The application of pest management involves two thresholds when the chemical control and biological control are adopted, respectively. Our purpose is to provide an appropriate balance between the chemical control and biological control. Therefore, a Smith predator-prey system for integrated pest management is established in this paper. In this model, the intensity of implementation of biological control and chemical control depends linearly on the selected control level (threshold). Firstly, the existence and uniqueness of the order-one periodic solution (i.e., OOPS) are proved by means of the subsequent function method to confirm the feasibility of the biological and chemical control strategy of pest management. Secondly, the stability of system is proved by the limit method of the successor points’ sequences and the analogue of the Poincaré criterion. Moreover, an optimization strategy is formulated to reduce the total cost and obtain the best level of pest control. Finally, the numerical simulation of a specific model is performed.
In the practical production, effective control of pests is a very important issue of the world, which catches attention of scholars for pest management method [
In mathematics, impulsive differential equations (IDES) is such a powerful tool to describe these phenomena that rapid changes in biological populations are caused by the variety of the pests control by artificial intervention [
It is of great practical significance to adopt biological and chemical control strategies based on the different pest thresholds. But an important issue in this process should be pointed out, in which the biological control is carried out when the density of pest denoted by
An outline of this paper is as follows. In next section, a pest management Smith model is formulated. Then the existence, uniqueness, and the asymptotically orbit stability of order-one periodic solution (OOPS) of system (
In biological mathematics, Logistic model [
By the control strategy, the following predator-prey Smith system is investigated in this paper:
In our paper,
In this section, we dynamically analyze system (
We first study the following continuous system of system (
The positive equilibrium point
At the point
When (I) holds, then
If
Let
By the method in [
The phase diagram of system (
For convenience, let
If
Assuming the intersection point of phase set
(i)
If
If
According to the continuity of subsequent function, there must be a point
(ii)
If
Now, the uniqueness of OOPS of system (
Assuming that
Assume
So
Thus
When
According to the proof above, we have
If
The existence of the OOPS of system (
If
(i) If
(ii) If
The existence of the OOPS of system (
The existence of the OOPS of system (
According to the discussion above, a unique OOPS exists in system (
If
We choose arbitrary point
The sequence
Similarly, we can use the above method to get an increasing point sequences
Thus the OOPS of system (
The orbitally asymptotically stability of the OOPS of system (
If
Let
Thus, when
In this section, the feasibility of our conclusions is verified by an example. Let
Let
Numerical simulations in case
Let
Numerical simulations in case
For the case of
Numerical simulations in case
The goal to investigate the existence of OOPS of system (
Assuming that unit cost of releasing predator is denoted by
The optimization problem is solved to yield the optimal pest level
The variety in the period
The change in the cost per unit time
A Smith prey-predator system with linear feedback control for integrated pest management is investigated in this paper. Integrated control strategy is more practical which can maximize the protection of the ecological environment and reduce the cost of pest management. First, the method of subsequent function and differential equation geometry theory are used to prove the existence, uniqueness, and stability of the OOPS of system (
We agree to share the data underlying the findings of the manuscript. Data sharing allows researchers to verify the results of an article, replicate the analysis, and conduct secondary analyses.
The authors declare that they have no conflicts of interest.
This work was supported by the National Natural Science Foundation of China (11371230 and 11501331), the SDUST Research Fund (2014TDJH102), Shandong Provincial Natural Science Foundation, China (ZR2015AQ001 and BS2015SF002), Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, the Open Foundation of the Key Laboratory of Jiangxi Province for Numerical Simulation and Emulation Techniques, Gannan Normal University, China, and SDUST Innovation Fund for Graduate Students (no. SDKDYC170225).