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LG–Holling type II diseased predator ecosystem with Lévy noise and white noise
Advances in Difference Equations volume 2018, Article number: 48 (2018)
Abstract
This paper considers an LG–Holling type II diseased predator ecosystem with Lévy noise and white noise. In this prey–predator system, assuming that the predator population is infected by disease and divided into two classes: susceptible predator and infected predator, we show that the system has a unique global positive solution. We investigate the persistence in the mean and extinction for the population, obtain threshold conditions of extinction, persistence in the mean by Itô’s formula and a comparison theorem for the stochastic system. In addition, we discuss the uniform boundedness of pth moment with \(p>0\) and reveal the stochastically ultimate boundedness of the system. Finally, some numerical simulations are introduced to illustrate our analytical findings.
1 Introduction
In ecology, prey–predator systems are always the important part, and the most classic model is Lotka–Volterra model. Out of the ordinary, Leslie introduced the following model on the premise that the carrying capacity of predator was proportional to the amount of prey [1, 2]:
here \(N_{1}\) and \(N_{2}\) represented the densities of prey and predator at time t, respectively. \(r_{1}\) and \(r_{2}\) were the intrinsic growth rates for prey and predator populations. \(b_{2}\) was conversion rate. Leslie assumed that the predator population still grew logistically with carrying capacity \(\frac{r_{2}N_{1}}{b_{2}}\).
Leslie assumed that the growth of predator population still satisfied the logistic rule, but the carrying capacity, which was distinguished from the constant in the common model, was determined by the density of prey population. Recently, there have been many experts and researchers studying or improving a Leslie type prey–predator model. Time delay, impulse, food chain, disease, environment pollution, reaction diffusion and so on, have been introduced into the model, see [3–8].
In [9], Sarwardi introduced a contagious disease among the predators in a predator-prey ecosystem, modeled via the following equations with nonnegative parameters:
The system incorporates a Holling type II functional response and a modified version of the Leslie–Gower one. In system (2), the following assumptions are made:
-
(A1)
\(x(t)\) represents the density of prey population at time t, and it is assumed that the disease spreads only among the predator, \(y(t)\) and \(z(t)\) denote the densities of susceptible predator and infected predator at time t, the total predator population is \(n(t)=y(t)+z(t)\). The disease transmission rate is λ.
-
(A2)
In the absence of predators, the prey population grows logistically with intrinsic growth rate \(a_{1}>0\) and carrying capacity \(a_{1}b_{1}^{-1}\). \(c_{1}\) and \(pc_{1}\) are the predation rates for susceptible and infected predator. Because of disease infection, the predation ability for the infected predator is a bit less than that for the susceptible predator; therefore, \(0< p<1\).
-
(A3)
Here \(a_{2}\), \(a_{3}\) are the per capita growth rates of each predator population. The parameters \(k_{1}\), \(k_{2}\) respectively represent the half saturation constants of prey and predator populations. In addition, the authors introduce intraspecific competition among the predators’ sound and infected subpopulations with parameters \(c_{2}\) and \(c_{3}\), for which \(c_{2}>c_{3}\).
The authors studied the stability of equilibrium point of system (2) and persistence.
Population growth in the natural world is inevitably affected by the environmental perturbations. In the past decades, white noise was introduced into the model to simulate the stochastic perturbation of environment. However, sudden environmental perturbation, such as earthquakes, hurricanes, the flood, etc., cannot be modeled by white noise, which is often used to describe those stable and continuous stochastic interferences. Bao et al. in [10, 11] firstly proposed that these phenomena could be described by Lévy noise. At present, there are many research papers considering Lévy noise [6, 12–17]. In this paper, we use white noise and Lévy noise to simulate the random change of environment. The following system with jumps is considered:
with the initial value \((x_{0},y_{0},z_{0})\in R^{3}_{+}\). For convenience, we use p̃ and β instead of p and λ in system (3), respectively. \(B_{1}(t)\), \(B_{2}(t)\) and \(B_{3}(t)\) are mutually independent Brownian motions defined on a complete probability space \((\Omega,\mathcal{F},\mathcal {P})\) with a filtration \({\{\mathcal{F}_{t}\}}_{t\in R_{+}}\) satisfying the usual conditions. \(\sigma_{i} ^{2}\) (\(i=1,2,3\)) denotes the intensity of white noise. In system (3), \(x(t^{-})\), \(y(t^{-})\) and \(z(t^{-})\) are the left limits of \(x(t)\), \(y(t)\) and \(z(t)\), respectively. N is a Poisson counting measure with characteristic measure λ on a measurable subset \(\mathbb{Y}\) of \((0,+\infty)\) with \(\lambda(\mathbb{Y})<\infty\), \(\tilde{N}(dt,du)=N(dt,du)-\lambda(du)\,dt\), \(\gamma_{i}:\mathbb{Y}\times \Omega\rightarrow\mathbb{R}\) is bounded and continuous with respect to λ, and is \(\mathcal{B}(\mathbb{Y})\times\mathcal{F}_{t}\)-measurable, \(i=1,2,3\).
For simplicity, we introduce the following notations:
Assumption 1
There exists a constant c satisfying
Throughout this paper, we assume that Assumption 1 is true.
2 The existence of global positive solution
Theorem 1
For any initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\), system (3) has a unique global solution \((x(t),y(t),z(t))\in R_{+}^{3}\) for \(t\geq0\), and the solution will remain in \(R_{+}^{3}\) almost surely.
Proof
At first, let us consider the following system:
on \(t\geq0\) with the initial data \((\ln x_{0},\ln y_{0}, \ln z_{0})\). Clearly, the coefficients of system (4) satisfy the local Lipschitz condition, then there is a unique local solution on \([0,\tau _{e})\), where \(\tau_{e}\) is the explosion time. Therefore, by Itô’s formula, \((\exp(u(t)),\exp(v_{1}(t)),\exp (v_{2}(t)))\) is the unique positive local solution of system (3) with the initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\). Whereafter, we will show that \(\tau_{e}=\infty\), namely the solution of system (3) is globally existent.
Consider the stochastic differential equations:
and
According to the comparison theorem for stochastic differential equations in [18], we can get that, for \(t\in[0,\tau_{e})\),
By virtue of Lemma 4.2 in [10], we can see that \(\Phi_{i}(t)\) and \(\Psi_{i}(t)\) (\(i=1,2,3\)) will not explode in any finite time, which means that \(\Phi_{i}(t)\) and \(\Psi_{i}(t)\) are existent on \(t\geq0\). Thus, we obtain \(\tau_{e}=+\infty\), the proof of Theorem 1 is completed. □
According to Lemma 4.4 in [10] and equations (5), we know that
for \(i=1,2,3\), when
Noting the limit \(\lim_{t\rightarrow\infty}\frac{\ln t}{t}=0\), \(\limsup_{t\rightarrow\infty} \frac{\ln\Phi_{i}(t)}{t} \leq0\) is obtained for \(i=1,2,3\), a.s. It then follows from the comparison theorem of [18] that
Thus, we have a lemma as follows.
Lemma 1
Assume that, for any \(t\geq0\) and \(i=1,2,3\),
Then, for any initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\), the solution \((x(t),y(t),z(t))\) of system (3) satisfies
3 Persistence in the mean and extinction
Lemma 2
(see Liu et al. [14])
Suppose that \(Z(t)\in C(\Omega\times[0,\infty), R_{+})\), and let \(\int_{\mathbb {Y}}[\ln(1+\gamma_{i}(u))]^{2}\lambda(du)< c\), \(i=1,2\), hold.
-
(I)
If there exist two positive constants T and \(\delta_{0}\) such that
$$\ln Z(t)\leq\delta t-\delta_{0} \int_{0}^{t}Z(s)\,ds +\alpha B(t)+\sum _{i=1}^{2}\delta_{i} \int_{0}^{t} \int_{\mathbb{Y}}\ln\bigl(1+\gamma_{i}(u)\bigr)\tilde {N}(ds,du),\quad \textit{a.s.} $$for all \(t\geq T\), where α, \(\delta_{1}\) and \(\delta_{2}\) are constants, then
$$ \left \{ \textstyle\begin{array}{l@{\quad}l} \limsup_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}Z(s)\,ds\leq\delta /\delta_{0}, &\textit{a.s.}, \textit{if }\delta\geq0;\\ \lim_{t\rightarrow\infty}Z(t)=0,&\textit{a.s.}, \textit{if }\delta< 0. \end{array}\displaystyle \right . $$ -
(II)
If there exist three positive constants T, δ and \(\delta _{0}\) such that
$$\ln Z(t)\geq\delta t-\delta_{0} \int_{0}^{t}Z(s)\,ds +\alpha B(t)+\sum _{i=1}^{2}\delta_{i} \int_{0}^{t} \int_{\mathbb{Y}}\ln\bigl(1+\gamma_{i}(u)\bigr)\tilde {N}(ds,du),\quad \textit{a.s.} $$for all \(t\geq T\), then \(\liminf_{t\rightarrow\infty}\frac{1}{t}\int _{0}^{t}Z(s)\,ds\geq\delta/\delta_{0}\), a.s.
Definition 1
Assume \(m(t)\) is the density of a population at time t:
-
(1)
If \(\lim_{t\rightarrow\infty}m(t)=0\), a.s., then species \(m(t)\) is said to be extinct.
-
(2)
If \(\lim_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}m(s)\,ds>0\), a.s., then species \(m(t)\) is said to be stable in the mean.
-
(3)
If \(\liminf_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}m(s)\,ds>0\), a.s., then species \(m(t)\) is said to be strongly persistent in the mean.
-
(4)
If \(\limsup_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}m(s)\,ds>0\), a.s., then species \(m(t)\) is said to be weakly persistent in the mean.
Obviously, if species \(m(t)\) is stable in the mean, it must be strongly or weakly persistent in the mean.
Making use of Itô’s formula to system (3) gives
For both sides of system (7), integrating from 0 to t and multiplying by \(\frac{1}{t}\), (7) equals
where \(M_{i}(t)=\int_{0}^{t}\int_{\mathbb{Y}}\ln(1+\gamma_{i}(u))\tilde {N}(ds,du)\), \(i=1,2,3\).
\(M_{i}(t)\) are local martingale, by Proposition 2.4 of [19] and Assumption 1, its quadratic variations are
According to the law of strong large numbers, we have
Obviously,
In the following, we discuss the persistence in the mean and extinction for prey and predator populations.
Theorem 2
-
(i)
If \(w_{i}<0\) (\(i=1,2,3\)), the prey population \(x(t)\) and both the predator populations \(y(t)\) and \(z(t)\) will be extinct with probability 1.
-
(ii)
If \(w_{1}<0\), \(w_{2}<0\) and \(w_{3}>0\), the prey population \(x(t)\) and the susceptible predator population \(y(t)\) will go to extinction, a.s., the infected predator population \(z(t)\) will be stable in the mean, i.e., \(\lim_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}z(s)\,ds=\frac {w_{3}k_{2}}{c_{3}}\).
-
(iii)
If \(w_{1}<0\), \(w_{2}>0\) and \(w_{3}<0\), the prey population \(x(t)\) will go to extinction, the growth of the predator population is divided into the following two cases:
-
(a)
when \(w_{3}+\frac{\beta k_{2}}{c_{2}}w_{2}<0\), then the susceptible predator \(y(t)\) will be stable in the mean, \(\lim_{t\rightarrow\infty }\frac{1}{t}\int_{0}^{t}y(s)\,ds=\frac{w_{2}k_{2}}{c_{2}}\), and the infected predator \(z(t)\) will be extinct, a.s.
-
(b)
when \(w_{3}+\frac{\beta k_{2}}{c_{2}}w_{2}>0\), \(w_{2}>\frac{(c_{2}+k_{2}\beta )(w_{3}c_{2}+\beta k_{2}w_{2})}{c_{2}c_{3}}\) and \(w_{3}+\beta Q>\frac{c_{3}w_{2}}{c_{2}}\), where \(Q=\frac{k_{2}}{c_{2}}[w_{2}-\frac {(c_{2}+k_{2}\beta)(w_{3}c_{2}+\beta k_{2}w_{2})}{c_{2}c_{3}}]\), the whole predator population \(y(t)\) and \(z(t)\) will be strongly persistent in the mean, a.s.
-
(a)
-
(iv)
If \(w_{1}>0\), \(w_{2}<0\) and \(w_{3}<0\), then the prey population \(x(t)\) will be stable in the mean, namely \(\lim_{t\rightarrow\infty}\frac{1}{t}\int _{0}^{t}x(s)\,ds= \frac{w_{1}}{b_{1}}\), a.s. Both the susceptible predator \(y(t)\) and the infected predator \(z(t)\) will die out, a.s.
-
(v)
Assume that, for any \(t\geq0\) and \(i=1,2,3\),
$$\sup_{t\geq0} \int_{0}^{t} \int_{\mathbb{Y}}\exp(s-t)\bigl[\gamma_{i}(u)-\ln \bigl(1+ \gamma_{i}(u)\bigr)\bigr]\lambda(du)\,ds< \infty, $$then if \(w_{1}>\frac{(w_{2}+w_{3})k_{2}c_{1}}{(c_{2}+c_{3}+k_{2}\beta)k_{1}}\), \(w_{2}>0\) and \(w_{3}>0\), then all the populations will be strongly persistent in the mean, where \(\liminf_{t\rightarrow\infty}[\frac{1}{t}\int_{0}^{t}x(s)\, ds]\geq\frac{w_{1}}{b_{1}}-\frac{(w_{2}+w_{3})k_{2}c_{1}}{b_{1}k_{1}(c_{2}+c_{3}+k_{2}\beta )}\) and \(\liminf_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}[y(s)+z(s)]\,ds \geq \frac{(w_{2}+w_{3})k_{2}}{c_{2}+c_{3}+k_{2}\beta}\), a.s.
Proof
Making use of (8), we get
(i) According to (11),
noting (9) and (10), by virtue of Lemma 2, we can get \(\lim_{t\rightarrow\infty}x(t)=0\) when \(w_{1}<0\). Then, for an arbitrary small constant \(\varepsilon_{1}\) which satisfies \(0<\varepsilon_{1}<-\frac{w_{3}}{\beta}\), there exists a sufficiently large number \(T_{1}\) (\(T_{1}>0\)) such that
for \(t\geq T_{1}\); therefore,
Because of \(w_{2}<0\), making use of Lemma 2, \(\lim_{t\rightarrow \infty}y(t)=0\), a.s. In this case, there is a sufficiently large constant \(T_{2}\) (\(T_{2}>T_{1}\)) satisfying \(-\varepsilon_{1}\leq y(t)\leq\varepsilon_{1}\), when \(t\geq T_{2}\), a.s. Thus,
Notice \(0<\varepsilon_{1}<-\frac{w_{3}}{\beta}\), it follows from Lemma 2 that \(\lim_{t\rightarrow\infty}z(t)=0\), a.s.
(ii) Similar to case (i), when \(w_{1}<0\), \(w_{2}<0\), it easily shows that \(\lim_{t\rightarrow\infty}x(t)=0\), \(\lim_{t\rightarrow\infty}y(t)=0\), and for an arbitrary small constant \(\bar{\varepsilon}_{1}>0\),
Making use of Lemma 2, \(w_{3}>0\) and the arbitrariness of \(\bar {\varepsilon}_{1}\), \(\limsup_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}z(s)\,ds\leq\frac {w_{3}k_{2}}{c_{3}}\), a.s. is obviously verified. In contrast,
Noting that \(\lim_{t\rightarrow\infty}y(t)=0\), a.s. and Lemma 2, we have
then we obtain \(\lim_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}z(s)\,ds= \frac{w_{3}k_{2}}{c_{3}}\), a.s.
(iii) It is easy to show that \(\lim_{t\rightarrow\infty}x(t)=0\), a.s. when \(w_{1}<0\). On the basis of (13) and Lemma 2, for \(t>T_{1}\), \(\limsup_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}y(s)\,ds\leq \frac{w_{2}(\varepsilon_{1}+k_{2})}{c_{2}}\), a.s. under the condition of \(w_{2}>0\). Considering the arbitrariness of \(\varepsilon_{1}\), \(\limsup_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}y(s)\,ds\leq\frac {w_{2}k_{2}}{c_{2}}\), a.s. Therefore, for an arbitrary constant \(\varepsilon_{3}>0\) (\(0<\varepsilon _{3}<\frac{1}{\beta}|w_{3}+\frac{\beta k_{2}}{c_{2}}w_{2}|\)), there exists a number \(T_{3}\) (\(T_{3}>T_{1}\)) such that
for \(t>T_{3}\). Making use of (12), then for \(t>T_{3}\), we have
(a) If \(w_{3}+\frac{\beta k_{2}}{c_{2}}w_{2}<0\), then
For the susceptible predator population,
Because of (9), (10), (14) and Lemma 2, we have \(\liminf_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}y(s)\,ds\geq\frac {w_{2}k_{2}}{c_{2}}\), a.s. Then \(\lim_{t\rightarrow\infty}\frac{1}{t}\int_{0}^{t}y(s)\,ds= \frac {w_{2}k_{2}}{c_{2}}\), a.s.
(b) If \(w_{3}+\frac{\beta k_{2}}{c_{2}}w_{2}>0\), by virtue of Lemma 2 and the arbitrariness of \(\varepsilon_{1}\) and \(\varepsilon_{3}\), we have
Then
Notice (15), for sufficiently large t, \(\frac{1}{t}\int _{0}^{t}z(s)\,ds\leq\frac{(w_{3}c_{2}+\beta k_{2}w_{2})k_{2}}{c_{2}c_{3}}+\varepsilon _{4}\), a.s., where \(\varepsilon_{4}\) is an arbitrary positive number. Hence, by (9), (10), Lemma 2 and the arbitrariness of \(\varepsilon_{4}\), we have
when \(w_{2}>\frac{(c_{2}+k_{2}\beta)(w_{3}c_{2}+\beta k_{2}w_{2})}{c_{2}c_{3}}\). Here we define \(Q=\frac{k_{2}}{c_{2}}[w_{2}-\frac{(c_{2}+k_{2}\beta)(w_{3}c_{2}+\beta k_{2}w_{2})}{c_{2}c_{3}}]\).
In this case,
For sufficiently large t, we have
If \(w_{3}+\beta Q>\frac{c_{3}w_{2}}{c_{2}}\), then by Lemma 2, we get
(iv) According to (11), we have
Making use of Lemma 2, (9) and (10), we have
and
Because of \(w_{2}<0\) and (9), (10), we get \(\limsup_{t\rightarrow\infty}\frac{\ln y(t)}{t}<0\), a.s., namely
Then
Considering (9), (10), (17) and \(w_{3}<0\), we have \(\limsup_{t\rightarrow\infty}\frac{\ln z(t)}{t}<0\); therefore,
At this time, for sufficiently large t,
Combining (19) with (16), we have \(\lim_{t\rightarrow \infty}\frac{1}{t}\int_{0}^{t}x(s)\,ds= \frac{w_{1}}{b_{1}}\), a.s.
(v) It is easy to have \(\limsup_{t\rightarrow\infty}\frac{1}{t}\int _{0}^{t}x(s)\,ds\leq\frac{w_{1}}{b_{1}}\), a.s. when \(w_{1}>0\) and
Then
namely
By virtue of the formula \((a+b)^{2}\geq2ab\) for \(a,b\in R_{+}\), we can get \(2\ln(a+b)\geq\ln2+\ln(ab)\), hence
then
By Lemma 2, we get
In this case, considering \(0<\tilde{p}<1\), we have
Taking the inferior limit, by virtue of Lemma 1 and (9), (10), we get
Hence,
□
4 Stochastically ultimate boundedness
From the biological point of view, the nonexplosion and positiveness property of solution in a population dynamical system is not enough. In this section we will show that the pth moment of solution (3) is bounded.
Definition 2
The solution \(X(t)\) of system (3) is called stochastically ultimate bounded if, for any \(\epsilon\in(0,1)\), there is a constant \(H:=H(\epsilon)\) such that, for any \(X(0)\in R_{+}^{3}\),
Theorem 3
Assume \((x(t),y(t),z(t))^{T}\) to be a solution to system (3) with the initial value \((x_{0},y_{0},z_{0})\in R_{+}^{3}\). If \(k_{2}>\frac{(1+a_{1})^{2}}{4b_{1}}\) for all \(p>0\), there exist constants \(\bar{K}_{i}(p)\) (\(i=1,2,3\)) such that
Proof
Making use of Itô’s formula and then taking expectations yield
Note for a function \(f(v)=v^{p}(a-bv)\), \(v\in[0,+\infty)\), \(p>0\), \(a>0\), \(b>0\), \(f(v)\) has a maximum value at \(\frac{ap}{(p+1)b}\), namely
Therefore,
where
Let \(t\rightarrow+\infty\), we have
Similarly, we can compute
Thus, taking expectations leads to
By virtue of (21), for arbitrary small \(\varepsilon_{0}>0\) (\(\varepsilon_{0}< k_{2}-\bar{K}_{1}(1)\)), there exists a positive constant \(T_{0}\) satisfying
for \(t>T_{0}\). Therefore,
If \(k_{2}>\bar{K}_{1}(1)\), then \(k_{2}>\bar{K}_{1}(1)+\varepsilon_{0}\). According to (20), the following result is available:
Here
where \(\bar{K}_{1}(1)=\frac{(1+a_{1})^{2}}{4b_{1}}\). Similarly, \(\limsup_{t\rightarrow\infty}E[y^{p}(t)]\leq K_{2}(p)\), here because of the arbitrariness of \(\varepsilon_{0}\), define
thus,
And
On the basis of Young’s inequality, we have
Notice \(\frac{p}{p+1}<1\), we have \(y(t)z^{p}(t)\leq\frac {y^{p+1}(t)}{p+1}+z^{p+1}(t)\), then
Considering (22), we can see that there exist positive constants \(\vec{K}_{2}(p)\) and \(\vec{T}_{2}\) (\(\vec{T}_{2}>T_{0}\)), for \(t>\vec {T}_{2}\), one can obtain that \(E[y^{p}(t)]\leq\vec{K}_{2}(p)\). Then, for \(t>\vec{T}_{2}\),
where
Take the superior limit results in
Notice the arbitrariness of \(\varepsilon_{0}\), let
clearly,
thus the proof of Theorem 3 is completed. □
By virtue of the formula of Lemma 2.1 in [10], we have
for \(\forall p>0\), \(x\in R_{+}^{n}\), where \(R_{+}^{n}:=\{x\in R^{n},x_{i}>0,1\leq i\leq n\}\).
For solution \(X(t)=(x(t),y(t),z(t))^{T}\in R_{+}^{3}\) and \(p>0\), we have
By Theorem 3,
As an application of Theorem 3, together with the Chebyshev inequality, we can also establish the following corollary.
Corollary 1
Under conditions of Theorem 3, the solution of system (3) is stochastically ultimate bounded.
5 Numerical simulation
In this section, we give some examples and numerical simulations to illustrate our analytical findings. These numerical simulations are given by the Euler scheme of [20].
For system (3), we choose the initial value \((x_{0},y_{0},z_{0})=(0.2,0.1,0.2)\), and
\(\mathbb{Y}=(0,+\infty)\), \(\lambda(\mathbb{Y})=1\). In the following, we change the values of part parameters to observe the asymptotic behavior of solution for system (3).
-
(1)
In Figure 1, let \(a_{1}=0.02\), \(a_{2}=0.03\), \(a_{3}=0.02\), \(\gamma _{1}(u)=\gamma_{3}(u)=0.2\), \(\gamma_{2}(u)=0.4\), the other parameters are the same as above, then
$$\begin{gathered} w_{1}= a_{1}-\frac{\sigma_{1}^{2}}{2}- \int_{\mathbb{Y}}\bigl[\gamma_{1}(u)-\ln \bigl(1+ \gamma_{1}(u)\bigr)\bigr]\lambda(du)=-0.1227< 0, \\ w_{2}= a_{2}-\frac{\sigma_{2}^{2}}{2}- \int_{\mathbb{Y}}\bigl[\gamma_{2}(u)-\ln \bigl(1+ \gamma_{2}(u)\bigr)\bigr]\lambda(du)=-0.1585< 0, \\ w_{3}= a_{3}-\frac{\sigma_{3}^{2}}{2}- \int_{\mathbb{Y}}\bigl[\gamma_{3}(u)-\ln \bigl(1+ \gamma_{3}(u)\bigr)\bigr]\lambda(du)=-0.1227< 0. \end{gathered} $$Thus, in accordance with case (i) in Theorem 2, all the species will go to extinction, Figure 1 confirms it.
-
(2)
In Figure 2, let \(a_{1}=0.02\), \(a_{2}=0.03\), \(a_{3}=0.5\), \(\gamma _{1}(u)=\gamma_{2}(u)=0.2\), \(\gamma_{3}(u)=0.4\), the other parameters remain the same as in (23), then
$$w_{1}=-0.1227< 0,\quad\quad w_{2}= -0.1127< 0,\qquad w_{3}= 0.3115>0. $$By the case (ii) of Theorem 2, the prey population and the susceptible predator population will be extinct, the infected predator population will be stable in the mean. Figure 2 shows it.
-
(3)
In Figure 3, we assume \(a_{1}=0.06\), \(a_{2}=0.6\), \(a_{3}=0.2\), \(\sigma _{3}=0.8\), \(\beta=k_{2}=0.2\), \(\gamma_{1}(u)=0.4\), \(\gamma_{2}(u)=\gamma_{3}(u)=0.2\), the other parameters are the same as in (23), at this time
$$\begin{gathered} w_{1}=-0.1285< 0,\qquad w_{2}= 0.4573>0,\\ w_{3}=-0.1377< 0,\qquad w_{3}+\frac{\beta k_{2}}{c_{2}}w_{2}=-0.1072< 0.\end{gathered} $$By the condition of (iii)(a), the prey population \(x(t)\) and the infected population \(z(t)\) will go to extinction, the susceptible predator population \(y(t)\) will be stable in the mean.
-
(4)
In Figure 4, we choose \(a_{1}=0.3\), \(a_{2}=0.03\), \(a_{3}=0.02\), \(\gamma _{1}(u)=0.2\), \(\gamma_{2}(u)=\gamma_{3}(u)=0.4\), the other parameters are the same as in (23), then
$$w_{1}=0.1573>0,\qquad w_{2}= -0.1585< 0,\qquad w_{3}= -0.1685< 0. $$On the basis of (iv) in Theorem 2, the prey population will be stable in the mean, the whole predator population will die out, which is consistent with Figure 4.
-
(5)
In Figure 5, let \(a_{1}=0.75\), \(a_{2}=0.7\), \(a_{3}=0.3\), \(c_{2}=0.3\), \(c_{3}=0.2\), \(k_{2}=0.6\), \(\beta=0.4\), \(\sigma_{3}=0.7\), \(\gamma_{1}(u)=0.4\), \(\gamma_{2}(u)=\gamma_{3}(u)=0.2\), the other parameters are the same as in (23), then
$$w_{1}=0.5615>0,\qquad w_{2}= 0.5573>0,\qquad w_{3}= 0.0373>0, $$and \(\frac{(w_{2}+w_{3})k_{2}c_{1}}{(c_{2}+c_{3}+k_{2}\beta)k_{1}}=0.3857\), \(w_{1}>\frac {(w_{2}+w_{3})k_{2}c_{1}}{(c_{2}+c_{3}+k_{2}\beta)k_{1}}\). By virtue of (v) in Theorem 2, all the populations will be strongly persistent in the mean.
In the following, we discuss the solution of deterministic system (2), assume that all the parameters are the same as in Figures 1, 4, 5. Figure 6 shows that the susceptible predator population will be extinct, but the prey and infected predator populations will be stable, which is different from Figure 1.
Figure 7 illustrates that the prey population and the infected predator population will be stable, and the susceptible predator population will go to extinction. Compare Figure 7 with Figure 4, the behavior of solution for system (2) is distinct from that for stochastic system (3). Figure 8 confirms it again, we give no more explanations.
These results show that, for a stochastic system, not only the growth rate but also the Lévy noise and Gauss white noise play an important role in the persistence of population. Sometimes, the stochastic perturbation of environment can make species extinct.
6 Conclusions
In this paper, we discuss an LG–Holling type II diseased predator ecosystem with Lévy noise and Gauss white noise. We show that the system admits a unique global positive solution; we also investigate persistence in the mean and extinction of all the populations, uniformly finite pth moment with \(p>0\) and stochastic ultimate boundedness. The threshold conditions of extinction or persistence in the mean for prey and predator populations illustrate that \(w_{i}\) is the key value, which means that \(a_{i}\) is advantageous to the population, white noise and Lévy noise will go against the persistence of population. From the numerical simulation, we find that Lévy noise and Gauss white noise have an important effect on persistence for a stochastic system; occasionally, it may be a determinant factor. Thus, we must consider random changes of environment in the mathematical model.
References
Leslie, P.H.: Some further notes on the use of matrices in population mathematics. Biometrika 35, 213–245 (1948)
Leslie, P.H.: A stochastic model for studying the properties of certain biological systems numerical methods. Biometrika 45, 16–31 (1958)
Pielou, E.C.: Mathematical Ecology. Wiley, New York (1977)
Korobeinikov, A.: A Lyapunov function for Leslie–Gower predator–prey models. Appl. Math. Lett. 14, 697–699 (2001)
Zhou, X., Cui, J., Shi, X., Song, X.: A modified Leslie–Gower predator–prey model with prey infection. J. Appl. Math. Comput. 33, 471–487 (2010)
Liu, M., Wang, K.: Dynamics of a Leslie–Gower Holling-type II predator-prey system with Lévy jumps. Nonlinear Anal. 85, 204–213 (2013)
Flores, J.D., Gonzalez-Olivares, E.: A modified Leslie–Gower predator–prey model with ratio-dependent functional response and alternative food for the predator. Math. Methods Appl. Sci. 40, 2313–2328 (2017)
Xu, Y., Liu, M., Yang, Y.: Analysis of a stochastic two-predators one-prey system with modified Leslie–Gower and Holling-type II schemes. J. Math. Anal. Appl. 7, 713–727 (2017)
Sarwardi, S., Haque, M., Venturino, E.: Global stability and persistence in LG–Holling type II diseased predator ecosystems. J. Biol. Phys. 37, 91–106 (2011)
Bao, J., Mao, X., Yin, G., Yuan, C.: Competitive Lotka–Volterra population dynamics with jumps. Nonlinear Anal. 74, 6601–6616 (2011)
Bao, J., Yuan, C.: Stochastic population dynamics driven by Lévy noise. J. Math. Anal. Appl. 391, 363–375 (2012)
Liu, M., Deng, M., Do, B.: Analysis of a stochastic logistic model with diffusion. Appl. Math. Comput. 266, 169–182 (2015)
Liu, Q., Chen, Q.: Asymptotic behavior of a stochastic non-autonomous predator-prey system with jumps. Appl. Math. Comput. 271, 418–428 (2015)
Liu, M., Wang, K.: Stochastic Lotka–Volterra systems with Levy noise. J. Math. Anal. Appl. 410, 750–763 (2014)
Zhao, Y., Yuan, S.: Optimal harvesting policy of a stochastic two-species competitive model with Levy noise in a polluted environment. Physica A 477, 20–33 (2017)
Liu, L., Meng, X., Zhang, T.: Optimal control strategy for an impulsive stochastic competition system with time delays and jumps. Physica A 477, 99–113 (2017)
Liu, Q., Jiang, D., Shi, N., Hayat, T., Alsaedi, A.: Stochastic mutualism model with Levy jumps. Commun. Nonlinear Sci. Numer. Simul. 43, 78–90 (2017)
Peng, S., Zhu, X.: Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations. Stoch. Process. Appl. 116, 370–380 (2006)
Kunita, H.: Itô’s stochastic calculus: its surprising power for applications. Stoch. Process. Appl. 120, 622–652 (2010)
Protter, P., Talay, D.: The Euler scheme for Lévy driven stochastic differential equations. Ann. Probab. 25, 393–423 (1997)
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This work is supported by the key research project of institutions of higher education of Henan Province in China (No. 18A110021).
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Li, S. LG–Holling type II diseased predator ecosystem with Lévy noise and white noise. Adv Differ Equ 2018, 48 (2018). https://doi.org/10.1186/s13662-018-1497-y
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DOI: https://doi.org/10.1186/s13662-018-1497-y