Abstract and Applied Analysis

Volume 2013 (2013), Article ID 908768, 11 pages

http://dx.doi.org/10.1155/2013/908768

Research Article

On the Multispecies Delayed Gurtin-MacCamy Model

Anna Poskrobko¹ and Antoni Leon Dawidowicz²

¹Faculty of Computer Science, Bialystok University of Technology, Ulica Wiejska 45A, 15-351 Białystok, Poland

²Faculty of Mathematics and Computer Science, Jagiellonian University, Ulica Łojasiewicza 6, 30-348 Kraków, Poland

Received 8 November 2012; Revised 9 March 2013; Accepted 29 March 2013

Academic Editor: Carlos Vazquez

Copyright © 2013 Anna Poskrobko and Antoni Leon Dawidowicz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The paper deals with the description of multispecies model with delayed dependence on the size of population. It is based on the Gurtin and MacCamy model. The existence and uniqueness of the solution for the new problem of populations dynamics are proved, as well as the asymptotical stability of the equilibrium age distribution.

1. Introduction

In this paper we consider mutual influence of populations. We assume that each population develops differently, affecting each other. Populations do not destroy each other. However, they share the same natural resources and the space. In our model denotes the density of the ecosystem consisting of different populations. Therefore is the vector function. Each component for denotes the th population density. The development of th population can be expressed by the following system of the equations:

with the initial condition

where

The rate

denotes the intensity of changing of the th population in time. In particular, if is differentiable then . The quantity is the total th population at time . We can express total population of the whole ecosystem at time by

Thus

We assume that

The birth and death processes of the th population are described by coefficients and . We assume that both processes depend on the population size not only at the moment , but also at any preceding period of time (so-called history segment). Introducing the delay parameter to the model has deep biological approach (see, for instance, [1]). All natural processes occur with some delay with respect to the moment of their initiation. We can take into consideration, for example, the period of pregnancy (constant delay) or morbidity (variable delay). In our model, considered processes occur with various delays, typical for populations. It is a novel approach to the population dynamics research. Moreover the system (1)–(4) consists of the general description of populations dynamics. If populations develop independently, then the functions and depend on th coordinate of the function only and the system consists of independent von Foerster equations. However, if there are populations relationships, we can consider three cases.

(1)  Competition for food: in this case is decreasing function of variables describing the numbers of species competing for food with the th one. For example, , where denotes total amount of food and is the set of species competing with the th one. (2)  Schema predator prey: if th species feeds on th species individuals, then and are increasing functions of variables and , respectively. The classical models show that balance state between two species competing for food is not possible. In this case the model does not have nonzero equilibrium age distribution. However, if we additionally consider predator feeding on one competing species, then we get balance between three considered species (see, for instance, [2–5]). (3)  Symbiosis: if th and th species live in symbiosis with each other, then can be an increasing function of (when the presence of th species individuals is the reproduction of th species favour) or is a decreasing function of (when the presence of th species protects th species individuals from death).

In this paper we develop the idea of Gurtin and MacCamy model [6] for one age-dependent population

This classical model has many generalizations [7–16]. Considerations of the dynamics of age-structured populations have received substantial treatment on various fields [17–22]. In particular our latest extension of this theme was presented in the article [10], where we considered an age-dependent population dynamics with a delayed dependence on the structure. The right-hand side of the equation in our model is not in the form but . Here is an operator that does not apply to the value of the function , as in the classical model, but to its restriction to some particular space:

In our paper [10] we present the proof of the existence and uniqueness of the new problem solution. Conditions of the exponential asymptotical stability for this model still remain to be formulated. On account of the significant difficulties with formulating conditions of the stability we consider slightly simpler version of the model than the one described in [10]. We resign from the delay in the structure during a deliberation on mortality. However, the delayed dependence of the structure still remains the element of the birth process description. In this paper we aim to state stability conditions for the ecosystem of populations. The plan of the paper is as follows. In Section 2 we formulate the problem in the terms of operator equations. Section 3 contains the proof of the existence and uniqueness of the solution. We also study equilibrium age distributions, that is, solutions to the problem which are independent of time. In Section 4 exponential asymptotic stability of the equilibrium age distributions is analyzed. We use generalized Laplace transform to study the stability.

We consider our model (1)–(4) under the following assumptions.

where for . for . The function is continuous. , where , ; the Fréchet derivatives of with respect to exist for all and . The components of the function   belong to ;  , respectively, for . The components of Fréchet derivative in the point as functions of belong to , . Here denotes the Banach space of all bounded linear operators from to . for . The functions and are bounded; that is, and where and are finite quantities. There exist and such that

                        for every and

                        for every .

2. Equivalent Formulation of the Problem

In this section we will formulate problem (1)–(4) in terms of operator equations. Thanks to this we will prove local and global existences of the presented problem solution. The next theorem is analogous to these well-known results, for example, von Foerster [23] or Gurtin and MacCamy models.

Theorem 1. Nonnegative continuous functions and , where

for , are the solutions of the problems (2) and (3) up to time if and only if the function is the solution of the age-dependent populations problem (1)–(4) on and is defined by the formula

where for each .

Proof. The idea of the proof is analogous to the result included in the paper [9], treating the theme of the age-dependent population problem for one species. Let be a solution of the problem up to time . Let , then we can rewrite (1) as the equation with the unique solution

Substituting , and , into (17) yields the formula (16). Applying (16) to (2) and (3) we obtain the operator equations (14) and (15).

To prove the second part of the theorem we should assume that and for are continuous functions on the interval fulfilling conditions (14) and (15). Let for each component be defined on by the formula (16). The function is nonnegative because of . An easy computation shows that (4) holds, and for . because , , and are continuous and . It follows from (14)–(16) that (2) and (3) are satisfied. To complete the proof, let us notice that (6) and (16) imply existing on and the equality (1) holds.

Let us make some estimation for the necessity of the next section. We turn back to the operator equation (14). By the assumptions and we get

Denoting

we obtain

and by Gronwall's inequality we get

3. Existence and Uniqueness

According to the theorem in Section 2 to solve the population problem up to time it is sufficient to find functions and satisfying the system of operator equations (14) and (15). We can notice that (14) is a Volterra equation of with a unique solution . Let us define on the new operator where

We define the previous operator using the system of (15) for with replaced by .

Theorem 2. The operator defined by (22) has a unique fixed point for any .

Proof. We prove that the operator is contracting, and then the assertion will be a consequence of the Banach fixed point theorem. Let us consider the Banach space with the Bielecki norm for any . Such norm is equivalent to classical supremum norm in . Choose . Applying the definition (22) we obtain

Estimating the quantities ,  , and we use the assumptions , , and , inequalities (21) and . Thus

From (14) and the definition of we have

Let us denote (26) by ; then

and hence, by Gronwall's inequality we have

Let us estimate

where and are positive constants and . Finally, we have

Therefore, we can choose sufficiently large constant for fixed that , , and are less than with the constant (independent of and ). This shows the contraction of and completes the proof.

According to the previous mentioned there exists the exact one solution for any interval so the solutions defined on two different intervals coincide on their intersection. By the extension property we have the existence and uniqueness of the populations problem solution for all times.

4. Stability of Equilibrium Age Distribution

A stationary solution of the model (1)–(4) satisfies the following system of the equations:

for.

The population of the whole ecosystem and its birthrate are constants. The quantity is the solution of the system (31), and it will be referred to the equilibrium age distribution. The probability that an individual of th population survives to age if the population of the ecosystem is on the constant size level can be expressed by

The quantity

is the number of offsprings expected to be born to an individual of th population when the population of the whole ecosystem equals . We can formulate the following theorem describing the connection between these three quantities.

Theorem 3. Let and for , and assume that ,   for each . Then

with

is a necessary and sufficient condition that an equilibrium age distribution exists. The unique equilibrium age distribution corresponding to is given by

where

Proof. The function (36) is the unique solution of (31)₁ with the initial condition . By (31)₂ we obtain the formula (37) for . An easy computation shows that (35) is equivalent to (31)₃ for each .

We now turn to the problem of the equilibrium age distribution stability. We consider “perturbations” and . Let us write

where for . And

where for . Our goal is to formulate relations for and which guarantee that and obey the basic equations (1)–(4). From (1) we obtain

for

where is the functional

We conclude from (2) that

where

Here is the functional. Moreover, we have

A trivial verification shows that (3) gives the condition for in the form

We consider the system (42)–(48) with the initial condition

where

We first express the solution of (42)–(48) in the form of the matrix equation. Let be the solution of (42) up to time . Let , , , , , and . Then we can rewrite (42) as the equation

with the unique solution

Substituting , for and , for yields the formulas

where

Furthermore, (45) implies that

It follows that

We conclude from (48) that

and hence

Conditions (57) and (59) imply that

where

satisfy the matrix equation

where and are block diagonal matrices. We have

for zero matrix , , and

Since is the functional so in the formula of we consider as the function with functional values. Furthermore, in the notation (63) looks like the ordinary matrix equation, but in fact it is the operator one. Moreover,

for

and , for .

We can notice that the last two elements of the expression defining equal zero for .

We return to the previous deliberation in the next theorem. We take some additional assumptions.

The Fréchet derivatives and the derivation for as functions of the variable belong to . and tend to zero as uniformly for ; here denotes the supremum norm in the Banach space . .

Denote by the formula . The exponential asymptotical stability of the model is established by the following theorem.

Theorem 4. Let , and let   +    . Let us assume that there exists some that the equation

has no solution with . Then there exist real numbers and such that for any initial data with , the corresponding solution of the population problem (1)–(4), if it exists for , satisfies

Proof. Let be the solution of the population problem (1)–(4) for . In the proof we will use the properties of the generalized Laplace transform. Let be the function with measure value. Define

If we define the convolution by the formula

where for , then we have the equality analogous to the property of classical Laplace transform, that is,

Using generalized Laplace transform to (63) we get

where

for and . From the equality (68) we conclude that the matrix has an analytic inverse for . Moreover, we have

so is analytic for .

The solution of the matrix equation (63) exists and is given by

where , for and

is the Laplace transform of the function and

where denotes the sum of moduli of the elements of the matrix . Here and in the whole proof , ,   denote positive constants.

By , (32), and (54)

For and we can estimate that

where ,

Let . By there exists such that

for . By the previous inequalities and the fact that , , and are finite, we can estimate that

We require that and . Hence . According to the previous remarks, we have

Thus (82)–(86) imply that

where . Gronwall's inequality yields

Let us choose and such that and . Assume that . In that case, from what has already been proved, it follows that , , and are . Therefore (41), (86)₂ imply (69) and (53)₂, (80), and (39) imply (70).

Example 5. Let . Let us consider

where

with ,  ,  ,   for . Let us define

for an arbitrary . From (32) and (33) we conclude that

By Theorem 3 there exists an equilibrium age distribution where if and only if . To investigate the stability of the equilibrium age distribution we consider (68). Let be the solution of the system of the equations

First, let us notice that

with the derivatives in the point . It is easy to notice that is bounded on half-plane for every real (also negative). Analogously

It is also possible to prove that for every real the function is bounded on half-plane . Consider the right-hand side of (68). All components can be presented in the form

where is bounded and . Let , and let :

Fix . For sufficiently large there exist some and that

For we have

In consequence for and sufficiently large, we can find that such that the modulus of the right-hand side of (68) is less than 1 for every , for which .

Acknowledgment

The first author acknowledges the support from Bialystok University of Technology (Grant no. S/WI/2/2011).

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