Research paper
Dynamics analysis of a Filippov pest control model with time delay

https://doi.org/10.1016/j.cnsns.2021.105865Get rights and content

Highlights

  • A Filippov prey–predator (pest-natural enemy) model with time delay is introduced to describe the IPM strategy with economic threshold.

  • The time delay represents the change of growth rate of the natural enemy before releasing it to eliminate pests.

  • Our findings show that, the time delay τ has a significant impact on the Filippov system.

  • A non-trivial periodic solution bifurcates from the positive equilibria.

Abstract

The most critical factor for increasing crop production is the successful resistance of pests and pathogens which has massive impacts on global food security. Therefore, Filippov systems have been used to model and grasp control strategies for limited resources in Integrated Pest Management (IPM). Extensive studies have been done on these systems where the evolution is governed by a smooth set of ordinary differential equations (ODEs). As far as we know the time delay has not been considered in these systems, which we mean that a set of delay differential equations (DDEs). With this motivation, a Filippov prey–predator (pest–natural enemy) model with time delay is introduced in this paper, where the time delay represents the change of growth rate of the natural enemy before releasing it to feed on pests. The threshold conditions for the stability of the equilibria are derived by using time delay as a bifurcation parameter. It is shown that when the time delay parameter passes through some critical values, a periodic oscillation phenomenon appears through Hopf bifurcation. Further, by employing Filippov convex method we obtain the equation of sliding motion and address the sliding mode dynamics. Numerically, we demonstrate that the time delay plays a substantial role in discontinuity-induced bifurcation. More precisely, one can get boundary focus bifurcation from boundary node bifurcation through variation of the value of the time delay. Moreover, the time delay is used as a bifurcation parameter to obtain sliding–switching and sliding–grazing bifurcations. In conclusion, a Filippov system with time delay can give new insights into pest control models.

Introduction

In natural and man-made systems, time delay is considered as a significant factor. Kuang [1] gave an example to show the importance of time delay. He said that animals need a time to predigest their food before further activities and responses occur. So, any model of species without time delay is an approximation at best [2]. Time delays arise in many systems and industrial processes, such as biological systems, metal forming, machining, thermal acoustics systems, and others [3], [4], [5], [6]. Moreover, dynamical systems with time delay demonstrates much more complex behaviors than those without time delay [7]. The presence of time delay in prey-predator models is attributed for two reasons [8], [9]. The first one is the time of gestation, and the other is the time of maturation. So, in order to ensure the realistic of the predator–prey models and to illustrate how the population dynamics of theses models depend on the past relevant information, it is indispensable to integrate delays into these models. In fact, time delays significantly impact the overall properties of dynamical systems. In the literature, many monographs have reported the theoretical analysis of the prey–predator with time delay, including hunting delay [10], the delay in dispersal [11], the gestation period for the predator [12] and intra-specific competition induced feedback delay [13].

Numerous problems in applied science have been modeled by ordinary differential equations (ODEs). However, instead of smooth ODEs, some of the applications are better designed using discontinuous ODEs [14], [15], [16]. These systems are distinguished by a discontinuity in their right-hand sides, which may be due to the system state reaching a discontinuous boundary or the discontinuities in evolution with respect to time [17], [18]. Thus, these systems can be used to describe many biological and physical processes which are distinguished by periods of smooth evolution being cut off by an instantaneous event or the systems whereby the physical states switch between two different states. Consequently, a different set of differential equations or maps can be used for describing each state when these states are modeled. In other words, the evolution of trajectories in each region can be defined by a smooth dynamical system and it loses its smoothness at the discontinuity due to instantaneous events [17], [18], [19]. Earlier studies were made on non-smooth systems [17], [20], [21], [22], [23], [24], [25]

Among the non-smooth types and the most important is the Filippov system where appears pervasively in real life models such as mechanical models [26], electrical circuits [27], [28] and biological models [29], [30], [31], [32], [33], [34], [35]. The Filippov system can be described in its simplest modelization, in which the discontinuity surface splits the state space into two regions. When the solution trajectory stays inside the same region and does not reach the discontinuity surface, then the dynamics behave as in a conventional smooth system. The question arises here, what will be happened when the trajectory reaches the discontinuity surface? There are two possible outcomes that can occur when the trajectory reaches the discontinuity surface. The trajectory either crosses it or stays on it. Therefore, in the latter case, it is required to describe the motion on the discontinuity surface which is called sliding motion [23].

The significant research is interested in the bifurcations of Filippov systems which focus on the bifurcations caused by the discontinuity surface, known as discontinuity-induced bifurcation [20] which cannot occur in smooth systems (ODEs). The discontinuity-induced bifurcation describes the interaction between a discontinuity boundary and an invariant set. Based on the kinds of invariant sets, discontinuity-induced bifurcations can be classified into pseudo-equilibrium bifurcations, boundary equilibrium bifurcations, sliding bifurcations, tangency point bifurcations and sliding homoclinic bifurcations, etc, for more details for each type see [36].

Meanwhile, there are many practical dynamical systems which are described by the time delay differential equations with discontinuous right-hand sides. Whenever the non-smoothness of the dynamical system is caused by the implementation of a switch, it can be expected that there may be a delay in the actual switch like delayed relay models [37]. Usually, due to the growth of this delay, sliding along the switching manifold can occur in a very short time interval of a large number of switches [37], [38]. Based on functional differential inclusions, Zhang et al. [39], [40], [41] introduced the generalized Filippov solutions and the definition of the stability for discontinuous systems with time delay. Moreover, they provided some results about the feedback stabilization and the L2-gain problem for these systems by using Lyapunov-Krasovskii theorem. Recently, Cai et al. investigated the existence of periodic solutions for Filippov systems with time delay [42], [43], [44].

To the best of the authors’ knowledge, Filippov systems with time delay have not been fully considered in the literature. Motivated by this, we introduce a time delay Filippov system and shed light on deeper theoretical questions:

  • It is well known that various complex phenomena arise as a result of the existence of a time delay in a nonlinear system such as bifurcation, multistability, chaos, hyperchaos, etc [4], [6], [10]. Therefore, what is the impact of the time delay on the dynamics of Filippov systems?

  • Filippov systems are characterized by exhibiting discontinuity-induced bifurcations that have not existed in smooth systems or delay systems. Can the time delay parameter play a significant role in the discontinuity-induced bifurcations?

In order to answer these questions, an example represents a delay Filippov model is considered in the next section. Section 3 presents a detailed analysis of the local stability of the equilibria. In Section 4, the existence of sliding segments and sliding mode dynamics for the delay Filippov model are investigated. In order to confirm our analytic findings, some numerical simulations are presented in Section 5. In Section 6, the local and global sliding bifurcations are analyzed by using the numerical simulations. Then, in the last section, the conclusions are given.

Section snippets

Presentation of the delay Filippov prey predator model

To improve the performances of global agricultural and associated food systems, it is necessary to take curtailment the crop pests and pathogens into consideration. In addition rising pests lead to devour crops worldwide, which affects the quality of harvested production and could cause future food shortages. Thus, pests must be resisted and the losses caused by it need to be tackled. Therefore, it is necessary to apply effective strategies to dominate pest outbreaks. Integrated pest management

Qualitative analysis of the subsystems FΠ1 and FΠ2

In this section, we investigate the equilibria of delay Filippov system (3) and their stability. Before doing that, we have the two important definitions which are related to the equilibria of Filippov system [17], [20], [22].

Definition 1

The regular equilibria of system (3) are the solutions of FΠi(X,Xτ)=0 and belong to Πi, where i=1,2.

Definition 2

The virtual equilibria of system (3) are the solutions of FΠi(X,Xτ)=0 and belong to Πj, where i,j=1,2,(ij).

Sliding regions and their dynamics

In this section, the existence of the sliding and the crossing segments of the delay Filippov system (3) are briefly stated. Then the dynamics of the sliding segment and its equilibria are discussed. The dynamics on the separating manifold Σ (crossing or sliding mode dynamics) for the delay Filippov system (3) is determined by the Filippov convex of vector fields FΠ1 and FΠ2 [17], [20], [22], [41].

Letσ(X,Xτ)=G(X),FΠ1(X,Xτ)G(X),FΠ2(X,Xτ),where · denotes the standard scalar product. Then,

Numerical simulations

In this section, some numerical simulations are carried out to illustrate our theoretical analysis. We choose ρ=3,k=8,α=4,β=3.5,η=5/6,δ=3,μ=1.2,γ=0.2 and vary ξ. The subsystem (4) has three equilibria which areE10=(0,0),E11=(5.61249,1.39582),E12=(8,0).In order to investigate the impact of the time delay τ on the oscillations of the delay Filippov system (3), we perform the numerical simulations in two steps. In the first step, we consider that the dynamic behavior of the delay Filippov system

Local and global sliding bifurcations of delay Filippov system

In this section, we investigate local and global sliding bifurcations of delay Filippov system (3).

Conclusions

The threat posed to crop production by plant pests and diseases is one of the key factors that could lead to instability in global food security, and the problem is forecasted to get worse which the scientists have warned against. Therefore, understanding and appropriate use of IPM strategies in dealing with pests can maintain environmental quality and keep the stability of agricultural systems, therefore stability in global food security. Thus in this paper, we proposed a Filippov

CRediT authorship contribution statement

Ayman A. Arafa: Conceptualization, Methodology, Investigation, Software, Writing - original draft. Soliman A.A. Hamdallah: Methodology, Investigation, Writing - original draft. Sanyi Tang: Supervision. Yong Xu: Writing - review & editing, Supervision. Gamal M. Mahmoud: Writing - review & editing, Supervision.

Declaration of Competing Interest

This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.

The authors have no affiliation with any organization with a direct or indirect financial interest in the subject matter discussed in the manuscript

Acknowledgements

We wish to thank the reviewer for his valuable comments, which helped us improve considerably the presentation of our results. The first author is very grateful to Northwestern Polytechnical University for the postdoctoral position provided to him.

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