Existence of Positive Periodic Solutions to -Species Nonautonomous Food Chains with Harvesting Terms

Abstract

By using Mawhin's continuation theorem of coincidence degree theory and some skills of inequalities, we establish the existence of at least positive periodic solutions for -species nonautonomous Lotka-Volterra type food chains with harvesting terms. An example is given to illustrate the effectiveness of our results.

1. Introduction

The dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. These problems may appear to be simple mathematically at first; sight, they are, in fact, very challenging and complicated. There are many different kinds of predator-prey models in the literature. For more details, we refer to [1, 2]. Food chain predator-prey system, as one of the most important predator-prey system, has been extensively studied by many scholars, many excellent results concerned with the persistent property and positive periodic solution of the system; see [3–13] and the references cited therein. However, to the best of the authors' knowledge, to this day, still no scholar study the -species nonautonomous case of Food chain predator-prey system with harvesting terms. Indeed, the exploitation of biological resources and the harvest of population species are commonly practiced in fishery, forestry, and wildlife management; the study of population dynamics with harvesting is an important subject in mathematical bioeconomics, which is related to the optimal management of renewable resources (see [14–16]). This motivates us to consider the following -species nonautonomous Lotka-Volterra type food chain model with harvesting terms:

(1.1)

where is the th species population density, is the growth rate of the first species that is the only producer in system (1.1), and stand for the th species intraspecific competition rate and harvesting rate, respectively, is the death rate of the th species, represents the th species predation rate on the th species, and stands for the transformation rate from the th species to the th species. In addition, the effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Therefore, the assumptions of periodicity of the parameters are a way of incorporating the periodicity of the environment (e.g, seasonal effects of weather, food supplies, mating habits, etc), which leads us to assume that and are all positive continuous -periodic functions.

Since a very basic and important problem in the study of a population growth model with a periodic environment is the global existence and stability of a positive periodic solution, which plays a similar role as a globally stable equilibrium does in an autonomous model, this motivates us to investigate the existence of a positive periodic or multiple positive periodic solutions for system (1.1). In fact, it is more likely for some biological species to take on multiple periodic change regulations and have multiple local stable periodic phenomena. Therefore, it is essential for us to investigate the existence of multiple positive periodic solutions for population models. Our main purpose of this paper is by using Mawhin's continuation theorem of coincidence degree theory [17], to establish the existence of positive periodic solutions for system (1.1). For the work concerning the multiple existence of periodic solutions of periodic population models which was done using coincidence degree theory, we refer to [18–21].

The organization of the rest of this paper is as follows. In Section 2, by employing the continuation theorem of coincidence degree theory and the skills of inequalities, we establish the existence of at least positive periodic solutions of system (1.1). In Section 3, an example is given to illustrate the effectiveness of our results.

2. Existence of at Least Positive Periodic Solutions

In this section, by using Mawhin's continuation theorem and the skills of inequalities, we shall show the existence of positive periodic solutions of (1.1). To do so, we need to make some preparations.

Let and be real normed vector spaces. Let be a linear mapping and be a continuous mapping. The mapping will be called a Fredholm mapping of index zero if dim and is closed in . If is a Fredholm mapping of index zero, then there exist continuous projectors and such that and and It follows that is invertible and its inverse is denoted by . If is a bounded open subset of , the mapping is called -compact on , if is bounded and is compact. Because Im is isomorphic to Ker , there exists an isomorphism .

The Mawhin's continuous theorem [17, page 40] is given as follows.

Lemma 2.1 (see [9]).

Let be a Fredholm mapping of index zero and let be -compact on . Assume that

• (a) for each , every solution of is such that ;

• (b) for each ;

• (c)

Then has at least one solution in

For the sake of convenience, we denote respectively; here is a continuous -periodic function.

For simplicity, we need to introduce some notations as follows:

(2.1)

where .

Throughout this paper, we need the following assumptions:

and

()

Lemma 2.2.

Let and for the functions and the following assertions hold:

• (1) and are monotonically increasing and monotonically decreasing on the variable respectively.

• (2) and are monotonically decreasing and monotonically increasing on the variable respectively.

• (3) and are monotonically decreasing and monotonically increasing on the variable respectively.

Proof.

In fact, for all we have

(2.2)

By the relationship of the derivative and the monotonicity, the above assertions obviously hold. The proof of Lemma 2.2 is complete.

Lemma 2.3.

Assume that and hold, then we have the following inequalities:

(2.3)

Proof.

Since

(2.4)

By assumptions Lemma 2.2 and the expressions of and we have

(2.5)

where that is Thus, we have The proof of Lemma 2.3 is complete.

Theorem 2.4.

Assume that and hold. Then system (1.1) has at least positive -periodic solutions.

Proof.

By making the substitution

(2.6)

system (1.1) can be reformulated as

(2.7)

where

Let

(2.8)

and define

(2.9)

Equipped with the above norm and are Banach spaces. Let

(2.10)

where , We put Thus it follows that is closed in and are continuous projectors such that

(2.11)

Hence, is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) is given by

(2.12)

Then

(2.13)

(2.14)

where

(2.15)

Obviously, and are continuous. It is not difficult to show that is compact for any open bounded set by using the Arzela-Ascoli theorem. Moreover, is clearly bounded. Thus, is -compact on with any open bounded set

In order to use Lemma 2.1, we have to find at least appropriate open bounded subsets of Corresponding to the operator equation we have

(2.16)

where Assume that is an -periodic solution of system (2.16) for some . Then there exist such that It is clear that From this and (2.16), we have

(2.17)

(2.18)

where

On one hand, according to (2.17), we have

(2.19)

namely,

(2.20)

which implies that

(2.21)

(2.22)

that is,

(2.23)

which implies that

(2.24)

By deducing for we obtain

(2.25)

namely,

(2.26)

which implies that

(2.27)

(2.28)

namely,

(2.29)

which implies that

(2.30)

In view of (2.21), (2.24), (2.27), and (2.30), we have

(2.31)

From (2.18), one can analogously obtain

(2.32)

By (2.31) and (2.32), we get

(2.33)

On the other hand, in view of (2.17), we have

(2.34)

namely,

(2.35)

which implies that

(2.36)

(2.37)

that is,

(2.38)

which implies that

(2.39)

By deducing for we obtain

(2.40)

that is,

(2.41)

which implies that

(2.42)

(2.43)

namely,

(2.44)

which implies that

(2.45)

It follows from (2.36), (2.39), (2.42), and (2.45) that

(2.46)

From (2.18), one can analogously obtain

(2.47)

By (2.33), (2.46), (2.47), and Lemma 2.3, we get

(2.48)

which implies that, for all

(2.49)

For convenience, we denote

(2.50)

Clearly, and are independent of For each , we choose an interval between two intervals and , and denote it as , then define the set

(2.51)

Obviously, the number of the above sets is We denote these sets as are bounded open subsets of Thus satisfies the requirement in Lemma 2.1.

Now we show that of Lemma 2.1 holds; that is, we prove when If it is not true, then when constant vector with , satisfies

(2.52)

where In view of the mean value theorem of calculous, there exist points such that

(2.53)

where Following the argument of (2.21)–(2.48), from (2.53), we obtain

(2.54)

Then belongs to one of This contradicts the fact that This proves that in Lemma 2.1 holds.

Finally, in order to show that in Lemma 2.1 holds, we only prove that for then it holds that To this end, we define the mapping by

(2.55)

here is a parameter and is defined by

(2.56)

where We show that for then it holds that Otherwise, parameter and constant vector satisfy that is,

(2.57)

where In view of the mean value theorem of calculous, there exist points such that

(2.58)

where Following the argument of (2.21)–(2.48), from (2.58), we obtain

(2.59)

given that belongs to one of This contradicts the fact that This proves holds. Note that the system of the following algebraic equations:

(2.60)

has distinct solutions since and hold, where or Similar to the proof of Lemma 2.3, it is easy to verify that

(2.61)

Therefore, uniquely belongs to the corresponding Since we can take A direct computation gives, for ,

(2.62)

Since then

(2.63)

So far, we have proved that satisfies all the assumptions in Lemma 2.1. Hence, system (2.7) has at least different -periodic solutions. Thus, by (2.6) system (1.1) has at least different positive -periodic solutions. This completes the proof of Theorem 2.4.

In system (1.1), if and are continuous periodic functions, then similar to the proof of Theorem 2.4, one can prove the following

Theorem 2.5.

Assume that and hold. Then system (1.1) has at least positive -periodic solutions.

Remark 2.6.

In Theorem 2.5, means that the th species does not prey the th species, thus That is to say, there is no relationship between the th species and the th species.

3. Illustrative Examples

Example 3.1.

Consider the following three-species food chain with harvesting terms:

(3.1)

In this case, and Since

(3.2)

taking then we have

(3.3)

Take then

(3.4)

Take then

(3.5)

Therefore, all conditions of Theorem 2.4 are satisfied. By Theorem 2.4, system (3.1) has at least eight positive -periodic solutions.