Semigroup Analysis of Structured Parasite Populations

Abstract

Abstract. Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a population is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral theory of semigroups to establish conditions for the existence of a positive equilibrium solution of our model. Then, we formulate conditions that guarantee that the linearised system is governed by a positive quasicontraction semigroup on the biologically relevant state space. We also show that the governing linear semigroup is eventually compact, hence growth properties of the semigroup are determined by the spectrum of its generator. In the case of a separable fertility function, we deduce a characteristic equation, and investigate the stability of equilibrium solutions in the general case using positive perturbation arguments.

Stationary solutions to a system of size-structured populations with nonlinear growth rate

“…If we find P * ∈ R N for which the operator A P * has zero eigenvalue, then letting φ * ∈ D(A P * ) be a corresponding eigenvector, a nonzero stationary solution φ is constructed by putting its ith component as φ i = P i * φ i * /(P φ * ) i . Such an approach was used recently for similar structured single-species population models by Borges et al [3] and Farkas et al [6].…”

“…Stability properties of stationary solutions have been investigated by semigroup theory by Farkas and Hagen [5]. Recently, Borges et al [3] and Farkas et al [6] have studied the existence problem of nonzero stationary solutions for certain structured population models by using zero eigenvalue problems. Also, Walker [10] has studied nonzero stationary solutions for age-and spatially structured population models by using a fixed-point problem.…”

Steady states in a structured epidemic model with Wentzell boundary condition

“…Therefore, to establish conditions which guarantee the existence of a positive steady state we utilise a combination of an operator theoretic framework (see e.g. [5,15]) and a fixed point approach. For basic definitions and results from linear semigroup theory used throughout this section we refer the reader to [2,10,13].…”

“…A similar operator theoretic framework was previously utilised for simpler problems (in particular with finite dimensional nonlinearities and classical boundary conditions), see e.g. [5,15]. The key idea to treat the steady state problem is to define a linear operator for a fixed environment (nonlinearity) and to study spectral properties of that operator.…”

“…Our main goal in this paper is to establish sufficient conditions for the existence of positive steady state solutions of model (1.1)-(1.3). We shall refer the interested reader to [5,8,15,17] where different size-structured models with distributed recruitment processes were investigated. The boundary condition (1.2) is the so called generalized Wentzell-Robin or dynamic boundary condition.…”