Analysis of a stochastic predator-prey system with mixed functional responses and Lévy jumps

1.   Introduction

Recently a lot of attention has been paid to the dynamic relation between two predators sharing a common prey species ^[1,2]. Three species predator-prey models are fundamental for building blocks of large scale ecosystems. In ^{[3,4,5,6,7,8]}, the authors considered a widely existing two-predator-prey population model. The three species predator-prey model with intraguild predation involves a prey, an intermediate predator which feeds upon only prey and a top predator (called intraguild predator) which feeds upon both prey and intermediate predator ^[9,10]. D. Sen ^[10] considered Holling type-II function responses between intermediate predator and top predator and other function responses were assumed to be linear which can be denoted by

where $x(t)$, $y(t)$ and $z(t)$ represent population size of each species at time $t$, respectively, $a_1 > 0$ stands for the growth rate of the species $x(t)$, $a_2$ and $a_3$ stands for the death rate of species $y(t)$ and $z(t)$, respectively, $b_{jj}(j = 1, 2, 3)$ is the intra-specific competition rate of the species $x(t), y(t)$ and $z(t)$. $b_{12}$ and $b_{13}$ represent the predation or consumption rate of intermediate and top predator feeding upon prey respectively, $c_1$ is the predation rate of the top predator feeding upon intermediate predator, $b_{21}$ and $b_{31}$ measure prey consumption into reproduction for intermediate predator and top predator respectively and $c_2$ measures intermediate predator consumption into reproduction for top predator. Parameters $a_i, b_{jj}$ and $c_i$ are positive constants for $i = 1, 2, j = 1, 2, 3$.

In fact, in the real world, most natural phenomena can not be explained by deterministic laws and are always affected by environmental noise, which is an inevitable property of any ecosystem dynamics (see ^{[11,12,13,14,15,16]}). So, modelling population dynamics with white noise and jumps has become an active and fruitful topic in mathematical biology. A large number of literatures show that many researchers introduce Brownian motion into the deterministic model to describe stochastic effects and establish predator-prey stochastic model. May ^[17] pointed out that due to environmental fluctuations, models involved in the birth rate or other parameters, bearing capacity, competition coefficient showed stochastic fluctuations more or less. X. H. Zhang ^[18] considered a stochastic Holling II one-predator two-prey system with jumps. Mao ^[19] pointed out that even a small amount of environmental noise can suppress a potential population explosion. Liu ^[20] established a random predator-prey model with stage structure in response to predator and Holling II functions, investigated the existence of uniqueness of traversal stable distribution, and obtained sufficient conditions for the extinction of predator populations.

In model (1.1), if the intrinsic growth rate of prey and the death rate of predators are not constants but are subject to environmental noise, then $a_j \; (j = 1, 2, 3)$ are stochastically perturbed with

where $\sigma _{1}^{2}, \sigma _{2}^{2}$ and $\sigma _{3}^{2}$ stand for the intensities of the white noise, $B(t)$ denotes the standard Brownian motion which is defined on a complete probability space $(\varOmega, \mathcal{F}, \left\{ \mathcal{F} \right\} _{t\ge 0}, \mathbb{P})$. Then we obtain the following stochastic three species predator-prey model with white noise and mixed functional responses

In mathematical modelling, some sudden environmental disturbances such as earthquake, epidemic diseases can't be described by white noise. These events are so strong that they can break the continuity of the sample path. Therefore, white noise cannot accurately describe these phenomena. In this case, introducing a Lévy jumps might be a reasonable approach. Based on the Brownian motion cases, jump process is introduced as the noise source by authors ^[21,22,23]. Bao et al. ^[21] initially used Lévy jumps to describe these phenomena and proposed stochastic competition model with jumps. Liu ^[23] investigated some asymptotic properties of a stochastic n-species Gilpin-Ayala competitive model and illustrated that these properties have close relationships with Lévy jumps. Inspired by the above discussion, we establish the following three predator-prey model with mixed responses and Lévy jumps:

with initial data

where $x\left(t^- \right)$, $y\left(t^- \right)$ and $z\left(t^- \right)$ represent the left limit of $x\left(t \right)$, $y\left(t \right)$ and $z\left(t \right)$, respectively. Parameters $a_j$, $b_{jj}$, $c_i$ and $\sigma _j, \left(i = 1, 2, j = 1, 2, 3 \right)$ are all positive constant. $N$ is a poisson counting measure with compensator $\tilde{N}$ and characteristic measure $\lambda$ on a measurable subset $\mathbb{Z}$ of $(0, \infty)$ with $\lambda(\mathbb{Z}) < \infty$ and $\tilde{N}(dt, du) = N(dt, du)-\lambda(du)dt$. Parameter $\gamma _j$ is the effect of Lévy noise on the $jth$ species. $\gamma _j(u) > 0$ represents the increasing of the species, and $\gamma _j(u) < 0$ represents the decreasing of the species. Therefore, it is reasonable to assume that $1+\gamma _j(u) > 0$ for $u \in \mathbb{Z}, j = 1, 2, 3$.

The organization of this paper is as follows. In Section 3, we obtain the global existence of positive unique solution of system (1.3). In Section 4, we discuss the asymptotical properties of system (1.3). In Section 5, we establish sufficient conditions for the asymptotic stability in distribution of (1.3). Finally, some numerical simulations are provided to illustrate our main results.

2.   Preliminaries

Throughout this paper, let $\left(\varOmega, \mathcal{F}, \left\{ \mathcal{F}_t \right\} _{t\ge 0}, \mathbb{P} \right)$ be a complete probability space with a filtration $\left\{ \mathcal{F}_t \right\} _{t\ge 0}$ satisfying the usual conditions (i.e. it is right continuous and $\mathcal{F}_0$ contains all $\mathbb{P}-null$ sets). Let $B\left(t \right)$($t\ge 0$) be a scalar standard Brownian motion defined on this probability space. We assume that $N$ and $B$ are independent. For biological reason, we suppose that $1+\gamma _j(u) > 0$, where $\gamma _j(u) > 0$ means the increasing of the species (e.g., planting) and $-1 < \gamma _j(u) < 0$ means the decreasing of the species (e.g., harvesting and epidemics), $u \in \mathbb{Z}, j = 1, 2, 3$.

For our discussion, some technical assumptions are given as follows.

Assumption 1 We assume

where $C_i(i = 1, 2)$ is positive constant, $j = 1, 2, 3.$

Assumption 2 $b_{22}-b_{21} > 0, b_{33}-b_{31} > 0$, which means that the influence of intraspecific competition is greater than the interaction between different species.

For convenience, we cite the following notions:

3.   Positive and global solution

In this section, under Assumption 1, we show the solution of system (1.3) is not only positive, but also will not explode to infinity at any finite time. That is, for any positive initial value, (1.3) has unique positive global solution.

Theorem 3.1. Under Assumption 1, system (1.3) has a unique solution $X(t) = (x(t), y(t), z(t))$ on $t\ge 0$ a.s. for any initial value $X(0) = (x(0), y(0), z(0)) \in \mathbb{R}_{+}^{3}$.

Proof. Since the coefficients of (1.3) are locally Lipschitz continuous, then for any initial value $X(0) = \left(x\left(0 \right), y\left(0 \right), z\left(0 \right) \right) \in \mathbb{R}_{+}^{3}$, there is a unique local solution $X(t) = \left(x\left(t \right), y\left(t \right), z\left(t \right) \right)$ on $t\in \left[ 0, \tau _e \right)$ a.s., where $\tau _e$ is the explosion time. To show this solution is global, we need to show that $\tau _e = \infty$ a.s. Let $k_0 > 0$ be sufficiently large such that $(x(0), y(0), z(0)) \in (\frac{1}{k_0}, k_0)$. For each integer $k\ge k_0$, define the stopping time

where $\inf {\emptyset} = \infty$ (as usual ${\emptyset}$ denotes the empty set). Clearly, $\tau _k$ is increasing as $k\rightarrow\infty$. Set $\tau _\infty = \lim_{k\rightarrow\infty}\tau _k$, whence $\tau _\infty \le \tau _e$ a.s. If we can show that $\tau _\infty = \infty$ a.s., then $\tau _e = \infty$ and therefore $\left(x\left(t \right), y\left(t \right), z\left(t \right) \right) \in \mathbb{R}_{+}^{3}$ a.s. In other words, to complete the proof we need to show that $\tau _\infty = \infty$ a.s. If this statement is false, there is a pair of constant $T > 0$ and $\epsilon \in (0, 1)$ such that $\mathbb{P}\left\{ \tau _{\infty}\le T \right\} > \epsilon$. Hence there is an integer $k_1 \ge k_0$ such that $\mathbb{P}\left\{ \tau _k\le T \right\} > \epsilon$ for all $k\ge k_1$.

We write $x(t) = x, y(t) = y, z(t) = z$ and define a $C^2$-function $V: \mathbb{R}_{+}^{3}\rightarrow \mathbb{R}$ by

where $\alpha = \frac{b_{12}}{b_{21}}$ and $\varrho = \frac{b_{13}}{b_{31}}$ are both positive constants. The nonnegativity of this function can be seen from

By the generalized Itô's formula, we yield

It is easy to see that there is a constant $G$ such that $\hat{\mathcal{G}}\left(x, y, z \right)\le G$, that is $LV(x, y, z)\le G$. We have

Integrating both sides of (3.1) from $0$ to $\tau _k\land T = \min \left\{ \tau _k, T \right\}$, and then taking expectations, yield

For $k\ge k_1$, let $\Omega_k = {\tau _k\le T}$. According to $\mathbb{P}(\tau _k\le T)\ge \epsilon$, then $\mathbb{P}(\Omega_k)\ge \epsilon$, and we obtain

where $I_{\varOmega _k}$ is the indicator function of $\Omega_k$. Letting $k\rightarrow\infty$ leads to the following contradiction

Then $\tau _{\infty} = \infty$ a.s., which means $\tau _{e} = \infty$ a.s. This completes the proof.

Lemma 3.1. (^[24]) For any $\beta, \varpi \in \mathbb{R}$ and any $s, r, \varepsilon > 0$, we have the variation of the Young's inequality

Lemma 3.2. Let $X(t) = (x(t), y(t), z(t))$ be a solution of system (1.3). Under Assumptions 2, for any initial value $X(0) = (x(0), y(0), z(0)) \in \mathbb{R}_{+}^{3}$, there exists $K_j(p) > 0, j = 1, 2, 3$ such that

That is, the p-th moment of the positive solution to (1.3) is upper bounded.

Proof. By the Itô's formula, we have

Integrating two sides of (3.3) and taking expectations leads to

Therefore, we have

Let $\hat{X}(t) = \mathbb{E}(x^p(t))$, then

Let $x\left(0 \right) < \frac{a_1+\frac{1}{2}\left(p-1 \right) \sigma _{1}^{2}+\frac{\tilde{\gamma}_1}{p}}{b_{11}}, p\ge1$, then

By the standard comparison argument, we have

Thus,

Next, we prove the boundedness of $y(t)$. Making use of the Itô's formula to $e^ty^p$, we have

Integrating the both sides from $0$ to $t$ and taking expectations yields

Using Lemma 3.1, we have

Let $\varGamma \left(y \right) = y^p \left(\left[ \frac{1+\tilde{\gamma}_2}{p}+\frac{1}{2}\left(p-1 \right) \sigma _{2}^{2}\right] -\left(b_{22}-b_{21} \right) y \right)$. To attain the maximum value of $\varGamma(y)$, we obtain $y = \frac{p\left[ \frac{1+\tilde{\gamma}_2}{p}+\frac{1}{2}\left(p-1 \right) \sigma _{2}^{2}\right]}{\left(1+p \right) \left(b_{22}-b_{21} \right)} > 0$. So the maximum value of $\varGamma \left(y \right)$ is given by

Hence, we obtain that

One can observe that for $t = 0$, $\mathbb{E}(y^p(t))\le y^p(0)$ and when $t\rightarrow \infty$,

Finally, we use the same method to prove the boundedness of $z(t)$. Obviously,

Integrating two sides of (3.4) and taking expectations leads to

As a result,

Under Assumption 2, it is clear that $K_j(p) > 0(j = 1, 2, 3).$ Therefore, the p-th moment of the positive solution to (1.3) is upper bounded. The proof is completed.

By Lemma 3.2, together with the Chebyshev inequality, we can obtain the following result.

Corollary 3.1. Under Assumption 2, the solution of (1.3) is stochastically ultimate bounded.

4.   Asymptotical property

In this section, we investigate the asymptotical property of (1.3).

Theorem 4.1. Under Assumption 1, the solution $X(t) = (x(t), y(t), z(t))$ of system (1.3) with any positive initial value has the property that

Particularly, if $r_1 < 0$, then $X(t) = (x(t), y(t), z(t))$ will go to extinction.

Proof. By Theorem 3.1, the solution $X(t) = (x(t), y(t), z(t))$ with initial value $X(0) = (x(0), y(0), z(0))\in \mathbb{R}_{+}^{3}$ remains in $\mathbb{R}_{+}$ with probability one. By the generalized Itô's formula, we derive from (1.3) that

where, for $j = 1, 2, 3$,

are local martingale with the quadratic variations $\left < N_{ji}(t)\right > : = \left < N_{ji}(t), N_{ji}(t)\right > , i = 1, 2$ ^[11], that is

By Lemma 3.1 in ^[21] and the strong law of large numbers, we obtain

From (4.2) and (4.5), we have $\limsup_{t\rightarrow \infty}\frac{\ln x\left(t \right)}{t}\le r_1\ a.s.$ If $r_1 < 0,$ then

Applying L'Hospital's rule, it follows that

Further, by combining (4.3), (4.5) and (4.6), we can get $\limsup_{t\rightarrow \infty}\frac{\ln y\left(t \right)}{t}\le -r_2+b_{21}\lim_{t\rightarrow \infty}\left < x\left(t \right) \right > = -r_2 < 0\ a.s.$ So

Similarly, we have $\lim_{t\rightarrow \infty}\left < y\left(t \right) \right > = 0\ a.s.$ Noting $\left < \frac{y\left(t \right)}{1+y\left(t \right)} \right > \le \left < y\left(t \right) \right >$, and from the positivity of $y(t)$, we have

Combining (4.4), (4.5) and (4.7), we get $\limsup_{t\rightarrow \infty}\frac{\ln z\left(t \right)}{t}\le -r_3+b_{31}\lim_{t\rightarrow \infty}\left < x\left(t \right) \right > +c_2\lim_{t\rightarrow \infty}\left < \frac{y\left(t \right)}{1+y\left(t \right)} \right > = -r_3 < 0\ a.s.$ Thus,

The proof is therefore completed.

Theorem 4.2. If $A_j > 0, D_j > 0 \; (j = 1, 2, 3)$, then for any given initial value $X(0) = (x(0), y(0), z(0))\in \mathbb{R}_{+}^{3}$, the solution $X(t) = (x(t), y(t), z(t))$ to system (1.3) has the following property

That is, all the populations in system (1.3) are persistent in mean.

Proof. It follows from (4.2) and (4.5) that

From Lemma 2 in ^[25], we obtain

From (4.3)–(4.5) and (4.9), we get

Thus, from Lemma 2 in ^[25], we have

From Theorem 4.1, we have $\limsup_{t\rightarrow \infty}\frac{\ln x\left(t \right)}{t} = 0$. By combining (4.2), (4.5), (4.10) and (4.11), then

Similarly, by Theorem 4.1, noting $\lim_{t\rightarrow \infty}\left < \frac{y\left(t \right)}{1+y\left(t \right)} \right > = 0\ a.s.$, we have

and

Therefore, under above conditions, all the populations of (1.3) are persistent in mean. This completes the proof.

Remark 4.1. Compared with the literature ^[26], the influence of Lévy jumps to system (1.3) is considered in this paper, while it is ignored in ^[26]. For the case without jump-diffusion coefficient, our main results are consistent with those in ^[26]. Therefore, the results of literature ^[26] are generalized in this paper.

5.   Asymptotic stability in distribution

In ^[27], for all parameter values, deterministic system (1.1) has a trivial equilibrium point $E_0 = (0, 0, 0)$ and an axial equilibrium point $E_1 = (\frac{a_1}{b_{11}}, 0, 0)$. In addition, when $b_{31}a_1-b_{11}a_3 > 0$ and $b_{21}a_1-b_{11}a_2 > 0$, respectively, deterministic system (1.1) has two boundary equilibria $E_2 = (\frac{b_{33}\; a_1+b_{13}\; a_3}{b_{11}\; b_{33}+b_{13}\; b_{31}}, 0, \frac{b_{31}\; a_1-b_{11}\; a_3}{b_{11}\; b_{33}+b_{13}\; b_{31}})$ and $E_3 = (\frac{b_{22}\; a_1+b_{12}\; a_2}{b_{11}\; b_{22}+b_{12}\; b_{21}}, \frac{b_{21}\; a_1-b_{11}\; a_2}{b_{11}\; b_{22}+b_{12}\; b_{21}}, 0)$. However, in the real world, population systems are often affected by environmental noise. So, when considering the introduction of Lévy jumps into population models, we are also interested in knowing how jumps affect the long-term dynamic behavior of species. That is what happens to the statistical characteristics of the species' long-term dynamic behavior. Thus, invariant distribution plays an important role in many actual ecosystems. In this section, our work aims to find sufficient conditions to obtain that the probability density function is asymptotically stable for the system (1.3) perturbed by Lévy noise. First we give the following definition and lemmas.

Definition 5.1. Let $X_1(t) = (x_1(t), y_1(t), z_1(t))$ be a positive solution of (1.3) with initial value $X_1(0) = (x_1(0), y_1(0), z_1(0))\in \mathbb{R}_{+}^{3}$. $X_1(t)$ is said to be globally asymptotically stable in expectation if for any other solution $X_2(t) = (x_2(t), y_2(t), z_2(t))$ of (1.3), we have

where $\mathbb{E}$ denotes the expectation of some stochastic variable.

Lemma 5.1. (^[28,29]) Let $X(t)$ be an n-dimensional stochastic process on $t\ge0$. Suppose that there exist positive constant $\alpha, \beta, \xi$ such that

Then there exists continuous modification $\tilde{X}\left(t \right)$ of $X(t)$, and almost every sample path of $\tilde{X}\left(t \right)$ is local but uniformly $H\ddot{o}der$ continuous with exponent $\kappa < \frac{\alpha}{\beta}$. In other words, the continuous modification $\tilde{X}\left(t \right)$ of $X(t)$ has the property that for every $\kappa \in (0, \frac{\alpha}{\beta})$,

Lemma 5.2. Assume that Assumptions 1 and 2 hold. Let $X(t) = (x(t), y(t), z(t))$ be a solution of (1.3) on $t\ge 0$ with initial data $X(0) = \left(x\left(0 \right), y\left(0 \right), z\left(0 \right) \right) \in \mathbb{R}_{+}^{3}$, then almost every sample path of $X(t)$ is uniformly continuous on $t\ge0$.

Proof. We rewrite $x(t)-x(0)$ as the following integral form:

Let $f_1 = x\left(t \right) \left[ a_1-b_{11}x\left(t \right) -b_{13}y\left(t \right) -b_{13}z\left(t \right) \right], g_1 = x\left(t \right) \sigma _1, h_1 = x\left(t \right) \gamma _1\left(u \right).$ By Lemma 3.2, there is a positive constant $K_1(p)$ such that $\mathbb{E}\left(x^p\left(t \right) \right) \le K_1\left(p \right)$ on $t\ge0$. Then we can derive that

and

We assume $p > 2$. For $0\le s < t < \infty$, using the moment inequality (cf. Friedman ^[30]) on (5.1) leads to

Under Assumption 2, with Kunita's first inequality (see Theorem 4.4.23, ^[31]), we have

Let $0 < s < T < \infty, t-s\le1, 1/p+1/q = 1$, then from (5.1)–(5.4), we obtain

where $Q\left(p \right) = \left(t-s \right) ^{\frac{p}{2}}Q_1(p)+\left[ \frac{p\left(p-1 \right)}{2} \right] ^{p/2}Q_2(p)+2^{p-1}C_{1}^{\frac{p}{2}}K_1\left(p \right) +2^{p-1}\left(t-s \right) ^{\frac{p}{2}}C_{1}^{\frac{p}{2}}K_1\left(p \right) < \infty$. We see from Lemma 5.1 that almost every sample path of $x(t)$ is locally but uniformly $H\ddot{o}der$ continuous with exponent $\kappa$ for every $\kappa \in (0, \frac{p-2}{2p})$. Therefore almost every sample path of $x(t)$ is uniformly continuous on $t\ge 0$. Similarly, we have

and

Let

then,

and

A similar discussion to $\mathbb{E}|y\left(t \right) -y\left(s \right) |^p$ and $\mathbb{E}|z\left(t \right) -z\left(s \right) |^p$, we can conclude that almost every sample path of $y(t)$ and $z(t)$ is uniformly continuous on $t\ge 0$. This completes the proof.

Lemma 5.3. Let $f(t)$ be a nonnegative function defined on $\left[0, \infty \right)$ such that $f(t)$ is integrable on $\left[0, \infty \right)$ and is uniformly continuous on $\left[0, \infty \right)$, then $lim_{t \to +\infty} f(t) = 0$.

For later proof, we give the following technical assumption.

Assumption 3

where

Lemma 5.4. If Assumption 2 and 3 hold, then system (1.3) is globally asymptotically stable in expectation.

Proof. Let $X_1(t) = \left(x_1\left(t \right), y_1\left(t \right), z_1\left(t \right) \right)$ and $X_2(t) = \left(x_2\left(t \right), y_2\left(t \right), z_2\left(t \right) \right)$ be any two solutions of system (1.3) with positive initial data. Consider a Lyapunov function $V(t)$ defined by

Making use of the Itô's formula with jumps, one can deduce that

Integrating from 0 to $t$ and taking expectations yields

Therefore,

By Lemma 3.2,

Therefore

By Assumption 3, then

It then follow from $V(t) \ge 0$ that $\mathbb{E}|x_1(t)-x_2(t)|\in L^1\left[ 0, \infty \right), \mathbb{E}|y_1(t)-y_2(t)|\in L^1\left[ 0, \infty \right)$ and $\mathbb{E}|z_1(t)-z_2(t)|\in L^1\left[ 0, \infty \right)$. Therefore,

Further, we can easily see from Lemma 5.2 that $|x_1(t)-x_2(t)|, |y_1(t)-y_2(t)|$ and $|z_1(t)-z_2(t)|$ are uniformly continuous with respect to $t$. So by Lemma 5.3 we easily obtain

This completes the proof.

Remark 5.1. From Lemma 5.4, it shows that the jump-diffusion coefficient $\gamma_j$ has influence on the globally asymptotically stable of system (1.3) in expectation. In other words, when Lévy noise intensity is too high then the system is not globally asymptotically stable in expectation.

Theorem 5.1. Under the conditions of Lemma 5.4, system (1.3) is asymptotically stable in distribution. That is, there exists a unique probability measure $\mu(\cdot)$ such that for any initial value $X(0) = (x(0), y(0), z(0)) \in \mathbb{R}_{+}^{3}$, the transition probability $p(t, X(0), \cdot)$ of $\tilde{X}(t)$ weakly converges to $\mu(\cdot)$ as $t\rightarrow \infty$.

Proof. Let $p(t, X(0), dY)$ denote the transition probability of the event ${X(t; X(0))\in \mathcal{B}}$, where $\mathcal{B}$ is a Borel measurable set of $\mathbb{R}_{+}^{3}$. Let $\mathcal{P}\left(\mathbb{R}_{+}^{3} \right)$ denote all probability measures on $\mathbb{R}_{+}^{3}$. For any $\mathcal{P}_1, \mathcal{P}_2 \in \mathcal{P}\left(\mathbb{R}_{+}^{3} \right)$, we define metric $d_{\mathcal{K}}$ as follow:

where $\mathcal{K} = \left\{ g:\mathbb{R}_{+}^{3}\rightarrow \mathbb{R}:|g\left(X \right) -g\left(Y \right) |\le ||X-Y||, |g\left(\cdot \right) \le 1 \right\}.$ First, we prove $p((t, X(0), dY):t\ge 0)$ is cauchy in the space $\mathcal{P}\left(\mathbb{R}_{+}^{3} \right)$ with metric $d_{\mathcal{K}}$. According to Lemma 3.2 and Chebyshev inequality, ${p((t, X(0), dY):t\ge 0)}$ is tight. For any $g\in\mathcal{K}$ and $t, s > 0$, we have

It follows from (5.6) that there is a constant $T\ge0$ such that

Thanks to the arbitrariness of $g$, we have

(5.7) is equivalent to

Therefore, the transition probability $p((t, X(0), \cdot):t\ge 0)$ of the solution of system (1.3) is cauchy in the space $\mathcal{P}\left(\mathbb{R}_{+}^{3} \right)$ with metric $d_{\mathcal{K}}$. So there is a unique $\mu(\cdot)$ such that

Then for any fix $X\left(0 \right) \in \mathbb{R}_{+}^{3}$, combining with (5.7) and (5.8), we have

That is,

The proof is completed.

Remark 5.2. According to the proof of Theorem 5.1, we can use the method presented in this paper to discuss the stability of stochastic systems driven by discrete time noises without Lévy jumps.

6.   Numerical simulations

In the section, we give some examples to demonstrate our main results by using the Milstein method ^[32]. We always choose $a_1 = 0.7, a_2 = 0.02, a_3 = 0.025, b_{11} = 0.5, b_{21} = 0.2, b_{31} = 0.06, b_{12} = 0.6, b_{22} = 0.4, b_{33} = 0.4, b_{13} = 0.12, c_1 = 0.065, c_2 = 0.05, \mathbb{Z} = (0, +\infty), \lambda(\mathbb{Z}) = 1$ with initial value $(x(0), y(0), z(0)) = (0.8, 0.35, 0.15)$.

First, we illustrate the effect of white noise on population dynamics. Let $\sigma _{1} = \sigma _{2} = \sigma _{3} = \gamma _1 = \gamma _2 = \gamma _2 = 0$, then (1.3) is a deterministic system. The dynamics of deterministic case is showed in Figure 1.

For the stochastic case, we choose $\sigma _{1} = 1.2, \sigma _{2} = 0.6, \sigma _{3} = 0.2, \gamma _1 = \gamma _2 = \gamma _2 = 0.48$. After a simple calculation, we have $r_1 = -0.1080 < 0$. By Theorem 4.1, system (1.3) is extinctive, see Figure 2(a). If $\sigma _{1} = 0.2, \sigma _{2} = 0.05, \sigma _{3} = 0.04, \gamma _1 = \gamma _2 = \gamma _3 = 0.3$, then

which means all conditions of Theorem 4.2 hold. Further, by computation we have

That is, (1.3) is persistence in mean, see Figure 2(b). The distributions of all species may see Figure 3.

Next, we demonstrate the effect of Lévy jumps on population dynamics. Let $\sigma _{1} = \sigma _{2} = \sigma _{3} = 0.2, \ \gamma _1 = \gamma _2 = 0.25, \ \gamma _3 = 0.05$. We can check that

Thus, Assumptions 1–3 are satisfied and $A_j > 0, D_j > 0(j = 1, 2, 3)$. By Theorem 4.2, we have

which may see Figure 4 (a).

If $\sigma _{1} = \sigma _{2} = \sigma _{3} = 0.2, \ \gamma _1 = 1.68, \gamma _2 = 0.45, \ \gamma _3 = 0.06$, then $r_1 = -0.0142 < 0$. Theorem 4.1 implies that $x(t), y(t)$ and $z(t)$ are all extinct a.s.(Figure 4(b)). Comparing Figure 4(a) with Figure 4(b), we can find that the Lévy jumps may suppress the survival of the species.

Remark 6.1. For the same set of parameter values, we note that in the absence of environmental noise the deterministic system is uniformly persistent (see Figure 1). In stochastic system, relatively smaller intensity of white noise and Lévy jumps can maintain the survival of species, while the species will be extinctive with larger intensity of white noise and Lévy jumps (see Figure 4). Therefore, whether the species of the stochastic system are persistence in mean or not depends on the intensity of the noise.

7.   Conclusions

In this paper, we discuss a stochastic two predator-one prey system with Lévy jumps and mixed functional responses, which contains ratio-dependent type (between intermediate predator and top predator) and linear functional responses. Firstly, we discuss the existence and the pth moment-boundedness of positive solution. Then under Assumption 2, we establish sufficient criteria for the extinction of system (1.3). The result reveals an important property of the Lévy jumps: They are unfavorable for the existence of species. Furthermore, we establish sufficient condition for asymptotically stable in distribution for system (1.3) under certain conditions. Finally, some numerical simulations are introduced to demonstrate the theoretical results.

Theorem 4.1 shows that the intensity of white noise and Lévy jumps can make prey extinction. This means that unpredictable events in nature are so severe and intense that they can dramatically change population size in a short period of time. The extinction of prey will lead to the extinction of intermediate and top predators. Theorem 4.2 shows that if the intensities $\gamma_1, \gamma_2$ and $\gamma_3$ are small, then all population will persist in mean. Theorem 5.1 shows the existence of a unique invariant probability measure for system (1.3).

Some interesting topics deserve further investigations. As done in ^[16,33], one can introduce time delays in stochastic system (1.3). Moreover, one can study deterministic system (1.1) with other perturbations, such as Markovian switching (see ^{[34,35,36,37,38]}) or second-order stochastic perturbation (see ^[39]). We leave these investigations for future work.

Acknowledgements

The author would like to thank the editor and anonymous referees for their careful reading of the manuscript and valuable suggestions to improve the quality of this paper. This work was supported by the National Natural Science Foundation of China (11861027).

Conflict of interest

The authors declare no conflict of interest.