New conditions on the existence and stability of positive periodic solutions for n-species Lotka-Volterra systems with deviating arguments
© Xu and He; licensee Springer. 2014
Received: 17 September 2013
Accepted: 17 February 2014
Published: 20 March 2014
In this paper, we study the existence and stability of positive periodic solutions for an n-species Lotka-Volterra system with deviating arguments, , , referred to as (E). By using Mawhin’s coincidence degree, matrix spectral theory, and some new estimation techniques for the prior bounds of unknown solutions to the equation , some new and interesting sufficient conditions are obtained guaranteeing the existence and global stability of positive periodic solutions of the above system. The model studied in this paper is more general, and it includes some known Lotka-Volterra type systems, such as competitive systems, predator-prey systems, and competitor-mutualist systems. Our new results are different from the known results in the previous literature.
where are w-periodic functions () with . They obtain one results as follows.
Theorem 1.1 Assume that the following conditions hold:
(A1) , ;
Then system (1.1) has at least one positive ω-periodic solution.
In the proof of Theorem 1.1, the author did not consider the deviating arguments in every terms , . Thus, Theorem 1.1 cannot be applied to system (E) when .
Compared to system (1.2), the front sign of coefficients of system (1.3) could change.
For the biological point of view, it is always assumed that is strictly positive.
It is not difficult to see that all the above mentioned models are special cases of this model. Thus, it is worth investigating the existence and stability of positive periodic solutions of system (1.4). To the best of our knowledge, very few authors have been concerned with employing matrix spectral theory to obtain the prior bounds for biological systems so far. In this paper, by combing matrix spectral theory with Mawhin’s coincidence degree theory, we manage to obtain a set of new and interesting conditions, which are very different from the known results in the literature.
The structure of this paper is as follows. In Section 2, some new and interesting sufficient conditions for the existence of positive periodic solutions of system (1.4) are obtained. In Section 3, we will explore the stability of positive periodic solution of system (1.4). Finally, an example is given to show that the results of this paper are easily applicable.
2 Existence of positive periodic solutions
In this section, we shall obtain some new sufficient conditions for the existence of a positive periodic solution of system (1.4).
For the matrix , denotes the transpose of G, and denotes the identity matrix of size n. represents a diagonal matrix with specified diagonal entries. A matrix or vector means that all entries of A are greater than or equal to zero. can be defined similarly. For matrices or vectors A and B, (resp., ) means that (resp., ). We denote the spectral radius of the matrix A by .
Lemma 2.1 ()
If satisfies and , , then the function has a unique inverse function satisfying and , .
Remark 2.1 If , and , , from Lemma 2.1, we have , , where is the inverse function of , thus .
Definition 2.1 ()
A real matrix is said to be an M-matrix if , , , and .
Lemma 2.2 ()
Let be an matrix and let . Then , where denotes the identity matrix of size n.
In order to use Mawhin’s continuation theorem, we recall this theorem first.
Let X, Y be real Banach spaces, let be a Fredholm operator with index zero. Here, denotes the domain of L. This means that ImL is closed in Y and . Consider the supplementary subspaces and such that , and let , be the natural projections. Clearly, , thus the restriction is invertible. Denote the inverse of L by .
Now, let Ω be an open bounded subset of X with , a map is said to be L-compact on , if is bounded and the operator is compact.
Lemma 2.3 (Mawhin )
, , ;
. Here is an isomorphism. Then the equation has at least one solution on .
(H1) The algebraic equation system
has a unique solution ;
(H2) , where and
where , , is defined by (2.1);
(H3) , , where , .
Then system (1.4) has at least one positive ω-periodic solution.
Obviously, system (1.4) has a positive ω-periodic solution if and only if system (2.3) has a ω-periodic solution.
Obviously, , . So ImL is closed in and , then the operator L is a Fredholm operator with index zero.
Clearly, is a constant independent of λ.
It follows from (H1) that the algebraic equation has a unique solution . Let . Then is a constant.
J is an identity mapping. Therefore, by using Lemma 2.3, we find that system (2.3) has at least one ω-periodic solution. By (2.2), system (1.4) has at least one positive ω-periodic solution. This completes the proof of Theorem 2.1. □
3 Global asymptotic stability of positive periodic solutions
Under the assumption of Theorem 2.1, we know that system (1.4) has at least one positive ω-periodic solution, denoted by . In this section, we always assume the existence of positive periodic solutions and we study the global stability of positive periodic solutions of (1.4).
We recall some facts which will be used in the proof.
is Lyapunov stable;
is globally attractive in the sense that for all .
Lemma 3.1 ()
Let f be a nonnegative function defined on such that f is integrable on and is uniformly continuous on . Then .
where is the inverse function of , . Then system (1.4) has a unique positive ω-periodic solution which is globally asymptotically stable.
From Definition 3.1, Theorem 3.1 follows. □
As an application, we consider the following example.
we know that , , , , , . So the result of the above example cannot be obtained by , which implies that the results of this paper are essentially new.
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