. The question of the existence of non-trivial steady states can be treated (at least!) in three different ways. One approach is to study spectral properties of a parametrized family of unbounded operators. The alternative approach, on which we focus here, is based on the reformulation of the problem as an integral equation. In this context we introduce a density dependent net reproduction rate and discuss its relationship to a biologically meaningful quantity. Finally, we briefly discuss a third approach, which is based on the finite rank approximation of the recruitment operator.
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