Probabilistic Analysis of a Marine Ecological System with Intense Variability

Abstract

This work seeks to simulate and examine the complex character of marine predation. By taking into account the interaction between phytoplankton and zooplankton, we present a sophisticated mathematical system with a general functional response describing the ecological competition. This system is disturbed by a novel category of perturbations in the hybrid form which simulates certain unstable climatic and environmental variations. We merge between the higher-order white noise and quadratic jumps to offer an excellent overview of the complexity induced in the ecosystem. Analytically, we offer a surrogate framework to get the sharp sill between stationarity and zooplankton eradication. Our analysis enriches and improves many works by proposing an unfamiliar form of perturbation and unifying the criteria of said asymptotic characteristics. Numerically, we probe the rigor of our sill in a non-standard case: cubic white noise and quadratic leaps. We demonstrate that the increased order of perturbation has a significant effect on the zooplankton living time. This result shows that the sources of intricate fluctuations carry out an active role in the transient dynamics of marine ecological systems.

1. Introduction

The oceans cover a vast surface and contain a large variety of species, for a massive part still undiscovered. All marine organisms are grouped into two large groups: benthos and pelagos [1]. Benthos includes organisms living on the seabed while pelagos is made up of those living in the water column. According to their swimming power, the pelagos individuals are classified into two enormous groups: nekton and plankton. Nekton individuals have the ability to swim against ocean currents and move freely, including marine mammals and pelagic fish. Conversely, individual plankton are unable to oppose currents and allow themselves to be carried by the movements of water masses. However, some of them have the ability to move vertically in the water column [2]. Plankton organisms that have a completely pelagic life cycle are referred to as holoplankton. Meroplankton, meanwhile, characterizes the planktonic individuals present in the water column on a temporary basis [3]. This is the case for some benthic larvae, which live in the water column before they attach to the substrate to continue their growth. More commonly, in the study of the functioning of ecosystems and food webs, we simply distinguish between plant plankton phytoplankton, and animal plankton zooplankton.

Phytoplankton is made up of unicellular prokaryotic (cyanobacteria) and eukaryotic organisms living mainly in the upper part of the water column, the euphotic zone, where light is sufficient to carry out photosynthesis (see Figure 1). Phytoplankton forms the basis of marine food chains, providing more than $45\%$ of the planet’s annual net primary production [4]. It thus caries out a substantial role in the production of energy for the entire food web and in the assimilation of $C{O}_2$ and its sequestration in the form of carbonaceous matter in the oceans, thus making it possible to regulate the climate on a global scale [5].

Zooplankton is a taxonomically diverse group of organisms at different levels of the marine food chain. Protozooplancton brings together the unicellular organisms of zooplankton and is mainly composed of heterotrophic flagellates, ciliates and certain heterotrophic dinoflagellates which feed mainly on pico and nano-phytoplankton as well as bacteria. Actually, zooplankton have a key position in the marine food web by carrying out secondary production. By grazing on phytoplankton and then being consumed by higher trophic levels, it helps connect primary producers with the rest of the food web [5].

Ecological modeling originated in the 1920s and is presently a well-established field of research. Marine environmental modeling relies on the quantitative measurement of an ecosystem to study its components and the processes that connect them [6]. Like any domain of research, marine modeling is constantly evolving. Models change, visions improve, and applications multiply [7]. The emergence and development of marine ecosystems are closely linked to advances in other areas of research, whether in mathematics, physics, computer science or others [8]. Specifically, mathematical formulations are a robust approach to scout the long-run behavior of the plankton expansion and the interplay in oceanic ecosystems [9]. Recently, in 2022, the authors of [10] proposed a prey-predator plankton populations model that takes into account plankton cell dimensions. They showed that the natural rivalry between prey (phytoplankton—${\mathfrak{m}}_p$) and predator (zooplankton—${\mathfrak{m}}_z$) can be expressed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \mathrm{d}{\mathfrak{m}}_p\left(t\right)=\left(\frac{a{\mathfrak{m}}_p\left(t\right)\left(1-\frac{{\mathfrak{m}}_p\left(t\right)}{\mathcal{K}}\right)}{{\eta }_1{a}^2+{\eta }_{2}a+{\eta }_3}-\frac{{\psi }_h{\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z\left(t\right)}{{\mathfrak{m}}_z\left(t\right)+\xi {\mathfrak{m}}_p\left(t\right)}{e}^{\left\{-\frac{1}{d_1}{(a-d_{2}b)}^2\right\}}\right)\mathrm{d}t,}\hfill \\ {\displaystyle \mathrm{d}{\mathfrak{m}}_z\left(t\right)=\left(\frac{\tau {\psi }_h{\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z\left(t\right)}{{\mathfrak{m}}_z\left(t\right)+\xi {\mathfrak{m}}_p\left(t\right)}{e}^{\left\{-\frac{1}{d_1}{(a-d_{2}b)}^2\right\}}-\beta {\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z\left(t\right)-\mu {\mathfrak{m}}_z\left(t\right)\right)\mathrm{d}t,}\hfill \\ {\mathfrak{m}}_p\left(0\right)>0,{\mathfrak{m}}_z\left(0\right)>0,\hfill \end{array}\right.\end{array}$$

(1)

where the positive parameters with their values and ranges are well defined in

Table 1. Definition of the positive parameters appearing in the phytoplankton-zooplankton model (1).

Parameter	Ecological Signification	First Test	Second Test
a	Phytoplanktondimension	0.35	0.35
$\mathcal{K}$	Ultimate ecological strength	2	2
${\eta }_1$	First empirical constant	0.02	0.02
${\eta }_2$	Second empirical constant	0.02	0.02
${\eta }_3$	Third empirical constant	0.08	0.08
${\psi }_h$	Maximum consuming average	2	2
$\xi$	Constant of ecological saturation	0.2	0.2
$d_1$	First ecological exhaustion degree	2	2
$d_2$	Second ecological exhaustion degree	0.5	0.5
b	Zooplankton shape measure	0.25	0.25
$\tau$	Ecological transformation rate of ${\mathfrak{m}}_z$	0.25	0.25
$\beta$	Toxin emission average	0.14	0.195
$\mu$	Natural death of of ${\mathfrak{m}}_z$	0.1	0.1

. For ease of reading the remaining parts of this manuscript, we outline the predation mechanisms of the above-mentioned model by the flow diagram shown in Figure 2. The ecological model (1) describes the impact of cell size on the metabolism and growth rate of phytoplankton ${\mathfrak{m}}_p$. For this reason, we consider the entry of ${\mathfrak{m}}_p$ to be a logistic growth that takes the form $\frac{a{\mathfrak{m}}_p\left(t\right)\left(1-\frac{{\mathfrak{m}}_p\left(t\right)}{\mathcal{K}}\right)}{{\eta }_1{a}^2+{\eta }_{2}a+{\eta }_3}$. Furthermore, system (1) simulates the ecological interaction between ${\mathfrak{m}}_p$ and ${\mathfrak{m}}_z$ according to cell dimensions of the plankton which takes the following form: $\frac{\tau {\psi }_h{\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z\left(t\right)}{{\mathfrak{m}}_z\left(t\right)+\xi {\mathfrak{m}}_p\left(t\right)}{e}^{\left\{-\frac{1}{d_1}{(a-d_{2}b)}^2\right\}}$. In Figure 3, we present the effect of the size of microalgae (phytoplankton lives in oceans and freshwater) and zooplankton on the level of plankton intensification. That is, the integration of the geometry of the cell and body dimensions of phytoplankton and zooplankton is very important in the marine modeling [10].

When exploring ecological models, other characteristics should be incorporated such as the sensitivity of zooplankton to nutrient concentrations and the marine ecological interference between the ecosystem components [11]. So, the selection of the functional feedback influences the behavior of marine predation dynamics [12,13]. For this purpose, the present work puts forward a new and general phytoplankton-zooplankton system with an interference function that includes many types of responses. In line with this framework, system (1) can be reformulated as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \mathrm{d}{\mathfrak{m}}_p\left(t\right)=\left\{f\left(a\right){\mathfrak{m}}_p\left(t\right)\left(1-\frac{{\mathfrak{m}}_p\left(t\right)}{\mathcal{K}}\right)-g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right){\mathfrak{m}}_z\left(t\right)\right\}\mathrm{d}t,}\hfill \\ {\displaystyle \mathrm{d}{\mathfrak{m}}_z\left(t\right)=\left\{\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right){\mathfrak{m}}_z\left(t\right)-\beta {\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z\left(t\right)-\mu {\mathfrak{m}}_z\left(t\right)\right\}\mathrm{d}t,}\hfill \\ {\mathfrak{m}}_p\left(0\right)>0,{\mathfrak{m}}_z\left(0\right)>0,\hfill \end{array}\right.\end{array}$$

(2)

where $f\left(a\right)=\frac{a}{{\eta }_1{a}^2+{\eta }_{2}a+{\eta }_3}$, $g(a,b)={\psi }_h{e}^{\left\{-\frac{1}{d_1}{(a-d_{2}b)}^2\right\}}$ and the general interplay function $\mathcal{Q}\in {\mathcal{C}}^2({\mathbb{R}}_{+}\times {\mathbb{R}}_{+},{\mathbb{R}}_{+})$ satisfies that $\mathcal{Q}(0,{\mathfrak{m}}_z)=0$$\forall {\mathfrak{m}}_z\ge 0$. In addition, we suppose that $\mathcal{Q}$ is increasing in ${\mathfrak{m}}_m$ and decreasing in ${\mathfrak{m}}_p$; and $\exists \Gamma$ such that $\frac{\partial \mathcal{Q}({\mathfrak{m}}_p,{\mathfrak{m}}_z)}{\partial {\mathfrak{m}}_p}\le \Gamma$, for all ${\mathfrak{m}}_p,{\mathfrak{m}}_z\ge 0$. For some analytical purposes, we also assume that $\mathcal{Q}$ follows the property of uniform continuity at ${\mathfrak{m}}_z=0$. All these properties are naturally verified in the case of the following typical examples: Beddington–DeAngelis, Crowley-Martin and the modified Crowley-Martin responses [14].

Modern probability applied in marine ecology is a burgeoning scientific field that integrates pure mathematics, differential equations, and modeling tools [15,16,17,18]. This merger seeks to provide insight into the dynamics of ocean organisms under certain environmental fluctuations [19,20,21,22,23,24]. Water salinity, temperature, lighting, pressure, and human intervention are concrete examples of factors that influence ocean plankton dynamics. To mathematically portray this variability, sundry analytical methods have been used to educe facts and prophesy the future of the studied phenomenon [25,26,27,28,29,30,31,32]. The most well-known approach claims that external variations can be formulated by integrating the Wiener process into the associated formulation [23,24]. By considering this type of perturbations in its linear form, the authors in [10], treated model (1) with two proportional white noises. They offered sufficient criteria for the permanence and the eradication of phytoplankton and zooplankton organisms.

Based on the fact that high amounts of extrinsic fluctuations can strongly affect population dynamics, in this research, we offer new insight into modeling these environmental complexities. Massive pollution, domestic and industrial sewage waste, sudden climatic fluctuations, high temperatures and melting ice are the common brutal and unstable changes that affect ecological competition and lead to some interruptions in the process of marine predation. Of course, these phenomena cannot be simulated by the probabilistic model presented in [10]. For this cause, we aim to give an alternative framework that well explains the said fluctuations. By using the Taylor expansion and quadratic jumps, we suggest a high perturbed model that extends the aforementioned studies and provides a general setting of the randomness. This form of noise has not been previously addressed in the literature due to a lack of modeling insight and some analytical complexities [33]. Since our proposed framework is new, threshold analysis is considered an intriguing and important issue. Thus, in this paper, we present the sill among the ergodicity and disappearance of zooplankton by using some non-classical mathematical tools.

The remainder of this research is ordered as the following arrangement: in Section 2, we present our novel perturbed model and the analytical approach. In Section 3, we demonstrate our theoretical outcomes. In Section 4, we numerically discuss the impact of higher-order noises on the phytoplankton-zooplankton attitude. Some conclusions are given in Section 5.

2. Materials and Methods

In this research, we simultaneously consider polynomial white noise and quadratic jumps. Specifically, system (2) can be enhanced as follows:

$$\begin{array}{ccc}& \left\{\begin{array}{c}\mathrm{d}{\mathfrak{m}}_p\left(t\right)=\stackrel{\mathrm{Non}-\mathrm{probabilistic}\mathrm{part}}{\overbrace{\left\{f\left(a\right){\mathfrak{m}}_p\left(t\right)\left(1-\frac{{\mathfrak{m}}_p\left(t\right)}{\mathcal{K}}\right)-g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right){\mathfrak{m}}_z\left(t\right)\right\}\mathrm{d}t}}+\stackrel{\mathrm{Hybrid}\mathrm{fluctuations}}{\overbrace{\mathrm{d}{\Sigma }_0\left(t\right)}},\hfill \\ \mathrm{d}{\mathfrak{m}}_z\left(t\right)=\left\{\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right){\mathfrak{m}}_z\left(t\right)-\beta {\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z\left(t\right)-\mu {\mathfrak{m}}_z\left(t\right)\right\}\mathrm{d}t+\mathrm{d}{\Sigma }_1\left(t\right),\hfill \\ {\mathfrak{m}}_p\left(0\right)>0,{\mathfrak{m}}_z\left(0\right)>0,\hfill \end{array}\right.\hfill & \hfill \end{array}$$

(3)

where

$$\begin{array}{cc}\hfill \mathrm{d}{\Sigma }_0\left(t\right)& =\stackrel{\mathrm{High}-\mathrm{order}\mathrm{difusion}}{\overbrace{{\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{0\ell }{\mathfrak{m}}_p^{\ell +1}\left(t\right)\mathrm{d}{\mathfrak{A}}_0\left(t\right)}}+\stackrel{\mathrm{Second}-\mathrm{order}\mathrm{pure}\mathrm{leaps}}{\overbrace{{\int }_{\mathcal{D}}{\displaystyle \underset{j=0}{\sum ^1}}{\Xi }_{0j}\left(r\right){\mathfrak{m}}_p^{j+1}\left(t_{-}\right){\mathfrak{M}}_0(\mathrm{d}t,\mathrm{d}r)}},\hfill \\ \hfill \mathrm{d}{\Sigma }_1\left(t\right)& ={\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{h+1}\left(t\right)\mathrm{d}{\mathfrak{A}}_1\left(t\right)+{\int }_{\mathcal{D}}{\displaystyle \underset{j=0}{\sum ^1}}{\Xi }_{1j}\left(r\right){\mathfrak{m}}_z^{j+1}\left(t_{-}\right){\mathfrak{M}}_1(\mathrm{d}t,\mathrm{d}r).\hfill \end{array}$$

Here, ${\mathfrak{m}}_p\left(t_{-}\right)$ and ${\mathfrak{m}}_z\left(t_{-}\right)$ are the left limits of ${\mathfrak{m}}_p\left(t\right)$ and ${\mathfrak{m}}_z\left(t\right)$, respectively. Let ${\Omega }^{\{\mathcal{E},\mathbb{P}\}}\equiv (\Omega ,\mathcal{E},{\left\{{\mathcal{E}}_t\right\}}_{t\ge 0},\mathbb{P})$ be a filtered probability space with ${\left\{{\mathcal{E}}_t\right\}}_{t\ge 0}$ verifying the usual conditions. In system (3), ${\mathfrak{A}}_0\left(t\right)$, ${\mathfrak{A}}_1\left(t\right)$ are two independent Wiener processes (WPs) defined on a ${\Omega }^{\{\mathcal{E},\mathbb{P}\}}$, and ${{\rm Y}}_{h\ell }>0$$(h=0,1)$$(\ell =0,1,2,\cdots ,L)$ are the polynomial order random intensities. ${\mathbb{M}}_0(·,·)$ and ${\mathbb{M}}_1(·,·)$ are two independent Poisson random measures with finite characteristic measures ${\varpi }_0(·)$ and ${\varpi }_1(·)$ defined on a measurable subset $\mathcal{D}$ of $(0,\infty )$, where ${\varpi }_i\left(\mathcal{D}\right)<\infty$$(i=0,1)$. The associated independent compensated Poisson random measures ${\mathfrak{M}}_0$ and ${\mathfrak{M}}_1$ are expressed by

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\mathfrak{M}}_0(\mathrm{d}t,\mathrm{d}u)=-{\varpi }_0\left(\mathrm{d}r\right)\mathrm{d}t+{\mathbb{M}}_0(\mathrm{d}t,\mathrm{d}r),\hfill \\ {\mathfrak{M}}_1(\mathrm{d}t,\mathrm{d}u)=-{\varpi }_1\left(\mathrm{d}r\right)\mathrm{d}t+{\mathbb{M}}_1(\mathrm{d}t,\mathrm{d}r),\hfill \end{array}\right.\end{array}$$

which are two $\left\{{\mathcal{E}}_t\right\}$-martingales. Furthermore, it is supposed that ${\mathbb{M}}_i$$(i=0,1)$ are independent to WPs ${\mathfrak{A}}_0$,${\mathfrak{A}}_1$; and the Lévy intensities ${\Xi }_{00}\left(r\right)$, ${\Xi }_{01}\left(r\right)$, ${\Xi }_{10}\left(r\right)$, ${\Xi }_{11}\left(r\right)$ are positive and continuous functions.

Defining ${\mathbb{R}}_{\circ ,+}^2=\{(a,b):a>0,b>0\}$ and persuming that ${\displaystyle {\int }_{\mathcal{D}}{\Xi }_{kj}^2\left(r\right){\varpi }_k\left(\mathrm{d}r\right)}$ is finite; $k,j=0,1$. By employing the identical techniques presented in [23], we can facilely demonstrate that for every positive started value, there is a single positive solution $\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right)\in {\mathbb{R}}_{\circ ,+}^2$ to the probabilistic system (3). This indicates that the model (3) is well-constructed scientifically.

The aim of this paper is to explore some mathematical outcomes on the long-run behavior of marine predation. Our goal is to provide the acute threshold between the pursuit of the zooplankton and its eradication. This threshold value gives an overview of the plankton dynamics. To deal with the new stochastic system (3), we propose an alternative method based on a second system very close to the equation of ${\mathfrak{m}}_p$. This new auxiliary system characterizes the prey-predator process in limited conditions when the predator is absent. Keeping the same probabilistic part, the auxiliary system is expressed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \mathrm{d}{\mathfrak{n}}_{\circ }\left(t\right)=f\left(a\right){\mathfrak{n}}_{\circ }\left(t\right)\left(1-\frac{{\mathfrak{n}}_{\circ }\left(t\right)}{\mathcal{K}}\right)\mathrm{d}t+{\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{0\ell }{\mathfrak{n}}_{\circ }^{\ell +1}\left(t\right)\mathrm{d}{\mathfrak{A}}_0\left(t\right)+{\int }_{\mathcal{D}}{\displaystyle \underset{j=0}{\sum ^1}}{\Xi }_{0j}\left(r\right){\mathfrak{n}}_{\circ }^{j+1}\left(t_{-}\right){\mathfrak{M}}_0(\mathrm{d}t,\mathrm{d}r),}\hfill \\ {\mathfrak{n}}_{\circ }\left(0\right)={\mathfrak{m}}_p\left(0\right)>0.\hfill \end{array}\right.\end{array}$$

(4)

The stochastic system (4) is ecologically well-posed and admits a unique positive solution ${\mathfrak{n}}_{\circ }\left(t\right)$. Moreover, ${\mathfrak{n}}_{\circ }\left(t\right)$ is a Markov process which satisfies the following nice analytical properties:

• ${\mathfrak{n}}_{\circ }\left(t\right)\ge {\mathfrak{m}}_p\left(t\right)$ a.s. (stochastic comparison result).
• If ${\displaystyle C=f\left(a\right)-0.5{{\rm Y}}_{00}^2-{\int }_{\mathcal{D}}\left({\Xi }_{00}\left(r\right)-ln\left(1+{\Xi }_{00}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)>0}$, then there exists a single invariant probability measure, named as ${\psi }^{{\mathfrak{n}}_{\circ }\left(t\right)}$.
• ${\mathfrak{n}}_{\circ }$ follows the ergodic property.
• If $C>0$, then the time average of ${\mathfrak{n}}_{\circ }\left(t\right)$ is

  $$\begin{array}{cc}\hfill {\displaystyle {\displaystyle \underset{t\to \infty }{lim}}\frac{1}{t}{\int }_0^t{\mathfrak{n}}_{\circ }\left(s\right)\mathrm{d}s}& {\displaystyle \le \frac{\mathcal{K}}{(f\left(a\right)+\mathcal{K}{{\rm Y}}_{00}{{\rm Y}}_{01})}\left\{f\left(a\right)-0.5{{\rm Y}}_{00}^2-{\int }_{\mathcal{D}}\left({\Xi }_{00}\left(r\right)-ln\left(1+{\Xi }_{00}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)\right\}={\varsigma }_1^{\star }>0}\hfill \end{array}$$

These properties can be demonstrated by a similar analysis presented in [34]. For some analytical purposes, throughout this research we always assume that $C>0$. Furthermore, we properly introduce the well-defined sill of the probabilistic system (3) which can be expressed in the following form:

$$\begin{array}{c}\hfill {\mathcal{S}}_0^{\mathfrak{m}}=\tau g(a,b){\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)-\mu -\frac{1}{2}{{\rm Y}}_{10}^2-{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right).\end{array}$$

In the next section, we will prove that the quantity ${\mathcal{S}}_0^{\mathfrak{m}}$ is regarded to be sufficient for having an excellent view of the long-run behavior of the marine ecological predation.

3. Results

As mentioned earlier, the focal question related to the analysis of marine ecosystems is to obtain the sharp sill between the continuation and eradication of the population. Thus, the main purpose of the following theorem is to deal with this query.

Theorem 1.

The marine ecological predation described by system (3) which has two possible scenarios:

1. 

The stationary state (${\mathcal{S}}_0^{\mathfrak{m}}>0$), that is, ecosystem (3) admits a single ergodic limiting distribution ${\pi }^{\Sigma }(·)$. In other words, phytoplankton and zooplankton persist.

2. 

The eradication state (${\mathcal{S}}_0^{\mathfrak{m}}<0$), that is, the zooplankton in the food chain will disappear with full probability.

Remark 1.

Biologically, the stationarity property (when ${\mathcal{S}}_0^{\mathfrak{m}}>0$) reveals that the perturbed model (3) has a limiting stable distribution that prophesies the continuation of the zooplankton. This means that the zooplankton group will tend to stay for a long time and feed on the persistent phytoplankton. In the extinction case, the quantity ${\mathcal{S}}_0^{\mathfrak{m}}$ contains linear random intensities, which are related to the zooplankton class. This designates that if ${\mathcal{S}}_0^{\mathfrak{m}}$ is strictly less than zero, the stochastic fluctuations help the inhibition of the zooplankton.

Proof. 

This proof is divided into two parts.

First point: we assume that ${\mathcal{S}}_0^{\mathfrak{m}}>0$.

The Itô differential operator ${\mathcal{L}}_{\mathrm{Itô}}$ related to the function $ln{\mathfrak{n}}_{\circ }\left(t\right)-ln{\mathfrak{m}}_p\left(t\right)$ is given by

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{Itô}}\left(ln{\mathfrak{n}}_{\circ }\left(t\right)-ln{\mathfrak{m}}_p\left(t\right)\right)& \le -\frac{f\left(a\right)}{\mathcal{K}}\left({\mathfrak{n}}_{\circ }\left(t\right)-{\mathfrak{m}}_p\left(t\right)\right)+\frac{g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right){\mathfrak{m}}_z\left(t\right)}{{\mathfrak{m}}_p\left(t\right)}\hfill \\ & -\frac{1}{2}{\left({\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{0\ell }{\mathfrak{n}}_{\circ }^{\ell }\left(t\right)\right)}^2+\frac{1}{2}{\left({\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{0\ell }{\mathfrak{m}}_p^{\ell }\left(t\right)\right)}^2\hfill \\ & +{\int }_{\mathcal{D}}\left\{ln\left(\frac{1+{\Xi }_{00}\left(r\right)+{\Xi }_{01}\left(r\right){\mathfrak{n}}_{\circ }\left(t\right)}{1+{\Xi }_{00}\left(r\right)+{\Xi }_{01}\left(r\right){\mathfrak{m}}_p\left(t\right)}\right)-{\Xi }_{01}\left(r\right)\left({\mathfrak{n}}_{\circ }\left(t\right)-{\mathfrak{m}}_p\left(t\right)\right)\right\}{\varpi }_0\left(\mathrm{d}r\right)\hfill \\ & \le -\frac{f\left(a\right)}{\mathcal{K}}\left({\mathfrak{n}}_{\circ }\left(t\right)-{\mathfrak{m}}_p\left(t\right)\right)+\Gamma g(a,b){\mathfrak{m}}_z\left(t\right)\hfill \\ & -\left({\mathfrak{n}}_{\circ }\left(t\right)-{\mathfrak{m}}_p\left(t\right)\right){\int }_{\mathcal{D}}\left\{\frac{{\Xi }_{01}\left(r\right)\left({\Xi }_{00}\left(r\right)+{\Xi }_{01}\left(r\right){\mathfrak{m}}_p\left(t\right)\right)}{1+{\Xi }_{00}\left(r\right)+{\Xi }_{01}\left(r\right){\mathfrak{m}}_p\left(t\right)}\right\}{\varpi }_0\left(\mathrm{d}r\right)\hfill \\ & \le -\frac{f\left(a\right)}{\mathcal{K}}\left({\mathfrak{n}}_{\circ }\left(t\right)-{\mathfrak{m}}_p\left(t\right)\right)+\Gamma g(a,b){\mathfrak{m}}_z\left(t\right).\hfill \end{array}$$

(5)

In order to simplify the notations and to provide a simple mathematical writing, we put

$$\begin{array}{c}\hfill {\left(\underset{\ell =0}{\sum ^L}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)\right)}^2=\underset{\ell =0}{\sum ^{2L}}\underset{{\gamma }_{\ell }}{\underbrace{\left(\sum _{z_1+z_2=\ell }{{\rm Y}}_{1z_1}{{\rm Y}}_{1z_2}\right)}}{\mathfrak{m}}_z^{\ell }\left(t\right)=\underset{\ell =0}{\sum ^{2L}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell }\left(t\right).\end{array}$$

Again, the application of ${\mathcal{L}}_{\mathrm{Itô}}$ on the second equation of (3) gives

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{Itô}}\left(-ln{\mathfrak{m}}_z\left(t\right)\right)& =-\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right)+\beta {\mathfrak{m}}_p\left(t\right)+\mu +\frac{1}{2}{\left({\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)\right)}^2\hfill \\ & -{\int }_{\mathcal{D}}\left\{ln\left(1+{\Xi }_{10}\left(r\right)+{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(t\right)\right)-\left({\Xi }_{10}\left(r\right)+{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(t\right)\right)\right\}{\varpi }_1\left(\mathrm{d}r\right).\hfill \end{array}$$

Then

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{Itô}}\left(-ln{\mathfrak{m}}_z\left(t\right)\right)& =-\tau g(a,b)\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(t\right),0\right)+\beta {\mathfrak{m}}_p\left(t\right)+\mu +\frac{1}{2}{{\rm Y}}_{10}^2+{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)\hfill \\ & +\tau g(a,b)\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(t\right),0\right)-\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right)-\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),0\right)\hfill \\ & +\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),0\right)+{{\rm Y}}_{10}{{\rm Y}}_{11}{\mathfrak{m}}_z\left(t\right)+\frac{1}{2}{\displaystyle \underset{\ell =2}{\sum ^{2L}}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)\hfill \\ & +{\int }_{\mathcal{D}}\left\{{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(t\right)-ln\left(1+\frac{{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(t\right)}{1+{\Xi }_{10}\left(r\right)}\right)\right\}{\varpi }_1\left(\mathrm{d}r\right).\hfill \end{array}$$

In line with the positivity of the solution, we have

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{Itô}}\left(-ln{\mathfrak{m}}_z\left(t\right)\right)& \le -\tau g(a,b)\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(t\right),0\right)+\mu +\frac{1}{2}{{\rm Y}}_{10}^2+{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)\hfill \\ & +\beta {\mathfrak{m}}_p\left(t\right)+\tau \Gamma g(a,b)\left({\mathfrak{n}}_{\circ }\left(t\right)-{\mathfrak{m}}_p\left(t\right)\right)+\left({{\rm Y}}_{10}{{\rm Y}}_{11}+{\int }_{\mathcal{D}}{\Xi }_{11}\left(r\right){\varpi }_1\left(\mathrm{d}r\right)\right){\mathfrak{m}}_z\left(t\right)\hfill \\ & +\frac{1}{2}{\displaystyle \underset{\ell =2}{\sum ^{2L}}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)+\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),0\right)-\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right).\hfill \end{array}$$

(6)

Based on the above calculation, we obtain

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{\mathrm{Itô}}\left(\frac{\tau \Gamma g(a,b)\mathcal{K}}{f\left(a\right)}\left(ln{\mathfrak{n}}_{\circ }\left(t\right)-ln{\mathfrak{m}}_p\left(t\right)\right)-ln{\mathfrak{m}}_z\left(t\right)\right)\hfill \\ & \le -\tau g(a,b)\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(t\right),0\right)+\mu +\frac{1}{2}{{\rm Y}}_{10}^2+{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)\hfill \\ & +\beta {\mathfrak{m}}_p\left(t\right)+\left({{\rm Y}}_{10}{{\rm Y}}_{11}+{\int }_{\mathcal{D}}{\Xi }_{11}\left(r\right){\varpi }_1\left(\mathrm{d}r\right)\right){\mathfrak{m}}_z\left(t\right)+\frac{\tau {\Gamma }^2{g}^2(a,b)\mathcal{K}}{f\left(a\right)}{\mathfrak{m}}_z\left(t\right)\hfill \\ & +\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),0\right)-\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right)+\frac{1}{2}{\displaystyle \underset{\ell =2}{\sum ^{2L}}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell }\left(t\right).\hfill \end{array}$$

Then, we get

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{\mathrm{Itô}}\left(\frac{\tau \Gamma g(a,b)\mathcal{K}}{f\left(a\right)}\left(ln{\mathfrak{n}}_{\circ }\left(t\right)-ln{\mathfrak{m}}_p\left(t\right)\right)-ln{\mathfrak{m}}_z\left(t\right)\right)\hfill \\ & =-\tau g(a,b){\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)+\mu +\frac{1}{2}{{\rm Y}}_{10}^2+{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)\hfill \\ & +\beta {\mathfrak{m}}_p\left(t\right)+\frac{1}{2}{\displaystyle \underset{\ell =2}{\sum ^{2\ell }}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)+\tau g(a,b)\left({\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)-\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(t\right),0\right)\right)\hfill \\ & +\left(\frac{\tau {\Gamma }^2{g}^2(a,b)\mathcal{K}}{f\left(a\right)}+{{\rm Y}}_{10}{{\rm Y}}_{11}+{\int }_{\mathcal{D}}{\Xi }_{11}\left(r\right){\varpi }_1\left(\mathrm{d}r\right)\right){\mathfrak{m}}_z\left(t\right)\hfill \\ & +\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),0\right)-\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right).\hfill \end{array}$$

Now, we choose a positive constant ${\chi }_{\star }$ that satisfies

$$\begin{array}{c}\hfill {\chi }_{\star }\ge \frac{1}{\mu }\left(\frac{\tau {\Gamma }^2{g}^2(a,b)\mathcal{K}}{f\left(a\right)}+{{\rm Y}}_{10}{{\rm Y}}_{11}+{\int }_{\mathcal{D}}{\Xi }_{11}\left(r\right){\varpi }_1\left(\mathrm{d}r\right)\right)\end{array}$$

To eliminate the term associated with ${\mathfrak{m}}_z$, we set

$$\begin{array}{c}\hfill {\mathcal{G}}^{\star }\left(t\right)=\frac{\tau \Gamma g(a,b)\mathcal{K}}{f\left(a\right)}\left(ln{\mathfrak{n}}_{\circ }\left(t\right)-ln{\mathfrak{m}}_p\left(t\right)\right)-ln{\mathfrak{m}}_z\left(t\right)+{\chi }_{\star }{\mathfrak{m}}_z\left(t\right).\end{array}$$

Applying ${\mathcal{L}}_{\mathrm{Itô}}$ to ${\mathcal{G}}^{\star }$ gives

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{Itô}}{\mathcal{G}}^{\star }\left(t\right)& \le -\tau g(a,b){\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)+\mu +\frac{1}{2}{{\rm Y}}_{10}^2+{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)\hfill \\ & +\tau g(a,b)\left({\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)-\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(t\right),0\right)\right)+\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),0\right)\hfill \\ & -\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right)+\beta {\mathfrak{m}}_p\left(t\right)+\frac{1}{2}{\displaystyle \underset{\ell =2}{\sum ^{2L}}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)+{\chi }_{\star }\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right){\mathfrak{m}}_z\left(t\right).\hfill \end{array}$$

Consequently

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{Itô}}{\mathcal{G}}^{\star }\left(t\right)& \le -{\mathcal{S}}_0^{\mathfrak{m}}+\tau \left({\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)-\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(t\right),0\right)\right)+\tau g(a,b)\left(\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),0\right)-\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right)\right)\hfill \\ & +\beta {\mathfrak{m}}_p\left(t\right)+\frac{1}{2}{\displaystyle \underset{\ell =2}{\sum ^{2L}}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)+{\chi }_{\star }\tau \Gamma g(a,b){\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z\left(t\right).\hfill \end{array}$$

Again, for all $\vartheta \in (0,1)$, we have

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{\mathrm{Itô}}\left(\frac{{\mathfrak{m}}_p^{\vartheta }\left(t\right)}{\vartheta }\right)\hfill \\ & ={\mathfrak{m}}_p^{\vartheta -1}\left(t\right)\left\{f\left(a\right){\mathfrak{m}}_p\left(t\right)\left(1-\frac{{\mathfrak{m}}_p\left(t\right)}{\mathcal{K}}\right)-g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right){\mathfrak{m}}_z\left(t\right)\right\}+\frac{1}{2}(\vartheta -1){\mathfrak{m}}_p^{\vartheta -2}\left(t\right){\left({\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{0\ell }{\mathfrak{m}}_p^{\ell +1}\left(t\right)\right)}^2\hfill \\ & +{\int }_{\mathcal{D}}\left\{\frac{{\left({\mathfrak{m}}_p\left(t\right)+{\Xi }_{00}\left(r\right){\mathfrak{m}}_p\left(t\right)+{\Xi }_{01}\left(r\right){\mathfrak{m}}_p^2\left(t\right)\right)}^{\vartheta }}{\vartheta }-\frac{{\mathfrak{m}}_p^{\vartheta }\left(t\right)}{\vartheta }-{\mathfrak{m}}_p^{\vartheta -1}\left(t\right)\left({\Xi }_{00}\left(r\right){\mathfrak{m}}_p\left(t\right)+{\Xi }_{01}\left(r\right){\mathfrak{m}}_p^2\left(t\right)\right)\right\}{\varpi }_0\left(\mathrm{d}r\right)\hfill \\ & \le f\left(a\right){\mathfrak{m}}_p^{\vartheta }\left(t\right)-\frac{1}{2}(1-\vartheta ){{\rm Y}}_{01}^2{\mathfrak{m}}_p^{\vartheta +2}\left(t\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{\mathrm{Itô}}\left(\frac{{\mathfrak{m}}_z^{\vartheta }\left(t\right)}{\vartheta }\right)\hfill \\ & ={\mathfrak{m}}_z^{\vartheta -1}\left(t\right)\left(\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right){\mathfrak{m}}_z\left(t\right)-\beta {\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z\left(t\right)-\mu {\mathfrak{m}}_z\left(t\right)\right)+\frac{1}{2}(\vartheta -1){\mathfrak{m}}_z^{\vartheta -2}\left(t\right){\left({\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell +1}\left(t\right)\right)}^2\hfill \\ & +{\int }_{\mathcal{D}}\left(\frac{{\left({\mathfrak{m}}_z\left(t\right)+{\Xi }_{00}\left(r\right){\mathfrak{m}}_z\left(t\right)+{\Xi }_{01}\left(r\right){\mathfrak{m}}_z^2\left(t\right)\right)}^{\vartheta }}{\vartheta }-\frac{{\mathfrak{m}}_z^{\vartheta }\left(t\right)}{\vartheta }-{\mathfrak{m}}_z^{\vartheta }\left(t\right)\left({\Xi }_{00}\left(r\right)+{\Xi }_{01}\left(r\right){\mathfrak{m}}_z\left(t\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)\hfill \\ & \le \tau \Gamma g(a,b){\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z^{\vartheta }\left(t\right)-\left(\mu +\frac{1}{2}(1-\vartheta ){{\rm Y}}_{10}^2\right){\mathfrak{m}}_z^{\vartheta }\left(t\right)-(1-\vartheta ){{\rm Y}}_{10}{{\rm Y}}_{11}{\mathfrak{m}}_z^{\vartheta +1}\left(t\right)\hfill \\ & -\frac{1}{2}(1-\vartheta ){{\rm Y}}_{11}^2{\mathfrak{m}}_z^{\vartheta +2}\left(t\right)-\frac{1}{2}(1-\vartheta ){\displaystyle \underset{\ell =2}{\sum ^{2\ell }}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell +\vartheta }\left(t\right)\hfill \\ & \le \frac{\tau g(a,b)\Gamma }{\vartheta +1}{\mathfrak{m}}_p^{\vartheta +1}\left(t\right)+\frac{\tau \vartheta g(a,b)\Gamma }{\vartheta +1}{\mathfrak{m}}_z^{\vartheta +1}\left(t\right)-\frac{1}{2}(1-\vartheta ){{\rm Y}}_{11}^2{\mathfrak{m}}_z^{\vartheta +2}\left(t\right)-\frac{1}{2}(1-\vartheta ){\displaystyle \underset{\ell =2}{\sum ^{2\ell }}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell +\vartheta }\left(t\right).\hfill \end{array}$$

We define the following new function

$$\begin{array}{c}\hfill {\tilde{\mathcal{G}}}^{\star }({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right))=x^{\star }{\mathcal{G}}^{\star }\left(t\right)+\frac{{\mathfrak{m}}_p^{\vartheta }\left(t\right)}{\vartheta }+\frac{{\mathfrak{m}}_z^{\vartheta }\left(t\right)}{\vartheta },\end{array}$$

where $x^{\star }>0$ verifies $2+{\mathsf{\Psi }}_{\vartheta }^{\star }\le x^{\star }{\mathcal{S}}_0^{\mathfrak{m}}$, and

$$\begin{array}{cc}\hfill {\mathsf{\Psi }}_{\vartheta }^{\star }=max\left\{{\displaystyle \underset{({\mathfrak{m}}_p,{\mathfrak{m}}_z)\in {\mathbb{R}}_{\circ ,+}^2}{sup}}\right\{& f\left(a\right){\mathfrak{m}}_p^{\vartheta }\left(t\right)+\frac{\tau g(a,b)\Gamma }{\vartheta +1}{\mathfrak{m}}_p^{\vartheta +1}\left(t\right)-\frac{1}{4}(1-\vartheta ){{\rm Y}}_{01}^2{\mathfrak{m}}_p^{\vartheta +2}\left(t\right)+\frac{\vartheta \tau g(a,b)\Gamma }{\vartheta +1}{\mathfrak{m}}_z^{\vartheta +1}\left(t\right)\hfill \\ & -\frac{1}{4}(1-\vartheta ){{\rm Y}}_{11}^2{\mathfrak{m}}_z^{\vartheta +2}\left(t\right)+\frac{x^{\star }}{2}{\displaystyle \underset{\ell =2}{\sum ^{2L}}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)-\frac{1}{2}(1-\vartheta ){\displaystyle \underset{\ell =2}{\sum ^{2L}}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell +\vartheta }\left(t\right)\},1\}.\hfill \end{array}$$

To deal with a non-negative Lyapunov function, we define

$$\begin{array}{c}\hfill {\tilde{\mathcal{G}}}^{\star }({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right))=x^{\star }\mathcal{G}\left(t\right)+\frac{{(1+{\mathfrak{m}}_p\left(t\right))}^{\vartheta }}{\vartheta }+\frac{{\mathfrak{m}}_z{\left(t\right)}^{\vartheta }}{p}-{\tilde{\mathcal{G}}}^{\star }({\displaystyle \underline{{\mathfrak{m}}_p}},{\displaystyle \underline{{\mathfrak{m}}_z}}),\end{array}$$

where the function ${\tilde{\mathcal{G}}}^{\star }({\mathfrak{m}}_p,{\mathfrak{m}}_z)$ attains the lower bound at a point $({\displaystyle \underline{{\mathfrak{m}}_p}},{\displaystyle \underline{{\mathfrak{m}}_z}})$ in ${\mathbb{R}}_{+}^2$, so

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{\mathrm{Itô}}{\tilde{\mathcal{G}}}^{\star }({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right))\hfill \\ & \le -x^{\star }{\mathcal{S}}_0^{\mathfrak{m}}+x^{\star }{\chi }_{\star }\tau g(a,b)\Gamma {\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z\left(t\right)+x^{\star }\tau g(a,b)\left(\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),0\right)-\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right)\right)+\frac{x^{\star }}{2}{\displaystyle \underset{\ell =2}{\sum ^{2L}}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)\hfill \\ & +f\left(a\right){\mathfrak{m}}_p^{\vartheta }\left(t\right)+\frac{\tau g(a,b)\Gamma }{\vartheta +1}{\mathfrak{m}}_p^{\vartheta +1}\left(t\right)-\frac{1}{4}(1-\vartheta ){{\rm Y}}_{01}^2{\mathfrak{m}}_p^{\vartheta +2}\left(t\right)+\frac{\vartheta \tau g(a,b)\Gamma }{\vartheta +1}{\mathfrak{m}}_z^{\vartheta +1}\left(t\right)\hfill \\ & -\frac{1}{4}(1-\vartheta ){{\rm Y}}_{11}^2{\mathfrak{m}}_z^{\vartheta +2}\left(t\right)-\frac{1}{2}(1-\vartheta ){\displaystyle \underset{\ell =2}{\sum ^{2L}}}{\gamma }_{\ell }{\mathfrak{m}}_z^{\ell +\vartheta }\left(t\right)+x^{\star }\tau g(a,b)\left({\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)-\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(t\right),0\right)\right)\hfill \\ & ={\mathcal{J}}^{\star }({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right))+x^{\star }\tau g(a,b)\left({\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)-\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(t\right),0\right)\right).\hfill \end{array}$$

Using the same techniques presented in the proof of (Theorem 3.2 [34]), and the uniform continuity of the function $\mathcal{Q}$, we can rapidly check that ${\mathcal{J}}^{\star }({\mathfrak{m}}_p,{\mathfrak{m}}_z)\le -1$, $\forall ({\mathfrak{m}}_p,{\mathfrak{m}}_z)\in {\mathbb{R}}_{\circ ,+}^2\setminus {\mathbb{V}}_{{\alpha }_{\star }}$, for a specific sufficiently small number ${\alpha }_{\star }>0$, where ${\mathbb{V}}_{{\alpha }_{\star }}=\left\{({\mathfrak{m}}_p,{\mathfrak{m}}_z)\in {\mathbb{R}}_{\circ ,+}^2|{\alpha }_{\star }\le {\mathfrak{m}}_p\le {\alpha }_{\star }^{-1},{\alpha }_{\star }\le {\mathfrak{m}}_z\le {\alpha }_{\star }^{-1}\right\}$. On the other hand, we can easily show that $\exists \tilde{\mathfrak{Z}}>0$ such that ${\mathcal{J}}^{\star }({\mathfrak{m}}_p,{\mathfrak{m}}_z)\le \tilde{\mathfrak{Z}}$, for all $({\mathfrak{m}}_p,{\mathfrak{m}}_z)\in {\mathbb{R}}_{\circ ,+}^2$. Hence, we get

$$\begin{array}{cc}\hfill -{\mathbb{E}}_{\{\Omega ,\mathbb{P}\}}\left\{{\tilde{\mathcal{G}}}^{\star }({\mathfrak{m}}_p\left(0\right),{\mathfrak{m}}_z\left(0\right))\right\}& \le {\mathbb{E}}_{\{\Omega ,\mathbb{P}\}}\left\{{\tilde{\mathcal{G}}}^{\star }({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right))\right\}-{\mathbb{E}}_{\{\Omega ,\mathbb{P}\}}\left\{{\tilde{\mathcal{G}}}^{\star }({\mathfrak{m}}_p\left(0\right),{\mathfrak{m}}_z\left(0\right))\right\}\hfill \\ & ={\int }_0^t{\mathbb{E}}_{\{\Omega ,\mathbb{P}\}}\left\{{\mathcal{L}}_{Itô}{\tilde{\mathcal{G}}}^{\star }\left({\mathfrak{m}}_p\left(s\right),{\mathfrak{m}}_z\left(s\right)\right)\right\}\mathrm{d}s\hfill \\ & \le {\int }_0^t{\mathbb{E}}_{\{\Omega ,\mathbb{P}\}}\left\{{\mathcal{J}}^{\star }({\mathfrak{m}}_p\left(s\right),{\mathfrak{m}}_z\left(s\right))\right\}\mathrm{d}s\hfill \\ & +x^{\star }\tau g(a,b){\mathbb{E}}_{\{\Omega ,\mathbb{P}\}}\left\{{\int }_0^t{\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)\mathrm{d}s-{\int }_0^t\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(s\right),0\right)\mathrm{d}s\right\}.\hfill \end{array}$$

In line with the ergodic characteristic of ${\mathfrak{n}}_{\circ }\left(t\right)$, we conclude that

$$\begin{array}{cc}\hfill 0& \le {\displaystyle \underset{t\to \infty }{lim\; inf}}\frac{1}{t}{\int }_0^t\left({\mathbb{E}}_{\{\Omega ,\mathbb{P}\}}\left\{{\mathcal{J}}^{\star }({\mathfrak{m}}_p\left(s\right),{\mathfrak{m}}_z\left(s\right)){\mathbb{1}}_{\{({\mathfrak{m}}_p\left(s\right),{\mathfrak{m}}_z\left(s\right))\in {\mathbb{V}}_{{\alpha }_{\star }}^c\}}\right\}+{\mathbb{E}}_{\{\Omega ,\mathbb{P}\}}\left\{{\mathcal{J}}^{\star }({\mathfrak{m}}_p\left(s\right),{\mathfrak{m}}_z\left(s\right)){\mathbb{1}}_{\{({\mathfrak{m}}_p\left(s\right),{\mathfrak{m}}_z\left(s\right))\in {\mathbb{V}}_{{\alpha }_{\star }}\}}\right\}\right)\mathrm{d}s\hfill \\ & \le {\displaystyle \underset{t\to \infty }{lim\; inf}}\frac{1}{t}{\int }_0^t\left(-\mathbb{P}\left(({\mathfrak{m}}_p\left(s\right),{\mathfrak{m}}_z\left(s\right))\in {\mathbb{V}}_{{\alpha }_{\star }}^c\right)+\tilde{\mathfrak{Z}}\mathbb{P}\left(({\mathfrak{m}}_p\left(s\right),{\mathfrak{m}}_z\left(s\right))\in {\mathbb{V}}_{{\alpha }_{\star }}\right)\right)\mathrm{d}s.\hfill \end{array}$$

Consequently,

$$\begin{array}{c}\hfill {\displaystyle \underset{t\to \infty }{lim\; inf}}\frac{1}{t}{\int }_0^t\mathbb{P}\left(({\mathfrak{m}}_p\left(s\right),{\mathfrak{m}}_z\left(s\right))\in {\mathbb{V}}_{{\alpha }_{\star }}\right)\mathrm{d}s\ge \frac{1}{1+\tilde{\mathfrak{Z}}}>0.\end{array}$$

Thus, we have checked that

$$\begin{array}{c}\hfill {\displaystyle \underset{t\to \infty }{lim\; inf}}\frac{1}{t}{\int }_0^t\mathbb{P}\left(({\mathfrak{m}}_p\left(0\right),{\mathfrak{m}}_z\left(0\right));s,{\mathbb{V}}_{{\alpha }_{\star }}\right)\mathrm{d}s\ge \frac{1}{1+\tilde{\mathfrak{Z}}}>0,\forall ({\mathfrak{m}}_p\left(0\right),{\mathfrak{m}}_z\left(0\right))\in {\mathbb{R}}_{\circ ,+}^2.\end{array}$$

Identical to the demonstration of (Lemma 3.2, [35]) and the mutually limited possibilities lemma [36], we establish the existence, uniqueness and ergodicity of a single invariant distribution for the perturbed model (3).

Second point: we assume that ${\mathcal{S}}_0^{\mathfrak{m}}<0$.

By employing Itô’s lemma, we directly get

$$\begin{array}{cc}\hfill \mathrm{d}ln{\mathfrak{m}}_z\left(t\right)& =(\tau g(a,b)\mathcal{Q}\left({\mathfrak{m}}_p\left(t\right),{\mathfrak{m}}_z\left(t\right)\right)-\beta {\mathfrak{m}}_p\left(t\right)-\mu -\frac{1}{2}{\left({\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)\right)}^2\hfill \\ & +{\int }_{\mathcal{D}}\left(ln\left(1+{\Xi }_{10}\left(r\right)+{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(t\right)\right)-\left({\Xi }_{10}\left(r\right)+{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(t\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right))\mathrm{d}t\hfill \\ & +{\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)\mathrm{d}{\mathfrak{A}}_1\left(t\right)+{\int }_{\mathcal{D}}ln\left(1+{\Xi }_{10}\left(r\right)+{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(t^{-}\right)\right){\mathfrak{M}}_1(\mathrm{d}t,\mathrm{d}r).\hfill \end{array}$$

In line with the stochastic comparison result, we conclude that

$$\begin{array}{cc}\hfill \mathrm{d}ln{\mathfrak{m}}_z\left(t\right)& \le (\tau g(a,b)\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(t\right),0\right)-\mu -\frac{1}{2}{\left({\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)\right)}^2\hfill \\ & +{\int }_{\mathcal{D}}\left(ln\left(1+{\Xi }_{10}\left(r\right)+{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(t\right)\right)-\left({\Xi }_{10}\left(r\right)+{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(t\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right))\mathrm{d}t\hfill \end{array}$$

$$\begin{array}{c}\hfill +{\displaystyle \underset{h=0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(t\right)\mathrm{d}{\mathfrak{A}}_1\left(t\right)+{\int }_{\mathcal{D}}ln\left(1+{\Xi }_{10}\left(r\right)+{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(t^{-}\right)\right){\mathfrak{M}}_1(\mathrm{d}t,\mathrm{d}r).\end{array}$$

(7)

Integrating from 0 to t, we obtain

$$\begin{array}{cc}\hfill \frac{1}{t}ln{\mathfrak{m}}_z\left(t\right)-\frac{1}{t}ln{\mathfrak{m}}_z\left(0\right)& \le \tau g(a,b)\frac{1}{t}{\int }_0^t\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(s\right),0\right)\mathrm{d}s-\mu -{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)\hfill \\ & +\frac{1}{t}{\int }_0^t{\int }_{\mathcal{D}}\left\{ln\left(1+\frac{{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(s\right)}{1+{\Xi }_{10}\left(r\right)}\right)-{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(s\right)\right\}{\varpi }_1\left(\mathrm{d}r\right)\mathrm{d}s\hfill \\ & +\frac{1}{t}\underset{=\varphi \left(t\right)}{\underbrace{\left({\int }_0^t\underset{\ell =0}{\sum ^L}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(s\right)\mathrm{d}{\mathfrak{A}}_2\left(s\right)-\frac{1}{2}{\int }_0^t{\left(\underset{\ell =0}{\sum ^L}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(s\right)\right)}^2\mathrm{d}s\right)}}\hfill \\ & +\frac{1}{t}{\int }_0^t{\int }_{\mathcal{D}}ln\left(1+{\Xi }_{10}\left(r\right)\right){\mathfrak{M}}_1(\mathrm{d}s,\mathrm{d}r)\hfill \end{array}$$

$$\begin{array}{c}\hfill +\frac{1}{t}{\int }_0^t{\int }_{\mathcal{D}}ln\left(1+\frac{{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(s^{-}\right)}{1+{\Xi }_{10}\left(r\right)}\right){\mathfrak{M}}_1(\mathrm{d}s,\mathrm{d}r).\end{array}$$

(8)

We set ${\displaystyle \mathcal{A}\left(t\right)={\int }_0^t{\int }_{\mathcal{D}}ln\left(1+{\Xi }_{10}\left(r\right)\right){\mathfrak{M}}_1(\mathrm{d}s,\mathrm{d}r)}$. Then, the quadratic variation of $\mathcal{A}\left(t\right)$ is equal to ${\displaystyle t{\int }_{\mathcal{D}}{\left(ln\left(1+{\Xi }_{10}\left(r\right)\right)\right)}^2{\varpi }_1\left(\mathrm{d}r\right)}$. Hence, we obtain $\frac{1}{t}\mathcal{A}\left(t\right)\to 0a.s.$, (as $t\to \infty$). To proceed, we use the exponential inequality for martingales, which implies that

$$\begin{array}{cc}\hfill \mathbb{P}\{{\displaystyle \underset{t\in [0,T_1]}{sup}}& ({\int }_0^t{\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(s\right)\mathrm{d}{\mathfrak{A}}_1\left(s\right)-\frac{1}{2}{\alpha }_1{\int }_0^t{\left({\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(s\right)\right)}^2\mathrm{d}s\hfill \\ & -{\int }_0^t{\int }_{\mathcal{D}}\left(\left(\frac{{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(s\right)}{1+{\Xi }_{10}\left(r\right)}\right)+ln\left(1+\frac{{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(s\right)}{1+{\Xi }_{10}\left(r\right)}\right)\right){\varpi }_1\left(\mathrm{d}r\right)\mathrm{d}s\hfill \\ & +{\int }_0^t{\int }_{\mathcal{D}}ln\left(1+\frac{{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(s^{-}\right)}{1+{\Xi }_{10}\left(r\right)}\right){\mathfrak{M}}_1(\mathrm{d}s,\mathrm{d}r))>\frac{2ln{T}_1}{{\alpha }_1}\}\le T_1^{-2},\hfill \end{array}$$

for all $0<{\alpha }_1<1$ and $T_1>0$. Due to Borel-Cantelli lemma, we confirm the existence of $T_{1,\omega }=T_1\left(\omega \right)$, $\forall \omega$ in $\Omega$ such that the inequality

$$\begin{array}{cc}\hfill & {\int }_0^t{\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(s\right)\mathrm{d}{\mathfrak{A}}_1\left(s\right)+{\int }_0^t{\int }_{\mathcal{D}}ln\left(1+\frac{{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(s^{-}\right)}{1+{\Xi }_{10}\left(r\right)}\right){\mathfrak{M}}_1(\mathrm{d}s,\mathrm{d}r)\hfill \\ & \le \frac{2ln{T}_1}{{\alpha }_1}+\frac{1}{2}{\alpha }_1{\int }_0^t{\left({\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(s\right)\right)}^2\mathrm{d}s+{\int }_0^t{\int }_{\mathcal{D}}\left(\left(\frac{{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(s\right)}{1+{\Xi }_{10}\left(r\right)}\right)+ln\left(1+\frac{{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(s\right)}{1+{\Xi }_{10}\left(r\right)}\right)\right){\varpi }_1\left(\mathrm{d}r\right)\mathrm{d}s,\hfill \end{array}$$

holds for all $T_1\ge T_{1,\omega }$ and $T_1-1<t\le T_1$ a.s. Under this setup, we can show that

$$\begin{array}{cc}\hfill & \frac{1}{t}\varphi \left(t\right)+\frac{1}{t}{\int }_0^t{\int }_{\mathcal{D}}ln\left(1+\frac{{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(s^{-}\right)}{1+{\Xi }_{10}\left(r\right)}\right){\mathfrak{M}}_1(\mathrm{d}s,\mathrm{d}r)\hfill \\ & \le \frac{2ln{T}_1}{{\alpha }_{1}t}+\frac{{\alpha }_1}{2t}{\int }_0^t{\left({\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(s\right)\right)}^2\mathrm{d}s-\frac{1}{2t}{\int }_0^t{\left({\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(s\right)\right)}^2\mathrm{d}s\hfill \\ & +\frac{1}{t}{\int }_0^t{\int }_{\mathcal{D}}\left(\left(\frac{{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(s\right)}{1+{\Xi }_{10}\left(r\right)}\right)-{\Xi }_{11}\left(r\right){\mathfrak{m}}_z\left(s\right)\right){\varpi }_1\left(\mathrm{d}r\right)\mathrm{d}s\hfill \\ & \le \frac{2ln{T}_1}{{\alpha }_1(T_1-1)}-\frac{1}{2}(1-{\alpha }_1)\frac{1}{t}{\int }_0^t{\left({\displaystyle \underset{\ell =0}{\sum ^L}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{\ell }\left(s\right)\right)}^2\mathrm{d}s\hfill \\ & \le \frac{2ln{T}_1}{\alpha (T_1-1)}-\frac{1}{2}(1-{\alpha }_1){{\rm Y}}_{10}^2.\hfill \end{array}$$

We take the superior limit on both sides of (8), then

$$\begin{array}{cc}\hfill & {\displaystyle \underset{t\to \infty }{lim\; sup}}\frac{1}{t}ln{\mathfrak{m}}_z\left(t\right)\hfill \\ & \le \tau g(a,b){\displaystyle \underset{t\to \infty }{lim}}\frac{1}{t}{\int }_0^t\mathcal{Q}\left({\mathfrak{n}}_{\circ }\left(s\right),0\right)\mathrm{d}s-\mu -{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)+{\displaystyle \underset{t\to \infty }{lim}}\frac{1}{t}\varphi \left(t\right)\hfill \\ & \le \tau g(a,b){\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)-\mu -{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)+{\displaystyle \underset{T_1\to \infty }{lim}}\frac{2ln{T}_1}{{\alpha }_1(T_1-1)}-\frac{1}{2}(1-{\alpha }_1){{\rm Y}}_{10}^2\hfill \\ & =\tau g(a,b){\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)-\mu -{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)-\frac{1}{2}(1-{\alpha }_1){{\rm Y}}_{10}^{2}a.s.\hfill \end{array}$$

Based on the arbitrariness of ${\alpha }_1\in (0,1)$, one obtains by letting ${\alpha }_1\to 0^{+}$ that

$$\begin{array}{c}\hfill {\displaystyle \underset{t\to \infty }{lim\; sup}}\frac{1}{t}ln{\mathfrak{m}}_z\left(t\right)\le \underset{={\mathcal{S}}_0^{\mathfrak{m}}}{\underbrace{\tau g(a,b){\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)-\mu -\frac{1}{2}{{\rm Y}}_{10}^2-{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)}}<0a.s.\end{array}$$

Since the exponential disappearance of zooplankton implies its almost surely full eradication, then ${\displaystyle \underset{t\to \infty }{lim}}{\mathfrak{m}}_z\left(t\right)=0$ a.s. This completes the proof. □

4. Discussion

This section is devoted to the discussion of our results using numerical illustrations. We seek to verify that ${\mathcal{S}}_0^{\mathfrak{m}}$ is the acute sill of the model (3). Additionally, we probe the complex effect of loud noises on the long-term behavior of the zooplankton. As an instance of the $\mathcal{Q}$ function, we use the Beddington–DeAngelis interference function ${\displaystyle \mathcal{Q}\left({\mathfrak{m}}_p,{\mathfrak{m}}_z\right)=\frac{{\mathfrak{m}}_p}{1+0.5{\mathfrak{m}}_p+0.5{\mathfrak{m}}_z}}$. We apply the Euler–Maruyama method for jump-diffusion noise to numerically deal with the probabilistic model (3). By selecting the parameters values appearing in Table 1 (taken from [10]), we treat two scenarios of the marine predation under heavy fluctuations. We mention that the sill of our model with the Beddington–DeAngelis interference function is expressed by

$$\begin{array}{cc}\hfill {\mathcal{S}}_0^{\mathfrak{m}}& =\tau g(a,b){\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)-\mu -\frac{1}{2}{{\rm Y}}_{10}^2-{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right),\hfill \\ \hfill & \stackrel{\mathrm{Ergodicity}\mathrm{of}{\mathfrak{n}}_{\circ }}{\overbrace{=}}{\displaystyle \underset{t\to \infty }{lim}}{t}^{-1}{\int }_0^t\frac{\tau g(a,b){\mathfrak{n}}_{\circ }\left(s\right)}{1+0.5{\mathfrak{n}}_{\circ }\left(s\right)}\mathrm{d}s-\mu -\frac{1}{2}{{\rm Y}}_{10}^2-{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right).\hfill \end{array}$$

The probability density function ${\psi }^{{\mathfrak{n}}_{\circ }}$ is associated with a (non-local) Fokker–Planck equation, which is easy to approximate it through Monte Carlo simulations. Via ergodic result, we can also estimate ${\displaystyle {\displaystyle \underset{T\to \infty }{lim}}{T}^{-1}{\int }_0^T\frac{{\mathfrak{n}}_{\circ }\left(s\right)}{1+{\mathfrak{m}}_1{\rho }_{\circ }\left(s\right)}\mathrm{d}s}$, for a large time T. This last technique is the one that we will use in our simulations. Since the equation of ${\mathfrak{n}}_{\circ }$ is disturbed by hybrid fluctuations, said limit will be changed according to the magnitude of the intensities and, consequently, the sill will also be modified. As an example, we treat a special case of system (3) with cubic white noises and quadratic jumps. Specifically, we numerically deal with the following probabilistic system:

$$\begin{array}{ccc}& \left\{\begin{array}{c}{\displaystyle \mathrm{d}{\mathfrak{m}}_p\left(t\right)=\left(\frac{a{\mathfrak{m}}_p\left(t\right)}{{\eta }_1{a}^2+{\eta }_{2}a+{\eta }_3}\left(1-\frac{{\mathfrak{m}}_p\left(t\right)}{\mathcal{K}}\right)-\frac{{\psi }_h{\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z\left(t\right)}{1+0.5{\mathfrak{m}}_p\left(t\right)+0.5{\mathfrak{m}}_z\left(t\right)}{e}^{\left\{-\frac{1}{d_1}{(a-d_{2}b)}^2\right\}}\right)\mathrm{d}t+\mathrm{d}{\Sigma }_0\left(t\right),}\hfill \\ {\displaystyle \mathrm{d}{\mathfrak{m}}_z\left(t\right)=\left(\frac{\tau {\psi }_h{\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z\left(t\right)}{1+0.5{\mathfrak{m}}_p\left(t\right)+0.5{\mathfrak{m}}_z\left(t\right)}{e}^{\left\{-\frac{1}{d_1}{(a-d_{2}b)}^2\right\}}-\beta {\mathfrak{m}}_p\left(t\right){\mathfrak{m}}_z\left(t\right)-\mu {\mathfrak{m}}_z\left(t\right)\right)\mathrm{d}t+\mathrm{d}{\Sigma }_1\left(t\right),}\hfill \\ {\mathfrak{m}}_p\left(0\right)=2.5,{\mathfrak{m}}_z\left(0\right)=1.8,\hfill \end{array}\right.\hfill & \hfill \end{array}$$

(9)

where

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\mathrm{d}{\Sigma }_0\left(t\right)=\stackrel{\mathrm{Cubic}\mathrm{white}\mathrm{noise}}{\overbrace{{\displaystyle \underset{\ell =0}{\sum ^2}}{{\rm Y}}_{0\ell }{\mathfrak{m}}_p^{\ell +1}\left(t\right)\mathrm{d}{\mathfrak{A}}_0\left(t\right)}}+\stackrel{\mathrm{Quadratic}\mathrm{leaps}}{\overbrace{{\int }_{\mathcal{D}}{\displaystyle \underset{j=0}{\sum ^1}}{\Xi }_{0j}\left(r\right){\mathfrak{m}}_p^{j+1}\left(t_{-}\right){\mathfrak{M}}_0(\mathrm{d}t,\mathrm{d}r)}},\hfill \\ {\displaystyle \mathrm{d}{\Sigma }_1\left(t\right)={\displaystyle \underset{\ell =0}{\sum ^2}}{{\rm Y}}_{1\ell }{\mathfrak{m}}_z^{h+1}\left(t\right)\mathrm{d}{\mathfrak{A}}_1\left(t\right)+{\int }_{\mathcal{D}}{\displaystyle \underset{j=0}{\sum ^1}}{\Xi }_{1j}\left(r\right){\mathfrak{m}}_z^{j+1}\left(t_{-}\right){\mathfrak{M}}_1(\mathrm{d}t,\mathrm{d}r),}\hfill \\ {{\rm Y}}_{00}=0.17,{{\rm Y}}_{01}=0.1,{{\rm Y}}_{02}=0.001,\hfill \\ {{\rm Y}}_{10}=0.13,{{\rm Y}}_{11}=0.112,{{\rm Y}}_{12}=0.001,\hfill \\ {\Xi }_{00}\left(r\right)=0.1,{\Xi }_{01}\left(r\right)=0.01,\hfill \\ {\Xi }_{10}\left(r\right)=0.1,{\Xi }_{11}\left(r\right)=0.01.\hfill \end{array}\right.\end{array}$$

The associated auxiliary system is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle \mathrm{d}{\mathfrak{n}}_{\circ }\left(t\right)=\frac{a{\mathfrak{n}}_{\circ }\left(t\right)\left(1-\frac{{\mathfrak{n}}_{\circ }\left(t\right)}{\mathcal{K}}\right)}{{\eta }_1{a}^2+{\eta }_{2}a+{\eta }_3}\mathrm{d}t+{\displaystyle \underset{\ell =0}{\sum ^2}}{{\rm Y}}_{0\ell }{\mathfrak{n}}_{\circ }^{\ell +1}\left(t\right)\mathrm{d}{\mathfrak{A}}_0\left(t\right)+{\int }_{\mathcal{D}}{\displaystyle \underset{j=0}{\sum ^1}}{\Xi }_{0j}\left(r\right){\mathfrak{n}}_{\circ }^{j+1}\left(t_{-}\right){\mathfrak{M}}_0(\mathrm{d}t,\mathrm{d}r),}\hfill \\ {\mathfrak{n}}_{\circ }\left(0\right)=2.5.\hfill \end{array}\right.\end{array}$$

(10)

Practically, we have the following two scenarios.

4.1. Scenario 1: Stationarity and Permanence of Zooplankton

By selecting the parameters values from Table 1 (Test 1) and choosing a sufficiently large number $T>0$, we obtain

$$\begin{array}{c}\hfill {\mathcal{S}}_0^{\mathfrak{m}}={\displaystyle \underset{t\to \infty }{lim}}{t}^{-1}{\int }_0^t\frac{\tau g(a,b){\mathfrak{n}}_{\circ }\left(s\right)}{1+0.5{\mathfrak{n}}_{\circ }\left(s\right)}\mathrm{d}s-\mu -\frac{1}{2}{{\rm Y}}_{10}^2-{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)=0.1083>0.\end{array}$$

Consequently, it follows from Theorem 1 that there exists a unique ergodic steady distribution of model (9). This signifies that the zooplankton is still present in the marine food chain. In Figure 4, we depict the permanence phenomenon by drawing trajectories and sketching the experimental two-dimensional density of $({\mathfrak{m}}_p,{\mathfrak{m}}_z)$.

4.2. Scenario 2: Disappearance of the Zooplankton

Now, we choose parameters values from Table 1 (Test 2) to move from persistence to the case of extinction. Then

$$\begin{array}{c}\hfill {\mathcal{S}}_0^{\mathfrak{m}}={\displaystyle \underset{t\to \infty }{lim}}{t}^{-1}{\int }_0^t\frac{\tau g(a,b){\mathfrak{n}}_{\circ }\left(s\right)}{1+0.5{\mathfrak{n}}_{\circ }\left(s\right)}\mathrm{d}s-\mu -\frac{1}{2}{{\rm Y}}_{10}^2-{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right)=-0.0074<0.\end{array}$$

It follows from Theorem 1 that the zooplankton ${\mathfrak{m}}_z$ in the marine food chain go to extinction with probability one and the phytoplankton ${\mathfrak{m}}_p$ persists. Figure 5 is the corresponding numerical simulation diagram.

4.3. The Influence of Cubic White Noise and Quadratic Jumps on the Long-Behavior of the Zooplankton

In this subsection, we numerically reveal that cubic diffusion and quadratic jumps have significant effects on the ecosystem dynamics. We take the fixed deterministic coefficients from Table 1 (Test 1) and the random intensities from Scenario 1 with varying cubic amplitudes. First we choose, $({{\rm Y}}_{02},{{\rm Y}}_{02})=(0.01,0.01)$. From Figure 6, we show that both phytoplankton and zooplankton persist. Now, we increase the values of ${{\rm Y}}_{02}$ and ${{\rm Y}}_{02}$ to $0.02$, we note that we are always in the persistence state but with a slight change in the form of the density function. However, when we choose $({{\rm Y}}_{02},{{\rm Y}}_{02})=(0.035,0.035)$, we get the extinction case. In other words, a dynamic transition occurs and a slight increase in cubic intensities completely changes the long-term behavior of the ecosystem. This dynamic bifurcation also occurs if the quadratic jump intensity is raised. From Figure 7, we conclude that heavy jumps lead to the disappearance of zooplankton. These huge fluctuations may lead to several problems of predation opportunities, which may directly limit the survival of zooplankton in the marine food web.

4.4. Long-Rung Modal Bifurcation (LRM-Bifurcation)

In the previous paragraph of this section, we numerically investigated the long-run bifurcation (LR-bifurcation). In this part, we show the long-rung modal bifurcation (LRM-bifurcation), which mainly depicts the geometric variations in the form of the steady probability density function associated with the dynamical system. Specifically, we numerically analyze the influence of noises on the form of the probability density function. Figure 6 and Figure 7 suggest that the intensities of noises have a significant role in changing the shape of the stationary distribution associated with model (3). Concerning the exact expression of the probability density function of a stochastic system (3), we mention that this density obeys a non-local Fokker–Planck equation, difficult to solve analytically. Alternatively, we can obtain estimates of the probability density function through some numerical methods. So, numerically, from Figure 6 and Figure 7, we conclude that its support and shape are changed at certain noise intensity values.

5. Conclusions

The mathematical formulation carries out a pivotal role in exploring the long-run behavior of the population in ecology. The regularly used systems provide deterministic predictions, that is to say, a strict attitude of the system studied, thus ignoring some environmental variations. We group these unpredictable variations under the name of stochasticity (or randomness), and the present study is devoted to the analysis of a marine ecological system under heavy stochasticity. The non-linearity and the complexity of the fluctuations pushed us to consider a general form of the probabilistic part. By assuming mutual interference between phytoplankton ${\mathfrak{m}}_p$ and zooplankton ${\mathfrak{m}}_z$, we have proposed a general ecological model for marine predation. Furthermore, we have unified the conditions of the stationarity and extinction by providing the following acute threshold value:

$$\begin{array}{c}\hfill {\mathcal{S}}_0^{\mathfrak{m}}=\tau g(a,b){\int }_0^{\infty }\mathcal{Q}\left(x,0\right){\psi }^{{\mathfrak{n}}_{\circ }}\left(\mathrm{d}x\right)-\mu -\frac{1}{2}{{\rm Y}}_{10}^2-{\int }_{\mathcal{D}}\left({\Xi }_{10}\left(r\right)-ln\left(1+{\Xi }_{10}\left(r\right)\right)\right){\varpi }_1\left(\mathrm{d}r\right).\end{array}$$

Specifically,

• The condition of the ergodicity is ${\mathcal{S}}_0^{\mathfrak{m}}>0$.
• The condition of the zooplankton disappearance is ${\mathcal{S}}_0^{\mathfrak{m}}<0$.

Numerically, we have treated a specific example of cubic white noise and quadratic jumps. Our main objective was to verify the clarity of the proposed threshold. Moreover, we pointed out that when increasing the order of the noises, the model changes its asymptotic behavior. That is to say, strong fluctuations have a passive influence on the survival of zooplankton.