1 Introduction

In recent years, the outbreak of crop diseases and pests has become more and more frequent, resulting in serious crop losses and even production failures, which have seriously hindered the development of China’s agriculture. A pest is a species that damages other valuable populations or interferes with human activities, so it is necessary to take measures to reduce pest damage to crops [1–5]. In the pest-natural enemy ecosystem, pests and natural enemies are interdependent and mutually restricted, and an appropriate amount of pests can maintain the ecological balance. If the pests are completely killed, the ecological imbalance will be caused, and the natural enemy population will be extinct due to lack of food. As a result, Integrated pest Management (IPM), a threshold control strategy that combines chemistry, economics, and biology, was developed. Integrated pest management (IPM) is implemented to keep pest populations below the economic harm level (EIL) rather than eliminate them completely, which benefits individual cropping systems and local ecosystems [3–5].

Discontinuity in the dynamics of animals’ population in the natural environment seems to be an universal property of the ecosystem, as the animal’s survivals and growth rate are subject to the impact of food resources, climatic conditions, seasonal change and human factors. To understand and to predict when these discontinuities may occur with high degree of accuracy, the systems are needed to be modelled by using non-smooth functional responses. Systems which exhibit non-smooth behavior can be broadly divided into three different types dependent on the degree of smoothness : i) non-smooth continuous system, ii) impulsive system and iii) Filippov system. Tang Sanyi et al. and other experts [1–5] studied the non-smooth continuous pest control under the integrated pest management system, by using the Impulsive Differential Equation to model the spraying of pesticide at fixed time intervals, and to release natural enemies intermittently dependent on the environmental conditions. Similar studies but using a general functional response and impulsive control, had been reported by the authors [3] in 2020 for the study of the extinction and permanence of the predator-prey system. Recent work that employed generalized functional response for modelling the population dynamics in the ecosystem has been further extended by the same authors for three species impulsive system [4], as well as a m-prey and n-predator impulsive system [5] under seasonal disturbance factors, had also been reported. One of the drawbacks for all the previous work has been the modelling of the ecosystem when the control strategy is instantaneously applied, and at the same time to deduce the effects of the control immediately after the control strategy is implemented. This methodology has modelled the reduction of pest population over a short period of time after the control strategy is applied, which may over estimate the number of pest deaths than it would actually happen in practice. To model closer in line with the real environment, it is necessary to introduce a continuity of control strategy, like the Filippov system [6–20], which allows the monitoring of the increase or the reduction of pest population before and after the control measures have been applied. In recent years, there are many encouraging reports that use Filippov system for pest control such as the work by Tang Sanyi et al. In 2019 Qin et al. [10] investigated the threshold control strategy for a non-smooth Filippov ecosystem which featured a group defense from the pest. In 2021 Arafa et al. [14] studied the effectiveness of population dynamics by using Filippov pest control model which incorporates with a time delay. The global dynamics of the Filippov predator-prey model which featured two independent thresholds for the integrated pest management (IPM) was discussed by Li et al. [8] in 2022. At the same time, Jiao et al. [8] probed the dynamics and bifurcations of the predator-prey system using Filippov Leslie-Gower response function to model the group defense of the pest with time delays.

As far as the authors aware, most if not all of the existing literatures including those mentioned above in [6–21], have assumed a single rate of feeding by the prey, i.e. the predation process is described by a single functional response for the entire period within the predator-prey system. This paper attempts to fill the gap by modelling a non-smooth Filippov predator-prey system with threshold strategies is investigated incorporating several different functional responses for the first time. Different from previous research, the present work develops the modelling for an integrated pest management (IPM) especially with several different functional responses such as Holling II functional response and ratio functional response etc, which should be selectively applied dependent on the population of the prey. In practice, the mutual competition amongst the predator is dependent on its population for a given number of prey in the environment. For example, in the predator-prey system, the thresholds about the preys are related to the population of predators. When the number of preys x(t) are in abundance, namely x(t) > ET > 0, where ET is the economic threshold, then the predator which has population of y(t) should not mutually compete for food as the result of sufficient of food for the predators. Thus the interactions of the prey-dependent can be described by functional responses such as Holling-I functional response , Holling-II functional response , Holling-III functional response and Ivlev functional response and so on. As an example we can adopt the Holling-II functional response to obtain the subsystem model of the predator-prey system

(1)

where denote the densities of the prey (pest) and predator (natural enemy) at time t₁, respectively. r > 0 is the intrinsic growth rate of the prey, K > 0 is the carrying capacity of the prey, D > 0 is the death rate of the predator. The e > 0 denotes the rate of converting the consumed preys into the growth of predators, and the is the Holling-II functional response which represents the rate of predation by the predator per-capita.

While the number of preys is declining to less than a certain multiple of the number of predators x(t) < ET, the mutual interference between predators will be triggered to take effect and the predator-dependent type will dominate the interactions between the predator and prey. In this case functional responses such as ratio-dependent , and others like the Beddington-DeAngelis functional response, Watt-type functional response and so on are more suitable to model the predator-prey system. The ratio-dependent functional response is selected here in the predator-prey models and the following subsystem can be obtained:

(2)

In the paper, we develop the Filippov predator-prey model by combining the above two subsystems:

(3)

that is:

(4)

in which

and

Note that the ET is set by certain threshold strategy. In order to simplify the system in (4), the parameters and variables can be defined as follows:

Then we can obtain:

(5)

in which

and

The aim of this study is to investigate the complex dynamics including bistabilities of the ecosystem when the relative populations of the prey and predator is substantially different, by modelling the non-smooth Filippov system with multiple switchable functional responses for the very first time, which highlights the significant role of the threshold in the process of pest controls: it is seen from this work that different types of equilibrium points can occur dependent on the choice of the economic threshold (ET). Finally, the resulting conclusion is given the corresponding biological explanation. The organization of this paper is outlined as follows: Sect 2 gives a summary of how various regimes such as the sliding region, crossing region and escaping region are defined, subsequently the five different kinds of equilibria that will be discussed in the following sections of the paper will be briefly introduced. In Sect 3, the dynamical behaviors of two subsystems (i.e. the system (1) and (2) as set out in the above paragraphs) and their dynamic behaviors on the discontinuity boundary Σ (see text in Sect 2) are derived. Subsequently their equilibria together with the existence of three regimes such as the sliding, escaping and crossing domains, are derived. The dynamics of the sliding mode and various forms of equilibria within the Filippov system (i.e. the system (5) in the above paragraph), and their global asymptotic stability are discussed in Sect 4. In Sect 5, we probe the bifurcation set of equilibria and their global stability of equilibria through numerical simulations. Subsequently the paper is concluded in Sect 6 and the theoretical results are discussed in the context of biological factors and practical viewpoints.

2 Preliminaries

Denote

Therefore, the system (5) can be rewritten as the following generalized system with discontinuous in the right-hand side as shown below:

(6)

in which

And the discontinuous boundary between the region G₁ and G₂ is defined as , so we have . The discontinuous boundary Σ can be classified as the following three different regions:

(i) The Sliding region and , which implies that once the trajectories of the system touch the boundary , it will stay in the same region.

(ii) The Crossing region , which implies that once the trajectories of the system touch the boundary , it will propagate to another region.

(iii) The Escaping region and F₂H<0}, which implies that once the trajectories of the system touch the boundary , it will propagate towards either region G₁ or G₂.

Definition 2.1. [8] Provided and ( and ), then E_R is a real equilibrium of system (6).

Definition 2.2. [8] Provided and ( and ), then is a virtual equilibrium of system (6).

Definition 2.3. [8] Provided and , where ,and , then E_p is a pseudo-equilibrium of system (6).

Definition 2.4. [8] Provided and , then E_b is a boundary equilibrium of system (6).

Definition 2.5. [8] Provided and , then E_T is the tangency point of system (6).

Definition 2.6. [8] Provided , and (or ), then E_T is an invisible (or visible) tangency equilibrium of subsystem (1). Similarly, provided , and (or ), then E_T is an invisible (or visible) tangency equilibrium of subsystem (2).

3 Qualitative analysis of the subsystem (1) and (2)

Lemma 3.1. Suppose that is any solution of system (5) with the initial value , , then Z(t) > 0, namely .

Proof: As

Thus Z(t) > 0 as long as the initial value satisfies . □

Lemma 3.2. Suppose to be the solution of system (5), then the set is positively invariant and attracting for any given initial values in R^2 +.

Proof: It follows that

By solving the above equation, we can obtain that:

which results in

Then we can get

Define the function , then:

which is the upper right derivative of along a solution of the system in (5) with respected to time and for we have

Thus there exists a positive constant number , such that

By solving the above equation it produces:

Hence is ultimately bounded by a constant, namely , thus Ω is positively invariant and attracting for any given initial values in R^2 +. □

3.1 Dynamical behaviors of the subsystem (1)

When x₁(t)>ET the Filippov system (6) in section 2 is qualitatively dependent on the subsystem (1). There are three equilibrium points in the subsystem(1) namely: E₀₁ = (0,0), E₁₁ = (1,0), in which

As , namely , we can obtain that , we can obtain that so we can get that .

Theorem 3.3. (i) The equilibrium E₀₁ = (0,0) is in a saddle and it is unstable all the time. (ii) The equilibrium point E₁₁ = (1,0) is in a saddle when , and it is locally asymptotically stable when . (iii) The interior equilibrium point is locally asymptotically stable when .

Proof: Consider the Jacobian matrix about the equilibrium point of the subsystem (1):

(i) Let , we can get the Then we obtain the eigenvalues of J(E₀₁) will be , so the equilibrium E₀₁ = (0,0) is saddle and unstable all the time.

(ii) Let , we can get the then we obtain the eigenvalues of J(E₁₁) is . When , then , so E₁₁ = (1,0) is saddle. Similarly, when , then , so E₁₁ = (1,0) is locally asymptotically stable.

(iii) Let , we can get the

then we obtain the result that all eigenvalues of J(E₁) are negative when tr(J(E₁))<0 and , that is, , the interior point is locally asymptotic stable. □

Theorem 3.4. The interior point is globally asymptotic stable when .

Proof: Define

where D₁ and D₂ are arbitrary positive constants. Then we can obtain:

(7)

Substituting the subsystem (1) into the above equation (7) which generates

Put D₁ = 1 and , then

When , then it has that . By employing the value of , we can get , namely . □

3.2 Dynamical behaviors of the subsystem (2)

When x₁(t)<ET, the Filippov system (6) is qualitatively dependent on the subsystem (2). There are three equilibrium points in the subsystem (1): E₂₀ = (0,0), E₂₁ = (1,0), in which

As , namely , we can obtain that , we can obtain that , so we can get that .

Theorem 3.5. (i) The equilibrium E₂₀ = (0,0) is in a saddle and it is unstable all the time. (ii) The equilibrium point E₂₁ = (1,0) is also in a saddle when , and it is locally asymptotically stable when . (iii) The interior point is locally asymptotically stable when and .

Proof: Consider the Jacobian matrix about the equilibrium point of the subsystem (2):

(i) Let , we can get the Then we obtain the eigenvalues of J(E₂₀) will be , therefore the equilibrium E₂₀ = (0,0) is in a saddle and it is unstable all the time.

(ii) Let , we can get the then we obtain the eigenvalues of J(E₂₁) will be . When , then , so E₂₁ = (1,0) is in a saddle. Similarly, when , then , so E₂₁ = (1,0) is locally asymptotically stable.

(iii) Let , we can get the

then we obtain the result that all eigenvalues of J(E₂) are negative when tr(J(E₂))<0 and , that is, and , the interior point is locally asymptotically stable. □

Theorem 3.6. The interior point is globally asymptotically stable when .

Proof: We select the Dulac fuction , then the following can be obtained:

Thus based on the Bendixson-Dulac criterion, when there are no any closed orbits in the region G₂, therefore the interior point is globally asymptotic stable. □

4 The dynamic behaviors on ΣSigma and equilibria

This section will be divided into three parts, firstly the existence of the sliding domain, escaping domain and crossing domain will be studied. Then the sliding mode dynamics will be focused and finally different kinds of equilibria points in Filippov system (6) will be examined in more details.

4.1 The existence of the sliding domain, escaping domain and crossing domain

Let’s consider the discontinuous boundary Σ and through simple evaluation on the boundary gives:

where and x₁(t) = ET. Hence, we can get that:

Next, consider the following three cases:

case 1: When , then , thus and , we can obtain that:

(i) Sliding region .

(ii) Crossing region

case 2: When and , thus and , we can obtain that:

(i) Sliding region .

(ii) Crossing region

case 3: If , then , thus y_s2<0 and , we can get that:

(i) Escaping region .

(ii) Crossing region .

According to the above results the following theorem can be produced:

Theorem 4.1. The sliding region and escaping region can not exist at the same time.

4.2 Sliding mode dynamics

By using the Utkin’s equivalent control method the following can be obtained:

When x₁(t) = ET and by solving the above equation:

then we can get:

4.3 Five kinds of equilibriums of Filippov system (6)

In this section five different kinds of equilibriums in the Filippov system (6) is discussed here. It follows from the section 3 that is the unique positive equilibrium of the subsystem (1), and is the unique positive equilibrium of the subsystem (2). The nature and types of the above two equilibriums can be studied as follow:

(i) When the condition that is satisfied, then E₁ is in a real equilibrium and hereby it is termed as . Otherwise E₁ is a virtual equilibrium and hereby termed as .

(ii) When the condition that is satisfied, then E₂ is in a real equilibrium which hereby termed as . If not, then E₂ is in a virtual equilibrium and hereby termed as .

Pseudo-equilibrium: There are two different ways to satisfy the pseudo-equilibrium condition as it is to be shown in the following: one way is to obtain the condition by solving the equation , where . The other is from the definition 2.3:

By solving the above equations the following can be obtained:

then by substituting the value of λ into the above equation it yields the pseudo-equilibrium such that:

According to and the stability theory of the ODE it can be shown that the pseudo-equilibrium E_p is locally asymptotic stable.

Theorem 4.2. (i) When , the pseudo-equilibrium E_p exists iff the real equilibrium and are both coexisted.

(ii) When and , the pseudo-equilibrium E_p exists iff the virtual equilibrium and are both coexisted.

Proof: Due to the fact that:

where , , .

Thus (i) When , then the pseudo-equilibrium E_p exists and the real equilibrium and are both coexisted.

(ii) When and , the pseudo-equilibrium E_p exists and then the virtual equilibrium and are both coexisted. □

Boundary equilibrium: It is from the definition 2.4 that the boundary equilibrium should respectively satisfy the following equation:

(8)

and

(9)

Therefore, the Eq (8) has the solution if and only if , then the boundary equilibrium can be obtained:

Similarly, the Eq (9) has the solution if and only if , then the boundary equilibrium can be obtained:

Tangent point: By putting F₁H = 0 and F₂H = 0, then the two tangent point can be computed:

Theorem 4.3. (i) When , then the tangent point is visible.

(ii) When , then the tangent point is visible.

Proof: (i) Through calculation we can get:

Thus, is visible

(ii) Similarly we can obtain:

Thus, is visible □

5 Numerical simulation and bifurcation analysis

In this section the bifurcation set of equilibria and the global stability of equilibria will be assessed by means of numerical simulation methods.

5.1 Bifurcation set of equilibria

It can be seen from the above that the dynamics of the Filippov system (5) are greatly dependent on the ET and equilibria points of the system. Moreover, the occurrence of various types of equilibrium points are also dependent on the value of ET and also the death rate of Predators D₁. Thus the bifurcation diagram as function of bifurcation parameters ET and D₁ is constructed in order to explore the richness of various possible dynamics of the Filippov system (5). Three critical curves are defined as follows:

By putting the and , then we can obtain the curves L₁ and L₂ can be plotted respectively to study the relationship between pseudo-equilibrium E_p and the sliding segment , namely or . Also on the left of the curve L₁ it depicts the interior equilibrium point E₁ which is labelled as , and it turns into on the right. Similarly, the curve L₂ is the dividing line between and . Also, the curve L₃ is the dividing line for the existence of the interior equilibrium point.

By fixing all other parameters , then the parameter plane (Fig 1) can be divided into five regions by these three curves, and the existence of possible equilibria is indicated in each area. It is seen that the pseudo-equilibrium E_p coexists in the region I₂ and I₄. It is worth mentioning that the boundary equilibria is located only along the curves. Moreover, if the parameter D₁ is fixed to 1, then we can see that as ET increases: and coexist exists and coexist exists and coexist. Thus it is evidenced that ET plays a key role in the analyses of bifurcations of the Filippov system (5).

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Fig 1. Bifurcation set of Filippov system (5)’s equilibria, where the values of other fixed parameters are .

https://doi.org/10.1371/journal.pone.0334425.g001

5.2 The global stability of equilibria

The global stability of the equilibria in the Filippov system (5) is validated in this section through numerical simulation. In here we consider two cases of when and :

(i) When the parameters are set as it can be shown through simple calculation that and indicating that . When ET is varied the interior equilibrium points E₁, E₂ and pseudo-equilibrium E_p changed as shown in Fig 2(a). For example when , then E₁ becomes and E₂ becomes such that E₁ becomes the only real equilibrium point where all solutions tend to converge into E₁ eventually, as it is shown in Fig 2(b). When , then E₁ is still and E₂ becomes . In this case both E₁ and E₂ are in real equilibrium and all solutions tend to converge into either E₁ or E₂ eventually, as it is shown in Fig 2(c). When , E₁ becomes and E₂ becomes . Then E₂ is the only real equilibrium in this case and all solutions tend to converge into E₂ eventually as it is shown in Fig 2(d).

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Fig 2. Illustrates the dynamical behavior of the Filippov system (5) which is plotted by using parameters : (a) Shows the existence of the interior equilibrium point (in solid blue plot) for subsystem (1), the dotted curves (in red and black) are the interior equilibrium point of subsystem (2).

(b) E₁ becomes the only real equilibrium point and all solutions tend to converge into E₁ eventually. (c) Both E₁ and E₂ are the real equilibrium points and all solutions tend to converge either into E₁ or E₂ eventually and (d) E₂ becomes the only real equilibrium point and all solutions tend to converge into E₂ eventually.

https://doi.org/10.1371/journal.pone.0334425.g002

(ii) Similarly let’s set the parameters and it can be shown by simple calculation that and which indicates that . As value of the ET is increased or decreased the interior equilibrium point E₁, E₂ and pseudo-equilibrium E_p are subsequently affected as shown in Fig 3. For example when , then E₁ and E₂ becomes and respectively. It can be seen that E₁ becomes the only real equilibrium and all solutions tend to converge into E₁ eventually as shown in Fig 3(b).When , E₁ still exists as and E₂ becomes . Then E₁ and E₂ are both the virtual equilibrium, we can clearly see that all solutions tend to deviate from both E₁ and E₂ and tend to converge into the pseudo equilibrium Ep, which exhibits as the global asymptotic stability as shown in Fig 3(c). However when , then E₁ becomes and E₂ becomes . In this case E₂ becomes the only real equilibrium and we can see that all solutions tend to converge into E₂ eventually as depicted in Fig 3(d).

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Fig 3. Illustrates the dynamical behavior of the Filippov system (5) similar to the above figure but using different parameters : (a) Shows the existence of the interior equilibrium point for subsystem (1) in solid blue plot, and the interior equilibrium point of subsystem (2) in red and black lines.

(b) E₁ becomes the only real equilibrium and all solutions tend to converge into E₁ eventually. (c) Both E₁ and E₂ are the virtual equilibrium points and all solutions tend to converge away from E₁ or E₂ eventually and to form a pseudo equilibrium Ep instead. (d) E₂ is the only real equilibrium and all solutions tend to converge into E₂ eventually.

https://doi.org/10.1371/journal.pone.0334425.g003

6 Conclusions and biological significance

Nowadays, the Filippov system has been found to be useful to describe the real-world problems and investigated in many fields, such as those in physics, ecology and many other multidisciplinary subjects like networked control systems, multi-agent systems, neural networks, mechanical systems as well as in the integrated pest management for the ecosystem modelling etc [6–23,24]. Despite great deal of reports on non-smooth Filippov predator-prey system [6–23,24], none of them have ever considered using functional responses together with the Filippov system simultaneously. This work explores the behavior of the Filippov predator-prey system by imposing threshold policy on the population of prey for the initiation of a mixture of functional response types to study the dynamic of the ecosystem for the very first time. In this study the Holling-II and ratio functional responses have been implemented in the model depending on the population of the prey. For example, the mutual interference of the predator will play an important role when the number of prey is below the economic threshold (ET), thus the ratio functional response has been adopted in this work to model the ecosystem by taking into account of the competition among the predators. On the other hand when the population of the prey reaches or exceeds ET, the mutual interference among predators becomes negligible thus the Holling-II functional response has been selected in this work to model the dynamics of the ecosystem.

So we make use of Filippov theories and qualitative techniques with numerical simulations to investigate dynamical behaviors of proposed system in detail, including global dynamics of subsystems, the existence of sliding mode and different types of equilibria, sliding mode dynamics and the global stability of equilibria.

Stability analysis (asymptotic analysis) provides critical support for integrated pest management (IPM) by studying the equilibrium states of systems under long-term dynamics [3–5,23,24]. By establishing Differential Equations, the asymptotic stability of the pest-predator-crop system is analyzed to predict the long-term equilibrium points after reducing chemical pesticide use. For example, predator-prey model in [23] demonstrated that when negative feedback mechanisms exist in the system, asymptotic stability can suppress pest population outbreaks.

Furthermore, the proposed Filippov system together with the switchable functional response had been validated through numerical simulations. It has been demonstrated that the real equilibrium and pseudo-equilibrium points can coexist when the population of the prey is less than that of the predator (i.e. ); and in the case of when the population of the prey is more than that of the predator (i.e. ), only the virtual equilibrium and pseudo-equilibrium can coexist. As the economic threshold (ET) increases and when , then the following equilibrium sequences can coexist: and coexist and coexist and coexist, as explicitly depicted in Fig 2 above. Since both E₁ and E₂ are both the real equilibrium, all solutions tend to either E₁ or E₂ eventually. This is an extremely interesting bistability phenomenon that can be seen in this switchable Filippov system. Similarly, when the ET increases and , then the following equilibrium sequences can coexist: and coexist and coexist and coexist, as it is demonstrated in Fig 3 above. According to our results it has also shown that the sliding and escaping regions cannot coexist under our proposed system. In particular, it is noted that all trajectories of the prey and predator’s population are eventually converging into certain equilibrium points as it is demonstrated in the numerical simulation in Sect 5. This implies that there exists global asymptotic stability of equilibrium points under the proposed system, in which the population of preys eventually reaches a steady state of density at the real equilibrium and pseudo-equilibrium points. This means we don’t need to take any action at this point.

This work also highlights the significant role of the threshold ET in the process of pest controls.So the reasonable control threshold (ET) can be effective for prevention and control of pests. Consequently, our findings are valuable for how to draw up strategies effectively and when to take measures.This paper enriches the theoretical and methodological framework of future system dynamics modeling, holding significant theoretical and practical implications.