Optimal Application Timing of Pest Control Tactics in Nonautonomous Pest Growth Model

and Applied Analysis 3 In order to analyze the dynamics of the pest population in system (4), the following subsystem is useful: dy (t) dt = −δ (t) y (t) , t ̸ = τk, y (τ + k ) = y (τk) + σk, t = τk (5) and we have the following main results for subsystem (5). Lemma 1. The subsystem (5) has a positive periodic solution y T (t) and for every solutiony(t) of (5) one has |y(t)−y(t)| → 0 as n → ∞, where y T (t) = {{{{{{{{{{{{{{{{{{{{{{{{{{ {{{{{{{{{{{{{{{{{{{{{{{{{{ { Y ∗ n exp [−∫ t nT δ (s) ds] , t ∈ (nT, nT + τ1] , .. Y ∗ n exp [−∫ t


Introduction
Recently, many ecologists are becoming increasingly concerned with the questions of pest control and designing the optimal control strategies.It is well known that the pesticides are still the main tactics for controlling pests, because the pesticides are relatively cheap and can be easily applied.But spraying insecticides for a long time may trigger "3R" questions (resistance, resurgence, and residue); then, the Department of Agricultural Ecology proposed the integrated pest management (IPM) strategies [1,2].
IPM is a long term management strategy, which includes chemical (insecticides), biological (releasing the natural enemy), agricultural control (crop rotation), and physics methods (utilizing light to trap and kill pest) [3,4].IPM which has been proved by experiment is more effective than any single control strategy.But when should we release the natural enemy and what proportion do we need to kill the pest by spraying pesticide?Undoubtedly, the mathematical models can help us to design the optimal control tactics and, in particular, help us to predict the density of pest population and to determine optimal application timing of IPM strategies (see [5][6][7][8]).
Volterra first proposed a simple predator-prey system, which has been extended and modified in many ways [9,10].
In recent years, continuous or discrete predator-prey systems concerning IPM strategies have been developed and investigated intensively in [11][12][13].Considering the interventions by human such as a periodic spraying pesticide and a constant periodic releasing for the predator, the impulsive differential equations with fixed moments were employed to model the interventions, and consequently the Lotka-Volterra system has been extended [14].However, one of the major assumptions in those publications was that all the growth rates of predator and prey are constants.However, many ecologists have shown that the growth of populations of various species is affected by the special living environment including the seasons, weather conditions, and food supply [15].Therefore, it is more realistic to consider the effects of periodic parameters on the dynamics of predator-prey models.Therefore, nonautonomous predator-prey systems with impulsive effect have been developed and investigated in [10,16,17].
However, those works mainly focused on the effects of periodic perturbations on the dynamics; the interesting questions concerning the effects of successful pest control, in particular how to apply the IPM tactics in periodic environment and to determine the optimal application timing, have not been addressed in more detail.

Autonomous ODE Model with Multi-Impulsive Effects.
Assume that the pest population follows the classical logistic growth system and that pest control by spraying pesticides and releasing natural enemies is implemented at some fixed times for each crop season.Denote the size of the pest and the natural enemy populations at time  by () and (), respectively.Assume that pests are killed by pesticides at a proportional rate   (0 ≤   < 1) and the natural enemy is released by a constant  at time   .Therefore, we have the following system with impulsive effects at fixed moments: where  is the intrinsic growth rate of pest population,  denotes the carrying capacity parameter,  is the attack rate of predator,  represents conversion efficiency, and  is the death rate of predator.System (1) is said to be a  periodic system if there exists a positive integer  such that  + =   ,  + =   + .This implies that, in each period ,  times of the pesticide applications are used and  times of the natural enemy releases are applied.The dynamical behavior and biological implications of system (1) have been extensively studied in [18].It follows from the literature [18] that if exp(−) < 1, then system (1) has a "pest-eradication" periodic solution (0,   ()) over the th time interval  <  ≤ ( + 1) with where The analytical formula defined above clearly shows how the key parameters affect the threshold value  0 , which can be used to design the optimal control strategies such that the threshold value  0 is the smallest.We will address those points in the following for more generalized model.

Nonautonomous ODE Model with
Multi-Impulsive Effects.However, it is well known that the growth of populations of various species is affected by several factors such as the seasons, weather conditions, and food supply, which can be described by using the periodic coefficients in model (1); that is, we have the following model: where (), (), (), (), and () are continuous  periodic functions.System (4) is said to be a  periodic system if there exists a positive integer  such that  + =   ,  + =   , and  + =   + .
In order to analyze the dynamics of the pest population in system (4), the following subsystem is useful: and we have the following main results for subsystem (5).
Proof.In any given time interval (, ( + 1)] (where  is a natural number), we investigate the dynamical behavior of system (5).In fact, integrating the first equation of system (5) from  to  +  1 yields At time  +  1 , the  1 natural enemy is released; thus, we have Again, integrating the first equation of system (5) from + 1 to  +  2 , one yields At time  +  2 , the  2 natural enemy is released, and By induction, we get for all  ∈ ( +   , ( + 1)].Therefore, we have Denote  +1 = (( + 1)); then, we have the following difference equation: which has a unique steady state Thus, there is a periodic solution of system (5), denoted by   (), which is given in (6).For the stability of   (), it follows from (12) and the formula of   () that and then the results of Lemma 1 follow.This completes the proof of Lemma 1.
Based on the conclusion of Lemma 1, there exists a "pestfree" periodic solution of system (4) over the th time interval  <  ≤ ( + 1), and we have the following threshold conditions.
Proof.Firstly, we prove the local stability.Define () = (), () =   () + V(); there may be written where Φ() satisfies and Φ(0) =  denoted the identity matrix.The resetting of the third and fourth equations of (4) becomes Hence, if both eigenvalues of have absolute value less than one, then the periodic solution (0,   ()) is locally stable.In fact, the two Floquet multiplies are thus according to Floquet theory (see [19][20][21] and the references therein), the solution that is, if (17) holds true, the solution of system ( 4) is locally stable.
Next, we can prove that () →   () as  → ∞.For any  > 0, there must exist a  1 > 0 such that 0 < () <  for  >  1 .Without loss of generality, we assume that 0 < () <  holds true for all  > 0; then, we have and consider the following impulsive differential equation: By using the same methods as those for system (23), it is easy to prove that system (29) has a globally stable periodic solution, denoted by V  () and ∈ (,  +  1 ] , . . .
with .
This completes the proof of global attractivity of (0,   ()).Then it is globally asymptotically stable.The proof of Theorem 2 is complete.

The Optimal Control Time with Different 𝑞
Assume that pesticide is sprayed and the natural enemy is released only at the time points  +   ( = 1, 2, . . .,  and 0 <  1 <  2 < ⋅ ⋅ ⋅ <   < ) during each period .It is well known that the size of the pest population at the end will be different if impulsive control occurs at different time.So, it is necessary to determine the optimal time to make sure that the pest can be eliminated quickly.

The Optimal Control Time with 𝑞 = 1.
In this subsection, we consider one-pulse controlling at time  +  1 in each period  (where  1 ∈ [0, ]) with aims to find the optimal time  +  1 such that the threshold value is the smallest.Therefore, if  = 1, then the threshold value  0 can be reduced as where ∈ ( +  1 , ( + 1) ] , (34) Thus, we have Taking the derivative of function  0 with respect to  1 , one obtains Letting  0 / 1 = 0, we can see that  1 min satisfies equation ( 1 )/ 1 +  1 ( 1 )/ 1 = 0; that is The second derivative of  0 with respect to  1 at  1 min can be calculated as follows: then  1 min is the minimal value point.According to the above discussion, we have the following theorem.
For example, if we let () =  0 +  cos(), () =  0 +  sin(), and () =  0 +  1 cos(), then by simple calculations we have and if  1 min satisfies then  1 min is the minimum value point.That is,  0 reaches its minimum value when  1 =  1 min .To confirm our main results obtained in this subsection, we fixed all parameters including (), (), (), (), (),  1 ,  1 , and initial values  0 ,  0 and carry out the numerical investigations.To find the optimal timing of applying IPM strategy, we consider  0 / 1 as a function with respect to  1 aiming to find the time point such that  0 / 1 = 0. Since  0 / 1 = − 0  1 = 0, we only need to plot the − 1 with respect to time  1 , as shown in Figure 1.By calculation, we have  1 min = 0.95, which satisfies inequality (41), and consequently  1 min is a minimum value point.Now, we can consider the effects of different timing of applying IPM strategies on the pest population, in particular the amplitudes of the pest population.To do this, we choose three different time points, denoted by  1 min ,  1 , and    1 , at which the one-time control action has been implemented.It follows from Figure 2 that the maximal value of the pest population is the smallest when we implement the one-time control action at time  1 min , which confirms that the results obtained here can help us to design the optimal control strategies.
In the following, we would like to address how the impulsive period , release quantity  1 , pest killing rate  1 , and death rate of the pest population () affect the threshold value  0 .To address this question, we fix the parameters concerning periodic functions (), (), (), and () and vary the impulsive period , the release quantity  1 , the killing rate  1 , and the death rate (), respectively.
In Figure 3, we see that the threshold value  0 is not monotonic with respect to time  and the effects of all four parameters (,  1 ,  1 , and  0 ) on threshold condition  0 are complex.Figure 3(a) shows the effect of impulsive period  on the  0 , and the results indicate that the larger the impulsive period  is, the larger the threshold value is, which will result in a more sever pest outbreak.Oppositely, in Figure 3(b), the results indicate that the smaller the release quantity  1 is, the larger the  0 is.Figures 3(c) and 3(d) clarify that slightly increasing the pest killing rate  1 and the death rate of pest () can reduce the quantity of threshold value  0 dramatically, and the results can be used to help the farmer to select appropriate pesticides.At the same time, we can see clearly how the periodic perturbations affect the threshold value  0 , as indicated in the bold curves in Figure 3.

The Optimal Control Time with 𝑞 = 2.
In this subsection, we consider two-pulse controlling at times + 1 and + 2 in each period , where 0 <  1 <  2 < .In the following, we focus on finding the optimal time points  +  1 and  +  2 such that the threshold value  0 is the smallest.
Therefore, if  = 2, then the threshold value  0 becomes as where
From the above argument, we have the following theorem.
In the following, we will discuss the effects of the pest killing rates ( 1 ,  2 ) and releasing quantities ( 1 ,  2 ) on the threshold value  0 .As we can see from Figure 5,  0 is quite sensitive to all four parameters ( 1 ,  2 ,  1 ,  2 ).According to these numerical simulations, the farmer can take appropriate measures to achieve successful pest control.

Discussion and Biological Conclusions
It is well known that the growth rate of the species is affected by the living environments, so it is more practical to consider the growth rates of predator and prey as the functions with respect to time  in the models with IPM strategies.Therefore, nonautonomous predator-prey systems with impulsive effects have been developed and investigated in the literatures [11][12][13].However, those works mainly focused on the dynamical behavior including the existence and stability of pest-free periodic solutions.From pest control point of view, one of interesting questions is to determine the optimal application timing of pest control tactics in such models and fall within the scope of the study.
In order to address this question, the existence and global stability of pest-free periodic solution have been proved in theory firstly.Moreover, the optimal application timings which minimize the threshold value for one-time pulse control, two-time pulse controls, and multipulse controls within a given period have been obtained, and most importantly the analytical formula of the optimal timings of IPM applications has been provided for each case.For examples, Figure 1 illustrates the existence of the minimum value point of the threshold value  0 , under which the maximum amplitude of pest population reaches its minimum value, and this is validated by comparison of different sizes of pest population at three different impulsive timings in Figure 2; Figure 3 clarifies how the four parameters (the impulsive period, the release quantity, the killing rate of pest, and the death rate of pest) affect the successful pest control.All those results obtained here are useful for the farmer to select appropriate timings at which the IPM strategies are applied.
Note that the complex dynamical behavior of model ( 4) can be seen in Figure 6, where the period  is chosen as a bifurcation parameter.As the parameter  increases, model (4) may give different solutions with period , 2, and eventually the model (4) undergoes a period-double bifurcation and leads to chaos.Moreover, the multiple attractors can coexist for a wide range of parameter, which indicates that the final stable states of pest and natural enemy populations depend on their initial densities.
In this paper, we mainly focus on the simplest preypredator model with impulsive effects.It is interesting to consider the evolution of pesticide resistance, which can be involved into the killing rate related to pesticide applications.Moreover, according to the definition of IPM, the control actions can only be applied once the density of pest populations reaches the economic threshold.Therefore, based on above facts, we would like to develop more realistic models in our future works.