Extinction and permanence of the predator-prey system with general functional response and impulsive control

Highlights

• Modelling the predator-prey ecosystem using a ‘generalized’ impulsive response function for the first time.

• Dynamical behaviors such as sufficient conditions for the stability of prey-eradication and the permanence have been given.

• The generalized solutions under asymptotically stable conditions are validated through various specific response functions.

• The correctness of the derived theoretical results are further validated using numerical simulations.

• Unlike previous research this paper also models the non-simultaneous occurrence of predator stocking and prey harvesting.

Abstract

Traditional approach for modelling the evolution of populations in the predator-prey ecosystem has commonly been undertaken using specific impulsive response function, and this kind of modelling is applicable only for a specific ecosystem under certain environmental situations only. This paper attempts to fill the gap by modelling the predator-prey ecosystem using a ‘generalized’ impulsive response function for the first time. Different from previous research, the present work develops the modelling for an integrated pest management (IPM) especially when the stocking of predator (natural enemy) and the harvesting of prey (pest) occur impulsively and at different instances of time. The paper firstly establishes the sufficient conditions for the local and the global stabilities of prey eradication periodic solution by applying the Floquet theorem of the Impulsive different equation and small amplitude perturbation under a ‘generalized’ impulsive response function. Subsequently the sufficient condition for the permanence of the system is given through the comparison techniques. The corollaries of the theorems that are established by using the ‘general impulsive response function’ under the locally asymptotically stable condition are found to be in excellent agreement with those reported previously. Theoretical results that are obtained in this work is then validated by using a typical impulsive response function (Holling type-II) as an example, and the outcome is shown to be consistent with the previously reported results. Finally, the implication of the developed theories for practical pest management is illustrated through numerical simulation. It is shown that the elimination of either the preys or the pest can be effectively deployed by making use of the theoretical model established in this work. The developed model is capable to predict the population evolutions of the predator-prey ecosystem to accommodate requirements such as: the combinations of the biological control, chemical control, any functional response function, the moderate impulsive period, the harvest rate for the prey and predator parameter and the incremental stocking of the predator parameter.

Introduction

Study of dynamic systems has been an important research subject due to its diverse applications across vast multi-disciplinary science and engineering such as for the modeling of biological systems, communication, control systems, networked systems, manufacturing and mechanical systems etc [1], [2]. The dynamic system is said to be hybrid when it exhibits continuous and discrete dynamics [3], [4], and it is termed as a switched system when it consists of a family of subsystems in which the dynamics is switched among the subsystems according to a logical rule [2], [5], [6], [7], [8], [9], [10]. The interactions of prey and predator, such as the study of the populations in the crops and pests system, is a simplified special case of the dynamic systems which has been investigated extensively in the ecological, agricultural and environmental research within the past few decades [11], [12], [13], [14], [15]. Particularly, the research on the pest control has been a popular topic as it imposes significant impacts on the social and economic stability of the region in question [16], [17], [18]. There are two commonly used approaches for pests control: chemically through pesticide spray and biological control through the introduction of natural enemies into the local environment. In some cases the integrated pest management (IPM) which utilizes both chemical and the environmentally friendly biological control [16], [17], [18], [19] is preferred, particularly when results are desired to be achieved rapidly. However, to maintain the long term existences of prey or the eliminations of the pest, it requires the delicate balance between when the cycles of pesticides should be applied and the exact timing for introducing natural enemies. This is needed to be evaluated through careful modeling or experiments due to the negative effect of pesticide to the natural enemies of the pest. Due to the difficulty to implement experiments effectively in the ecological systems, mathematical modeling has been an alternative promising way to solve practical problems [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38]. Considering the discontinuity of the spraying pesticides and natural enemies, the description of the short-term rapid changes of their populations is commonly modeled through impulsive differential equations (IDE) [8], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. Most previous work modelled the simultaneously spraying pesticides and natural enemies impulsively at the same fixed moments [8], [22], [27]. It is not reasonable due to the effects of pesticides on natural enemies. Thus the modelling of the predator-prey system using non-simultaneous timings of impulsive control strategies is more realistic and relevant for practical applications.

The predator-prey relationship between pests and natural enemies is quite well-known. Typical models involve a characteristic predators rate of feeding on the prey, which is known as the functional response, to model the change of predation rate as a function of the prey density through a set of differential equations. Depending on the nature of ecosystems concerned, many different kinds of functional responses such as the Holling type-I (ϕ₁), the Holling type-II (ϕ₂), the Holling type-III (ϕ₃) [15], [19], [20], [21] have been applied for various applications. The Holling type ϕ₁, ϕ₂ and ϕ₃ are commonly in the form of:

$${\varphi }_1\left(x\right)=\{\begin{array}{c}\frac{b}{a}x0<x<a\hfill \\ bx>a\hfill \end{array},{\varphi }_2\left(x\right)=\frac{rx}{a+bx},{\varphi }_3\left(x\right)=\frac{r{x}^2}{a+b{x}^2}.$$

The Ivlev-type functional response ϕ₄ is in the form of [22], [30]:

$${\varphi }_4\left(x\right)=r\left(1-e^{-\alpha x}\right).$$

where a, b, r, α are constants and x(t) is prey density. Note that the above functional responses are modelled for scenarios when the predator-prey interaction depends on prey densities only. In the real world both the prey and predator densities affect the predator-prey interactions, so more complex functional response such as the Beddington-DeAngelis functional response ϕ₅ has been proposed [8]:

$${\varphi }_5\left(x,y\right)=\frac{mx}{a+bx+cy}.$$

where y(t) represents the density of the predator, and alternative form of the ratio-dependent functional response ϕ₆ has also been suggested [15], [23], [25]:

$${\varphi }_6\left(\frac{x}{y}\right)=\frac{\frac{cx}{y}}{m+\frac{x}{y}}=\frac{cx}{my+x}.$$

where a,b,m,c are constants. Due to the complicated natural environment which compounds by various degree of human interferences, a plethora of functional responses has been proposed for modelling the dynamical relationships between the predator and the prey, for example [26], [27], [28], [29], [31]:

the Watt-type functional response ${\varphi }_7=exp\frac{-cx}{y^m}$;

the Hassell-Varley functional response ${\varphi }_8=\frac{cx\left(t\right)}{m{y}^{\gamma }+x}$;

the Square-Root functional responses ${\varphi }_9=\frac{\beta \sqrt{x\left(t\right)}}{1+\alpha \sqrt{x\left(t\right)}}$;

the Monod-Haldance functional response ${\varphi }_{10}=\frac{mx\left(t\right)}{a+x^2\left(t\right)}$ etc.

Instead of modelling the predator-prey system using a specific type of functional response like that performed by previous research, this paper attempts for the first time to model the predator-prey system by using a general functional response. This paper gives a generalized local and global stability of prey eradication periodic solution under sufficient conditions, and the solution is then illustrated by using a specific functional response as an example.

Previous work employed the Holling II functional response [24], the Monod-Haldance functional response [29], [31], the Ivlev-type functional response [30], the ratio-dependent functional response [15], [23], the Beddington-DeAngelis functional response [8] and the Hassell-Varley functional response etc [27], for modelling the predator-prey system using these impulsive control strategies. Although these previous studies had provided the local asymptotic stability of prey-eradication periodic solution and the permanence of system under sufficient conditions, they could be applied only for specific scenarios and therefore their impacts to the real world is limited.

Thus it is necessary to generalize previous findings such as that reported in [24], [29], [30] by using a general functional response for modelling the predator-prey system, especially when the predator is stocked and the prey is harvested at different instances of time. It is believed that this work may represent a first study of its kind for modelling the predator-prey system by using the general function g(x, y) as the system in (1) below:

$$\{\begin{array}{c}\begin{array}{c}\frac{\mathrm{d}x\left(t\right)}{\mathrm{d}t}=x\left(t\right)\left(r-\frac{r}{K}x\left(t\right)\right)-g\left(x,y\right)x\left(t\right)y\left(t\right)\hfill \\ \frac{\mathrm{d}y\left(t\right)}{\mathrm{d}t}=y\left(t\right)\left(-D+kg\left(x,y\right)x\left(t\right)\right)\hfill \end{array}\}t\ne \left(n+l-1\right)T,t\ne nT\hfill \\ \begin{array}{c}\Delta x\left(t\right)=-p_{1}x\left(t\right)\hfill \\ \Delta y\left(t\right)=-p_{2}y\left(t\right)\hfill \end{array}\}t=\left(n+l-1\right)T\hfill \\ \begin{array}{c}\Delta x\left(t\right)=0\hfill \\ \Delta y\left(t\right)=\mu \hfill \end{array}\}t=nT\hfill \end{array}$$

where x(t), y(t) denote the densities of the prey (pest) and predator (natural enemy) at time t, respectively. T is the period of the impulsive harvesting and stocking. r > 0 is the intrinsic growth rate of the prey, K > 0 is the carrying capacities of the prey, D > 0 is the death rate of the predator. The k > 0 denotes the rate of converting consumed preys into the growth of predator, and the x(t)g(x, y) is the general functional response of the prey which represents the rate of predation by the predator per-capita. The g(x, y) is assumed to be monotonous decreasing with respect to x and y respectively, which satisfied the Lemma 2.3 in Section 2, and y(t)g(x, y) is the monotonous increasing density of y which represents the main functional responses of the ecosystem. Also, $\Delta x\left(t\right)=x\left(t^{+}\right)-x\left(t\right);\Delta y\left(t\right)=y\left(t^{+}\right)-y\left(t\right);x\left(t^{+}\right)=\underset{t\to t^{+}}{lim}x\left(t\right);y\left(t^{+}\right)=\underset{t\to t^{+}}{lim}y\left(t\right).$ And 0 ≤ p₁, p₂ < 1 is the harvest rate of the prey and the predator at time $\left(n+l-1\right)T$ respectively, where l ∈ (0, 1), n ∈ N. μ > 0 is the incremental stocking of the predator at time nT.

This paper is organized as follows: Section 1 (this section) outlines the problem statement and the objectives of this paper. The preliminaries including definitions and lemmas are given in Section 2. In Section 3, the stability of the prey-eradication periodic solution, including the local asymptotic stability and global asymptotically stability of the predator-prey system modelled by using a ‘general’ functional response is given. Section 4 investigates the boundedness of solutions and the sufficient conditions that are required for the permanence of the system (1). To illustrate the established theoretical results, numerical simulations of the systems in section 3 have been performed by using a specific response function as an example and the results are presented in Section 5. Section 6 summarizes the methodology of the present work to model the prey and predator system using a generalized impulsive response function, and the suggestions to apply the results of this work for practical pest management are subsequently given.