A Note on the Linear Stability of the Steady State of a Nonlinear Renewal Equation with a Parameter
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DOI: 10.1007/s13226-020-0438-0
A NOTE ON THE LINEAR STABILITY OF THE STEADY STATE OF A NONLINEAR RENEWAL EQUATION WITH A PARAMETER Suman Kumar Tumuluri University of Hyderabad, Gachibowli, India e-mail: [email protected] (Received 29 March 2018; accepted 6 May 2019) In this article we consider a variant of age-structured nonlinear Lebowitz-Rubinow equation. We study the linear stability of this equation near the nontrivial steady state by analyzing the corresponding characteristic equation. In particular, we provide some sufficient conditions under which the nonzero steady state is linearly stable. Key words : Existence of steady states; linear stability; nonlinear renewal equation. 2010 Mathematics Subject Classification : 35B35, 35L04, 35L60.
1. I NTRODUCTION Age structured models play a central role in the study of population dynamics. The mathematical theory of linear age structured population models was originally studied by Lotka and Sharpe (see [8, 11]). Among the age structured models the simplest one is due to McKendrick–Von Foerster (see [8, 11, 16]). Mathematical study of this model can be found in [5, 16]. In this model, the mortality rate d and the fertility rate B for the population are just age dependent functions. So this model does not take into account the aging process which is very important for cell population (see [2]). Moreover, in the McKendrick–Von Foerster model it is assumed that the birth process does not change aging properties of off-spring. However, experimental results show that different cells, under identical conditions, have exhibited highly variable intermitotic intervals (see [16]). Therefore, Lebowitz et al. have proposed a modified McKendrick-Von Foerster model, which is widely known as the Lebowitz-Rubinow model, where population is structured not only in age x but also in terms of generation time a. Existence, uniqueness and long time behavior of a solution to the Lebowitz-Rubinow model has been discussed in [10].
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SUMAN KUMAR TUMULURI However, this model or McKendrick-Von Foerster model do not incorporate competition among
cells to acquire/utilize resources (like nutrients). This competition term induces nonlinearity in vital rates (see [4]). For existence, uniqueness results and asymptotic behavior of solution to the nonlinear McKendrick-Von Foerster equations, one can refer [5, 12, 16]. In this article, we consider the following partial differential equation which is a variant of a Lebowitz-Rubinow model with competition term in the fertility rate, i.e., ∂ ∂ ∂t u(t, x; a) + ∂x u(t, x; a) + d(x; a)u(t, x; a) = 0, t > 0, x > 0, a ≥ 0, Z ∞ u(t, 0; a) = B(x, S(t); a)u(t, x; a)dx, t > 0, a ≥ 0, 0 u(0, x; a) = u0 (x; a) ≥ 0, x > 0, a ≥ 0, with a coupling
Z S(t; a) =
(1)
∞
ψ(x; a)u(t, x; a)dx, t > 0, x > 0.
(2)
0
Throughout this article we assume that the vital rates B and d satisfy the following conditions. We assume that B is a smooth, positive, bounded and integrable function for every nonnegative parameter a. Moreover, we assume
Z
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