Threshold dynamics and pulse control of a stochastic ecosystem with switching parameters

https://doi.org/10.1016/j.jfranklin.2020.10.035Get rights and content

Abstract

This paper studies the stochastic dynamic behaviors and pulse control strategy for a vegetation ecosystem with switching parameters. First, a novel stochastic switching ecosystem is built by the variable perturbation, where the parameters are assumed to be time-varying functions. Then, based on the stochastic Ito^ formula and the theory of switched system, a sufficient condition of vegetation extinction in mean sense is established in detail, which provides a warning of the ecological degradation. Furthermore, in order to prevent ecological degradation, an achievable pulse control scheme is proposed, and the effectiveness of the pulse control is proved. Several numerical simulations are given to illustrate the developed threshold theorems and pulse control scheme.

Introduction

The worldwide ecological environmental issues, such as global warming, reduced biodiversity and land desertification, have become increasingly serious since the 1960s [1], [2]. As a result, ecological protection has gradually attracted the attention of the international community [3]. Theoretically, ecological models are vital for implementing and optimizing the dynamical schemes to curb the ecological degradation [4]. There have been a large number of models of developed so far for various ecosystems, see [5], [6], [7], [8] and the references therein. The following is a basic dimensionless vegetation ecosystem [6] which includes two state variables: the vegetation biomass x and the soil water volume y:x˙=ρxyρx2Kβxx+1,y˙=Rαyλxy,where the parameter ρ is the maximum biomass growth rate, K is the carrying capacity of biomass, and β is the maximum consumption rate by herbivores or other factors. The parameters R,α and λ are all positive real numbers representing the rainfall, the soil water loss rate and the consumption rate of water by biomass, respectively. This mathematical model may be simplified, but it is useful for understanding the underlying dynamical schemes driving the evolution of the vegetation.

Much work regarding the regime shifts of this vegetation ecosystem has been done, and many extended models have also been developed to analyze the dynamics of the ecosystem [9], [10], [11]. Most existing model parameters are assumed to be constants. However, they are actually time-varying due to the environmental changes. For example, the rainfall R is different and alternating in summer months and in winter months, and the maximum consumption rate β also varies in time because of the seasonal changes. This naturally leads to the introduction of the switching parameters. Some work has been done about switching models [12], [13], [14], [15], which give us some inspiration to study the switching ecosystem.

Switching systems, as a typical class of hybrid systems, combine continuous subsystems and discrete events via a switching rule [16], [17]. These types of systems have many practical applications such as population systems, biological models and engineering technology [13], [18], [19], [20]. The dynamics of a switching system are often different from the dynamics of its subsystems [21], [22].

On the other hand, stochastic events such as the extreme weather, the drought-fire interactions and the pest outbreaks often happen in real ecosystems [23], [24], [25]. It is more realistic and compatible to introduce stochasticity to ecological models, compared with their deterministic counterpart. Stochastic vegetation ecosystems have been investigated by many researchers [26], [27]. To the best of our knowledge, very little research has been done about the stochastic ecological models with switching parameters.

The main benefit of studying ecosystem dynamics is to achieve the control of ecological balance. Physically, there have been many control schemes about dynamical switching systems including adaptive fuzzy control [28] and output feedback control [29]. Ecologically, however, the vegetation ecosystems are generally controlled by artificial rainfall, planting and pest control, etc. Most of them are applied to the ecosystem in a short time, which can be considered as an effective impulse control strategy in mathematics [30]. Well-studied examples of such control include quantum computation [31], [32] and epidemic models [14], [15], [33].

The objective of this paper is to investigate the threshold dynamics and pulse control strategy of a novel stochastic vegetation ecological model with switching parameters. The rest of this paper is organized as follows. In Section 2, the governing equation of the vegetation biomass and soil water with the switching parameters and stochastic noise is proposed, and the threshold criteria is established to examine the vegetation extinction in detail from the perspective of stochastic switching dynamics. A pulse control strategy is applied to the stochastic switching ecosystem in Section 3, and then a sufficient condition for the control scheme model is derived to suppress the vegetation extinction. Numerical simulations are given in Section 4 to illustrate the established threshold theorems and the feasibility of the pulse control. Finally, this paper ends with some conclusions and future perspectives in Section 5.

Section snippets

Switched model and threshold dynamics

Assume that the mean rainfall is modeled as a periodic switching parameter Rσ. Here, the switching rule σ{1,2,,m} is a piecewise constant function about time and left continuous in a period ω with ω=ω1+ω2++ωm and ωk=tktk1 with switching time tk. More specifically, the kth subsystem is activated when t(hω+tk1,hω+tk], h=0,1,2,, that is, Rσ=Rk. Meanwhile, we assume that the consumption rate by herbivores is also governed by a switching parameter βσ with the same switching rule σ, and this

Pulse control strategy

In this section, we will exert pulse control in the ecosystem to suppress the extinction of the vegetation biomass. Assume that the vegetation biomass instantly increases by px(t) after passing a short time due to human control (e.g. regular planting). In fact, the control strength p should be small, because the large strength means the high control costs. More precisely, assume that the switching times are also the pulse times. This leads to a new stochastic impulsive model with switching

Numerical simulations and ecological explanation

As the above theorems demonstrate, a deeper understanding of the behavior of the switching ecosystem with pulse control is crucial. We therefore give quantitative analysis about theoretical results by four examples, of course, for the practical examples that may not meet the conditions of the theorem, it is necessary to find other methods to investigate the stability of vegetation. In following examples, we only assume that constant parameters of the ecosystem are ρ=0.5,K=1.0,α=0.5,λ=0.12. Let t

Conclusions

In this paper, we have proposed a more realistic stochastic ecological model with time-varying rainfall and vegetation consumption rate and studied in detail its stochastic dynamics and a pulse control scheme. For a stochastic switched vegetation ecosystem with both stable and unstable subsystems, we have established a new threshold criterion on vegetation extinction by utilizing the switched system and stochastic analysis techniques. Moreover, we have developed a pulse control strategy to

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (Grant Nos. 11872305 and 11532011), the Doctoral Innovation Fund of Northwestern Polytechnical University, and NSERC Canada.

References (34)

  • S. Li et al.

    Adaptive fuzzy control of switched nonlinear time-varying delay systems with prescribed performance and unmodeled dynamics

    Fuzzy Sets Syst.

    (2019)
  • X. Liang et al.

    Multiple kinds of optimal impulse control strategies on plant–pest–predator model with eco-epidemiology

    Appl. Math. Comput.

    (2016)
  • L. Duan et al.

    Suppressing environmental noise in quantum computation through pulse control

    Phys. Lett. A

    (1999)
  • M. Liu et al.

    Persistence, extinction and global asymptotical stability of a non-autonomous predator–prey model with random perturbation

    Appl. Math. Model.

    (2012)
  • Y. Hautier et al.

    Anthropogenic environmental changes affect ecosystem stability via biodiversity

    Science

    (2015)
  • T. Wernberg et al.

    Climate-driven regime shift of a temperate marine ecosystem

    Science

    (2016)
  • S.M. Sundstrom et al.

    Detecting spatial regimes in ecosystems

    Ecol. Lett.

    (2017)
  • Cited by (9)

    • Stationary distribution, density function and extinction of stochastic vegetation-water systems

      2023, Communications in Nonlinear Science and Numerical Simulation
    View all citing articles on Scopus
    View full text