On a System Modelling a Population with Two Age Groups

and Applied Analysis 3


Introduction
In this paper, we will study the following problem: where , , , and  ∈ (R, (0, ∞)) are -periodic functions and  and  are positive constants.
We are interested in dividing the individuals within a population into two age groups.The first group contains all newborns in addition to all young individuals who are unable to produce newborns; such group will be referred to as the juvenile group.The second group, which we will call the adult group, contains all individuals who can produce newborns in addition to old individuals who may not be able to produce newborns.The functions  and V represent, respectively, the total number of individuals who belong to the juvenile and adult groups.As adults give birth to juveniles, the function  corresponds to the birth rate of the population.Juveniles are lost both through death and through becoming adults; the function  corresponds to this overall loss.The function  gives the rate at which juveniles become adults and the function  corresponds to the death rate of adult population.The terms −[ + V] and −V[ + V] correspond to decrease in population size due to competition for limited resources.
In natural environments the number of individuals of a population changes in time in different ways.Many observations show that the number of individuals of a population can have large oscillations in nature.In the earlier models the population is characterized by its size which is the total number of individuals within the population or the total biomass.One of such models is the Malthus model for the human population growth.P. F. Verhulst in 1838 introduced another model which is known as the realistic model; see [1,2].Models presenting qualitatively this type of behavior are density dependent unstructured population models; the most well-known model for interspecific competition has been proposed by Lotka and Volterra and has been studied extensively by Bonhoeffer, Borrelli, and Murray; see also [1][2][3].In fact, up to the mid of the 20th century most models characterize the population by its size or total biomass.In such models the population is considered as homogeneous; that is, the models do not distinguish between the individuals within the population.Models that involve structured population are called structured population models.A structured model describes how individuals move in time among different groups and thus describes the dynamics of population groups and as a result it describes the dynamics of the whole population.For other related results, we refer the readers to [4][5][6][7] and the references therein.
Recently, a model for the growth of a population of two age groups (adult and juvenile) in which there is competition for limited resources has been considered in [8], where the authors assumed that the population is homogenous with common birth rate, common death rate, and common inhibiting constants.They established a time-invariant structure under general conditions and discussed the stability of 2 Abstract and Applied Analysis the equilibrium points.More concretely, at time  the net rate of change in the populations of the two groups is modeled by the system (2) the authors showed in [8] that if  < , then (2) has a unique positive equilibrium point value in addition to the trivial equilibrium point  = V = 0.
Obviously, in [8], since , , , and  are positive constants, the solutions of (2) can be explicitly given and some estimates can be carried out easily.However, when , , , and  are positive functions, the method of [8] cannot be applied to deal with the system (1) any more.If , , , and  are not constants, whether the system (1) has a positive solution or not is a natural question.Inspired by above considerations, in the present paper, we will first establish the lower and upper solutions method for more general system where  and  : R × R → R are continuous functions and ,  are -periodic continuous functions, and then we will prove the existence of positive solutions for system (1) by applying above method.
Our main results can be stated as below.
Theorem 1. Suppose that the functions , V, , V ∈  ∩  1 (R, R). (, V) and (, V) are ordered coupling lower and upper solutions of systems (3); the following condition is hold (H).There exists  1 ,  1 > 0 such that, for any Then, the problem (3) has at least one solution Theorem 2. There exists a positive periodic solution of systems (1) if and only if  1 () < 0.
Remark 3. To overcome the difficulties caused by the spatially heterogeneous, we discuss the system (3) by lower and upper solutions method established in Theorem 1 and obtain the necessary and sufficient conditions for the existence of positive periodic solutions of (1) in terms of the principal eigenvalue of the associated linear system.For other related results on the study of differential systems via lower and upper solutions method, we refer the readers here to [9][10][11] and the references listed therein.
The rest of the paper is organized as follows.In Section 2, we establish the lower and upper solutions methods for the system (3).In Section 3, we obtain the necessary and sufficient conditions for the existence of a positive periodic solution of (1).

Lower and Upper Solutions Method
In this section, we will develop lower and upper solutions method for system (3).
Let  be a Banach space defined as Definition 4. Assume that the functions , V, , V ∈  ∩  1 (R, R).Then, (, V) and (, V) are called ordered coupling lower and upper solutions of systems (3), respectively, if  ≤  and V ≤ V satisfying Proof of Theorem 1.By the condition (H), there exist  ≥  1 and  ≥  1 such that For any ,  ∈ , we consider the following linear problem: +  ()  +  =  (, ) + , It is well known that the system (7) is equivalent to the equation where and, consequently, Abstract and Applied Analysis It is easy to see that  :  2 →  2 is completely continuous. 2is also completely continuous.We will show that  :  → .

Existence and Nonexistence of Positive Periodic Solutions
We consider the system where , , , and  are  periodic functions, ,  ∈ (R[0, ∞)).
Proof.The result follows from Lemma 5 by choosing  0 = V 0 =  where  is any positive number.Lemma 7. The system (14) has a principal eigenvalue; that is, there exists Λ ∈ R and functions , V ∈ ∩ 1 (R, R) such that , V > 0 and and define the matrix () by By essentially the same argument as in [12,Lemma 12], if  > 0 is sufficiently large, then  −  +  is an invertible operator such that ( −  + ) −1 is compact.If, moreover,  is chosen sufficiently large to ensure that () + () −  < 0 and () + () −  < 0, it follows from Corollary 6 that ( −  + ) −1 is strongly positive.
We will denote the principal eigenvalue of  −  by  1 ().
System (1) can be rewritten as where () = ( −() () () −() ) and  :  2 →  2 such that Although () is a cooperative matrix, system (1) is not a cooperative system.We can give necessary and sufficient conditions for the existence of a positive solution.