On the asymptotic behavior of a size-structured model arising in population dynamics

. We study Perron’s theorem of a size-structured population model with delay when the nonlinearity is small in some sense. The novelty in this work is that the operator governing the linear part of the equation does not generate a compact semigroup unlike in the results present in literature. In such a case the spectrum does not consist wholly of eigenvalues but also has a non-trivial component called Browder’s essential spectrum. To overcome the lack of compactness, we give a localization of Browder’s essential spectrum of the operator governing the linear part and we use the Perron-Frobenius spectral analysis adapted to semigroups of positive operators in Banach lattices to investigate the long time behavior of the system


Introduction
Many areas of applied mathematics involve delay partial differential equations.Dynamical systems found in biology, physics, or economics depend not only on the present state of the dynamic but also on the past states.One of the simplest delay models describing a population of species struggling for a common food is the logistic model [13,19] Ṅ (t) = γ 1 − N (t − r) K N (t). (1.1) The delay r here is the production time of food resources.The food resources at time t are determined by the population number at time t − r.The constant γ is related to the reproduction of species, and represents the difference between birth and death rates.Usually, γ is called the Maltus coefficient of linear growth.The constant On the asymptotic behavior of a size-structured model arising in population dynamics K is the average population number, and is related to the ability of the environment to sustain the population.At the same time, Equation (1.1) can be used to study hatching periods, pregnancy duration, egg-laying, etc.However, individuals in every biological population differ in their physiological characteristics.This gives an importance to structured partial differential equations to understand the dynamics of such populations.We refer the interested reader to the monographs [20] for basic concepts and results in the theory of structured populations, and [24,30] for the theory of structured populations models using the semigroup approach.
In this work, we study the asymptotic behavior of the following size structured population model: when the nonlinear perturbation f is small in some sense.To achieve this task, we will use a functional analytic approach involving semigroups of operators.
The theory of strongly continuous semigroups of operators have been applied with great success to partial differential equations with delay.This idea goes back to N. Krasovskii [21], who showed that solutions of delay differential equations generate a semigroup of operators on an appropriate function space, known as history or phase space.J. Hale [15] and S. N. Shimanov [28] were the first to formulate a general theory.Subsequently, using semigroup theory, J. Hale and S. Verduyn Lunel [16] described the asymptotic properties of the solution in the finite-dimensional case.Other works in this direction include [1,5,10,18,29].The idea is to rewrite delay partial differential equations in the following form: where A is a linear (unbounded) operator acting on a Banach space X, x t is the history function and L is a linear operator acting on the delay space with values in X.If X is finite dimensional and L = 0, then A is a matrix and Equation (1.3) is an ordinary differential equation.If X is infinite dimensional, then the operator A is usually considered to be unbounded and generates a strongly continuous semigroup of operators (T (t)) t≥0 [12].The so called Perron's Theorem for the asymptotic behavior of solutions of differential equations have been the subject of many studies, see [3,4,7,23,[25][26][27].For ordinary differential equations, we refer the reader to the books [8,9,11,17].Let us recall the original Perron's Theorem for ordinary differential equations.
Theorem.[9] Consider the following ordinary differential equation where A is an n × n constant complex matrix and ) is a solution of Equation (1.4), then either where λ 0 is one of the eigenvalues of A.
In [26], the author proved a Perron's Theorem for Equation (1.3), when A = 0, with a finite delay and the space X is finite dimensional.In [22], the authors studied the case when X is infinite dimensional and the delay is infinite.They assumed that the operator A is the infinitesimal generator of a compact strongly continuous semigroup on X.A typical example of such an operator A is the differential operator in reaction diffusion equations on bounded regular domains Ω.
The aim of this work is to investigate the asymptotic behavior of the semilinear partial differential equation (1.2).Unlike in most models of semilinear reaction diffusion equations, the linear part of our equation is governed by a semigroup which is not compact.In such a case the spectrum does not consist wholly of eigenvalues but also has a non-trivial component called the essential spectrum.In the literature there are many different ways of looking at the essential spectrum, but a notable result in this area is that due to Nussbaum and (independently) Lebow and Schechter: the radius of the essential spectrum is the same for all the commonly used definitions of essential spectrum.To overcome the lack of compactness in our system, we will first give a localization of Browder's essential spectrum of the operator governing the linear part.This allows us to investigate the asymptotic behavior of the semilinear equation via a spectral decomposition by splitting the spectrum of the linear part with vertical lines iR + ρ, ρ ∈ R "far" from the essential spectrum.Finally, we give a sufficient condition for extinction of the population in terms of the coefficients of the system.To achieve this task, we use the semigroup version of the Perron-Frobenius theory of positive operators in Banach lattices [2,14].
This work is organized as follows: In Section 2, we give a localization of Browder's essential spectrum of the linear model.In Section 3, we will study the effect of small nonlinear perturbations on the original linear model.Moreover, we give a sufficient condition for extinction of the population using a Perron-Frobenius type theory of positive operators.

The linear model: localization of the essential spectrum
We consider a population of individuals that are distinguished by their individual size.Therefore, the density of population of size s at time t can be described by the number u(t, s).More precisely s2 s1 u(t, s)ds is the number of individuals that at time t have size s between s 1 and s 2 .As time passes, the following processes are supposed to take place in this population: • Individuals grow linearly in time at constant rate γ > 0.
• Individuals are subject to a size-dependent mortality denoted by µ.
• It is assumed that individuals may have different sizes at birth, and therefore β(σ, s, b) gives the rate at which an individual of size b produces offspring of the size s.This process is assumed to occur with a continuous time delay smaller than r (e.g.pregnancy duration).
• The population is subject to a density-dependent migration process with continuous time lags smaller then r represented by the term 0 −r ν(s, σ)u(t + σ, s)dσ.From those assumptions the following evolution equation can be derived: (2.1) On the asymptotic behavior of a size-structured model arising in population dynamics In the sequel, we assume that: (2.2) An example of such function is given by where β 1 and β 2 are bounded functions respectively on [−r, 0] and R + .
To write this equation in an abstract form, we introduce the Banach lattice X = L 1 (R + ) and the operator A defined on X by The operator A generates a c 0 -semigroup on X the given by (2.4) We introduce the delay operator Φ : If we write u(t, .)= u(t), then system (2.1) is written on the Banach lattice X = L 1 (R + ) as follows: To rewrite this equation as an abstract equation, we introduce the product space X = X × L 1 ([−r, 0], X) and the function In this case we have Further, on this product space we define the following operator where d dσ denotes the derivative with respect to σ.
The following result is a consequence of [5, Corollary 3.5]: Nadia Drisi et al.
Proposition 2.1.Equation (2.6) is equivalent to the following abstract Cauchy problem To show that A generates a c 0 -semigroup on X , we split it as where The following result is a consequence of [5, Theorem 3.25]: Proposition 2.2.The operator A 0 generates a c 0 -semigroup given explicitly by the following formula: where One can see that the perturbation operator A Φ is bounded.Moreover, we can see that the semigroup (T (t)) t≥0 (see (2.4)) and the delay operator Φ (see (2.5)) are positive.Thus from [12, Theorem 1.10], we have the following result.
For a bounded subset B of a Banach space Z, the Kuratowski measure of noncompactness α (B) is defined by α (B) := inf {d > 0 : there exist finitely many sets of diameter at most d which cover B} .
Moreover, for a bounded linear operator K on Z, we define α (K) by The essential radius of C is defined by

On the asymptotic behavior of a size-structured model arising in population dynamics
We recall some important facts about c 0 -semigroups.Let (R (t)) t≥0 be a c 0 -semigroup on a Banach space Z and A R its infinitesimal generator.Definition 2.5.[12,30] The growth bound ω 0 (R) of the c 0 -semigroup (R (t)) t≥0 is defined by Definition 2.6.[30] The essential growth bound (or α-growth bound) ω ess (R) of the c 0 -semigroup (R (t)) t≥0 is defined by: (2.9) The relation between r ess (R (t)) and ω ess (R) is given by the following formula ([30, Proposition 4.13 ]) r ess (R (t)) = e tωess(R) and e tσess(A R ) ⊂ σ ess (R (t)) . (2.10) Let A R be the generator of (R (t)) t≥0 .Then This means that if λ ∈ σ (A R ) and Reλ > ω ess (R), then λ does not belong to σ ess (A R ).Therefore λ is an isolated eigenvalue of A R ([30, Proposition 4.11]).The spectral bound s (A R ) of the infinitesimal generator A R is defined by: Recall the following formula [30] Consider the operator Φ λ defined on X for each λ ∈ C and z ∈ X by The following result gives a localization of Browder's essential spectrum of the operator A.
Theorem 2.9.Let λ 0 be the unique real solution of the following equation: Proof.Consider the following decomposition where Φ 1 is defined for each φ ∈ W 1,1 ([−r, 0] , X) and s ≥ 0 by Note that condition (2.2) implies that Φ 2 is bounded.
Remark.The growth rate γ does not have an effect on the asymptotic behavior of the c 0 -semigroup (T 1 (t)) t≥0 .

Nonlinear small perturbations
Consider the following model: Assume that f : R + × R → R satisfies the following hypotheses: Nadia Drisi et al.
• For all t ≥ 0 and z ∈ L • For all (t, z), • f is globally Lipschitz with respect to the second variable.
An example of such a function is f (t, x) = e −t x 1 + x 2 .We write (3.1) in the space X = X × L 1 ([−r, 0], X) in the following form where U(t) := u(t) u t , F(t, U(t)) = F (t, u(t)) 0 and F (t, u(t))(s) := f (t, u(t, s)) for all s ≥ 0. It follows that the nonlinear function F : R + × X → X is continuous and globally Lipschitz with respect to the second variable.Thus we have the following result [31] Theorem 3.1.Equation (3.1) has a unique solution U defined on R + .
In reality individuals with large sizes cannot give birth, then without loss of generality we can assume that the birth function component β 2 (σ, s) vanishes for s ≥ m where m is the maximal size of fertility.Thus Condition (2.2) becomes sup We state the first main result of this section: Theorem 3.2.Let λ 0 be the unique real solution of the following equation Assume that the solution U does not vanish for sufficiently large t.Then, we have either for each 1 ≤ j ≤ n, where γ j is a positively oriented closed curve in C enclosing the isolated singularity λ j , but no other points of σ(A) (see Figure 2).Then Π j is a projection in X and Π j Π h = 0 for j ̸ = h.Let U j := R(Π j ) be the range of Π j , then A restricted to U j is a bounded operator with spectrum consisting of the single point λ j . Let Then P 1 and P 2 are projections on U ρ and S ρ respectively and and U ρ and S ρ are closed subspaces of X which are invariant under the semigroup (T (t)) t≥0 .Let Π Uρ := P 1 and Π Sρ := P 2 .The subspace U ρ is finite-dimensional.Moreover, for every sufficiently small ε > 0, there exists For more details, we refer the reader to [30,Proposition 4.15].
In what follows, T Uρ (t) and T Sρ (t) denote the restrictions of T (t) on U ρ and S ρ respectively.Then T Uρ (t) t∈R is a group of operators and (3.9) We deduce from (3.8) that there exists a constant C ρ > 0 such that for each t ≥ 0 We introduce the new norm defined on X by where The corresponding operator norms T Sρ (t) T and T Uρ (−t) T satisfy T Sρ (t) T ≤ e ρ1t and T Uρ (−t) T ≤ e −ρ2t for t ≥ 0. (3.12) Lemma 3.4.Let U be the solution of Equation (3.1).Then for any ε > 0, there exists a constant C (ε) ≥ 1 such that In particular, there exists a constant C 1 ≥ 0 such that for m ∈ N and m ≤ t ≤ m + 1, we have Proof.Using the variation of constants formula, we have for 0 ≤ σ ≤ t Let ε > 0.Then, there exists C (ε) ≥ 1 such that where Proposition 3.5.Let U be the solution of Equation (3.1).If U(t) does not vanish for sufficiently large t, then we have Remark.It is clear from Lemma 3.4 that if U(t 0 ) = 0 for some t 0 ≥ 0, then U(t) = 0 for all t ≥ t 0 .
Proof of Proposition 3.5.Let ε > 0, from Lemma 3.4, we deduce that for t ≥ 0 Since t 0 p (s) ds t → 0 as t → ∞, then by taking t → ∞ in (3.18), we obtain that Now by letting ε → 0 in (3.19) we obtain the desired estimation.■ We fix a real number ρ such that ρ > λ 0 and σ (A) where ρ 1 and ρ 2 are the real numbers defined by (3.9).

23). ■
In what follows, we assume that the solution U does not vanish for sufficiently large t.We have the following Lemma.

Definition 2 . 4 .
[6] Let C be a closed linear operator with dense domain in a Banach space Z.Let σ (C) denote the spectrum of the operator C. The Browder's essential spectrum of C denoted by σ ess (C) is the set of λ ∈ σ (C) such that one of the following conditions holds: (i) Im (λI − C) is not closed, (ii) the generalized eigenspace M λ (C) := k≥1 Ker (λI − C)k is of infinite dimension, (iii) λ is a limit point of σ (C) .

Figure 2 :
Figure 2: Spectrum of the operator A