Proof. To establish the local stability of the periodic solution
for Equation (
1), we introduce new variables and define
,
,
. This leads to a linearly similar system for Equation (
1) with a single periodic solution
:
Acquiring the fundamental matrix is a simple endeavor:
The linearization of Equation (
1) for the fifth, sixth, seventh, and eighth terms results in the following:
The linearization of Equation (
1) for the ninth, tenth, eleventh, and twelfth terms yields the following:
The linearization of Equation (
1) for equations involving the thirteenth, fourteenth, fifteenth, and sixteenth terms is as follows:
The stability of the periodic solution
is determined by the eigenvalues of the system, i.e.,
which are
where
,
, and condition (
15) holds. In accordance with conditions (
15), (
16), and the Floquet theory [
22], if
then
and
Consequently, the local stability of the pest eradication boundary periodic solution
of (
1) is ensured.
In the subsequent analysis, we will demonstrate the global attraction property. By utilizing condition (
16), we are able to select
such that
Upon examining the second and fourth equations of (
1), it becomes evident that
Based on this observation, we consider the subsequent impulsive comparative differential equations:
Based on Lemma 3 and the comparison theorem of impulsive equations (refer to Theorem 3.1.1 in [
23]), the following inequalities hold:
,
, and as
t approaches infinity,
converges to
and
converges to
. Consequently, we can conclude that
for sufficiently large values of
t. To simplify the analysis, we can consider that Equation (
18) holds for all
t greater than or equal to zero. By combining Equations (
1) and (
18), we obtain
Therefore, for each , we have . Consequently, it follows that , and as n approaches infinity, tends to 0 . As a result, approaches 0 for each as t tends to infinity.
Next, we aim to demonstrate the convergence of
to
as
t approaches infinity, where
i takes the values 1 and 2. Let
be a positive value. It follows that there exists a
such that
holds for all
. Without loss of generality, we can assume that
for all
. Considering system (
1), we obtain the following expression:
Consequently, it follows that
,
. Furthermore, as
t approaches infinity, we have the convergence of
to
,
to
,
to
, and
to
. Here,
and
represent the solutions of Equation (
17) and
respectively.
where
and
are determined as
The definitions of
and
are as follows:
and
are defined as
where
Given any
, there exists a value
such that for all
, we have
and
By letting
approach 0, we obtain the following result
and
for sufficiently large values of
t, it can be deduced that
approaches
and
approaches
as
t approaches infinity. This conclusion signifies the completion of the proof. □
Proof. Firstly, according to Lemma 1, uniform boundedness is ensured. By utilizing Equation (
1) and invoking Theorem 1, we can deduce that
, for a
that is small enough. Therefore, it suffices to find
and
such that
for sufficiently large
t.
Suppose the opposite is true, and let us assume
for all
, where
is selected to be sufficiently small, satisfying
holds. By utilizing condition (
26) and selecting
to be small enough, we can establish that this assumption is invalid.
Taking
where
,
, and
are defined in accordance with Equations (
30)–(
32) below.
According to Lemma 3, it follows that
, with
converging to
and
converging to
as
t approaches infinity. Here,
represents the solution of
with
where
and
are determined as
and
,
are defined as
and
,
are defined as
where
Therefore, there exist
and
meeting
and
Then,
for
, let
and
. By taking the integral of Equation (
33) over the interval
for
, we have
Consequently, we have goes to infinity, which contradicts the boundedness of and . Thus, there exists a positive constant satisfying . This concludes the proof. □