^{1}

^{1}

^{2}

^{1}

^{1}

^{2}

The main purpose of this paper is to study the periodicity and global asymptotic stability of a generalized Lotka-Volterra’s competition system with delays. Some sufficient conditions are established for the existence and stability of periodic solution of such nonlinear differential equations. The approaches are based on Mawhin’s coincidence degree theory, matrix spectral theory, and Lyapunov functional.

In the past few decades, differential equations have been used in the study of population dynamics, ecology and epidemiology, malaria transmission, and so forth (see, e.g., [

The structure of this paper is as follows. In Section

In this section, we will obtain some sufficient conditions for the existence of periodic solution of system (

For convenience, we introduce some notations, definitions, and lemmas. If

If

Let

Let

Let

for each

for each

Then

A real

Let

In what follows, we will introduce some function spaces and their norms, which are valid throughout this paper. Denote

Assume that the following conditions hold:

the system of algebraic equations

has finite solutions

Then system (

Note that every solution

On the other hand, we prove that

Therefore, by the generalized Arzela-Ascoli theorem, we have that

Take

So far, we have shown that the open subset

Under the assumption of Theorem

Let

Let

We need a lemma which can follow immediately from Theorem 2.1 in Xia et al. [

Suppose that

then system (

If

Assume that _{1})–(H_{4}) hold, then system (

By Lemma

To illustrate the generality of our results, we will give a corollary in this section. Now recall that, for a given matrix

In addition to (H_{1}), (H_{2}), and (H_{4}), if one further supposes that there exist positive constants

then system (

For any matrix norm

The corollary implies that the conditions given in terms of the spectral radius are much better than the classic norms.

Now we consider a special case of system (

In this case, Theorem

As we know, dynamic systems are often classified into two categories of either continuous-time or discrete-time systems. However, many real-world phenomena are neither purely continuous-time nor purely discrete-time. This leads to the development of dynamic systems with impulses, which display a combination of characteristics of both the continuous-time and discrete-time systems and hence provide a more natural framework for mathematical modeling of many real-world phenomena. Whether the new method proposed in this paper can be applied to study the existence and global asymptotic stability of the LV systems with impulses remains open.

In this section, some examples and their simulations are presented to illustrate the feasibility and effectiveness of our results.

Consider the two-species competitive system

In this example, one can observe that, though the spectral

Consider the three-species competitive system

In this example, one can observe that though the spectral

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China under Grant no. 11271333 and ZJNSFC.