# Paradox of Enrichment in a Stochastic Predator-Prey Model

Authors: Jawdat Alebraheem
Journal: Journal of Mathematics
 (2020)

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## Abstract

We propose a stochastic predator-prey model to study a novel idea that involves investigating random noises effects on the enrichment paradox phenomenon. Existence and stochastic boundedness of a unique positive solution with positive initial conditions are proved. The global asymptotic stability is studied to determine the occurrence of the enrichment paradox phenomenon. We show theoretically that intensive noises play an important role in the occurrence of the phenomenon, where increasing intensive noises lead to occurrence of the paradox of enrichment. We perform numerical simulations to verify and demonstrate the theoretical results. The new results in this study may contribute to increasing attention to study the random noise effects on some ecological and biological phenomena as the paradox of enrichment.


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## Body

## 1. Introduction

Theoretical ecology has grown to the extent that we regularly see papers in mathematical ecology discussing models in purely mathematical terms that from time to time bear little resemblance to real ecological processes. There is a desire to reexamine the situation and study the diverse constructing blocks once more. Mathematical modeling is a beneficial tool to reveal how a process works and predict how it will progress [1]. However, the difficulty in determining ecological principles makes formulating ecological problems a complicated process [2, 3].Differential equations are a central tool that are used to describe many ecological problems mathematically. Predator-prey interaction is one of the most significant topics in applied mathematics and mathematical biology [4–8]. Different styles of mathematical models have been used to study predator-prey interaction. Logistic predator-prey models have been used by some researchers due to their realistic descriptions of the growth rates of species [1]. Logistic models display a carrying capacity term that represents limiting species sustainable by the environment. In this paper, we use logistic models of predator and prey equations, in which the carrying capacity of predator is a proportional to the available amount of prey.Paradoxical phenomena have attracted much attention than normal observations. One of the most important paradoxes in ecology is the paradox of enrichment, which was firstly mentioned by Rosenzweig [9]. The paradox of enrichment states that coexistence equilibrium point will destabilize when the carrying capacity is increased, and the destabilization may lead to stochastic extinction for one of the species or all species. Mathematically, destabilization can be explained by the oscillations closer to one of the axes or both axes of phase space. Although the enrichment paradox phenomenon is interpreted as the variance between the real world and the mathematical construction in many experimental studies, the paradox of enrichment has occurred as explained in some recent experimental studies [10–12].Stochastic process has many applications in different sciences [13–17]. Random noises are a ubiquitous characteristic of ecological systems, since nearly all environments are subject to some unexpected factors, which have an important role in an ecosystem component [18–23]. Random noises are important in ecology with regard to focusing on the variability of ecological systems and moving away from thinking in terms of equilibrium dynamics. Some recent studies investigated stochastic predator-prey models to study the random noise effects on the dynamic behaviors such as [24–27]. We highlight the main points that have been focused in the literature on stochastic predator-prey models topic as follows: firstly, using different stochastic predator-prey models; secondly, establishing existence and uniqueness of global solution of these systems; thirdly, establishing sufficient conditions for the existence of a unique ergodic stationary distribution; finally, obtaining the persistence and extinction conditions.Alebraheem [28] studied the occurrence of the paradox of enrichment with deterministic models. Motivated by the results in the previous study, we study a novel idea in this paper that involves investigating stochastic concept to study the random noise effects on the occurrence of the paradox of enrichment in a predator-prey model. In order to study our idea, we propose a stochastic predator-prey model and prove the global stability of the stochastic model by constructing suitable Lyapunov function and using Ito^ formula, which determine the occurrence of the phenomenon. This study may contribute to increasing attention to study the random noise effects on some ecological and biological phenomena as the paradox of enrichment.The rest of this paper is organized as follows. In the next section, we introduce a stochastic predator-prey model. In Section3, we prove existence and stochastic boundedness of a unique positive solution with positive initial conditions. Section 4 presents the global asymptotic stability and the occurrence of the enrichment paradox phenomenon in stochastic and deterministic models, in addition to numerical simulations that verify and demonstrate the theoretical results. Finally, in Section 5, discussion and conclusions are presented.


## 2. Stochastic Predator-Prey Model

We investigate the random noises in a predator-prey model to study its effects on the dynamic behaviors of this model, which have effects on prey and predator species.The standard Ito^ stochastic differential equation has the following form [21]:(1)dXt=FXt,tdt+GXt,tdWt,Xt0=X0,fort≥t0,where the function Ft,Xt is called the drift and Gt,Xt is the diffusion matrix. W is a standard Wiener or Brownian motion processes.We use stochastic differential equations to describe the random noises of continuous-time model that have been applied by some studies [23–27].We define(2)ℝ+n=X=X1,X2,⋯,Xn∈ℝn:Xi>0,1≤i≤n.LetΩ,ℱ,ℙ be a complete probability space with a filtration ℱtt≥0, and suppose that the constant initial value X0∈ℝ+n.The differential operatorL of equation (1) is defined by the following formula [21]:(3)L=∂∂t+∑i=1nfiXt,t∂∂Xi+12∑i,j=1ngXt,t,gXt,tTij∂2∂Xi∂Xj.IfL acts on a function V∈C2,1ℝ+n,ℝ+, then(4)LVXt,t=VtXt,t+VXXt,tfXt,t+12tracegTXt,tVXXXt,tgXt,t,where Vt=∂V/∂t, VX=∂V/∂X1,∂V/∂X2,…,∂V/∂Xn, and VXX=∂2V/∂Xi∂Xjn×n. By Ito^’s formula, we can attain(5)dVXt,t=LVXt,tdt+VXXt,tgXt,tdWt.In this paper, we use the continuous-time predator-prey model with stochastic perturbations as follows:(6a)dX=Xρ1−Xk−αYdt+σ1XdW1,(6b)dY=Y−u+eαX−eαYdt+σ2YdW2.Subjecting to initial conditions,(7)X0=X0>0,Y0=Y0>0,where σi represent the strength of noise and dWi is a standard Wiener or Brownian motion process for i=1,2.The meaning of variables and parameters of model ((6a) and (6b)) are summarized as follows:X: density of prey speciesY: density of predator speciesρ: inherent growth rate of preyk: carrying capacity of the environmentu: death rates of predatorα: attack rate of predatore: efficiency of converting consumed prey into predator birth


## 3. Existence, Uniqueness, and Boundedness of the Positive Solution

Since the model ((6a) and (6b)) is a biological model, the following theorem shows the existence and uniqueness of the global positive solution to the model ((6a) and (6b)). For a model of population dynamics, we have a unique global (i.e., no explosion in a finite time) solution for any given initial value which means that there is a finite population size at a finite time [29] while the solution is positive because of biological feasibility.Theorem 1.
If any initial conditionX0,Y0∈R+2, then there is a unique positive solution Xt,Yt of the model ((6a) and (6b)) almost surely for all t≥0 with probability one.Proof.
We assume the solution of the model ((6a) and (6b)) is Xt,Yt for t∈0,τe, where τe indicates the explosion time.
Let us considerq1t=lnXt and q2=lnYt. We apply Ito^ formula on the system ((6a) and (6b)) and then the equation (6a) becomes as follows:(8)dlnXt=∂q1∂t+∂q1∂XXρ1−Xk−αY+12∂2q1∂X2σ1X2dt+∂q1∂Xσ1XdW1=0+∂q1∂XXρ1−Xk−αY−121X2σ12X2dt+1Xσ1XdW1=ρ−ρXk−αY−12σ12dt+σ1dW1,dq1=ρ−ρkeq1−αeq2−12σ12dt+σ1dW1.
Similarly, for equation (6b), we obtain(9)dq2=−u+eαeq1−eαeq2−12σ12dt+σ2dW2,with initial values q10=lnX0 and q20=lnY0.
The transformed system becomes(10a)dq1=ρ−ρkeq1−αeq2−12σ12dt+σ1dW1,(10b)dq2=−u+eαeq1−eαeq2−12σ12dt+σ2dW2,with q10=lnX0 and q20=lnY0.
The coefficients of the system ((10a) and (10b)) satisfy local Lipschitz condition; this means that the system ((10a) and (10b)) has unique local solution q1t,q2t for t∈0,τe. By using the Ito formula, the unique positive solution of the system ((6a) and (6b)) with initial value X0,Y0 is Xt,Yt=eq1t,eq2t for t∈0,τe.
We show that the solution is global if we verifyτe=∞ a.s.
Since the solution is positive on0,τe, we find(11)dX≤ρXdt+σ1XdW1.
Letξ1t be the unique solution of the equation(12)dξ1=ρξ1dt+σ1ξ1dW1,with ξ10=X0.
LetU=1/ξ1t. Now, we apply Ito^ formula, and we obtain(13)dU=−ρξ1ξ12+σ12ξ12ξ13dt−σ1ξ1ξ12dW1=−ρξ1+σ12ξ1dt−σ1ξ1dW1=−ρ+σ12Udt−σ1UdW1,dU=−ρ+σ12Udt−σ1UdW1,with U0=1/X0.
The unique solution of stochastic differential equation (13) is(14)U=1X0e−ρ+σ12t−σ1W1,i.e.ξ1=X0eρ−σ12t+σ1W1.∴Xt≤ξ1t.
We have from the second equation of the system ((6a) and (6b)) that(15)dY=Y−u+eαX−eαYdt+σ2YdW2,dY≤eαXYdt+σ2YdW2.
Letξ2t be the unique solution of the equation(16)dξ2=eαξ1ξ2dt+σ2ξ2dW2,with ξ20=Y0.
Applying the Ito^ formula, as doing the same procedure for first equation of the system ((6a) and (6b)), we obtain the unique solution of stochastic differential equation (16) as(17)ξ2t=Y0eeα∫0tξ1sds+σ2W2.∴Yt≤ξ2t.
Furthermore, for the first equation of the system ((6a) and (6b)), we have(18)dX≥Xρ1−Xk−αYdt+σ1XdW1.
We can say that(19)dX≥Xρ1−Xk−αξ2dt+σ1XdW1.
AssumeH1t to be the unique solution of the equation(20)dH1t=H1tρ1−H1tk−αξ2dt+σ1H1tdW1,with H10=X0.
By applying the Ito formula, we have the unique solution of stochastic differential equation (20) as(21)H1t=eρ−σ12t−α∫0tξ2sds+σ1W11/X0+ρ/k∫0teρ−σ12s−α∫0tξ2vdu+σ1W1dv.∴Xt≥H1t.
Through comparison theorems for stochastic differential equations [30], we get from (14) and (21) that(22)H1t≤Xt≤ξ1ta.s.fort∈0,τe.
In the same manner, for the second equation of the system ((6a) and (6b)), we have(23)dY≥−uYdt+σ2YdW2.
Suppose thatH2t be the unique solution of the equation(24)dH2t=−uH2dt+σ2H2dW2,with H20=Y0.
By applying Ito^ formula, we have the unique solution of stochastic differential equation (24) as(25)H2t=Y0e−u+σ22t+σ2W2.∴Yt≥H2t.
Also, through comparison theorems for stochastic differential equations [30], we obtain from (17) and (25) that(26)H2t≤Yt≤ξ2ta.s.fort∈0,τe.
It follows thatξ1t,ξ2t,Η1t, and Η2t exist for all t≥0 and τe=∞ and we can conclude that Xt,Yt globally exists. This proves the theorem.
Through the previous proof, we have the following theorem.Theorem 2.
If the unique positive solutionXt,Yt of the system ((6a) and (6b)) has any initial condition X0,Y0∈R+2, then there exist the functions ξ1t,ξ2t,Η1t, and Η2t defined to satisfy(27)Η1t≤Xt≤ξ1t,Η2t≤Yt≤ξ2t,a.s.,forallt≥0.


## 4. Global Asymptotic Stability and Paradox of Enrichment

### 4.1. Theoretical Analysis

In this section, we study the global stability of the stochastic model ((6a) and (6b)) that is an equivalent of the stationary distribution and ergodicity to stochastic models by constructing suitable Lyapunov function and using Ito formula [31]. There is not positive time-independent equilibrium point as deterministic systems. In this paper, the global asymptotic stability of the stochastic system ((6a) and (6b)) (i.e., ergodic property) is studied to determine the occurrence of the enrichment paradox phenomenon.Definition 1.
(see [32]). The trivial solution of stochastic differential equation (1) is defined to be the following:(i)
Stochastically stable∀ε∈0,1 and r>0, ∃ a δ=δε,r>0 such that(28)ℙXt,X0<r∀t≥0≥1−ε.(ii)
Stochastically asymptotically stable if it is stochastically stable and, moreover,∀∊∈0,1, ∃δ0=δ0ε>0 such that(29)ℙlimt⟶∞Xt,X0=0≥1−ε,wheneverX0<δ0.(iii)
Globally stochastically asymptotically stable if it is stochastically stable and, moreover,∀X0∈ℝ with(30)ℙlimt⟶∞Xt,X0=0=1.
If we assumeσ1=0 and σ2=0, then the model ((6a) and (6b)) becomes a deterministic model as given by Alebraheem [28], but in this paper, inherent growth rate of prey ρ is an unknown value, and it was fixed to be 1 in [28]:(31)dXdt=ρX1−Xk−αXY,dYdt=−uY+eαXY−eαY2.
The coexistence equilibrium point of the model (31) is(32)E2=X,Y=ku+ρeeαk+e,eραk−ueα2k+eα.
The coexistence equilibrium point exists under the following condition:(33)eραk>u.Theorem 3.
Ifk<ρX−X˙2/1/2eX˙σ12+Y˙σ22−αY−Y˙2and for any initial condition X0,Y0∈R+2, then there is a unique positive solution Xt,Yt of the system ((6a) and (6b)) for all t≥0, which is globally asymptotically stable almost surely (a.s.).Proof.
Define Lyapunov functions as(34)V1=X−X˙−X˙lnXX˙,V2=Y−Y˙−Y˙lnYY˙.
We apply Ito formula and so we have(35)LV1=1−XX˙dX+12X˙X2dX2=X−X˙ρ1−Xk−αYdt+σ1dW1+12X˙σ12dt=X−X˙ρ−ρXk−αYdt+σ1dW1+12X˙σ12dt=X−X˙ρX˙k+αY˙−ρXk−αYdt+σ1dW1+12X˙σ12dt=−ρkX−X˙2−αX−X˙Y−Y˙+12X˙σ12dt+σ1X−X˙dW1.
Similarly, we obtain(36)LV2=1−YY˙dY+12Y˙Y2dY2=Y−Y˙−u+eαX−eαYdt+σ2dW2+12Y˙σ22dt=Y−Y˙eαX−eαX˙−eαY+eαY˙dt+σ2dW2+12Y˙σ22dt=Y−Y˙eαX−X˙−eαY−Y˙dt+σ2dW2+12Y˙σ22dt=eαX−X˙Y−Y˙−eαY−Y˙2+12Y˙σ22dt+σ2Y−Y˙dW2.
Now, we define(37)V=X−X˙−X˙lnXX˙+ΒY−Y˙−Y˙lnYY˙,LV=LV1+BLV2≔LVdt+σ1X−X˙dW1+σ2Y−Y˙dW2,where(38)LV=−ρkX−X˙2−eαY−Y˙2−αX−X˙Y−Y˙+eαX−X˙Y−Y˙+12X˙σ12+12Y˙σ22.We select B=1/e to simplify the mathematical analysis; then,(39)LV=−ρkX−X˙2−αY−Y˙2+12eX˙σ12+12eY˙σ22.
Therefore, the number of terms is reduced.
Ifk<ρX−X˙2/1/2eX˙σ12+Y˙σ22−αY−Y˙2, then this implies LV<0 along all trajectories.
Through the previous analysis and by removing stochastic terms, the dynamics of deterministic model (31) always coexist in a globally stable state and there is no bifurcation under any conditions. Therefore, the paradox of enrichment does not arise with deterministic model (31) as it was proved by Alebraheem [28].Corollary 1.
Ifk<ρX−X˙2/1/2eX˙σ12+Y˙σ22−αY−Y˙2 is violated, then coexistence equilibrium point E2=X,Y destabilizes.Corollary 2.
The paradox of enrichment arises with the stochastic model ((6a) and (6b)).We conclude that the intensive noises affect the stability as shown through Theorem3; this leads to occurrence of the paradox of enrichment in the stochastic model ((6a) and (6b)) as shown through Corollary 2. The occurrence of the enrichment paradox phenomenon depends on the carrying capacity in one side and stochastic terms in another side as it is shown in Theorem 3 and Corollary 1.


### 4.2. Numerical Simulations

We perform numerical simulations to show the effects of stochastic process on the occurrence of the enrichment paradox phenomenon. The stochastic Runge–Kutta method based on Ito^ process is used to solve the model ((6a) and (6b)). We use a command “StochasticRungeKuttaScalarNoise” in MATHEMATICA 11.3 program as a method to execute the numerical simulations, according to Wolfram website [33]. The command uses stage Rossler Stochastic Runge–Kutta method for scalar noise that has order 3/2. The values of the parameters are selected to satisfy condition (33) with regard to the coexistence of prey and predator. The values of the parameters and initial conditions are fixed. However, we use different values of noise strength. The values are as follows:(40)r=1.5,k=3,α=1.5,e=1.0,u=0.65,X0=0.7,Y0=0.3.Model (31) presents stable dynamic behavior as shown in Figure 1 when σ=0, which refers to deterministic model (31). In our simulations, we observe that the dynamic behavior changes from stable to oscillated when investigating the random noises to be σ=0.4, σ=1.0, and σ=3.0, as shown in Figures 2–4, respectively. However, the difference between the second, third, and fourth cases is the size of the oscillations; with increased values of σ, the size of the oscillations increases. Then, we obtain that the random noises have effects on the occurrence of the paradox of enrichment, moving from stable dynamic behavior to oscillated dynamic behavior.Figure 1
Dynamics model (31) when σ=0.0: (a) time series; (b) phase plane.
(a)(b)Figure 2
Dynamics model (6a) and (6b) whenσ=0.4: (a) time series; (b) phase plane.
(a)(b)Figure 3
Dynamics model (6a) and (6b) whenσ=1.0: (a) time series; (b) phase plane.
(a)(b)Figure 4
Dynamics model (6a) and (6b) whenσ=3.0: (a) time series; (b) phase plane.
(a)(b)


## 4.1. Theoretical Analysis

In this section, we study the global stability of the stochastic model ((6a) and (6b)) that is an equivalent of the stationary distribution and ergodicity to stochastic models by constructing suitable Lyapunov function and using Ito formula [31]. There is not positive time-independent equilibrium point as deterministic systems. In this paper, the global asymptotic stability of the stochastic system ((6a) and (6b)) (i.e., ergodic property) is studied to determine the occurrence of the enrichment paradox phenomenon.Definition 1.
(see [32]). The trivial solution of stochastic differential equation (1) is defined to be the following:(i)
Stochastically stable∀ε∈0,1 and r>0, ∃ a δ=δε,r>0 such that(28)ℙXt,X0<r∀t≥0≥1−ε.(ii)
Stochastically asymptotically stable if it is stochastically stable and, moreover,∀∊∈0,1, ∃δ0=δ0ε>0 such that(29)ℙlimt⟶∞Xt,X0=0≥1−ε,wheneverX0<δ0.(iii)
Globally stochastically asymptotically stable if it is stochastically stable and, moreover,∀X0∈ℝ with(30)ℙlimt⟶∞Xt,X0=0=1.
If we assumeσ1=0 and σ2=0, then the model ((6a) and (6b)) becomes a deterministic model as given by Alebraheem [28], but in this paper, inherent growth rate of prey ρ is an unknown value, and it was fixed to be 1 in [28]:(31)dXdt=ρX1−Xk−αXY,dYdt=−uY+eαXY−eαY2.
The coexistence equilibrium point of the model (31) is(32)E2=X,Y=ku+ρeeαk+e,eραk−ueα2k+eα.
The coexistence equilibrium point exists under the following condition:(33)eραk>u.Theorem 3.
Ifk<ρX−X˙2/1/2eX˙σ12+Y˙σ22−αY−Y˙2and for any initial condition X0,Y0∈R+2, then there is a unique positive solution Xt,Yt of the system ((6a) and (6b)) for all t≥0, which is globally asymptotically stable almost surely (a.s.).Proof.
Define Lyapunov functions as(34)V1=X−X˙−X˙lnXX˙,V2=Y−Y˙−Y˙lnYY˙.
We apply Ito formula and so we have(35)LV1=1−XX˙dX+12X˙X2dX2=X−X˙ρ1−Xk−αYdt+σ1dW1+12X˙σ12dt=X−X˙ρ−ρXk−αYdt+σ1dW1+12X˙σ12dt=X−X˙ρX˙k+αY˙−ρXk−αYdt+σ1dW1+12X˙σ12dt=−ρkX−X˙2−αX−X˙Y−Y˙+12X˙σ12dt+σ1X−X˙dW1.
Similarly, we obtain(36)LV2=1−YY˙dY+12Y˙Y2dY2=Y−Y˙−u+eαX−eαYdt+σ2dW2+12Y˙σ22dt=Y−Y˙eαX−eαX˙−eαY+eαY˙dt+σ2dW2+12Y˙σ22dt=Y−Y˙eαX−X˙−eαY−Y˙dt+σ2dW2+12Y˙σ22dt=eαX−X˙Y−Y˙−eαY−Y˙2+12Y˙σ22dt+σ2Y−Y˙dW2.
Now, we define(37)V=X−X˙−X˙lnXX˙+ΒY−Y˙−Y˙lnYY˙,LV=LV1+BLV2≔LVdt+σ1X−X˙dW1+σ2Y−Y˙dW2,where(38)LV=−ρkX−X˙2−eαY−Y˙2−αX−X˙Y−Y˙+eαX−X˙Y−Y˙+12X˙σ12+12Y˙σ22.We select B=1/e to simplify the mathematical analysis; then,(39)LV=−ρkX−X˙2−αY−Y˙2+12eX˙σ12+12eY˙σ22.
Therefore, the number of terms is reduced.
Ifk<ρX−X˙2/1/2eX˙σ12+Y˙σ22−αY−Y˙2, then this implies LV<0 along all trajectories.
Through the previous analysis and by removing stochastic terms, the dynamics of deterministic model (31) always coexist in a globally stable state and there is no bifurcation under any conditions. Therefore, the paradox of enrichment does not arise with deterministic model (31) as it was proved by Alebraheem [28].Corollary 1.
Ifk<ρX−X˙2/1/2eX˙σ12+Y˙σ22−αY−Y˙2 is violated, then coexistence equilibrium point E2=X,Y destabilizes.Corollary 2.
The paradox of enrichment arises with the stochastic model ((6a) and (6b)).We conclude that the intensive noises affect the stability as shown through Theorem3; this leads to occurrence of the paradox of enrichment in the stochastic model ((6a) and (6b)) as shown through Corollary 2. The occurrence of the enrichment paradox phenomenon depends on the carrying capacity in one side and stochastic terms in another side as it is shown in Theorem 3 and Corollary 1.


## 4.2. Numerical Simulations

We perform numerical simulations to show the effects of stochastic process on the occurrence of the enrichment paradox phenomenon. The stochastic Runge–Kutta method based on Ito^ process is used to solve the model ((6a) and (6b)). We use a command “StochasticRungeKuttaScalarNoise” in MATHEMATICA 11.3 program as a method to execute the numerical simulations, according to Wolfram website [33]. The command uses stage Rossler Stochastic Runge–Kutta method for scalar noise that has order 3/2. The values of the parameters are selected to satisfy condition (33) with regard to the coexistence of prey and predator. The values of the parameters and initial conditions are fixed. However, we use different values of noise strength. The values are as follows:(40)r=1.5,k=3,α=1.5,e=1.0,u=0.65,X0=0.7,Y0=0.3.Model (31) presents stable dynamic behavior as shown in Figure 1 when σ=0, which refers to deterministic model (31). In our simulations, we observe that the dynamic behavior changes from stable to oscillated when investigating the random noises to be σ=0.4, σ=1.0, and σ=3.0, as shown in Figures 2–4, respectively. However, the difference between the second, third, and fourth cases is the size of the oscillations; with increased values of σ, the size of the oscillations increases. Then, we obtain that the random noises have effects on the occurrence of the paradox of enrichment, moving from stable dynamic behavior to oscillated dynamic behavior.Figure 1
Dynamics model (31) when σ=0.0: (a) time series; (b) phase plane.
(a)(b)Figure 2
Dynamics model (6a) and (6b) whenσ=0.4: (a) time series; (b) phase plane.
(a)(b)Figure 3
Dynamics model (6a) and (6b) whenσ=1.0: (a) time series; (b) phase plane.
(a)(b)Figure 4
Dynamics model (6a) and (6b) whenσ=3.0: (a) time series; (b) phase plane.
(a)(b)


## 5. Discussion and Conclusions

In this paper, we investigated a stochastic predator-prey model to study the random noise effects on the enrichment paradox phenomenon. To the best of our knowledge, this is the first study to discuss the random noise effects on the occurrence of the enrichment paradox phenomenon. We proved the existence, uniqueness, and boundedness of the model as shown in Theorems1 and 2. The global stability was studied to determine the occurrence of the enrichment paradox phenomenon; therefore, we proved the global stability of the stochastic model by constructing suitable Lyapunov function and using Ito^ formula as demonstrated in Theorem 3. This is an equivalent of the stationary distribution and ergodicity to stochastic models.The theoretical results showed that intensive noises play an important role in occurrence of the paradox of enrichment, where increasing intensive noises lead to occurrence of the paradox of enrichment as demonstrated in Theorem3 and Corollaries 1 and 2. Finally, we performed the numerical simulations to verify and demonstrate the theoretical results. In addition, our simulations showed that the size of the oscillations increases with increased values of intensive noises.Overall, this study corresponds with the literature, in which the random noises focus on the variability of ecological systems and moving away from thinking in terms of equilibrium dynamics. We proved theoretically and showed by the numerical simulations that the random noises make our model destabilize with increased values of intensive noises, although with no noises, our model is stable without conditions. Hence, the paradox of enrichment arises with the random noises. This study may contribute to increasing attention to study the random noise effects on some ecological and biological phenomena as the paradox of enrichment for future work.



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