Existence of Positive Periodic Solutions for a Time-Delay Biological Model ()


1. Introduction
The plants can survive independently and insect pollination can improve the growth rate of plants in [1] and [2]. According to this phenomenon, based on the classical Lotka-Volterra model, we establish a model of two populations of Lotka-Volterra which cannot survive independently, finally he analyzes the stability of the model.
There is still less research work of the model which cannot exist independently. The existing researches basically are the autonomous models (see [3] and [4]). In this paper, we establish a Lotka-Volterra model with time delay which a species cannot survive independently. The main aim is to discuss existence of periodic positive solution for the model.
Suppose that there are two plant populations (A and B) living in their natural environment, which are free from other interference factor. Let
and
are the population density of plant A and plant B,
are continuous functions with periodic
, and
. The constants
are stimulations of living environment. By the thought of [1]-[4], we could have got the following Lotka-Volterra model with time delay which a species cannot survive independently.
(1)
The main aim of the paper is to discuss existence of periodic positive solution for the model.
2. Lemma 1 and Lemma 2
Assume X and Z are normed vector space,
and
are linear mappings. If L is Fredholm mapping which Zero is index, and there are continuous projection
and
, such that
and
, we can get that
is reversible. If Inverse mapping
is tight, we call N is tight on
.
Lemma 1 (Continuation theorem) [5] Let L be the mapping of Fredholm with zero index, collection N is tight on collection
. Suppose the following: for any
, the solution
of equation
; for any
and
. Then, there is at least a solution for
on
.
Lemma 2
is positive invariant set of model (1).
Proof: By formula (2), we have
(2)
Since formula (2) is always true for
, lemma 2 is proved.
For the convenience of discuss, we give following notations.
,
, ![]()
,
,
,
.
3. Existence of Periodic Solutions
In order to apply Continuation theorem to system (3), we define
![]()
and
![]()
,
then X, Z is Banach space under the norm
(see [6] and [7]).
Let
,
, the Equation (1) can be turns into
(3)
Since
is periodic, we know that
![]()
and
![]()
are continuous function with the periodicity
.
Let
,
,
, for
,
, for![]()
, then
,
is closed set in set Z,
, and P, QP and Q is the continuous projection, such that
,
. Thus there is the inverse mapping of L
and
.
So that we get
,
![]()
It is obvious that
and
is continuous.
We assume that
is bounded open set. It is obvious that
is bounded. We have that
is compact set by Arzela-Ascoli theorem, so we get N is L-tight on
.
The corresponding operator equation is
with
,we have the following formula
(4)
We assume that
is the solution for system (4) with
,by integral we get the following formula (5)
(5)
To move term from one side of an algebraic equation to the other side, reversing its sign to maintain equality, we get the following
(6)
(7)
From formula (5), formula (6) and formula (7), we have
(8)
and
(9)
From formula (8) and formula (9) we can get
(10)
(11)
From formula (10) and formula (11) we get
![]()
So that
![]()
Similarly, we have
![]()
Using formula (7) we get
![]()
Thus
(12)
Similarly, we have
(13)
From formula (8) and formula (13) we get
![]()
From formula (9) and formula (12) we get
![]()
Since
, it exists
, such that
(14)
By formula (12), formula (13) and formula (14), we get
, ![]()
,
.
So that
,
.
It is obvious that
has nothing with choose of
. Thus the following formula (15) has a unique positive solution
.
(15)
Let
, where
is sufficiently large and
. Then,
, then
satisfying the first
condition for Lemma 1. When
,
![]()
So that
,
satisfying all the conditions for Lemma 1. We have that there is at least a solution with
on
from Lemma 1.
Let
,
, when
,
is positive periodic solution for system (1) which the length of
. Hence, there is at least a positive periodic solution for system (1).
Theorem If
, then
is a positive periodic solution for system (1). In other word, there is at least one positive periodic solution for system (1).
Acknowledgements
This research was financially supported by the National Science Foundation of Zhejiang Province (LY12A01010) and by the College Students’ Scientific and Technological Innovation of Zhejiang Province (2015R411035).
NOTES
![]()
*Corresponding author.