Stochastic Analysis of a Hassell-Varley Type Predation Model

Abstract

We investigate a Hassell-Varley type predator-prey model with stochastic perturbations. By perturbing the growth rate of prey population and death rate of predator population with white noise terms, we construct a stochastic differential equation model to discuss the effects of the environmental noise on the dynamical behaviors. Applying the comparison theorem of stochastic equations and Itô’s formula, the unique positive global solution to the model for any positive initial value is obtained. We find out some sufficient conditions for stochastically asymptotically boundedness, permanence, persistence in mean and extinction of the solution. Furthermore, a series of numerical simulations to illustrate our mathematical findings are presented. The results indicate that the stochastic perturbations do not cause drastic changes of the dynamics in the deterministic model when the noise intensity is small under some conditions, but while the noise intensity is sufficiently large, the species may die out, which does not happen in the deterministic model.

1. Introduction

It is well known that predator-prey interaction is one of basic interspecies relations for ecosystems, and it is also the basic block of more complicated food chain, food web, and biophysical network structure [1]. Because of the universal existence of predator and prey and their importance in ecology, the dynamical relationship between them has long been and will continue to be one of the dominant themes [2, 3].

The classical predator-prey model has received extensive attentions from mathematicians as well as ecologists [4–7], and it can be expressed by a model of nonlinear ordinary differential equations as follows: 𝑑𝑁(𝑁)()𝑑𝑡=𝑓𝑁−𝑏𝑔𝑁,𝑃𝑃,𝑑𝑃(),𝑑𝑡=𝑃𝑐𝑔𝑁,𝑃−𝑑 (1) where 𝑁=𝑁(𝑡) and 𝑃=𝑃(𝑡) denote the density of prey and predator population at time 𝑡, respectively. Parameters 𝑏, 𝑐, and 𝑑 are positive constants. 𝑏 stands for capturing rate of prey by predator, 𝑐 is conversion rate of prey into predator, and 𝑑 is the natural death rate of the predator. The function 𝑓(𝑁) represents the density-dependent specific growth rate of prey in absence of predator. The amount of prey biomass consumed by each predator per unit of time is described by the functional response 𝑔(𝑁,𝑃).

In this paper, we consider the usual logistic form of the growth function for prey in the absence of predator as 𝑓(𝑁)𝑁=𝑟1−𝐾, (2) where 𝑟(>0) is the natural growth rate of prey and 𝐾(>0) is the environmental carrying capacity. The functional response 𝑔(𝑁,𝑃) is taken as 𝑔()=𝑁𝑁,𝑃𝑁+𝑚𝑃^𝛼, (3) which is called the Hassell-Varley type functional response and 𝛼∈(0,1) is the Hassell-Varley constant [8] and 𝑚(>0) stands for half capturing saturation constant. The predator-prey model with Hassell-Varley type functional response has been studied in the ecological literature [6, 9–11].

For more biological motivation in population dynamics, we take into account the density-dependence of predator population. And the corresponding Hassell-Varley type predator-prey model is described by the following form: 𝑑𝑁𝑁𝑑𝑡=𝑟𝑁1−𝐾−𝑏𝑁𝑃𝑁+𝑚𝑃^𝛼,𝑑𝑃𝑑𝑡=𝑃𝑐𝑁𝑁+𝑚𝑃^𝛼,𝑁(0)−𝑑−ℎ𝑃=𝑁₀>0,𝑃(0)=𝑃₀>0, (4) where ℎ𝑃 stands for the density-dependence of the predator population and ℎ>0.

On the other hand, most natural phenomena do not follow strictly deterministic laws but rather oscillate randomly about some average. So that the population density never attains a fixed value with the advancement of time but rather exhibits continuous oscillation around some average values [12, 13]. In fact, there are many benefits to be gained by using stochastic models because real life is full of random fluctuations (i.e., the effects of noise), which undeniably arise from either environmental variability or internal species. The basic mechanism and factors of population growth like resources and vital rates—birth, death, immigration, and emigration—change nondeterministically due to continuous fluctuations in the environment (e.g., variation in intensity of sunlight, temperature, water level, etc.) [2, 3, 14]. Recent advances in stochastic differential equations enable a lot of authors to introduce noise into the model of physical phenomena, whether it is a random noise in the system of differential equations or environmental fluctuations in parameters [15–31]. So far as our knowledge is concerned, the work of a modified Hassell-Varley type predator-prey model with stochastic perturbations seems rare. Motivated by these, we attempt to study the stochastic behaviors of the modified Hassell-Varley type predation model in a random fluctuating environment.

The organization of this paper is as follows. In Section 2, we present a stochastic model corresponding to the deterministic model (4) and discuss it in detail. In Section 3, we use numerical simulations to reveal the influence of noise on the dynamical behaviors of the model. A brief discussion is given in Section 4.

2. The Stochastic Model and Analysis

In this section, we investigate the effects of fluctuating environments on the dynamical behaviors of model (4). Assuming that random fluctuations in the environment would display themselves as fluctuations in the growth rate of prey population 𝑁 and in the death rate of predator population 𝑃, then the parameters 𝑟 and 𝑑 in model (4) can be replaced by 𝑟⟶𝑟+𝜎₁𝐵̇₁(𝑡),−𝑑⟶−𝑑+𝜎₂𝐵̇₂(𝑡). (5) In this way, model (4) will be reduced to the following form: 𝑟𝑑𝑁=𝑁𝑟−𝐾𝑁−𝑏𝑃𝑁+𝑚𝑃^𝛼𝑑𝑡+𝜎₁𝑁𝑑𝐵₁(𝑡),𝑑𝑃=𝑃𝑐𝑁𝑁+𝑚𝑃^𝛼−𝑑−ℎ𝑃𝑑𝑡+𝜎₂𝑃𝑑𝐵₂(𝑡), (6) where 𝜎²₁ and 𝜎²₂ are known as the intensities of environmental noise and 𝐵̇_𝑖(𝑡)(𝑖=1,2) is a standard white noise; that is, 𝐵_𝑖(𝑡)(𝑖=1,2) is a Brownian motion defined in a complete probability space (Ω,ℱ,𝐏) with a filtration {ℱ_𝑡}_𝑡∈𝑅+ satisfying the usual conditions (i.e., it is right continuous and increasing while ℱ₀ contains all 𝐏-null sets) [14].

2.1. Positive and Global Solution

For model (6), there is a positive local solution.

Lemma 1. There is a unique local solution (𝑁(𝑡),𝑃(𝑡)) for 𝑡∈[0,𝜏_𝑒) to model (6) almost surely for initial value (𝑁₀,𝑃₀)∈𝑅²₊, where 𝜏_𝑒 is the explosion time.

The proof of this lemma is rather standard and hence is omitted.

Lemma 1 only tells us that there is a unique positive local solution to model (6). Next, we show that this solution is global which is more interesting.

In particular, let us consider the one-dimensional stochastic population model 𝑑𝑁(𝑡)=𝑟𝑁(𝑡)1−𝑁(𝑡)𝐾𝑑𝑡+𝜎₁𝑁(𝑡)𝑑𝐵₁(𝑡),𝑡≥0,𝑁(0)=𝑁₀; (7) there is an explicit solution 𝑁(𝑡)𝜎=exp𝑟−²¹2𝑡+𝜎₁𝐵₁(𝑡×1)𝑁₀+𝑟𝐾^𝑡₀𝜎exp𝑟−²¹2𝑠+𝜎₁𝐵₁(𝑠)𝑑𝑠⁻¹. (8) From model (6), we have (𝑡)(𝑡)𝑁(𝑡)𝑑𝑁≤𝑟𝑁1−𝐾𝑑𝑡+𝜎₁𝑁(𝑡)𝑑𝐵₁(𝑡). (9) By the comparison theorem of stochastic equations [14], we have 𝑁(𝑡)≤𝑁(𝑡) a.s. 𝑡∈[0,𝜏_𝑒).

Besides, for the following equation 𝑑𝑁(𝑡)=𝑁(𝑡)𝑏𝑟−𝑚−𝑟𝐾𝑁(𝑡)𝑑𝑡+𝜎₁𝑁(𝑡)𝑑𝐵₁(𝑡),𝑁(0)=𝑁₀, (10) there is a unique solution as 𝑁(𝑡)𝑏=exp𝑟−𝑚−𝜎²₁2𝑡+𝜎₁𝐵₁(𝑡×1)𝑁₀+𝑟𝐾^𝑡₀𝑏exp𝑟−𝑚−𝜎²₁2𝑠+𝜎₁𝐵₁(𝑠)⁻¹. (11) In model (6), for 𝛼∈(0,1), we can get (𝑡)(𝑡)𝑏𝑑𝑁≥𝑁𝑟−𝑚−𝑟𝐾𝑁(𝑡)𝑑𝑡+𝜎₁𝑁(𝑡)𝑑𝐵₁(𝑡); (12) then 𝑁(𝑡)≥𝑁(𝑡) a.s. 𝑡∈[0,𝜏_𝑒).

Consequently, we obtain 𝑁(𝑡)≤𝑁(𝑡)≤𝑁(𝑡)a.s.𝑡∈0,𝜏_𝑒. (13)

On the other hand, the equation 𝑑𝑃(𝑡)=𝑃(𝑡)𝑐−𝑑−𝑐𝑚−ℎ𝑃(𝑡)𝑑𝑡+𝜎₂𝑃(𝑡)𝑑𝐵₂(𝑡),𝑃(0)=𝑃₀, (14) has a unique solution as follows: 𝑃(𝑡)𝜎=exp𝑐−𝑑−𝑐𝑚−²²2𝑡+𝜎₂𝐵₂(𝑡×1)𝑃₀+ℎ^𝑡₀𝜎exp𝑐−𝑑−𝑐𝑚−²²2𝑠+𝜎₂𝐵₂(𝑠)𝑑𝑠⁻¹. (15) Considering the predator population 𝑃(𝑡) in model (6), we have 𝑑𝑃(𝑡)≤𝑃(𝑡)(𝑐−𝑑)𝑑𝑡+𝜎₂𝑃(𝑡)𝑑𝐵₂(𝑡),𝑑𝑃(𝑡)=𝑃(𝑡)𝑐−𝑑−𝑐𝑚𝑃^𝛼(𝑡)𝑁(𝑡)+𝑚𝑃^𝛼(𝑡)−ℎ𝑃(𝑡)𝑑𝑡+𝜎₂𝑃(𝑡)𝑑𝐵₂(𝑡)≥𝑃(𝑡)(𝑐−𝑑−𝑐𝑚−ℎ𝑃(𝑡))𝑑𝑡+𝜎₂𝑃(𝑡)𝑑𝐵₂(𝑡). (16) By the comparison theorem, we obtain 𝑃(𝑡)≥𝑃(𝑡) a.s. 𝑡∈[0,𝜏_𝑒); then 𝑃(𝑡)≤𝑃(𝑡)≤𝑃₀𝜎exp𝑐−𝑑−²²2𝑡+𝜎₂𝐵₂(𝑡=)𝑃(𝑡)a.s.𝑡∈0,𝜏_𝑒. (17)

From the representation of solutions 𝑁(𝑡), 𝑁(𝑡), 𝑃(𝑡), and 𝑃(𝑡), we can see that they are all existence for 𝑡∈[0,∞); that is, 𝜏_𝑒=∞. Therefore, we have the following theorem to show that the positive solution of model (6) is global, which is essential for a population system.

Theorem 2. There is a unique positive solution (𝑁(𝑡),𝑃(𝑡)) of model (6) almost surely for any initial value (𝑁₀,𝑃₀)∈𝑅²₊. Moreover there exist functions 𝑁(𝑡), 𝑁(𝑡), 𝑃(𝑡), and 𝑃(𝑡) defined as (8), (11), (15), and (17) such that 𝑁(𝑡)≤𝑁(𝑡)≤𝑁(t),𝑃(𝑡)≤𝑃(𝑡)≤𝑃(𝑡)a.s.𝑡≥0. (18)

2.2. Stochastic Boundedness

In this subsection, we show that the solution (𝑁(𝑡),𝑃(𝑡)) of model (6) with any positive initial value is uniformly bounded in mean.

Theorem 3. The solution (𝑁(𝑡),𝑃(𝑡)) of model (6) with any positive initial value has the property that limsup_𝑡→∞𝐄[𝑁(𝑡)]≤𝐾,limsup_𝑡→∞𝐄[𝑃(𝑡≤())]𝑐𝐾𝑟+𝑑².4𝑟𝑏𝑑 (19)

Proof. From (7), we obtain limsup_𝑡→∞𝐄𝑁(𝑡)≤𝐾; (20) combining 𝑁(𝑡)≤𝑁(𝑡), then limsup_𝑡→∞𝐄[𝑁(𝑡)]≤𝐾. (21)
Set 𝑀(𝑡)(𝑡)+𝑏=𝑁𝑐𝑃(𝑡); (22) then (𝑡)=(𝑡)𝑁(𝑡)𝑑𝑀𝑟𝑁1−𝐾−𝑏𝑐𝑃(𝑡)(𝑑+ℎ𝑃(𝑡))𝑑𝑡+𝜎₁𝑁(𝑡)𝑑𝐵₁(𝑡)+𝜎₂𝑏𝑐𝑃(𝑡)𝑑𝐵₂(𝑡)=()𝑁(𝑡)−𝑟𝑟+𝑑𝐾𝑁²(𝑡)−𝑏ℎ𝑐𝑃²(𝑡)−𝑑𝑀(𝑡)𝑑𝑡+𝜎₁𝑁(𝑡)𝑑𝐵₁(𝑡)+𝜎₂𝑏𝑐𝑃(𝑡)𝑑𝐵₂(𝑡). (23) Integrating the above equation from 0 to 𝑡, we obtain 𝑀(𝑡)=𝑀(0)+^𝑡₀)𝑁(𝑠)−𝑟(𝑟+𝑑𝐾𝑁²(𝑠)−𝑏ℎ𝑐𝑃²(𝑠)−𝑑𝑀(𝑠)𝑑𝑠+𝜎₁^𝑡₀𝑁(𝑠)𝑑𝐵₁(𝑠)+𝜎₂𝑏𝑐^𝑡₀𝑃(𝑠)𝑑𝐵₂(𝑠), (24) and taking expectations leads to 𝐄[𝑀(𝑡)]=𝑀(0)+^𝑡₀𝐄)𝑁(𝑠)−𝑟(𝑟+𝑑𝐾𝑁²(𝑠)−𝑏ℎ𝑐𝑃²(𝑠)−𝑑𝑀(𝑠)𝑑𝑠; (25) then 𝑑𝐄[𝑀(𝑡)]=()𝐄[𝑁(𝑡−𝑟𝑑𝑡𝑟+𝑑)]𝐾𝐄𝑁²(𝑡)−𝑏ℎ𝑐𝐄𝑃²(𝑡[𝑀(𝑡≤()𝐄[𝑁(𝑡−𝑟)−𝑑𝐄)]𝑟+𝑑)]𝐾(𝐄[𝑁(𝑡)])²−𝑏ℎ𝑐(𝐄[𝑃(𝑡)])²[𝑀(𝑡≤𝐾()−𝑑𝐄)]𝑟+𝑑²[𝑀(𝑡.4𝑟−𝑑𝐄)] (26) By the comparison theorem, we can get limsup_𝑡→∞𝐄[𝑀(𝑡)]=limsup_𝑡→∞𝐄[𝑁(𝑡+𝑏)]𝑐𝐄[𝑃(𝑡≤𝐾())]𝑟+𝑑².4𝑟𝑑 (27) Therefore, we obtain limsup_𝑡→∞𝐄[𝑃(𝑡≤())]𝑐𝐾𝑟+𝑑².4𝑟𝑏𝑑 (28) This completes the proof.

2.3. The Long Time Behavior

It is well known that the property of permanence is more desirable since it means the long time survival in a population dynamics. Now, the definition of stochastic permanence will be given below [32, 33].

Definition 4. The solution (𝑁(𝑡),𝑃(𝑡)) of model (6) is said to be stochastically permanent, if, for any 𝜀∈(0,1), there exists a pair of positive constants 𝛿=𝛿(𝜀) and 𝜒=𝜒(𝜀) such that, for any initial value (𝑁₀,𝑃₀)∈𝑅²₊, the solution (𝑁(𝑡),𝑃(𝑡)) to model (6) has the properties that liminf_𝑡→∞𝐏{|𝑁(𝑡),𝑃(𝑡)|≥𝛿}≥1−𝜀,liminf_𝑡→∞𝐏|𝑁(𝑡),𝑃(𝑡)|≤𝜒≥1−𝜀. (29)

Lemma 5. For any initial value (𝑁₀,𝑃₀)∈𝑅²₊, the solution (𝑁(𝑡),𝑃(𝑡)) satisfies that limsup_𝑡→∞𝐄𝑁²+𝑃^2−𝜃/2≤𝐶𝑘, (30) where 𝐶=𝐶(𝜃) is a positive constant and 𝜃,𝑘 are arbitrary positive constants satisfying 𝑏𝜃min𝑟−𝑚>𝜃(),𝑐−𝑑𝜃+12max𝜎²₁,𝜎²₂+𝑘. (31)

Proof. Set a function 𝑉()=1𝑁,𝑃,𝑁+𝑃 (32) for (𝑁(𝑡),𝑃(𝑡))∈𝑅²₊; using Itô’s formula, we have 𝑑𝑉=−𝑉²𝑁𝑟𝑟−𝐾𝑁−𝑏𝑃𝑁+𝑚𝑃^𝛼+𝑃𝑐𝑁𝑁+𝑚𝑃^𝛼−𝑑−ℎ𝑃𝑑𝑡+𝑉³𝜎²₁𝑁²+𝜎²₂𝑃²𝑑𝑡−𝑉²𝜎₁𝑁𝑑𝐵₁+𝜎₂𝑃𝑑𝐵₂. (33) Choosing a positive constant 𝜃 and by Itô’s formula, we get 𝐋(1+𝑉)^𝜃=𝜃(1+𝑉)^𝜃−1−𝑉²𝑁𝑟𝑟−𝐾𝑁−𝑏𝑃𝑁+𝑚𝑃^𝛼+𝑃𝑐𝑁𝑁+𝑚𝑃^𝛼−𝑑−ℎ𝑃+𝑉³𝜎²₁𝑁²+𝜎²₂𝑃²+𝜃(𝜃−1)2𝑉⁴(1+𝑉)^𝜃−2𝜎²₁𝑁²+𝜎²₂𝑃². (34)
Let 𝑘>0 be sufficiently small such that it satisfies (31); by Itô’s formula, then 𝐋e^𝑘𝑡(1+𝑉)^𝜃=e^𝑘𝑡(1+𝑉)^𝜃−2𝑘(1+𝑉)²−𝜃𝑉²×𝑁𝑟𝑟−𝐾𝑁−𝑏𝑃𝑁+𝑚𝑃^𝛼+𝑃𝑐𝑁𝑁+𝑚𝑃^𝛼−𝑑−ℎ𝑃−𝜃𝑉³𝑁𝑟𝑟−𝐾𝑁−𝑏𝑃𝑁+𝑚𝑃^𝛼+𝑃𝑐𝑁𝑁+𝑚𝑃^𝛼−𝑑−ℎ𝑃+𝜃𝑉³𝜎²₁𝑁²+𝜎²₂𝑃²+𝜃(𝜃−1)2𝑉⁴𝜎²₁𝑁²+𝜎²₂𝑃². (35) Based on the following inequality, 𝑉³𝜎²₁𝑁²+𝜎²₂𝑃²≤max𝜎²₁,𝜎²₂𝑉. (36) Therefore, we obtain 𝐋e^𝑘𝑡(1+𝑉)^𝜃≤e^𝑘𝑡(1+𝑉)^𝜃−2𝑘(1+𝑉)²−𝜃𝑉²𝑁𝑟𝑟−𝐾𝑁−𝑏𝑃𝑁+𝑚𝑃^𝛼+𝑃𝑐𝑁𝑁+𝑚𝑃^𝛼−𝑑−ℎ𝑃−𝜃𝑉³𝑁𝑟𝑟−𝐾𝑁−𝑏𝑃𝑁+𝑚𝑃^𝛼+𝑃𝑐𝑁𝑁+𝑚𝑃^𝛼𝜎−𝑑−ℎ𝑃+𝜃𝑉max²₁,𝜎²₂+𝜃(𝜃−1)2𝑉²max𝜎²₁,𝜎²₂=e^𝑘𝑡(1+𝑉)^𝜃−2𝑘+2𝑘−𝜃𝑉²×𝑟𝑟𝑁−𝐾𝑁²−𝑏𝑁𝑃𝑁+𝑚𝑃^𝛼+𝑐𝑁𝑃𝑁+𝑚𝑃^𝛼−𝑑𝑃−ℎ𝑃²+𝜃max𝜎²₁,𝜎²₂𝑉−𝜃𝑟𝑟𝑁−𝐾𝑁²−𝑏𝑁𝑃𝑁+𝑚𝑃^𝛼+𝑐𝑁𝑃𝑁+𝑚𝑃^𝛼−𝑑𝑃−ℎ𝑃²−𝜃(𝜃−1)2max𝜎²₁,𝜎²₂𝑉²≤e^𝑘𝑡(1+𝑉)^𝜃−2×𝑘+𝜃𝑉²𝑟𝐾𝑁²+ℎ𝑃²+2𝑘+𝜃𝑉²𝑟𝐾𝑁²+ℎ𝑃²−𝜃𝑉²×𝑟𝑁−𝑏𝑁𝑃𝑁+𝑚𝑃^𝛼+𝑐𝑁𝑃𝑁+𝑚𝑃^𝛼𝜎−𝑑𝑃+𝜃max²₁,𝜎²₂𝑉−𝜃𝑟𝑁−𝑏𝑁𝑃𝑁+𝑚𝑃^𝛼+𝑐𝑁𝑃𝑁+𝑚𝑃^𝛼−𝜃()−𝑑𝑃𝜃−12max𝜎²₁,𝜎²₂−𝑘𝑉²≤e^𝑘𝑡(1+𝑉)^𝜃−2𝑟𝑘+𝜃max𝐾+𝑟,ℎ2𝑘+𝜃max𝐾𝑏,ℎ−𝜃min𝑟−𝑚𝜎,𝑐−𝑑+𝜃max²₁,𝜎²₂𝑉−𝑏𝜃min𝑟−𝑚−𝜃(),𝑐−𝑑𝜃−12max𝜎²₁,𝜎²₂−𝑘𝑉². (37) There exists a positive constant 𝐶₀ such that 𝐋e^𝑘𝑡(1+𝑉)^𝜃≤𝐶₀e^𝑘𝑡; then 𝐄e^𝑘𝑡(1+𝑉)^𝜃≤(1+𝑉(0))^𝜃+𝐶₀𝑘e^𝑘𝑡. (38) So, we can get limsup_𝑡→∞𝐄𝑉^𝜃(𝑡)≤limsup_𝑡→∞𝐄(1+𝑉(𝑡))^𝜃≤𝐶₀𝑘. (39)
In addition, we know that (𝑁+𝑃)^𝜃≤2^𝜃(𝑁²+𝑃²)^𝜃/2; consequently, limsup_𝑡→∞𝐄𝑁²+𝑃^2−𝜃/2≤2^𝜃limsup_t→∞𝐄𝑉^𝜃(𝑡)≤2^𝜃𝐶₀𝑘≜𝐶𝑘. (40) The proof is complete.

Based on the results of Theorem 3, Lemma 5, and the Chebyshev inequality [14], we can obtain the following theorem.

Theorem 6. Assume that max{𝜎²₁,𝜎²₂}<2min{𝑟−𝑏/𝑚,𝑐−𝑑}; the solution of model (6) is stochastically permanent.

In a view of ecology, the coexistence of species may be a good situation. In the following, we consider the stochastic persistence (i.e., stochastic persistence in mean) of the species.

Theorem 7. Assume that 𝑟−𝜎²₁/2>𝑏/𝑚 holds, for any initial value 𝑁₀>0; then the solution 𝑁(𝑡) to model (6) has the property 𝐾𝑟−𝑏/𝑚−𝜎²₁/2𝑟≤liminf_𝑡→∞1𝑡^𝑡₀𝑁(𝑠)𝑑𝑠≤limsup_𝑡→∞1𝑡^𝑡₀𝑁(𝑠)𝐾𝑑𝑠≤𝑟−𝜎²₁/2𝑟𝑎.𝑠. (41)

Proof. Denoting 𝑉(𝑁)=ln𝑁 and by Itô’s formula, we obtain 𝑟𝑑𝑉=𝑟−𝐾𝑁(𝑡)−𝑏𝑃(𝑡)𝑁(𝑡)+𝑚𝑃^𝛼(𝑡)−𝜎²₁2𝑑𝑡+𝜎₁𝑑𝐵₁(𝑡). (42) Then, we have ln𝑁(𝑡)=ln𝑁₀+𝜎𝑟−²¹2𝑟𝑡−𝐾^𝑡₀𝑁(𝑠)𝑑𝑠−𝑏^𝑡₀𝑃(𝑠)𝑁(𝑠)+𝑚𝑃^𝛼(𝑠)𝑑𝑠+𝜎₁𝐵₁(𝑡). (43) And we get 𝑟𝐾^𝑡₀𝑁(𝑠)𝑑𝑠≤−ln𝑁(𝑡)+ln𝑁₀+𝜎𝑟−²¹2𝑡+𝜎₁𝐵₁(𝑡). (44) Dividing 𝑡 on both sides of the previously mentioned inequality yields 𝑟𝐾1𝑡^𝑡₀𝑁(𝑠)(𝑡)𝑑𝑠≤−ln𝑁𝑡+ln𝑁₀𝑡+𝜎𝑟−²¹2+𝜎₁𝐵₁(𝑡)𝑡. (45)
Letting 𝑡→∞, we know that lim_𝑡→∞ln𝑁(𝑡)𝑡=0a.s.; (46) Then we obtain limsup_𝑡→∞1𝑡^𝑡₀𝑁(𝑠)𝐾𝑑𝑠≤𝑟−𝜎²₁/2𝑟a.s. (47)
On the other hand, 𝑟𝐾^𝑡₀𝑁(𝑠)𝑑𝑠≥−ln𝑁(𝑡)+ln𝑁₀+𝜎𝑟−²¹2𝑏𝑡−𝑚𝑡+𝜎₁𝐵₁(𝑡); (48) dividing 𝑡 on both sides and letting 𝑡→∞, then liminf_𝑡→∞1𝑡^𝑡₀𝑁(𝑠)𝐾𝑑𝑠≥𝑟−𝑏/𝑚−𝜎²₁/2𝑟. (49) From the above results, inequality (41) holds.

Theorem 8. Assume that 𝑐−𝑑−𝜎²₂/2>0 holds and that (𝑁(𝑡),𝑃(𝑡)) is the solution of model (6) for any initial value (𝑁₀,𝑃₀)∈𝑅²₊; then liminf_𝑡→∞1𝑡^𝑡₀𝑁(𝑠)𝑁(𝑠)+𝑚𝑃^𝛼(𝑠)𝑑𝑠≥𝑑+𝜎²₂/2𝑐,limsup_𝑡→∞1𝑡^𝑡₀𝑃^𝛼(𝑠)𝑁(𝑠)+𝑚𝑃^𝛼(𝑠)𝑑𝑠≤𝑐−𝑑−𝜎²₂/2,𝑐𝑚 (50) which are stable in time average.

Proof. Denoting 𝑉(𝑃)=ln𝑃 and by Itô’s formula, we have (𝑡)𝑑𝑉=𝑐𝑁𝑁(𝑡)+𝑚𝑃^𝛼(𝑡)(𝑡)−𝜎−𝑑−ℎ𝑃²²2𝑑𝑡+𝜎₂𝑑𝐵₂(𝑡). (51) Then, 𝑐^𝑡₀𝑁(𝑠)𝑁(𝑠)+𝑚𝑃^𝛼(𝑠)(𝑡)𝑑𝑠=ln𝑃−ln𝑃₀+𝜎𝑑+²²2𝑡+ℎ^𝑡₀𝑃(𝑠)𝑑𝑠−𝜎₂𝐵₂(𝑡)≥ln𝑃(𝑡)−ln𝑃₀+𝜎𝑑+²²2𝑡−𝜎₂𝐵₂(𝑡). (52) Dividing 𝑡 on both sides yields 𝑐𝑡^𝑡₀𝑁(𝑠)𝑁(𝑠)+𝑚𝑃^𝛼(𝑠)≥(𝑡)𝑑𝑠ln𝑃𝑡−ln𝑃₀𝑡𝜎+𝑑+²²2−𝜎₂𝐵₂(𝑡)𝑡; (53) we obtain liminf_𝑡→∞1𝑡^𝑡₀𝑁(𝑠)𝑁(𝑠)+𝑚𝑃^𝛼(𝑠)𝑑𝑠≥𝑑+𝜎²₂/2𝑐. (54)
Furthermore, 𝑑𝑉=𝑐−𝑐𝑚𝑃^𝛼(𝑡)𝑁(𝑡)+𝑚𝑃^𝛼(𝑡)(𝑡)−𝜎−𝑑−ℎ𝑃²²2𝑑𝑡+𝜎₂𝑑𝐵₂(𝑡); (55) then we can get 𝑐𝑚^𝑡₀𝑃^𝛼(𝑠)𝑁(𝑠)+𝑚𝑃^𝛼(𝑠)(𝑡)𝑑𝑠=−ln𝑃+ln𝑃₀+𝜎𝑐−𝑑−²²2𝑡−ℎ^𝑡₀𝑃(𝑠)𝑑𝑠+𝜎₂𝐵₂(𝑡)≤−ln𝑃(𝑡)+ln𝑃₀+𝜎𝑐−𝑑−²²2𝑡+𝜎₂𝐵₂(𝑡). (56) Dividing 𝑡 on both sides, we obtain limsup_𝑡→∞1𝑡^𝑡₀𝑃^𝛼(𝑠)𝑁(𝑠)+𝑚𝑃^𝛼(𝑠)𝜎𝑑𝑠≤𝑐−𝑑−²₂/2.𝑐𝑚 (57)

2.4. Extinction

From (17), if 𝑐−𝑑−𝜎²₂/2<0, then lim_𝑡→∞𝑃(𝑡)=0 a.s. Moreover, from Theorem 7 and (49), we know that if 𝑟−𝑏/𝑚−𝜎²₁/2>0 holds, then liminf_𝑡→∞1𝑡^𝑡₀𝑁(𝑠)𝐾𝑑𝑠≥𝑟−𝑏/𝑚−𝜎²₁/2𝑟a.s., (58) which implies that there are a 𝑇₀>0 and a positive constant 𝑛₀ such that 𝑁(𝑡)>𝑛₀ a.s. for 𝑡≥𝑇₀. Besides, for all 𝜀>0, there are 𝑇>𝑇₀ and Ω_𝜀 such that 𝐏(Ω_𝜀)≥1−𝜀 and 𝑏𝑃(𝑡)/𝑁(𝑡)≤𝜀 for 𝑡≥𝑇. Then we obtain (𝑡)(𝑡)𝑟𝑑𝑁=𝑁𝑟−𝐾𝑁(𝑡)−𝑏𝑃(𝑡)𝑁(𝑡)+𝑚𝑃^𝛼(𝑡)𝑑𝑡+𝜎₁𝑁(𝑡)𝑑𝐵₁(𝑡)(𝑡)𝑟≥𝑁𝑟−𝐾𝑁(𝑡)−𝑏𝑃(𝑡)𝑁(𝑡)𝑑𝑡+𝜎₁𝑁(𝑡)𝑑𝐵₁(𝑡)(𝑡)𝑟≥𝑁𝑟−𝐾𝑁(𝑡)−𝜀𝑑𝑡+𝜎₁𝑁(𝑡)𝑑𝐵₁(𝑡), (59) which implies that liminf_𝑡→∞1𝑡^𝑡₀𝑁(𝑠)𝐾𝑑𝑠≥𝑟−𝜀−𝜎²₁/2𝑟>0. (60)

From Theorem 7 and (47), we have limsup_𝑡→∞1𝑡^𝑡₀𝑁(𝑠)𝐾𝑑𝑠≤𝑟−𝜎²₁/2𝑟a.s. (61) Therefore, by the arbitrary of 𝜀, we get lim_𝑡→∞1𝑡^𝑡₀𝑁(𝑠)𝐾𝑑𝑠=𝑟−𝜎²₁/2𝑟a.s. (62)

Combining the above arguments, we can get the theorem as follows.

Theorem 9. Let (𝑁(𝑡),𝑃(𝑡)) be the solution of model (6) with any initial value (𝑁₀,𝑃₀)∈𝑅²₊. If 𝑟−𝑏/𝑚−𝜎²₁/2>0 and 𝑐−𝑑−𝜎²₂/2<0, then lim_𝑡→∞1𝑡^𝑡₀𝑁(𝑠)𝐾𝑑𝑠=𝑟−𝜎²₁/2𝑟,lim_𝑡→∞𝑃(𝑡)=0. (63)

Furthermore, set 𝑢(𝑡)=ln𝑁(𝑡) and 𝑣(𝑡)=ln𝑃(𝑡); for the first equation of model (6) we have (𝑡)=𝑟𝑑𝑢𝑟−𝐾e^𝑢(𝑡)−𝑏e^𝑣(𝑡)e^𝑢(𝑡)+𝑚e^{𝛼𝑣(𝑡)}−𝜎²₁2𝑑𝑡+𝜎₁𝑑𝐵₁(𝑡)≤𝜎𝑟−²¹2𝑑𝑡+𝜎₁𝑑𝐵₁(𝑡). (64) Taking the comparison theorem of stochastic equations and the theory of diffusion processes [14], then lim_𝑡→∞𝑢(𝑡)=−∞ a.s.; that is, lim_𝑡→∞𝑁(𝑡)=0a.s. (65)

Similarly, we obtain lim_𝑡→∞𝑃(𝑡)=0a.s. (66) If not, then there is a positive constant 𝐻 such that limsup_𝑡→∞𝑃(𝑡)=𝐻>0a.s. (67) Hence, for any given 𝜀>0, there exist 𝑡₀ and a set Ω_𝜀 such that 𝐏(Ω_𝜀)≥1−𝜀 and 𝑐𝑁(𝑡)/(𝑁(𝑡)+𝑚𝑃^𝛼(𝑡))≤𝜀 for 𝑡≥𝑡₀. Therefore, −𝑃(𝑡)(𝑑+ℎ𝑃(𝑡))𝑑𝑡+𝜎₂𝑃(𝑡)𝑑𝐵₂(𝑡)≤𝑑𝑃(𝑡)≤𝑃(𝑡)(−𝑑+𝜀)𝑑𝑡+𝜎₂𝑃(𝑡)𝑑𝐵₂(𝑡),−(𝑡)+𝜎𝑑+ℎ𝑃²²2𝑑𝑡+𝜎₂𝑑𝐵₂(𝑡)(𝑡)≤𝜎≤𝑑𝑣−𝑑+𝜀−²²2𝑑𝑡+𝜎₂𝑑𝐵₂(𝑡). (68) By the same reasoning as previously stated, we can get lim_𝑡→∞𝑣(𝑡)=−∞ a.s.; that is, lim_𝑡→∞𝑃(𝑡)=0a.s. (69) There is a contradiction; hence (66) is true.

Based on the above, we obtain the following theorem which means that if the noise satisfies some conditions, then both species 𝑁 and 𝑃 of model (6) will die out.

Theorem 10. Let (𝑁(𝑡),𝑃(𝑡)) be the solution of model (6) with any initial value (𝑁₀,𝑃₀)∈𝑅²₊. If 𝑟−𝑏/𝑚−𝜎²₁/2<0, then lim_𝑡→∞𝑁(𝑡)=0,lim_𝑡→∞𝑃(𝑡)=0. (70)

3. Numerical Simulations

In this section, we perform some numerical simulations for model (6) with environmental noise to illustrate the previously mentioned analytical findings by referring to the method mentioned in Higham [34]. Next, we consider the discretization equations 𝑁_𝑖+1=𝑁_𝑖+𝑁_𝑖𝑟𝑟−𝐾𝑁_𝑖−𝑏𝑃_𝑖𝑁_𝑖+𝑚𝑃_𝑖^𝛼Δ𝑡+𝛼𝑁_𝑖√Δ𝑡𝜉_1𝑖+𝛼²2𝑁²_𝑖𝜉²_1𝑖𝑃−1Δ𝑡,_𝑖+1=𝑃_𝑖+𝑃_𝑖𝑐𝑁_𝑖𝑁_𝑖+𝑚𝑃_𝑖^𝛼−𝑑−ℎ𝑃_𝑖Δ𝑡+𝛽𝑃_𝑖√Δ𝑡𝜉_2𝑖+𝛽²2𝑃²_𝑖𝜉²_2𝑖−1Δ𝑡, (71) where 𝜉_1𝑖 and 𝜉_2𝑖(𝑖=1,2,…,𝑛) are the Gaussian random variables 𝐍(0,1).

When choosing the values of parameters 𝑟=2, 𝐾=0.15, 𝑏=0.8, 𝑐=0.5, 𝑑=0.1, 𝑚=0.1, 𝛼=0.2, and ℎ=0.1 for model (4), which has three equilibria in the positive quadrant, where 𝐸₀=(0,0) (total extinct) and 𝐸₁=(0.15,0) (extinct of the predator or prey only) are saddle points, 𝐸^∗=(0.023,0.2025) (coexistence of the prey and predator) is globally asymptotically stable. The trajectory of the prey and predator population of model (4) is shown in Figure 1.

Figure 1 

Phase portrait of model (4). Other parameters are taken as 𝑟=2, 𝐾=0.15, 𝑏=0.8, 𝑐=0.5, 𝑑=0.1, 𝑚=0.1, 𝛼=0.2, and ℎ=0.1. The horizontal axis is prey population 𝑁 and the vertical axis is predator population 𝑃. 𝐸0=(0,0) and 𝐸1=(0.15,0) are two saddle points; 𝐸∗=(0.023,0.2025) is stable.

Figure 2 shows the time-series plots of model (6) with different noise intensities 𝜎²₁ and 𝜎²₂. When choosing 𝜎₁=𝜎₂=0.045 (Figure 2(a)) and 𝜎₁=0.12, 𝜎₂=0.3 (Figure 2(b)), from Theorem 6, we know that the positive solution of model (6) is stochastically permanent, which means that stochastic perturbations do not change the permanence of the deterministic model (4). Moreover, from Theorem 7, the model will be stochastically persistent in mean. In other words, we can use the deterministic model (4) describing the dynamics of the stochastic model when the noise intensities 𝜎²₁ and 𝜎²₂ are small. From Figure 2(b), we can observe that the violent fluctuations appear as the noise intensities further increase. It means that noise has strong destabilizing effects on the model and the amplitude of the fluctuations in population density of prey and predator species increases obviously, implying instability of the coexisting equilibrium point in the stochastic environment.

(a)

(b)

Figure 2 

Solutions of model (6) with different noise and other parameters are the same as those of Figure 1 and initial condition (𝑁0,𝑃0)=(0.05,0.23). (a) 𝜎1=𝜎2=0.045 and (b) 𝜎1=0.12, 𝜎2=0.3.

In Figure 3, we continue to choose different noise intensities 𝜎²₁ and 𝜎²₂ to consider the effects of noise to model (6). When choosing 𝜎₁=0.15 and 𝜎₂=0.9 (Figure 3(a)), the conditions of Theorem 9 are satisfied; then we can find that prey population 𝑁(𝑡) of model (6) is permanent and predator population 𝑃(𝑡) will die out. Choosing 𝜎₁=0.918 and 𝜎₂=0.6 in Figure 3(b), which satisfies the conditions of Theorem 10, both species 𝑁 and 𝑃 in model (6) become extinct. That is to say, big noise can make the two species die out. From the above numerical results and by Theorems 3, 6, 7, 9, and 10, we conclude that for some noise intensities 𝜎²₁ and 𝜎²₂ the dynamical behaviors of stochastically ultimately boundedness, permanence, persistence in mean, and extinction can be observed in model (6).

(a)

(b)

Figure 3 

Solutions of model (6) with different noise and other parameters are the same as those of Figure 1. (a) 𝜎1=0.15, 𝜎2=0.9 and (b) 𝜎1=0.918, 𝜎2=0.6.

4. Conclusions and Remarks

In this paper, we consider a stochastic Hassell-Varley type predator-prey model. The value of this study lies in twofolds. First, it verifies some relevant properties of the corresponding stochastic model (6), which shows the global existence, boundedness and stochastic permanence, persistence in mean, and extinction of the positive solution. Second, it illustrates the dynamics of the model via numerical simulations, which shows that if the noise is not large and satisfies some conditions, the stochastic perturbations do not cause drastic changes of the dynamics in the deterministic model (4), while if the noise is sufficiently large and satisfies some conditions, it will force two species in the model to die out.

In order to study the stochastic model (6), we perturb the deterministic model (4) by incorporating white noise terms in the growth rate of prey population and in the death rate of predator population. Establishing a Lyapunov function, there is a unique positive solution to the model for any positive initial value. Applying Itô’s formula, we derive that, under some conditions, the solution of model (6) is stochastically bounded, permanent, and stochastic persistent in mean and extinct. For the fixed parameters 𝑟, 𝐾, 𝑏, 𝑐, 𝑑, 𝑚, 𝛼, and ℎ, some conditions depend on the intensities of noise 𝜎²₁ and noise 𝜎²₂. When the noise intensities satisfy some conditions of Theorem 9, we can find that prey population 𝑁(𝑡) of model (6) is permanent and predator population 𝑃(𝑡) will die out (see Figure 3(a)), while with the noise intensities increasing which satisfy the conditions of Theorem 10, from Figure 3(b), two species 𝑁 and 𝑃 will die out. In other words, when the noise satisfies some conditions of Theorems 6 and 7 and is not sufficiently large, the populations 𝑁 and 𝑃 may be stochastically permanent and persistent in mean, while large noise satisfying the conditions of Theorems 9 and 10 will force the population to become extinct. Our complete analysis of the stochastic model will give some suggestions to the studies of the population dynamics.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the editor and the anonymous referees for the very helpful suggestions and comments which led to improvement of their original paper. This research was supported by the National Science Foundation of China (no. 11301263).