Transient Dynamics Analysis of a Predator-Prey System with Square Root Functional Responses and Random Perturbation

Abstract

In this paper, we study the asymptotic and transient dynamics of a predator–prey model with square root functional responses and random perturbation. Firstly, the mean square stability matrix is obtained from the stability theory of stochastic systems, and three stability indexes (root-mean-square resilience, root-mean-square reactivity and root-mean-square amplification envelope) of the ecosystem response to stochastic disturbances are calculated. We find that: (1) no matter which population is disturbed, increasing the intensity of disturbance improves the ability of the system leaves steady state and thus decreases the stability. The root-mean-square amplification envelope rises with increasing disturbance intensity, (2) the system is more sensitive to the disturbance of the predator than disturbance to prey, (3) ${\rho }_{max}$ and $t_{max}$ are important indicators, which represent the intensity and time of maximum amplification by disturbance. These findings are helpful for managers to take corresponding management measures to reduce the disturbances, especially for predators, thereby avoiding the possible change of the structure and functions of the ecosystem.

1. Introduction

Holling proposed three types of functional response functions to better reveal predator–prey interactions. The Holling functional response function is a function of prey density, representing the amount of prey consumed per predator, and is usually a primary or quadratic function of prey density. In reality, prey in nature usually gather together to resist predators. The idea of population defense in ecology was proposed by Freedman and Wolkowicz [1]. In general, the behavior of a population defense known as the Holling functional response is insufficient. Later, Ajraldi et al. [2] believed that for a prey–predator interaction that exhibited group behavior, it was appropriate to use the square root of prey density to model the functional response function. Then Bera [3] used the square root of prey density instead of prey density to generate a new functional response function, and introduced a new model with this functional response.

Ecosystem stability is an important corollary of sustain ability. For a long time, people focused on the long-term dynamic behavior of the equilibrium point of the system [4]. With the deepening of the research, ecologists are more and more interested in the transient dynamic behavior of the systems [5,6,7]. In 1977, Pimm and Lawton [8] first proposed the definition of resilience in a biological sense, which is the asymptotic decay rate of disturbance. For the concept of resilience, with the deepening of research, scholars have given different concepts and calculation methods to different fields; the traditional definition of resilience is given in [9]. It concentrates on stability near an equilibrium, where resistance to disturbance and speed of return to the equilibrium, following a perturbation, are used to measure the property. This view of resilience provides one of the foundations for economic theory as well and may be termed engineering resilience. Another definition of resilience emphasizes conditions far from any equilibrium, where instabilities can flip a system into another regime of behavior, i.e., to another stability domain [10]. In this case, the measurement of resilience is the magnitude of disturbance that can be absorbed before the system changes its structure by changing the variables and processes that control behavior. This second view may be termed ecological resilience [11] and clearly involves properties of the basin of attraction. Lundström and Mitra et al. [12,13] present methods for measuring resilience through the basin of attraction, and a concrete application can be found in [14].

In general, most systems will be far away from the equilibrium point and deviate from the initial orbit, but eventually they will return to the stable state. For example, infectious diseases eventually become extinct or endemic diseases, but the epidemic is initially exposed to external influences [15]. Therefore, if we only consider the long-term dynamic behavior of the system and ignore the instantaneous behavior, it is conceivable that both nature and human society will suffer heavy losses.

In the past few decades, researchers have made remarkable achievements in studying the transient dynamics of ecosystems. For example, Wu and Hua studied transient phenomena in immune-randomized tumor growth models [16]. Hastings and Abbott et al. have summarized the contents of the transient [17]. Neubert and Caswell [18] proposed a complementary concept of resilience, called reactivity, which represents the maximum initial rate at which disturbances can grow and characterizes the transient response of ecosystems. Resilience and reactivity describe the two extremes of the dynamic behavior of the solution: $t\to \infty$ and $t\to 0$. To describe the transient situation at $0<t<\infty$, Neubert and Caswell also put forward the concept of amplification envelope $\rho \left(t\right)$, which is the maximum amplification of the solution that can be caused by any initial value disturbance at time t. Accordingly, the time at which ${\rho }_{max}$ appears is denoted by $t_{max}$. The ecosystem in nature is constantly disturbed by the external environment. In order to fit the reality, it is more reasonable to explore the random biological model. Buckward and Kelly [19] extended the three measurement indicators of resilience, reactivity and amplification envelope to the predator–prey model with random disturbance, called the root-mean-square resilience, root-mean-square reactivity and root-mean-square amplification envelope. The above three indicators can help the situation of the reaction stability change.

However, the transient dynamics of a predator–prey model with square root functional responses and random perturbations is still not clear. Therefore, the contents of this paper are as follows: Section 2 briefly introduces the research model and some basic concepts; in Section 3, the stochastic perturbation is introduced into the deterministic model, and the mean square stability matrix of the stochastic predator-prey system is calculated. In Section 4, numerical simulations are carried out to demonstrate the effects of stochastic perturbations on the stability and transient dynamics of the system.

2. Preliminaries

2.1. Predator–Prey Model with Square Root Functional Responses

First, the traditional deterministic square-root functional response predator–prey model is as follows:

$$\left\{\begin{array}{c}\frac{dX}{dt}=rX(1-\frac{X}{N})-\frac{\alpha \sqrt{X}Y}{1+t_h\alpha \sqrt{X}},\hfill \\ \frac{dY}{dt}=-sY+\frac{c\alpha \sqrt{X}Y}{1+t_h\alpha \sqrt{X}},\hfill \end{array}\right.$$

(1)

where X and Y are the density of prey and predator, respectively, r is the growth rate of prey, and N is the carrying capacity of the environment. $t_h$ is the average handling time of predators to prey, $\alpha$ is the search efficiency of predators to prey, c is the conversion rate from prey to predator, and s is the natural mortality rate of predators in the absence of prey. In this paper, we study a model in which the predator’s processing time to the prey is zero ($t_h=0$).

2.2. Concepts and Definitions

The definitions and properties of predator–prey models with square root functional responses and random perturbations are as follows:

Definition 1

([19]). For linear (linearized) stochastic systems:

$$dX\left(t\right)=AX\left(t\right)dt+\sum _{i=1}^m{B}_{i}X\left(t\right)d{W}_i,t>0,$$

(2)

where $A\in {\mathbb{R}}^{d\times d},$$B\in {\mathbb{R}}^{d\times d},$ and $W={(W_1,W_2,\dots ,W_m)}^T$ is an m dimensional Wiener process defined on the filtered probability space $(\mathsf{\Omega },\mathcal{F},\mathcal{F}{\left(t\right)}_{t\ge 0},\mathbb{P}),$ where $\mathcal{F}{\left(t\right)}_{t\ge 0}$ is the natural filtration generated by W.

Its mean square stability matrix is defined as follows:

$$S=I_A\otimes A+A\otimes I_d+\sum _{i=1}^m{B}_i\otimes B_i.$$

(3)

Definition 2

([19]). The Kronecker product of an $m\times n$ matrix A and a $p\times q$ matrix B is the $mp\times nq$ matrix defined by

$$A\otimes B=\left(\begin{array}{ccc}{a}_{11}B& \cdots & a_{1n}B\\ ⋮& \ddots & ⋮\\ a_{m1}B& \cdots & a_{mn}B\end{array}\right).$$

(4)

Definition 3

([15]). The equilibrium position of $\left(2\right)$ is said to be stochastically asymptotically stable (or asymptotically stable in probability) if it is stochastically stable and

$$\begin{array}{c}\hfill \underset{c\to 0}{lim}P\{\underset{t\to \infty }{lim}{X}_t\left(c\right)=0\}=1.\end{array}$$

Definition 4

([15]). The equilibrium position of $\left(2\right)$ is said to be mean-square stable if for every $\epsilon >0$, there exists a $\delta >0$, such that for all $\left|c\right|<\delta$, $su{p}_{t_0\le t\le \infty }E{\left|X_t\left(c\right)\right|}^2=0$. If in addition for all c in a neighborhood of $X=0$, $li{m}_{t\to \infty }E{\left|X_t\left(c\right)\right|}^2=0$, the equilibrium position is said to be mean-square asymptotically stable.

Definition 5

([19]). Considering the predator–prey system with square root functional responses and stochastic disturbance, the three indexes of a system response to random disturbance are as follows:

(1) 

The root-mean-square resilience is $-\frac{1}{2}\alpha \left(S\right);$ we denote spectral abscissa $\alpha \left(S\right)$ is the maximum of ${\lambda }_i$, ${\lambda }_i$ are the eigenvalues of S;

(2) 

The root-mean-square reactivity is $\frac{1}{2}{\lambda }_{max}\left(\frac{S+S^T}{2}\right);$

(3) 

The root-mean-square amplification envelope is $\sqrt{\parallel e^{St}\parallel }$, where $\parallel ·\parallel$ is the matrix norm.

3. Stability Analysis of Predator–Prey System with Square Root Functional Responses

3.1. Stability of the Deterministic Model

We use a nondimensionalized method to simplify the deterministic model.

Setting

$$\begin{array}{c}\hfill x=\frac{X}{N},y=\frac{\alpha Y}{r\sqrt{N}},t_{new}=r{t}_{old},\\ \hfill c_{new}=c_{old}\frac{\alpha \sqrt{N}}{r},s_{new}=\frac{s_{old}}{r},a=t_h\alpha \sqrt{N}.\end{array}$$

The new system is as follows ($t_h=0$)

$$\left\{\begin{array}{c}\frac{dx}{dt}=x(1-x)-\sqrt{x}y,\hfill \\ \frac{dy}{dt}=-sy+c\sqrt{x}y,\hfill \end{array}\right.$$

(5)

where s represents the mortality rate of predators, and c represents the conversion rate from prey to predator. The system has three equilibrium points:

$$O(x=0,y=0),E_1(x=1,y=0),E_2(x=x^{*},y=y^{*}),\mathrm{where}{x}^{*}=\frac{s^2}{c^2},y^{*}=\frac{s(c^2-s^2)}{c^3}.$$

Linearize the system (5) and obtain the Jacobian matrix:

$$A=\left(\begin{array}{cc}1-2x-\frac{y}{2\sqrt{x}}& -\sqrt{x}\\ \frac{cy}{2\sqrt{x}}& -s+c\sqrt{x}\end{array}\right).$$

The stability analysis of the deterministic system of the equilibrium point $E_1$ and $E_2$ is as follows (See Reference [20]):

Theorem 1.

When $s>c$, the equilibrium point $E_1(1,0)$ is stable when $s<c$, $E_1$ is the unstable equilibrium point.

Theorem 2.

When $\frac{c}{\sqrt{3}}<s<c$, the coexistence equilibria $E_2$ is stable equilibria, when $0<s<\frac{c}{\sqrt{3}}$, $E_2$ is an unstable equilibrium point.

3.2. Stochastic Stability Analysis of Equilibrium Points

In order to explore the influence of random environmental fluctuations on the equilibrium point of the system, the stability and transient dynamic behavior of $E_1(1,0)$ and positive equilibrium point $E_2(\frac{s^2}{c^2},\frac{s(c^2-s^2)}{c^3})$ of system (5) under white noise random disturbances are studied. These disturbances are proportional to the deviation of the solution $(x,y)$ from the equilibrium point $(x^{*},y^{*})$. This is equivalent to studying the effect of random disturbances on stability. Let $W_1$ and $W_2$ be independent Wiener processes and consider the stochastic system:

$$\left\{\begin{array}{c}dx=(x(1-x)-\sqrt{x}y)dt+{\sigma }_1(x-x^{*})d{W}_1,\hfill \\ dy=(-sy+c\sqrt{x}y)dt+{\sigma }_2(y-y^{*})d{W}_2,\hfill \end{array}\right.$$

(6)

next, system (6) is linearized at the equilibrium point and $(x-x^{*},y-y^{*})$ is replaced by $(X,Y)$ to obtain a linearized predator–prey model (7) with square root functional responses and random disturbance:

$$\begin{array}{cc}\hfill d\left(\begin{array}{c}X\\ Y\end{array}\right)& =\left(\begin{array}{cc}1-2x-\frac{y}{2\sqrt{x}}& -\sqrt{x}\\ \frac{cy}{2\sqrt{x}}& -s+c\sqrt{x}\end{array}\right)\left(\begin{array}{c}X\\ Y\end{array}\right)dt\hfill \\ \hfill & +\left(\begin{array}{cc}{\sigma }_1& 0\\ 0& 0\end{array}\right)\left(\begin{array}{c}X\\ Y\end{array}\right)d{W}_1+\left(\begin{array}{cc}0& 0\\ 0& \sigma {}_2\end{array}\right)\left(\begin{array}{c}X\\ Y\end{array}\right)d{W}_2.\hfill \end{array}$$

(7)

According to Definition 4, we can obtain the mean square stability matrix S:

$$\left(\begin{array}{cccc}2(1-2x-\frac{y}{2\sqrt{x}})+{\sigma }_1^2& -\sqrt{x}& -\sqrt{x}& 0\\ \frac{cy}{2\sqrt{x}}& 1-2x-\frac{y}{2\sqrt{x}}-s+c\sqrt{x}& 0& -\sqrt{x}\\ \frac{cy}{2\sqrt{x}}& 0& 1-2x-\frac{y}{2\sqrt{x}}-s+c\sqrt{x}& \sqrt{x}\\ 0& \frac{cy}{2\sqrt{x}}& \frac{cy}{2\sqrt{x}}& -2(s-c\sqrt{x})+{\sigma }_2^2\end{array}\right).$$

In the next section, an appropriate parameter range s will be selected to simulate the root-mean-square resilience, the root-mean-square reactivity, and the root-mean-square amplification envelope of the system when the equilibrium points are disturbed.

4. Numerical Simulation and Analysis

In this section, the transient dynamics and stability of the system under random perturbation are analyzed based on numerical simulation. The fixed parameter is $c=1$, and the mortality rate of predator s is the variable parameter. In this paper, we use the mathematical software MATLAB 2015b to carry out numerical simulation.

4.1. Transient Dynamics Analysis of Equilibrium Point $E_1(1,0)$

(1)

The system with no environment disturbances $({\sigma }_1=0,{\sigma }_2=0)$

Figure 1 shows the variation of the resilience and reactivity (in the mean square sense) of (7) with respect to the equilibrium point $E_1$ in the absence of random environmental fluctuations for a given range of parameters. It can be used as a comparison diagram to analyze the effects of random disturbances on the system. In Figure 1, the system (7) is mean-square stable when $s\ge c=1.$ The root-mean-square resilience increases as the parameter s goes up, and reaches the maximum value 2 until $s=2.$ The root-mean-square reactivity shows a monotonically decreasing trend and until $s=1.24$ and the reactivity disappeared.

(2)

Disturbance to prey $({\sigma }_1=0.4,{\sigma }_1=0.8$ and ${\sigma }_2=0)$

Figure 2 depicts random environmental fluctuations acting only on prey, with perturbations of ${\sigma }_1=0.4,{\sigma }_1=0.8$ and ${\sigma }_2=0$, respectively. Comparing Figure 1 and Figure 2, (1) when there is no random disturbance, the system itself is also reactive. The range of parameters s for the existence of the root mean square reactivity becomes larger from $s\in (0,1.25)$ to $s\in (0,1.26)$ with the strength of the disturbance increasing from ${\sigma }_1=0.4$ to ${\sigma }_1=0.8$, that is, the ability of the system leaves the steady state to enhance. (2) The trend of the root-mean-square resilience and the stability region have not changed, but the resilience is decreasing. In other words, as the stability of the system becomes weaker, the root-mean-square resilience is on behalf of the self-recovery ability.

(3)

Disturbance to predator $({\sigma }_2=0.4,{\sigma }_2=0.8$ and ${\sigma }_1=0)$

Figure 3 describes random environmental fluctuations acting only on predators with ${\sigma }_2=0.4,{\sigma }_2=0.8$ and ${\sigma }_1=0$, respectively. (1) When the predator suffers the same perturbation, the parameter range of the existence of the root-mean-square reactivity changes more obviously than that of the prey’s. (2) With the increase of the disturbance intensity, the parameter range of the mean square stability also decreases correspondingly from $s\ge 1.08$ to $s\ge 1.32.$

From Figure 1, Figure 2 and Figure 3, we find that both the changes of resilience and reactivity show a linear trend, which is worth thinking about. To explore the truth of this situation, we return to the theoretical analysis of system stability. The eigenvalues of the Jacobian matrix at the equilibrium point $E_1$ are found to be linear with respect to the parameter s, and its characteristic values are $-2,2-2s,-s,-s.$ This is consistent with the linear trend in the graph. However, in order to demonstrate the influence of random disturbance on the stability of the system more intuitively, we use a numerical model and drawing form to show the change of stability. At the same time, we also found that when s reaches a certain value, the reactivity disappears. The reason for this is that as the parameter s increases, the resilience (the ability of the system returns to the steady state) is gradually greater than the reactivity (the ability of the system leaves the steady state), so the reactivity will decay all the way to zero.

No matter which population is disturbed, the root-mean-square reactivity demonstrates a decreasing trend; however, the change of root-mean-square reactivity is more obvious when the predator is disturbed. Thus, the system is more sensitive to disturbances to the predator.

4.2. The Transient Dynamics Analysis of Coexistence Equilibrium Point $E_2(\frac{s^2}{c^2},\frac{s(c^2-s^2)}{c^3})$

(1)

The system with no environment disturbances $({\sigma }_1=0,{\sigma }_2=0)$

Figure 4 shows the variation of resilience and reactivity (in the mean square sense) of the coexistence equilibrium point in the given range of parameters with no random environmental fluctuations. It can be used as a comparison diagram to analyze the effects of random disturbances on the system. From Figure 4, we can observe that the stability region is $s\in [0.577,1]$, which is in accordance with the range of stable parameters determined by Theorem $3.2$. The root-mean-square reactivity first decreases from $0.6$ to $0.12$ and then increases to $0.20$ with the increase of s.

(2)

Disturbance to prey $({\sigma }_1=0.4,{\sigma }_1=0.8$ and ${\sigma }_2=0)$

It can be observed from Figure 5: (1) as the disturbance strength ${\sigma }_1$ increases, the root-mean-square reactivity increases gradually. (2) The range of the mean-square stability parameter s decreases from [0.60, 1] to [0.66, 1], indicating that the mean-square stability of the system decreases. (3) With the increase in ${\sigma }_1$, even when $\frac{1}{\sqrt{3}}<s<1$ (the equilibrium point $E_2$ is stable when there is no random disturbance), the equilibrium point $E_2$ is no longer stable, implying that the stability of the system is damaged by the random disturbance and the dynamic behavior is changed.

(3)

Disturbance to predator $({\sigma }_2=0.4,{\sigma }_2=0.8$ and ${\sigma }_1=0)$

In Figure 6, (1) as the disturbance intensity ${\sigma }_2$ increases, the root-mean-square reactivity increases, the ability of the system leaves the steady state rises. (2) The range of the mean square stable parameter s decreases from [0.6, 0.92] to [0.64, 0.90]. (3) With the increase in ${\sigma }_2$ even when $\frac{1}{\sqrt{3}}<s<1$ (the range of stability parameters of the determinate system), the root-mean-square resilience of the system cannot restore the system to the stable equilibrium point, and $E_2$ is no longer stable.

According to the conclusion in Figure 5 and Figure 6, no matter which species is disturbed, the root-mean-square reactivity first decreases to the minimum and then goes up with the increase in disturbance intensity from $0.4$ to $0.8$. For the coexistence equilibrium point, the stability region of the system becomes smaller and smaller with the increase in disturbance intensity. The stability region of the system changes significantly when the predator is disturbed versus that of prey, in which the system is more sensitive to the predator.

4.3. Root-Mean-Square Amplification Envelope of Coexistence Equilibrium Point $E_2$

In Figure 7 and Figure 8, ($E_2$ is a stable positive equilibrium point when $c=1$ and $s=0.8$); (1) when there is no random disturbance or the disturbance is small, the disturbance will eventually decay to zero, although when being amplified, the system will return to the steady state. (2) When the disturbance ${\sigma }_1\in [0,0.7]$ of prey, the disturbance will eventually decay to zero; however, when ${\sigma }_2=0.7$, the amplification envelope is magnified infinitely, the stability of the system is destroyed and the dynamic behavior is changed.

Figure 9 shows how to reach the equilibrium state when the coexistence equilibrium point $E_2$ is not disturbed with the initial value $(0.5,0.5)$. The first figure in Figure 9 is the time history diagram. It can be clearly observed that when both predators and prey are not disturbed, they will eventually return to the equilibrium state despite a brief fluctuation, which is consistent with the solid line $({\sigma }_1=0, {\sigma }_2=0)$ in Figure 8. Figure 10 show the phase diagram and time history of the system when the prey is not disturbed, while the predator is disturbed by an intensity of 0.4. Comparing Figure 9 and Figure 10, the random disturbance causes more fluctuations in the system, but it can still return to the equilibrium point, which is consistent with Figure 8. Readers can easily understand the transient phenomenon through the phase diagram and time history diagram in Figure 9 and Figure 10.

No matter which population is disturbed, the greater the disturbance intensity, the greater the amplification envelope. When the predator is disturbed, the system becomes more sensitive and responds more strongly to the random disturbances.

5. Conclusions

In this paper, we study 3 key stochastic stability indexes (root-mean-square resilience, root-mean-square reactivity and root-mean-square amplification envelope) of a predator–prey model with root-mean-square functional response under random perturbations. For the equilibrium point $E_1$ and the coexistence equilibrium point $E_2$, the changes of the three indexes with increasing disturbance to prey and predator are analyzed, respectively. No matter which species is disturbed, the root-mean-square reactivity goes up with the increase in disturbance intensity.

As for the root-mean-square amplification envelope, when the perturbation ${\sigma }_1=0.7$ for prey, the system can recover the steady state, but for ${\sigma }_2=0.7$ for predator, the stability of the system is destroyed and the dynamic behavior of the system is changed even if the parameter s in the stable parameter range of the deterministic system. By comparing the changes of the three indexes, it can be found that the system is more sensitive to the perturbation of predator than prey. We also found that ${\rho }_{max}$ and $t_{max}$ are important indicators, which represent the intensity and time of the maximum disturbance amplification.

For biological models with population defense and the functional response in the square root of prey, scholars mainly study the asymptotic dynamics and analyze the long term equilibrium state of the system. In this paper, we study the transient dynamics of the model, and obtain how the system reacts in the initial stage of disturbance, which provides reference for management scientists and biologists. They are helpful for managers to take corresponding management measures to reduce the disturbances, especially for predators, thereby avoiding the possible change of the structure and functions of the ecosystem.