Dynamical analysis of a ratiodependent predator–prey model with Holling III type functional response and nonlinear harvesting in a random environment
 Guijie Lan^{1},
 Yingjie Fu^{1},
 Chunjin Wei^{1} and
 Shuwen Zhang^{1}Email author
https://doi.org/10.1186/s1366201816258
© The Author(s) 2018
Received: 18 January 2018
Accepted: 28 April 2018
Published: 24 May 2018
Abstract
The objective of this paper is to study the dynamics of the stochastic ratiodependent predator–prey model with Holling III type functional response and nonlinear harvesting. For the autonomous system, sufficient conditions for globally positive solution and stochastic permanence are established. Then, applying comparison theorem for stochastic differential equation, sufficient conditions for extinction and persistence in the mean are obtained. In addition, we prove that there exists a unique stationary distribution and it has ergodicity under certain parametric restrictions. For the periodic system, we obtain conditions for the existence of a nontrivial positive periodic solution. Finally, numerical simulations are carried out to substantiate the analytical results.
Keywords
1 Introduction
Renewable resources (such as fisheries and forestry resources) are considered to be inexhaustible at all times, but excessive exploitation will actually exhaust them. The optimal management of renewable resources, which has a direct relationship to sustainable development, has been studied extensively by many authors (see [1, 2] and the references cited therein). Xiao [1] pointed out that the aim is to determine how much we can harvest without dangerously altering the harvested population. According to Clark in [2], the management of renewable resources has been based on the maximum sustainable yield, with the property that any larger harvest rate will lead to the depletion of the population. Thus, it is important to investigate the reasonable exploitation of renewable resources and their effective utilization to obtain the maximum revenue.
The rest of the paper is organized as follows. In Sect. 2, we give the existence and uniqueness of global positive solution and the solution is stochastically ultimately bounded. Moreover, we obtain that stochastic system (1.2) is stochastically permanent in Sect. 3. In Sect. 4, sufficient conditions for extinction, stochastic persistence in the mean of the population are established. In Sect. 5, we show the existence of a unique stationary distribution and ergodicity. The existence of a positive periodic solution for nonautonomous periodic solution is also obtained in Sect. 6. Finally, the conclusions are given and our main results are illustrated through numerical simulations.
2 Existence and uniqueness of globally positive solution
If \(f(t)\) is integrable, we define \(\langle f(t) \rangle_{T}=\frac {1}{T}\int_{0}^{T} f(t)\,dt\), \(T>0\). And if \(f(t)\) is bounded, we define \(f^{u}=\sup_{t\in[0,+\infty)} f(t)\), \(f^{l}=\inf_{t\in[0,+\infty)}f(t)\).
As we know, for a stochastic differential equation in order to have a unique global (i.e., no explosion in a finite time) solution for any given initial value, the functions involved in the stochastic system are generally required to satisfy the linear growth condition and the local Lipschitz condition [31]. However, the functions of system (1.2) do not satisfy the linear growth condition. So the solution of system (1.2) may explode at a finite time. In this section, we first show that there exists a unique positive local solution of system (1.2) and then, by constructing some suitable Lyapunov function, we prove that this solution is global. Explanation for ‘explosion time’ used in the following lemma can be found in [31].
Lemma 2.1
For \((x(0),y(0) )\in R^{2}_{+}\), there is a unique positive local solution \((x(t),y(t) )\) of system (1.2) for \(t\in[0,\tau_{e})\) a.s., where \(\tau_{e}\) is the explosion time.
Proof
Now we are in a position to show that this unique solution is not only a local solution but a global solution. To prove this, we need to show that \(\tau_{e}=\infty\) a.s.
Theorem 2.1
For any given initial value \((x(0),y(0) )\in R_{+}^{2}\), there is a unique solution \((x(t),y(t) )\) to system (1.2), and the solution will remain in \(R^{2}_{+}\) with probability 1, that is, \((x(t),y(t) )\in R_{+}^{2}\) for all \(t\geq0\) almost surely.
Proof
So we must have \(\tau_{\infty}=\infty\). The conclusion is confirmed. □
Theorem 2.1 shows that the solutions to system (1.2) will remain in \(R_{+}^{2}\). The property makes us continue to discuss how the solution varies in \(R_{+}^{2}\) in more detail. We first present the definition of stochastic ultimate boundedness which is one of the important topics in population dynamics.
Definition 2.1
([32])
Theorem 2.2
The solutions of system (1.2) are stochastically ultimately bounded for any initial value \(X(0)= (x(0),y(0) )\in R^{2}_{+}\).
Proof
3 Stochastic permanence
Generally speaking, the nonexplosion property, the existence, and the uniqueness of the solution are not enough, but the property of permanence is more desirable since it means the long time survival in population dynamics. Now, the definition of stochastic permanence will be given below [32].
Definition 3.1
([32])
Theorem 3.1
Proof
The proof is completed. □
Theorem 3.2
If system (1.2) satisfies \(\min\{r_{1},r_{2}\} \max\{b_{1}+f_{2},f_{1}\}\frac{(\theta+1)\max\{\sigma_{1}^{2},\sigma_{2}^{2}\} }{2}>0\), where \(0<\theta<2\), then Eq. (1.2) is stochastically permanent.
The proof is the application of the wellknown Chebyshev inequality, Theorems 2.2 and 3.1. Here it is omitted.
4 Stochastic persistence in the mean and extinction
Let us continue to discuss the long time behavior of stochastic model (1.2). From the point of view of the optimal management of renewable resources, how much can we harvest without dangerously altering the harvested population? On the other hand, how much will larger harvest rate lead to the depletion of the population? In this section, we will show that stochastic system (1.2) may preserve some important dynamics of the original deterministic system without harvesting terms when the intensities of noises and the catchability coefficient are small. On the contrary, if the catchability coefficient is sufficiently large, the populations will become extinct with probability one. Now, we present the definition of persistence in the mean and extinction.
Definition 4.1
([33])
Definition 4.2
([33])
Lemma 4.1
Proof
Lemma 4.2
Moreover, if \(r_{1}>\frac{\sigma_{1}^{2}}{2}\) is satisfied, then \(\Phi(t)\) is persistent in the mean a.s.
Proof
The proof is completed. □
Theorem 4.1
Suppose \(f_{1}< r_{1}\frac{b_{1}}{2\sqrt {m}}\frac{\sigma_{1}^{2}}{2}\), \(f_{2}< r_{2}\frac{\sigma_{2}^{2}}{2}\) are satisfied, and \(x(t)\), \(y(t)\) is the positive solution to Eq. (1.2) with initial value \(x(0)>0\), \(y(0)>0\), then the system is persistent in the mean.
Proof
The proof is completed. □
Theorem 4.2
 (i)
If \(r_{1}<2\sqrt{\frac{a_{1}f_{1}}{w_{1}}}\frac{a_{1}}{w_{1}}\), \(r_{2}>f_{2}+\frac{\sigma_{2}^{2}}{2}\), then x is extinct and y is persistent in the mean a.s.
 (ii)
If \(r_{1}<2\sqrt{\frac{a_{1}f_{1}}{w_{1}}}\frac{a_{1}}{w_{1}}\), \(r_{2}<2\sqrt{\frac{b_{2}f_{2}}{k_{2}w_{2}}}\frac{b_{2}}{k_{2}w_{2}}\), then both the populations x and y are extinct a.s.
Proof
The proof is completed. □
5 A sufficient condition for stationary distribution
In this section, we prove the existence of stationary distribution of the prey and predator populations. The stationary solution means that it is a stationary Markov process which shows that the prey and predator can be persistent and will not die out in the population. For this purpose, we find the stationary distribution for solutions of system (1.2), which in turn imply the stability in stochastic sense. Before proving the main theorem related to the stationary distribution, we state a useful lemma from [30] which will be useful to prove the theorem.
Definition 5.1
 \((P_{1})\) :

In the domain U and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix \(A(x)\) is bounded away from zero.
 \((P_{2})\) :

If \(x\in E_{l} \setminus U\), the mean time τ at which a path emerging from x reaches the set U is finite, and \(\sup_{x\in K}E_{x}\tau<\infty\) for every compact subset \(K\subset E_{l}\).
Lemma 5.1
To validate \((P_{2})\), it is enough to show that there exist some neighborhood U and a nonnegative \(C^{2}\)function V such that, for any \(x\in E_{l} \setminus U\), \({LV}(x)\) is negative.
Theorem 5.1
Assume \(f_{1}< r_{1}\sigma_{1}^{2}\frac {b_{1}}{2\sqrt{m}}\), \(f_{2}< r_{2}\sigma_{2}^{2}\). Then, for any initial value \((x_{0},y_{0})\in R^{2}_{+}\), there exists a unique stationary distribution \(\mu(\cdot)\) for system (1.2), and it has ergodic property.
Proof
6 The existence of periodic solution of nonautonomous system
In what follows, we first recall a basic definition and introduce a lemma which gives criteria for the existence of a periodic Markov process (see Khasminskii [30]).
Definition 6.1
([30])
A stochastic process \(\xi(t)=\xi(t,\omega)\) (\(\infty< t<+\infty\)) is said to be Tperiodic if for every finite sequence of numbers \(t_{1},t_{2},\ldots, t_{n}\), the joint distribution of random variables \(\xi (t_{1}+h),\xi(t_{2}+h),\ldots,\xi(t_{n}+h)\) is independent of h, where \(h=kT\) (\(k=1,2,\ldots\)).
Lemma 6.1
([30])
 \((Q_{1})\) :

\(\inf_{x>R}V(t,x)\rightarrow\infty\),
 \((Q_{2})\) :

\({LV}(t,x)\leq1 \) outside some compact set.
Then system (6.1) has at least a Tperiodic Markov process.
Theorem 6.1
If \(\langle r_{1}(t)f_{1}(t)\sigma _{1}^{2}(t)\frac{b_{1}(t)}{2\sqrt{m(t)}}\rangle_{T}>0\), \(\langle r_{2}(t)f_{2}(t)\sigma_{2}^{2}(t) \rangle_{T}>0\), then system (1.3) has one positive Tperiodic solution.
Proof
By the same way as in Theorem 2.1 one can see that, for any initial \((x,y)\in R^{2}_{+}\), system (1.3) has a unique global positive solution \((x,y)\in R^{2}_{+}\), we only need to verify the conditions \((Q_{1})\), \((Q_{2})\) of Lemma 6.1.
7 Numerical simulations and conclusion
In this paper, we have considered the basic features of a ratiodependent predator–prey model with Holling III type functional response and nonlinear harvesting in presence of white noise terms to understand the dynamics in presence of environmental driving forces. Although we are considering a predator–prey model, the survival of predator species in absence of the prey population is justified as we have assumed that the predators have alternative food source and their growth follows the logistic growth law. For the autonomous system, we have established the existence of positive global solution of the stochastic model. Moreover, we show that the positive solutions are stochastically bounded. The sufficient conditions for stochastic permanence, stochastic persistence in the mean, and extinction are established. Then, by constructing some suitable Lyapunov function, the existence of stationary distribution for both populations is established under certain parametric restrictions. These parametric restrictions reflect the idea that large amplitude environmental noise can destabilize the system, and in that situation one cannot find any stationary distribution. The result shows that stationary distribution does not rely on the existence and the stability of the positive equilibrium in the deterministic system. There is a periodic phenomenon in a nonautonomous periodic system: when \(\langle r_{1}(t)f_{1}(t)\sigma _{1}^{2}(t)\frac{b_{1}(t)}{2\sqrt{m(t)}}\rangle_{T}>0\), \(\langle r_{2}(t)f_{2}(t)\sigma_{2}^{2}(t) \rangle_{T}>0\) hold, it follows from Theorem 6.1 that there exists at least one Tperiodic solution, which means that the two species of prey and predator will coexist and exhibit periodicity in the long run. Obtained analytical results are verified with supportive numerical simulations.
Results of simulations run reveal that the forcing intensity of fluctuating environment and catchability play a crucial role behind the survival of prey and predator species.
Declarations
Acknowledgements
This research was supported by the Fujian provincial Natural science of China (No. 2016J01667, 2016J05012).
Authors’ contributions
SZ and CW suggested the model, helped in result interpretation, manuscript evaluation, and supervised the development of work. GL and YF helped to evaluate, revise, and edit the manuscript. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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