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Threshold analysis of an algae-zooplankton model incorporating general interaction rates and nonlinear independent stochastic components

  • The stochastic nature of ecological systems is fundamental to their modeling and understanding. In this paper, we introduce a comprehensive algae-zooplankton model that incorporates general interaction rate and second-order independent stochastic components. Our model's perturbation component encompasses both white noise and jump processes, enabling us to account for various sources of variability and capture a wide range of potential fluctuations in the system. By utilizing an auxiliary equation, we establish a global threshold for the stochastic system, distinguishing between scenarios of extinction and ergodicity. This threshold serves as a critical determinant of the system's long-term behavior and sheds light on the delicate balance between population persistence and decline in ecological communities. To elucidate the impact of noise on the dynamics of algae and zooplankton, we present a series of numerical illustrations. Through these simulations, we highlight how noise influences not only the extinction time but also the shape of the stationary distribution. Our findings underscore the significant role of stochasticity in shaping ecological dynamics and emphasize the importance of considering noise effects in ecological modeling and management practices.

    Citation: Yassine Sabbar, Aeshah A. Raezah. Threshold analysis of an algae-zooplankton model incorporating general interaction rates and nonlinear independent stochastic components[J]. AIMS Mathematics, 2024, 9(7): 18211-18235. doi: 10.3934/math.2024889

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  • The stochastic nature of ecological systems is fundamental to their modeling and understanding. In this paper, we introduce a comprehensive algae-zooplankton model that incorporates general interaction rate and second-order independent stochastic components. Our model's perturbation component encompasses both white noise and jump processes, enabling us to account for various sources of variability and capture a wide range of potential fluctuations in the system. By utilizing an auxiliary equation, we establish a global threshold for the stochastic system, distinguishing between scenarios of extinction and ergodicity. This threshold serves as a critical determinant of the system's long-term behavior and sheds light on the delicate balance between population persistence and decline in ecological communities. To elucidate the impact of noise on the dynamics of algae and zooplankton, we present a series of numerical illustrations. Through these simulations, we highlight how noise influences not only the extinction time but also the shape of the stationary distribution. Our findings underscore the significant role of stochasticity in shaping ecological dynamics and emphasize the importance of considering noise effects in ecological modeling and management practices.



    Stochasticity is inherent in ecological systems, manifesting in myriad ways that shape the destiny of their residents. Stochasticity, characterized by its inherent randomness and unpredictability, is a prime example of this variability, playing a pivotal role in ecological dynamics [1]. The occurrence of stochastic events, such as unpredictable fluctuations in environmental conditions or chance encounters with predators, introduces a layer of uncertainty that can profoundly impact the survival and evolution of species [2]. For example, consider a population that, under deterministic conditions, would persist indefinitely due to stable environmental factors and abundant resources. Despite apparent resilience, the introduction of stochastic events can dramatically alter the population's trajectory. Sudden environmental disturbances, disease outbreaks, or shifts in predator-prey dynamics can swiftly tip the balance, leading to population declines or even extinction [3].

    Generally, the effects of stochasticity are not uniform across species or ecosystems. Some species may possess traits or behaviors that buffer them against the impacts of random events, while others may be more susceptible to stochastic shocks [4]. Additionally, the scale and frequency of stochastic events vary spatially and temporally, further complicating their influence on ecological systems [5]. Understanding the interplay between deterministic and stochastic factors is crucial for comprehending the resilience and vulnerability of ecosystems. While deterministic processes provide a framework for understanding general patterns and trends, stochastic events inject the element of chance that can shape the outcomes in unpredictable ways. Therefore, a holistic approach that considers both deterministic and stochastic forces is essential for effectively managing and conserving biodiversity in a dynamic and uncertain world [6].

    The addition of random perturbations in the algae-zooplankton model is motivated by the inherent variability and unpredictability present in natural ecosystems [7]. Environmental fluctuations, such as changes in temperature, light, nutrient availability, and water quality, significantly impact the growth and interactions of both algae and zooplankton populations. Incorporating random perturbations helps capture these environmental variabilities. Additionally, immigration and emigration processes, which are stochastic in nature, can influence population sizes through currents and other physical processes. The interactions between algae and zooplankton are complex and influenced by random predation events and changes in prey availability. Stochastic elements account for the noise and uncertainty in real-world data due to measurement errors and sampling inconsistencies. Understanding how populations respond to random disturbances is crucial for assessing their stability and resilience. Therefore, incorporating random perturbations into the model provides a realistic and comprehensive framework for understanding and predicting the dynamics of these populations in fluctuating environments. In ecological modeling, various methods are employed to introduce variability into ecosystems, each serving different purposes and mimicking distinct aspects of natural dynamics. Among the most common techniques are white noise and Lévy jumps, which play crucial roles in capturing different types of variability within ecological systems.

    White noise is a stochastic process characterized by random fluctuations with equal intensity across all frequencies. In ecological contexts, white noise is often used to simulate natural disturbances driven by environmental changes [8]. These disturbances can include fluctuations in temperature, precipitation, or resource availability, as well as disturbances caused by human activities such as logging, pollution, or habitat fragmentation [9]. By incorporating white noise into ecological models, researchers can replicate the unpredictability and variability inherent in natural ecosystems, providing insights into how species and communities respond to environmental fluctuations over time [10]. Building on this approach, certain papers have explored the dynamics between algae and zooplankton under white noise perturbations. For instance:

    In [11], the author introduced a novel approach to modeling the dynamics of algae-zooplankton systems, incorporating stochasticity through the inclusion of random environmental perturbations represented as white noise. His work yielded insights into the conditions for stochastic extinction, persistence, and the establishment of an ergodic stationary distribution within the system.

    In [12], the authors presented a nutrient-algae model enriched with the incorporation of white noise, aiming to capture the stochastic nature of ecological processes. Their study not only provided insights into the necessary conditions for extinction, persistence, and ergodicity within the model but also shed light on the dynamic interplay between nutrient availability and algae abundance in fluctuating environments.

    In [13], the authors investigated the dynamics of a stochastic system involving toxin-producing algae and fish, incorporating harvesting practices. Their study yielded valuable insights into the system's behavior, offering sufficient conditions for both extinction and persistence.

    In [14], the authors explored the dynamics of a stochastic model concerning algae-zooplankton interactions, taking into account variations in algae cell size and zooplankton body size. The authors derived criteria for both extinction and ergodicity within the model.

    On the other hand, pure Lévy jumps represent a type of stochastic process characterized by infrequent but intense changes in a system's state [15]. Unlike white noise, which generates random fluctuations of relatively constant magnitude, pure Lévy jumps simulate abrupt and extreme events that can have significant impacts on ecological dynamics. These events may include rare but severe disturbances such as wildfires, hurricanes, or disease outbreaks, as well as sudden shifts in population dynamics due to predator-prey interactions or competitive pressures [16]. By incorporating pure Lévy jumps into ecological models, researchers can explore the effects of these rare but influential events on the resilience and stability of ecosystems, shedding light on how ecosystems respond to and recover from extreme perturbations. Expanding upon this methodology, specific studies have delved into the interplay between algae and zooplankton dynamics, incorporating the concept of Lévy jumps. For example:

    In [17], the authors introduced a stochastic model of nutrient-plankton food chains, incorporating Lévy jumps into their framework. They established necessary conditions for both the persistence and extinction of plankton populations within the model.

    In [18], the authors introduced an ecological system subjected to high-order white noise and quadratic Lévy jumps. They delineated the conditions governing the extinction and ergodic behavior of the model, offering valuable insights into the long-term dynamics and stability of the system under stochastic perturbations.

    By integrating both white noise and pure jumps into ecological modeling frameworks, researchers can capture a wide range of variability and complexity in natural systems. This comprehensive approach enables scientists to investigate how different types of disturbances and environmental changes shape the dynamics of ecological communities, helping to inform conservation and management strategies in an ever-changing world. Additionally, understanding the interplay between different sources of variability can provide valuable insights into the adaptive capacity of species and ecosystems, aiding efforts to predict and mitigate the impacts of global environmental change.

    In this paper, we present a general ecological model that incorporates two distinct sources of jumps, each tailored to capture specific aspects of stochasticity within natural systems. Firstly, we integrate second-order white noises, meticulously designed to account for the subtle yet impactful fluctuations that characterize small-scale variations in environmental conditions. These fluctuations, although seemingly minor, can have profound effects on ecosystem dynamics, influencing everything from population dynamics to community structure. Secondly, we introduce second-order pure jumps, strategically employed to capture the abrupt and intense changes that typify heavy perturbations within ecological systems. These jumps represent the sudden shifts and disturbances that can arise from various factors, such as extreme weather events, sudden changes in resource availability, or outbreaks of disease.

    Crucially, we introduce a parameter ranging between 0 and 1, offering the flexibility to personalize the perturbation characteristics and providing insight into both subtle and heavy variations. This parameterization allows us to systematically explore the effects of different levels of perturbation intensity on ecosystem dynamics, offering a nuanced understanding of how ecological systems respond to environmental variability across a spectrum of conditions.

    A notable feature of our study is the establishment of a global threshold delineating the boundary between extinction and ergodicity within the model. This threshold plays a pivotal role in distinguishing between two critical scenarios: The sustained persistence of ecological communities (ergodicity) and their eventual decline and disappearance (extinction). By identifying and elucidating this threshold, we offer valuable insights into the underlying mechanisms governing the long-term stability and resilience of algae-zooplankton communities.

    The remainder of this paper is organized as follows: Section 2 introduces the algae-zooplankton model and its stochastic formulation. Following this, we present some preliminary concepts and lemmas. In Section 3, we unveil the main theorems that establish the global threshold between extinction and ergodicity within the stochastic model. Section 4 provides numerical illustrations to examine the impact of noise on the dynamics of algae and zooplankton. Finally, we conclude with a discussion of our findings and a consideration of our limitations.

    Drawing upon the model outlined in [14], we proceed to introduce a refined algae-zooplankton system that builds upon and extends previous findings in the field. Our system encapsulates the intricate interplay between algae y0 and zooplankton y1 dynamics, aiming to elucidate the underlying mechanisms governing their interactions within aquatic ecosystems. The model under perturbation is structured as follows:

    {dy0(t)=(ry0(t)s1r2+s2r+s3(1y0(t)L)β0y0(t)y1(t)γy0(t)+y1(t)e{1ζ1(rζ2u)2})dt+ϖ00y0(t)dS0(t),dy1(t)=(αβ0y0(t)y1(t)γy0(t)+y1(t)e{1ζ1(rζ2u)2}β1y0(t)y1(t)d1y1(t))dt+ϖ10y1(t)dS1(t),y1(0)R+,y1(0)R+, (2.1)

    where r is the size of the algae cells, s1, s2, and s3 are some experimental coefficients, L is the significant ecological carrying capacity, β0 is the peak consumption rate, γ is the saturated coefficient, ζ1 and ζ2 are the usage rate coefficients, u is the metric for zooplankton morphology, α is the transition rate of y1, β1 is the toxic emission, d1 is the mortality rate of y1, and ϖ00 and ϖ10 represent the intensity of noise for the independent standard Brownian motions S0(t) and S1(t), respectively.

    In the complex ecosystems of aquatic environments, the predation dynamics between zooplankton and phytoplankton are closely tied to the size disparities among these tiny organisms. A key principle recognized by ecologists is that larger-bodied zooplankton typically prey on smaller-bodied phytoplankton [19,20,21]. However, as the ecosystem becomes more populated with multiple species of both phytoplankton and zooplankton, each with varying body sizes, the predation dynamics become increasingly intricate. In these multifaceted scenarios, interactions among planktonic organisms are often complicated by mutual interference, resulting in complex webs of predation and resource use. Consequently, the incidence rates that describe these interactions must take on specific forms to accurately reflect the nuanced relationships at play. Understanding these specific forms of incidence rates is crucial for elucidating the detailed mechanisms governing predator-prey interactions and species coexistence within aquatic ecosystems.

    In the model described in [14], the authors focused on the nonlinear interaction function β0y0(t)y1(t)γy0(t)+y1(t). However, real-world scenarios and certain modeling requirements may necessitate a more versatile approach. In order to accommodate a broader range of situations, this paper will explore a generalized functional interaction function f(y0(t),y1(t)). By doing so, we aim to encompass a wider spectrum of epidemiological dynamics and provide a more flexible framework for modeling ecological competition in various contexts. The generalized form of the system (2.1) is presented as follows:

    {dy0(t)=(ry0(t)s1r2+s2r+s3(1y0(t)L)f(y0(t),y1(t))y1(t)e{1ζ1(rζ2u)2})dt+ϖ00y0(t)dS0(t),dy1(t)=(αf(y0(t),y1(t))y1(t)e{1ζ1(rζ2u)2}β1y0(t)y1(t)d1y1(t))dt+ϖ10y1(t)dS1(t),y0(0)R+,y1(0)R+, (2.2)

    where fC2(R+×R+,R+) is an uniformly continuous function such that

    f(y0(t),y1(t))y1(t)0f(y0(t),y1(t))y0(t)ϑ,

    for a given positive constant ϑ. This generalized function, as described under the aforementioned conditions, includes several specific cases outlined in Table 1 (see, for example, [22,23,24,25,26]).

    Table 1.  List of some examples of the general interference function f.
    Name Expression
    Bilinear f(y0,y1)=y0,
    Saturated f(y0,y1)=y0m0+y0, (m0>0)
    Crowley-Martin f(y0,y1)=y0(m0+y0)(m1+y1), (m0,m1>0)
    Beddington-DeAngelis f(y0,y1)=y01+m0y0+m1y1, (m0,m1>0)
    Modified Crowley-Martin f(y0,y1)=y0m0+m1y0+m2y1+m3y0y1, (m0,m1,m2,m3>0)

     | Show Table
    DownLoad: CSV

    Having expanded the incidence rate to a more generalized form, we are now poised to introduce our novel model, enriched with second-order perturbations [26,27]. Consider the following system:

    {dy0(t)=(ry0(t)s1r2+s2r+s3(1y0(t)L)f(y0(t),y1(t))y1(t)e{1ζ1(rζ2u)2})dt+p1i=0ϖ0iyi+10(t)dS0(t)+(1p)R2{0}1j=0ν0j(ξ)yj+10(t)D0(dt,dξ),dy1(t)=(αf(y0(t),y1(t))y1(t)e{1ζ1(rζ2u)2}β1y0(t)y1(t)d1y1(t))dt+p1i=0ϖ1iyi+11(t)dS1(t)+(1p)R2{0}1j=0ν1j(ξ)yj+11(t)D1(dt,dξ),y0(0)R+,y1(0)R+. (2.3)

    Here, Sk(t)(k=0,1) denotes two mutually independent Standard Brownian motions (SBMs) of strengths ϖ0i>0(i=0,1) and ϖ1i>0(i=0,1), respectively. These SBMs are essentially defined on a filtered probability triple (stochastic basis) (Ω,FΩ,(F{Ω,t})t0,P) equipped with a filtration satisfying the usual criteria [4]. yk(t)(k=0,1) is the left limits of yk(t)(k=0,1). ˉDk(k=0,1) are two independent stochastic Poisson processes related to the measures mk(k=0,1) defined on a measurable set R2{0} as follows:

    mk(A)=R2{0}R+ααs1set1A(tx)dtRk(dx),AB(R2{0}), (2.4)

    where 1 denotes the indicator function, αs(0,2), and Rk is the Rosiński measure defined on R2{0} such that Rk(0)=0, (k=0,1), with R2{0}(||x||2||x||αs)Rk(dx)<,αs(0,2) (for more details, please see [28]). Dk(k=0,1) are two independent compensated Poisson random measures such that Dk(dt,dξ)=ˉDk(dt,dξ)mk(dξ)dt. νk:R2{0}R are the jumps size functions which are postulated to be positive and continuous on R2{0}, for all i,k=0,1. It is assumed that R2{0}ν2ik(ξ)mi(dξ) is finite. The variable p is a constant between 0 and 1. This is included in order to have the possibility of personalizing the influence of the noise part (from a modeling perspective). If p=1, system (2.3) simplifies to a model featuring solely second-order white noise, effectively simulating continuous and Gaussian variations commonly observed in ecological systems. Conversely, when p=0, system (2.3) transforms into a model characterized by unexpected jumps with heavy tails, resulting in abrupt and drastic changes in species dynamics.

    Based on the theoretical framework outlined in references [15,29], it becomes evident that the system described by Eq (2.3) possesses a unique positive global solution. This observation not only confirms the ecological validity of the model but also establishes its mathematical robustness and well-posedness.

    As outlined in the introduction, the central focus of this paper revolves around delineating the global threshold that delineates the balance between the sustenance and extinction of zooplankton populations. To address the intricacies posed by the new stochastic system (2.3), we propose a new approach based on the ergodicity of an auxiliary equation closely resembling the equation of y0. Based on the stochastic comparison theorem [30], we present the following equation:

    {d˜y0(t)=(r˜y0(t)s1r2+s2r+s3(1˜y0(t)L))dt+p1i=0ϖ0i˜yi+10(t)dS0(t)+(1p)R2{0}1j=0ν0j(ξ)˜yj+10(t)D0(dt,dξ),˜y0(0)=y0(0)>0. (2.5)

    Based on Lemma 2.2 in [31], it is evident that Eq (2.5) maintains its well-posed nature. Additionally, ˜y0(t) emerges as a stochastic process endowed with the following properties:

    (1) P(˜y0(t)y0(t)0)=1.

    (2) If

    ˜d0=rs1r2+s2r+s3p2ϖ2002R2{0}((1p)ν00(ξ)log(1+(1p)ν00(ξ)))m0(dξ)>0, (2.6)

    then there exists a unique ergodic stationary distribution ˜η0 for Eq (2.5); and

    limt1tt0˜y0(s)dsL˜d0(rs1r2+s2r+s3+L(1p)2ϖ00ϖ01).

    For analytical coherence, it is imperative to maintain the assumption of positivity for ˜d0 throughout this study. Additionally, we rigorously define the following quantities, crucial in delineating the global threshold within the framework of system (2.3):

    θ0=αe{1ζ1(rζ2u)2}R+f(y,0)˜η0(dy),θ1=d1+p2ϖ2102+R2{0}((1p)ν10(ξ)log(1+(1p)ν10(ξ)))m0(dξ).

    As elucidated earlier, the crux of ecosystem examination lies in accurately demarcating the threshold that distinguishes between population persistence and decline. Therefore, the paramount aim of the forthcoming theorems is to tackle this pivotal inquiry head-on, while simultaneously enriching our comprehension of algae-zooplankton system dynamics with additional insights and refinements.

    Theorem 3.1. Suppose that θ0θ1. Under this condition, for any initial positive data (y0(0),y1(0)), the stochastic system described by Eq (2.3) asymptotically tends towards a state of exponential decline and

    P(limty1(t)=0)=1.

    Proof. Utilizing Itô's formula, we readily obtain

    dlogy1(t)=(αe{1ζ1(rζ2u)2}f(y0(t),y1(t))β1y0(t)d1p22(1i=0ϖ1iyi1(t))2+R2{0}(log(1+(1p)ν10(ξ)+(1p)ν11(ξ)y1(t))m1(dξ)(1p)R2{0}(ν10(ξ)+ν11(ξ)y1(t))m1(dξ))dt+p1i=0ϖ1iyi1(t)dS1(t)+R2{0}log(1+(1p)ν10(ξ)+(1p)ν11(ξ)y1(t))D1(dt,dξ).

    In accordance with the stochastic comparison theorem [30], we deduce that

    dlogy1(t)(αe{1ζ1(rζ2u)2}f(˜y0(t),0)β1y0(t)d1p22(1i=0ϖ1iyi1(t))2+R2{0}(log(1+(1p)ν10(ξ)+(1p)ν11(ξ)y1(t))m1(dξ)(1p)R2{0}(ν10(ξ)+ν11(ξ)y1(t))m1(dξ))dt+p1i=0ϖ1iyi1(t)dS1(t)+R2{0}log(1+(1p)ν10(ξ)+(1p)ν11(ξ)y1(t))D1(dt,dξ). (3.1)

    By integrating the above inequality from 0 to t, we acquire

    logy1(t)tαte{1ζ1(rζ2u)2}t0f(˜y0(s),0)dsd1R2{0}((1p)ν10(ξ)log(1+(1p)ν10(ξ)))m1(dξ)+1tt0R2{0}{log(1+(1p)ν11(ξ)y1(s)1+(1p)ν10(ξ))(1p)ν11(ξ)y1(s)}m1(dξ)ds+1t(pt01i=0ϖ1iyi1(s)dS2(s)p22t0(1i=0ϖ1iyi1(s))2ds)+1tt0R2{0}log(1+(1p)ν10(ξ))D1(ds,dξ)+logy1(0)t+1tt0R2{0}log(1+(1p)ν11(ξ)y1(s)1+(1p)ν10(ξ))D1(ds,dξ). (3.2)

    We let M0(t)=t0R2{0}log(1+ν10(ξ))D1(ds,dξ). Then, we get

    M0(t),M0(t)=tR2{0}(log(1+ν10(ξ)))2m1(dξ)<.

    Referring to the strong law of large numbers, we derive P(limt1tM0(t)=0)=1. Now, we select 0<ϱ1<1 and consider the following quantities:

    M1(t)=pt01i=0ϖ1iyi1(s)dS1(s)p22t0(1i=0ϖ1iyi1(s))2ds,M2(t)=pt01i=0ϖ1iyi1(s)dS1(s)p2ϱ12t0(1i=0ϖ1iyi1(s))2ds,M3(t)=t0R2{0}(((1p)ν11(ξ)y1(s)1+(1p)ν10(ξ))+log(1+(1p)ν11(ξ)y1(s)1+(1p)ν10(ξ)))m1(dξ)ds,M4(t)=t0R2{0}log(1+(1p)ν11(ξ)y1(s)1+(1p)ν10(ξ))D1(ds,dξ).

    To continue, we employ the exponential inequality designed for martingales, we obtain

    P{supt[0,T1](M2(t)M3(t)+M4(t))>2ϱ11logϱ2}ϱ22,for allϱ2>0.

    Thanks to the Borel-Cantelli lemma, we establish the presence of ϱ2(ω) for all ωΩ, ensuring the validity of the following inequality:

    pt01i=0ϖ1iyi1(s)dS1(s)+M4(t)2ϱ11logϱ2+p2ϱ12t0(1i=0ϖ1iyi1(s))2ds+M3(t),

    verifies for any ϱ2 satisfying ϱ2ϱ2(ω) and P(t]ϱ21,ϱ2[)=1. Consequently,

    M1(t)t+M4(t)t2ϱ11logϱ2t+p2ϱ12tt0(1i=0ϖ1iyi1(s))2dsp22tt0(1i=0ϖ1iyi1(s))2ds+1tt0R2{0}(((1p)ν11(ξ)y1(s)1+(1p)ν10(ξ))(1p)ν11(ξ)y1(s))m1(dξ)ds2ϱ11logϱ2ϱ21p2(1ϱ1)2tt0(1i=0ϖ1iyi1(s))2ds2ϱ11logϱ2ϱ21p2(1ϱ1)ϖ2102.

    We compute the upper limit on both sides of Eq (3.2), and then we get

    lim suptlogy1(t)tαe{1ζ1(rζ2u)2}limt1tt0f(˜y0(s),0)dsd1R2{0}((1p)ν10(ξ)log(1+(1p)ν10(ξ)))m1(dξ)+limt1tϕ(t)θ0d1R2{0}((1p)ν10(ξ)log(1+(1p)ν10(ξ)))m1(dξ)+limϱ22ϱ11logϱ2ϱ21p2(1ϱ1)ϖ2102.

    Therefore,

    lim suptlogy1(t)tθ0d1p2(1ϱ1)ϖ2102R2{0}((1p)ν10(ξ)log(1+(1p)ν10(ξ)))m1(dξ).

    By considering the arbitrary nature of 0<ϱ1<1, one can deduce that as ϱ1 approaches 0,

    P(lim suptlogy1(t)tθ0θ1<0)=1.

    Put differently, for any arbitrarily small positive quantity ˜θ satisfying ˜θ<min{0.5,0.5(θ1θ0)}, we can infer the existence of ϱ3=ϱ3(ω) associated with Ωϱ3 such that P(Ωϱ3)1ϱ3, where logy1(t)0.5(θ1θ0)t for all tϱ3 and ωΩϱ3. As a result, y1(t)exp(0.5(θ1θ0)t) and

    P(lim supty1(t)0)=1.

    This, coupled with the positivity of the solution (y0,y1), suggests

    P(limty1(t)=0)=1.

    This indicates that the exponential decrease in the population of y1 implies its almost certain complete eradication. With this, the proof is concluded.

    Corollary 3.1. (case of second-order Brownian motions) If p=1, and

    αe{1ζ1(rζ2u)2}R+f(y,0)˜η0(dy)d1+ϖ2102,

    then, the extinction of the population y1 occurs almost surely.

    Corollary 3.2. (case of second-order jumps) If p=0, and

    αe{1ζ1(rζ2u)2}R+f(y,0)˜η0(dy)d1+R2{0}(ν10(ξ)log(1+ν10(ξ)))m0(dξ),

    then, the extinction of the population y1 occurs almost surely.

    Theorem 3.2. Suppose that θ0>θ1. Under this condition, for any initial positive data (y0(0),y1(0)), the stochastic system described by Eq (2.3) asymptotically tends towards a stationary state and

    P(lim inft1tt0y(s)ds=R2+yη(y0,y1)>0)=1,=0,1,

    where η(,) is the unique ergodic stationary distribution associated with system (2.3).

    Proof. Utilizing the Itô formula for the function F0(t)=log(˜y0(t)y0(t)) yields:

    LF0(t)rL(s1r2+s2r+s3)(˜y0(t)y0(t))+1y0(t)e{1ζ1(rζ2u)2}f(y0(t),y1(t))y1(t)12(p1i=0ϖ0i˜yi0(t))2+12(p1i=0ϖ0iyi0(t))2+R2{0}{log(1+(1p)ν00(ξ)+(1p)ν01(ξ)˜y0(t)1+(1p)ν00(ξ)+(1p)ν01(ξ)y0(t))(1p)ν01(ξ)(˜y0(t)y0(t))}m0(dξ).

    Based on the properties of the function f, it follows that:

    LF0(t)rL(s1r2+s2r+s3)(˜y0(t)y0(t))+ϑe{1ζ1(rζ2u)2}y1(t)(˜y0(t)y0(t))R2{0}{(1p)2ν01(ξ)(ν00(ξ)+ν01(ξ)y0(t))1+(1p)ν00(ξ)+(1p)ν01(ξ)y0(t)}m0(dξ).

    By virtue of the positive property of the solution, we obtain:

    LF0(t)rL(s1r2+s2r+s3)(˜y0(t)y0(t))+ϑe{1ζ1(rζ2u)2}y1(t). (3.3)

    Once more, employing the operator L on the second equation of (2.3) yields:

    Llog(1y1(t))=αe{1ζ1(rζ2u)2}f(y0(t),y1(t))+β1y0(t)+d1+12(p1i=0ϖ1iyi1(t))2R2{0}{log(1+(1p)ν10(ξ)+(1p)ν11(ξ)y1(t))(1p)(ν10(ξ)+ν11(ξ)y1(t))}m1(dξ).

    Consequently,

    Llog(1y1(t))=αe{1ζ1(rζ2u)2}f(˜y0(t),0)+β1y0(t)+d1+p2ϖ2102+R2{0}((1p)ν10(ξ)log(1+(1p)ν10(ξ)))m1(dξ)+αe{1ζ1(rζ2u)2}f(˜y0(t),0)αe{1ζ1(rζ2u)2}f(y0(t),y1(t))αe{1ζ1(rζ2u)2}f(y0(t),0)+αe{1ζ1(rζ2u)2}f(y0(t),0)+pϖ10ϖ11y1(t)+p22ϖ211y21(t)+R2{0}{(1p)ν11(ξ)y1(t)log(1+(1p)ν11(ξ)y1(t)1+(1p)ν10(ξ))}m1(dξ).

    Consistent with the positivity of the solution, we find

    Llog(1y1(t))αe{1ζ1(rζ2u)2}f(˜y0(t),0)+d1+p2ϖ2102+R2{0}((1p)ν10(ξ)log(1+(1p)ν10(ξ)))m1(dξ)+β1y0(t)+αϑe{1ζ1(rζ2u)2}(˜y0(t)y0(t))+(pϖ10ϖ11+(1p)R2{0}ν11(ξ)m0(dξ))y1(t)+p22ϖ211y21(t)+αe{1ζ1(rζ2u)2}f(y0(t),0)αe{1ζ1(rζ2u)2}f(y0(t),y1(t)). (3.4)

    Now, let us examine the following function:

    F1(t)=Lαϑr(s1r2+s2r+s3)e{1ζ1(rζ2u)2}F0(t)+log(1y1(t)).

    By applying the operator L to F1, we obtain

    LF1(t)αe{1ζ1(rζ2u)2}f(˜y0(t),0)+d1+p2ϖ2102+R2{0}((1p)ν10(ξ)log(1+(1p)ν10(ξ)))m1(dξ)+β1y0(t)+(pϖ10ϖ11+(1p)R2{0}ν11(ξ)m0(dξ))y1(t)+L(s1r2+s2r+s3)αϑ2re2{1ζ1(rζ2u)2}y1(t)+αe{1ζ1(rζ2u)2}f(y0(t),0)αe{1ζ1(rζ2u)2}f(y0(t),y1(t))+p22ϖ211y21(t).

    By rearranging certain terms, we achieve

    LF1(t)αe{1ζ1(rζ2u)2}R+f(y,0)˜η0(dy)+d1+p2ϖ2102+R2{0}((1p)ν10(ξ)log(1+(1p)ν10(ξ)))m1(dξ)+β1y0(t)+p22ϖ211y21(t)+αe{1ζ1(rζ2u)2}(R+f(y,0)˜η0(dy)f(˜y0(t),0))+(L(s1r2+s2r+s3)αϑ2re2{1ζ1(rζ2u)2}+pϖ10ϖ11+(1p)R2{0}ν11(ξ)m0(dξ))y1(t)+αe{1ζ1(rζ2u)2}f(y0(t),0)αe{1ζ1(rζ2u)2}f(y0(t),y1(t)).

    Once more, let us examine this new function:

    F2(t)=F1(t)+cy1(t).

    Here, c denotes a positive constant that fulfills:

    c1d1(L(s1r2+s2r+s3)αϑ2re2{1ζ1(rζ2u)2}+pϖ10ϖ11+(1p)R2{0}ν11(ξ)m1(dξ)).

    Upon applying the operator L to F2, we acquire

    LF2(t)αe{1ζ1(rζ2u)2}R+f(y,0)˜η0(dy)+d1+p2ϖ2102+R2{0}((1p)ν10(ξ)log(1+(1p)ν10(ξ)))m1(dξ)+αe{1ζ1(rζ2u)2}(R+f(y,0)˜η0(dy)f(˜y0(t),0))+αe{1ζ1(rζ2u)2}f(y0(t),0)αe{1ζ1(rζ2u)2}f(y0(t),y1(t))+β1y0(t)+p22ϖ211y21(t)+cαe{1ζ1(rζ2u)2}f(y0(t),y1(t))y1(t).

    Then

    LF2(t)θ1θ0+αe{1ζ1(rζ2u)2}(R+f(x,0)˜η0(dx)f(˜y0(t),0))+αe{1ζ1(rζ2u)2}(f(y0(t),0)f(y0(t),y1(t)))+β1y0(t)+p22ϖ211y21(t)+cαe{1ζ1(rζ2u)2}f(y0(t),y1(t))y1(t).

    Let c(0,1), then we define a new function as follows:

    F3(t)=1c(yc0(t)+yc1(t)).

    By applying the operator L to F3, we get

    LF3(t)=yc10(t){ry0(t)s1r2+s2r+s3(1y0(t)L)e{1ζ1(rζ2u)2}f(y0(t),y1(t))y1(t)}+(c1)2yc20(t)(p1i=0ϖ0iyi+10(t))2+R2{0}{(y0(t)+(1p)ν00(ξ)y0(t)+(1p)ν01(ξ)y20(t))ccyc0(t)c}m0(dξ)R2{0}(1p)yc10(t)(ν00(ξ)y0(t)+ν01(ξ)y20(t))m0(dξ)+yc11(t)(αe{1ζ1(rζ2u)2}f(y0(t),y1(t))y1(t)βy0(t)y1(t)d1y1(t))+(c1)2yc21(t)(p1i=0ϖ1iyi+11(t))2+R2{0}((y1(t)+(1p)ν10(ξ)y1(t)+(1p)ν11(ξ)y21(t))ccyc1(t)c)m1(dξ)R2{0}(1p)yc1(t)(ν00(ξ)+ν01(ξ)y1(t))m0(dξ)rs1r2+s2r+s3yc0(t)(1c)2ϖ201yc+20(t)+αe{1ζ1(rζ2u)2}ϑc+1yc+10(t)+αce{1ζ1(rζ2u)2}ϑc+1yc+11(t)(1c)2ϖ211yc+21(t).

    We amalgamate functions F2 and F3 in the following manner:

    F4(t)=hF2(t)+F3(t).

    Here, h>0 satisfies 2+Kh(>0θ0θ1)0, where K=max{A,1}, and

    A=sup(y0,y1)R2,+{rs1r2+s2r+s3yc0(t)(1c)4ϖ201yc+20(t)+αe{1ζ1(rζ2u)2}ϑc+1yc+10(t)+αce{1ζ1(rζ2u)2}ϑc+1yc+11(t)(1c)4ϖ211yc+21(t)}.

    To guarantee the positivity of our Lyapunov function, we introduce this new function:

    F5(t)=F4(t)F_4,

    where, F_4 represents the lower bound of function F4. When we apply the operator L to F5, we obtain

    LF5(t)F6(t)+hαe{1ζ1(rζ2u)2}(R+f(y,0)˜η0(dy)f(˜y0(t),0)), (3.5)

    where

    F6(t)=h(<0θ1θ0)+hcαe{1ζ1(rζ2u)2}ϑy0(t)y1(t)+β1y0(t)+hp22ϖ211y21(t)+rs1r2+s2r+s3yc0(t)(1c)4ϖ201yc+20(t)+αe{1ζ1(rζ2u)2}ϑc+1yc+10(t)+αce{1ζ1(rζ2u)2}ϑc+1yc+11(t)(1c)4ϖ211yc+21(t).

    Utilizing the methodologies outlined in the proof of Theorem 3.2 in [31], along with the properties of the function f, we can promptly verify F6(t)1, (y0,y1)R2+Hx=Hcx, x>0, where

    Hx={(y0,y1)R2+|xy0x1,xy1x1}.

    Alternatively, it's evident that there exists a positive constant κ such that F6(t)κ for all (y0,y1)Hx. Then, from (3.5), we derive

    0t0EF6(t)ds+hαe{1ζ1(rζ2u)2}×E(t0R+f(y,0)˜η0(dy)dst0f(˜y0(s),0)ds).

    By virtue of the ergodicity property of the process ˜y, we obtain

    0lim inft1tt0(EF6(t)1{(y0,y1)Hx}+EF6(t)1{(y0,y1)Hcx})ds.

    So,

    0lim inft1tt0(κP((y0,y1)Hx)P((y0,y1)Hcx))ds=lim inft1tt0(κP((y0,y1)Hx)1+P((y0,y1)Hx))ds.

    Consequently,

    lim inft1tt0P((y0,y1)Hx)ds(1+κ)1>0.

    By employing the lemma of mutually constrained possibilities [32], we infer the existence of a unique ergodic stationary distribution η(,). Hence, incorporating the ergodic property into consideration, we can firmly conclude that

    P(lim inft1tt0y0(s)ds=R2+y0η(y0,y1)>0)=1,P(lim inft1tt0y1(s)ds=R2+y1η(y0,y1)>0)=1.

    This implies that the system asymptotically converges to the persistent state over time. Thus, the proof is concluded.

    Corollary 3.3. (case of second-order Brownian motions) If p=1, and

    αe{1ζ1(rζ2u)2}R+f(y,0)˜η0(dy)>d1+ϖ2102,

    then, the permanence of the populations y1 and y1 occurs almost surely.

    Corollary 3.4. (case of second-order jumps) If p=0, and

    αe{1ζ1(rζ2u)2}R+f(y,0)˜η0(dy)>d1+R2{0}(ν10(ξ)log(1+ν10(ξ)))m0(dξ),

    then, the permanence of the populations y1 and y1 occurs almost surely.

    Remark 3.1. From Theorem 3.1, we infer that the zooplankton population tends to extinction rapidly. This rapid decline suggests that under certain conditions, the zooplankton cannot sustain its population over time, likely due to insufficient resources, high predation pressure, or adverse environmental factors. The extinction of zooplankton can disrupt the balance of the aquatic ecosystem, affecting nutrient cycling and the food web structure. In contrast, Theorem 3.2 implies the persistence and stationarity of both algae and zooplankton populations. This means that under a different set of conditions, both populations are able to maintain stable levels over time, without the risk of extinction. The persistence of these populations indicates a balanced interaction between the species, where the growth rates of algae and zooplankton, along with other ecological factors, allow for long-term coexistence.

    In this section, we carry out a series of numerical examples to validate the findings presented in our study, utilizing the data outlined in [14]. We take r=0.3, u=0.25, s1=0.02, s2=0.02, s3=0.08, L=(2.5;3.5), ζ1=2, ζ2=0.5, α=0.4, β1=0.2, and d1=0.6. About the dual incidence functions g1 and g2, we consider the following general nonlinear incidence:

    f(y0,y1)=β0y01+k0y0+k1y1+k2y0y1,

    where β0=4.5, k0=0.16, k0=0.11 and k2=0.012. To simulate the stochastic component, we present a method for simulating jumps and tempered stable processes [28]. Firstly, we examine the sequences {S1,j}j1, {S2,j}j1, and {S3,j}j1, representing independent and identically distributed (i.i.d.) random variables taking values in the real numbers, following the distribution (2.4). Additionally, we define {S2,j}j1 and {S3,j}j1 as i.i.d. sequences of uniform random variables within the intervals (0,1) and (0,T), respectively. Furthermore, consider {S4,j}j1 and {S5,j}j1 as i.i.d. sequences of random variables following an exponential distribution with a rate coefficient of 1. It is assumed that all the aforementioned random variables are mutually independent. Then, we let {S6,j}=jk=1{S5,k}. Noticeably, {S6,j} can be regarded as a Poisson point process on the interval R+ with random intensity measure. In reference to the theory outlined in [28], if αs(0,2), then

    mt=+j=1S1,j1{S3,jt}|S1,j|((αsS6,jT||ρ||)α1s(Sα1s2,jS4,j|S1,j|)),

    converges almost surely and uniformly for t within the interval [0,T] to a Lévy process, where ||ρ||=R2{0}|x|αsRk(dx). In practice, we can devise a method for generating a jump process with specified parameters at discrete time instances ti. Here, tii[0,I] denotes a partition of the interval [0,T], characterized by uniformly sized sub-intervals and a mesh size Δt=T/I, where IN. Subsequently, we employ the following algorithm:

    (1) Choose a specific time duration, denoted as T, and partition the time interval [0,T] into I equal-sized segments.

    (2) Specify and fix a numerical parameter, denoted as N.

    (3) Generate numerical replications or simulations of independent quantities Si,j, where i ranges from 1 to 6, with a sample size of N.

    (4) Calculate the value of mt.

    Let us now direct our attention to the following system:

    {dy0(t)=(ry0(t)s1r2+s2r+s3(1y0(t)L)β0y0(t)y1(t)e{1ζ1(rζ2u)2}1+k0y0(t)+k1y1(t)+k2y0(t)y1(t))dt+p1i=0ϖ0iyi+10(t)dS0(t)+(1p)R2{0}1j=0ν0j(ξ)yj+10(t)D0(dt,dξ),dy1(t)=(αβ0y0(t)y1(t)e{1ζ1(rζ2u)2}1+k0y0(t)+k1y1(t)+k2y0(t)y1(t)β1y0(t)y1(t)d1y1(t))dt+p1i=0ϖ1iyi+11(t)dS1(t)+(1p)R2{0}1j=0ν1j(ξ)yj+11(t)D1(dt,dξ),y0(0)=2.5,y1(0)=1.8. (4.1)

    By incorporating the ergodic property of Eq (2.5), we define the parameters θ0 and θ1 as follows:

    θ0=R+αβ0ye{1ζ1(rζ2u)2}1+k1y˜η0(dy)=limt1tt0αβ0e{1ζ1(rζ2u)2}˜y0(s)1+k0˜y0(s)ds,θ1=d1+p2ϖ2102+R2{0}((1p)ν10(ξ)log(1+(1p)ν10(ξ)))m0(dξ).

    Firstly, we consider L=2.5, p=0.5 and we take the noise parameters as follows: ϖ00=0.02, ϖ01=0.01, ϖ10=0.021, ϖ11=0.012, ν00=0.01, ν01=0.009, ν10=0.02 and ν11=0.007. In this case, we get θ1=0.0526 and θ2=0.07492. According to Theorem 3.1, it is highly probable that the population y1 will become extinct. This outcome is visually exemplified in Figure 1, which provides a numerical illustration of the aforementioned result.

    Figure 1.  Conducting numerical simulations for solving system (4.1) in the presence of second-order noise. In this case, we take L=2.5 and p=0.5. The white noise parameters are selected as follows: ϖ00=0.02, ϖ01=0.01, ϖ10=0.021 and ϖ11=0.012. The jump intensities are defined as follows: ν00=0.01, ν01=0.009, ν10=0.02 and ν11=0.007. Here, we get θ1=0.0526 and θ2=0.118.

    Secondly, we take p=1 and we establish specific values for the white noise parameters ϖ00=0.02, ϖ01=0.01, ϖ10=0.021 and ϖ11=0.012. In this case, we get θ1=0.0526 and θ2=0.1894. As a result, we have verified that the prerequisite outlined in Corollary 3.1 has been met and the extinction of y1 occurs almost surely. This result is visually depicted in Figure 2.

    Figure 2.  Conducting numerical simulations for solving system (4.1) in the presence of second-order white noise. In this case, we take L=2.5 and p=1. The white noise parameters are selected as follows: ϖ00=0.02, ϖ01=0.01, ϖ10=0.021 and ϖ11=0.012.

    Now, we set p=0. The jump intensities are defined as follows: ν00=0.01, ν01=0.009, ν10=0.02 and ν11=0.007. Here, we get θ1=0.0526 and θ2=0.118. As a result, we have verified that the prerequisite outlined in Corollary 3.2 has been met. This result is visually depicted in Figure 3.

    Figure 3.  Conducting numerical simulations for solving system (4.1) in the presence of second-order jumps. In this case, we take L=2.5 and p=0. The jump intensities are defined as follows: ν00=0.01, ν01=0.009, ν10=0.02 and ν11=0.007. Here, we get θ1=0.0526 and θ2=0.118.

    Upon scrutinizing the aforementioned figures, a discernible pattern emerges: The extinction time is contingent upon the nature of the noise. Notably, a significant presence of noise hastens the extinction process considerably. This observation underscores the critical influence of noise intensity on the dynamics of extinction within the system.

    We consider L=3.5, p=0.5 and we take the noise parameters as follows: ϖ00=0.001, ϖ01=0.001, ϖ10=0.004, ϖ11=0.0012, ν00=0.001, ν01=0.001, ν10=0.01 and ν11=0.003. In this case, we get θ1=2.1982 and θ2=0.9862. As per Theorem 3.2, there is a strong likelihood of the population y0 and y1 persisting, alongside the existence of a stationary distribution. This assertion is vividly illustrated in Figure 4.

    Figure 4.  A three-dimensional visualization showcasing the stationary distribution, complemented by a two-dimensional contour plot illustrating the dynamics of system (4.1). In this case, L=3.5 and p=0.5. For noise intensities, we take ϖ00=0.001, ϖ01=0.001, ϖ10=0.004, ϖ11=0.0012, ν00=0.001, ν01=0.001, ν10=0.01 and ν11=0.003.

    We take p=1 and we establish specific values for the white noise parameters ϖ00=0.001, ϖ01=0.001, ϖ10=0.004, ϖ11=0.0012. In this case, we get θ1=2.1982 and θ2=1.3412. Consequently, we have confirmed the fulfillment of the condition outlined in Corollary 3.3, ensuring the almost certain persistence of y0 and y1. This outcome is graphically illustrated in Figure 5, providing a visual representation of the verified result.

    Figure 5.  A three-dimensional visualization showcasing the stationary distribution, complemented by a two-dimensional contour plot illustrating the dynamics of system (4.1). In this case, L=3.5 and p=1. For noise intensities, we take ϖ00=0.001, ϖ01=0.001, ϖ10=0.004 and ϖ11=0.0012.

    Now, we set p=0. The jump intensities are defined as follows: ν00=0.001, ν01=0.001, ν10=0.01 and ν11=0.003. Here, we get θ1=2.1982 and θ2=1.2981. Henceforth, we have substantiated the fulfillment of the prerequisite delineated in Corollary 3.4. This validation is visually elucidated through the representation provided in Figure 6.

    Figure 6.  A three-dimensional visualization showcasing the stationary distribution, complemented by a two-dimensional contour plot illustrating the dynamics of system (4.1). In this case, L=3.5 and p=0.5. For noise intensities, we take ν00=0.001, ν01=0.001, ν10=0.01 and ν11=0.003.

    Expanding upon the insights drawn from the aforementioned figure, it becomes apparent that the characteristics of the noise exert a discernible influence on the magnitude of variation. Specifically, we observe a pronounced effect: as the intensity of the noise increases, there is a notable augmentation in the amplitude of variation. This correlation underscores the pivotal role played by noise levels in amplifying the variability within the system dynamics, thereby emphasizing the significance of noise intensity in shaping the observed outcomes.

    Plankton size is a critical variable because it influences various biological and ecological processes. For example, smaller plankton may have faster reproduction rates but are more susceptible to predation, while larger plankton might be more resilient but have slower population growth. By studying different plankton sizes, we can observe how size-specific traits affect the overall population dynamics and stationary distribution. In instances of persistence, we explore the influence of plankton size and the parameter p on the morphology of the stationary distribution, as illustrated in Figure 7. This detailed analysis illuminates the interplay between plankton size and system parameters, providing insights into how these factors shape the stationary distribution under varying conditions. By examining different plankton sizes and adjusting p, we can observe the impact on population dynamics and structure, ultimately gaining a deeper understanding of the conditions that favor particular plankton sizes and how populations adapt over time. This comprehensive examination is crucial for predicting plankton responses to environmental changes and for effective marine ecosystem management.

    Figure 7.  Numerical demonstration elucidating the profound impact of plankton size variations and parameter p fluctuations on the intricate morphology of the stationary distribution.

    The stochastic dynamics inherent in marine ecosystems are of paramount importance, shaping the stability and resilience of these intricate environments. In this paper, we presented a novel nonlinear algae-zooplankton model, that offers a comprehensive framework for understanding the dynamics influenced by general interaction rates and second-order noise. Specifically, we distinguish between white noise and jump components, allowing for a detailed examination of their respective impacts on population dynamics.

    Central to our investigation is the delineation of the global threshold that governs the delicate balance between species extinction and persistence. Through rigorous analysis, in Theorems 3.1 and 3.2, we probed the intricate interplay of various ecological factors, shedding light on the fundamental mechanisms underlying ecosystem dynamics.

    In the numerical simulation section, we investigate the repercussions of noise on the long-term behavior of algae and zooplankton populations. By systematically varying parameters and plankton sizes, we unraveled the nuanced effects of noise intensity on population dynamics and stationary distributions. Through visual representations and quantitative analyses, we provided insights into how different types of noise shape the ecological trajectories of these crucial marine species, offering valuable implications for ecosystem management and conservation efforts.

    The analysis described in this paper can be extended to describe the dynamics of a mycoloop model in aquatic food webs. This model involves interactions between phytoplankton, chytrids, and zooplankton [33]. By incorporating additional hypotheses and exploring more complex interactions, the model can provide deeper insights into the ecological processes governing mycoloop dynamics. We aim to pursue this idea in our future work, further enhancing our understanding of aquatic ecosystem dynamics.

    Yassine Sabbar: Conceptualization, Writing Original Draft, Software, Formal Analysis; Aeshah A. Raezah: Conceptualization, Software, Validation. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP 2/174/45.

    The authors declare no conflict of interest.



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