Existence of solutions for an age-structured insect population model with a larval stage

In this paper we particularly draw attention to the existence of solutions which describe the maturation rates of an age-structured insect population model. Such models commonly divide the population into immature and mature individuals. Immature individuals are defined as individuals of age less than some threshold age τ, while adults are individuals of age exceeding τ. The model is represented by the nonlinear neutral differential equation with variable coefficients. The conditions, which guarantee that the population size tends to a nonnegative constant or nonconstant function are also established.


Introduction
In the paper [11] authors Gourley and Kuang study the age structured population model.Such model is suitable for describing the dynamics of an insect population with larval and adult phases.As example may be the periodical cicada.The life phases of cicadas and other types of insect are fully described in [11].Gourley It is supposed that the total number of adults M(t) obeys the equation dM(t) dt = u(t, τ) − d(M(t)). (1.1) The term u(t, τ) is the maturation rate and d(M) is the adult mortality function which is strictly increasing in M and d(0) = 0.The authors in [11] consider the equation ∂u ∂t with a constant linear death rate µ and where b(a) ≥ 0 is the birth rate function and the initial condition u(0, a) = u 0 (a) ≥ 0, a ≥ 0. (1.3) In [11] it is considered that where H(a) is the Heaviside function and δ(a) the Dirac delta function.It is assumed that individuals of age less than τ produce b 0 eggs per unit time, those of age greater than τ produce b 1 eggs per unit time and each individual lays b 2 eggs on reaching maturation age τ.
For the adult population there is derived in [11] for t ≥ τ the neutral delay differential equation For 0 ≤ t ≤ τ and u(t, τ) = u 0 (τ − t)e −µt the M(t) is governed by the equation Further in the paper the authors consider the case when b 0 = 0.For more details about model we refer readers to [11] and for related models see [3,10,12,13].Qualitative properties of differential equations with delay are studied, for example, in [1,2,[4][5][6][7][8]15]. Instead of constants b 1 , b 2 in the function b(a) we will consider positive bounded functions b 1 (M(t)), b 2 (t), t ≥ 0, since b 1 is dependent on the total numbers of mature.Then following [11] we get and for u(t, 0) it follows For u(t, 0) we obtain The solution of (1.2) with respect to (1.3) and u(t, 0) = N(t) is given by For t > τ with regard to (1.4) we get equation (1.1) implies that Thus we get Then the equation (1.1) has the form which is the neutral delay differential equation.For t ≤ τ, u(t, τ) = u 0 (τ − t)e −µt and with respect to equation (1.1) we obtain In the rest of this paper we will assume that b 0 = 0. Motivated by the discussion above, we will consider the differential equations where It is reasonable to assume that the size of population is bounded.In this paper we consider a population which size is bounded by the functions w(t), v(t).The function w(t), for example, can depend on food resources, seasonal conditions, the size of territory in which the population lives, etc.We will focus on the existence of positive solutions for the neutral differential equation (1.6), since this problem is not solved in [11].The conditions which guarantee that the population size tends to nonnegative constant are also established.We also study the case when the size of population tends to nonconstant function.The main contribution of [11] is the Theorem 4. But this theorem we cannot apply to the Example 3.2.
The following fixed point theorem will be used to prove the main results in the next section.
Lemma 1.1 (see [9,16] Krasnoselskii's fixed point theorem).Let X be a Banach space, let Ω be bounded closed convex subset of X, and let S 1 , S 2 be maps of Ω into X such that S 1 x + S 2 y ∈ Ω for every pair x, y ∈ Ω.If S 1 is a contraction and S 2 is completely continuous then the equation

Existence theorems
The aim of the paper is to show the correctness of the model, that is, to show that the equation which represents the model, has a positive solution.
and there exist bounded functions v, w Then equation (1.6) has a positive solution which is bounded by functions v, w.
Proof.Let C([0, ∞), R) be the Banach space of all continuous bounded functions with the norm x = sup t≥0 |x(t)|.We define a closed, bounded and convex subset We now define two maps S 1 and S 2 : We will show that S 1 x + S 2 y ∈ Ω for any x, y ∈ Ω.For t ≥ τ and every x, y ∈ Ω, applying (2.4) we derive Furthermore using (2.4) for t ≥ τ we get According to inequality above and (S 1 x)(τ) + (S 2 y)(τ) ≥ v(τ), for t ∈ [0, τ] and any x, y ∈ Ω we get Thus we have proved that S 1 x + S 2 y ∈ Ω for any x, y ∈ Ω.
We will show that S 1 is a contraction mapping on Ω.For x, y ∈ Ω and t ≥ τ we derive This yields that Such inequality is also valid for t ∈ [0, τ].With regard to (2.1) S 1 is a contraction mapping on Ω.
We now show that S 2 is completely continuous.First we will show that S 2 is continuous.This means that S 2 is continuous.
We will show that S 2 Ω is relatively compact.It is sufficient to show by the Arzelà-Ascoli theorem that the family of functions {S 2 x : x ∈ Ω} is uniformly bounded and equicontinuous on every finite subinterval of [0, ∞).The uniform boundedness follows from the definition of Ω.For the equicontinuity we only need to show, according to Levitan's result [14], that for any given ε > 0 the interval [0, ∞) can be decomposed into finite subintervals in such a way that on each subinterval all functions of the family have a change of amplitude less than ε.Then with regard to condition (2.7), for x ∈ Ω and any ε > 0 we choose t * ≥ τ large enough so that Thus there exists δ = ε/B, where Consequently {S 2 x : x ∈ Ω} is uniformly bounded and equicontinuous on [0, ∞) and hence S 2 Ω is relatively compact subset of C([0, ∞), R).By Lemma (1.1) there is an x 0 ∈ Ω such that S 1 x 0 + S 2 x 0 = x 0 .Thus x 0 (t) is a positive solution of equation (1.6).The proof is completed.
Corollary 2.2.Suppose that (2.1) holds and there exist bounded functions v, w 2), (2.4) hold and Then equation (1.6) has a positive solution which is bounded by the functions v, w.
Proof.We only need to prove that the condition (2.8) implies (2.3)Let t ∈ [0, τ] and set Then with regard to (2.8) it follows that this implies that Thus all conditions of Theorem (2.1) are satisfied. where We will show that the conditions of Corollary 2.5 are satisfied.Condition b 2 e −µτ < 1 obviously holds.For K = 0 and w(t) = e −t we get Then x(t) = w(t) = e −t is the solution of (3.1) for t ≥ τ and tends to zero as t → ∞.For function Example 3.2.Consider the neutral differential equation where We will show that for w(t) = k + e −rt , k > 0 and K = k 1 − b 2 e −µτ the conditions of Corollary (2.5) are satisfied.Consequently we get This implies that (2.1) holds and K > 0. For the condition (2.9) we get Since Thus the conditions of Corollary 2.5 are satisfied and equation (3.2) has the solution x(t) = w(t) = k + e −rt , t ≥ τ, such that lim t→∞ x(t) = k.For function The applicability of Theorem 2.6 and hence also the Corollary 2.2 is illustrated in the following example.
We will show that the conditions of Theorem 2.6 are satisfied.Condition (2.11) has a form For condition (2.12) we get exp(−pk 2 ) + exp(pk 2 (t We obtain

Conclusions
In this paper, we present a study of a nonlinear nonautonomous neutral delay differential equation, which represents a population model.This model may be suitable for describing the development of an insect population with larval and adult phases.The article is not about finding solutions of this type of equations, but about investigating the conditions for the existence of a solution.The existence of positive solutions for the equation (1.6) is treated.Let us mention that we cannot apply Theorem 4 [11] to the equation (3.2).To clarify the detailed comparison we state the mentioned theorem.We remark that for equation (3.2) the conditions of Corollary 2.5 are satisfied and equation (3.2) has solution x(t) = k + e −rt , k > 0, t ≥ τ and lim t→∞ x(t) = k.
In addition in this paper we consider the equations with variable coefficients and the correctness of the model is supported by the existence Theorem 2.1.As far as the authors know, there are no other existence results for the insect population model with larval and adult phases.This confirmed that the results are new.