Research paper
Stability analysis and pattern selection of a plankton system with nonlocal predation

https://doi.org/10.1016/j.cnsns.2022.106969Get rights and content

Highlights

  • An algal system with nonlocal predation is modeled by integro-differential equations.

  • The Turing bifurcation and the weak/strong Turing-Hopf instability are discussed.

  • Weak nonlinear analysis is applied to the nonlocal system for pattern selection.

  • The more dispersedly the zooplankton hunt, the higher the concentration of plankton.

Abstract

A plankton micro-ecosystem composed of phytoplankton (prey) and zooplankton (predators with nonlocal predation) is researched. First, the Hopf bifurcation, the Turing bifurcation, and the strong/weak Turing-Hopf (TH) instability are gained with the help of linear stability analysis. Then, by the weakly nonlinear analysis, the amplitude equations for the pattern selection are calculated and show some types of patterns: spot patterns, stripe patterns and mixed patterns of the two. At last, the numerical simulations not only verify the analysis but also show the effect of the hunter function on the pattern selection. The results show that if the zooplankton hunt more dispersedly or diffuses faster, the plankton would gather and form denser patches. Besides, our analysis extends the application of the weakly nonlinear analysis and applies it to the nonlocal reaction-diffusion models.

Introduction

Plankton, one of the most important species, if not the most important, are the basis of the marine food webs [1]. They release oxygen, supporting the tremendous marine ecosystem, so the evolution of plankton deeply affects the marine and terrestrial biosphere [2], [3]. The plankton are divided into phytoplankton (tiny plants) and zooplankton (tiny animals that feed on phytoplankton) [4]. Phytoplankton, plants in water, absorb sunlight and nutrients to produce carbohydrates and oxygen. If the conditions are met, they will increase rapidly, leading to algae blooms [5], [6]. Then the zooplankton will rise and gradually inhibit the increase of phytoplankton [7], [8], [9]. Some phytoplankton produce toxins that can poison the zooplankton as well as other species to resist predation [10], [11]. Therefore, the interaction within the planktonic ecosystem is vital to humans.

Many scholars have devoted themselves to researching the properties of plankton, especially phytoplankton. There are two mainstreams: one is to study the physiological natures or ecological data of plankton, and the other is mathematical dynamics. On the first road, some scientists monitored phytoplankton changes in the ocean [12], some investigated energy and materials flow [13], [14], and some laboratories cultivated phytoplankton to study the effects of light and nutrients [15], [16]. What is more, the Latest AI techniques have been also applied to predict the evolution of phytoplankton [17], [18]. On the second road, the scholars have tried to explain and predict the variation of the population of the plankton through mathematical analysis. First of all, they should know the mathematical model depicting the interaction within the plankton community, and one of the most important relationships is the predation between zooplankton and phytoplankton. So some biologists studied the predation functions [19], [20], [21], but there is not an appropriate general functional response yet [22]. The bio-mathematicians further employed these predation functions to establish mathematical models of the plankton community containing phytoplankton and zooplankton, and then analyzed the systems’ dynamical behaviors. Some scholars treated the phytoplankton/zooplankton as a whole and described them through ordinary differential equations (ODE) [23], [24], [25], [26], and others further developed the models by adding nutrients, fishes or some other factors in the food chains [27], [28]. Naturally, some researchers have noticed the limitations of the ODE models and considered the spatiotemporal ones [29], [30], [31]. Several teams considered 3D models and studied the effects of depth, sunlight, sinking or some other factors [32], [33]. However, more researchers preferred the two-dimensional models. They ignored the depth and only studied the two-dimensional distribution of plankton [34], [35] because the drift layer of plankton in the surface water is too thin. As a result, the reaction-diffusion equations are often used to model the algae ecosystem containing phytoplankton and zooplankton.

Most of the previous functional responses are local, which implies that zooplankton can only hunt phytoplankton next to them. However, zooplankton randomly wander and prey in a sphere centering on one point, so they hunt in the radial domain, and the magnitude function of predation is radial. Besides, some zooplankton can use tools, such as feeding current and mucus net, to hunt in larger areas [36]. In a word, the nonlocal predation is closer to the reality. Therefore, it is necessary to study the reaction-diffusion model with nonlocal predation.

The motivation of this work is to analyze the stability of the plankton ecosystem and give the influence of nonlocal predation on pattern selection. Though nonlocal predation is common, few scholars have studied the plankton systems with nonlocal phenomenon. Nonlocal predation, as well as diffusion, affects the spatial stability of the plankton system. Some scholars have studied the influence of the nonlocal predation or the nonlocal competition on the stability of the predator-prey system, but they rarely involve the strong/weak TH instability as well as the pattern selection of the two-dimensional planktonic system. Therefore, the planktonic system with nonlocal predation is worth studying.

In this article, we provide a spatiotemporal model for the plankton communities with nonlocal predation. Some scholars recently created some nonlocal predation functions, such as nonlocal Holling II type functional responses [37], [38]. Inspired by their works, we build the nonlocal predation function based on the Arditi-Ginzburg type II functional response [19]. Besides, we also consider the allelopathy [10], [11]. Phytoplankton produce toxins and store them inside to kill the zooplankton which swallow them, so the poison function is in the same form as the predation function. The other terms remain the same as the general local model. Under such a nonlocal model, we analyze the stability of the integral-differential equations, get the Hopf bifurcation and the Turing bifurcation, and distinguish the strong/weak TH instability. Then, the weakly nonlinear analysis is applied to the model to obtain the conditions for various patterns of self-organization. We enlarge the tuning parameter, across and far away from the Turing bifurcation, and find that the patterns change in such an order: homogeneous steady state spot patterns mixed patterns stripe patterns. Finally, the simulations show the influence of the hunter function on the pattern formation.

This paper is organized as follows. In Section 2, we introduce the integral-differential equations to model the plankton community with nonlocal predation. Then, in Section 3, we get the Hopf bifurcation, Turning bifurcation, and the TH bifurcation, and distinguish the weak/strong TH instability. Besides, in Section 4, through the weakly nonlinear analysis, we get patterns of three kinds, such as spot patterns, stripe patterns and mixed patterns of the two. Finally, in Section 5, numerical simulations validate our analysis and give the effect of the nonlocal hunter function on the plankton system.

Section snippets

Spatiotemporal model

In this paper, we would consider a spatiotemporal planktonic model with nonlocal predation Pt(x)=rP(x)1P(x)KaP(x)ΦZ(x)ΦZ(x)+ahP(x)+D12P(x),x[0,L]×[0,L],Zt(y)=eZ(y)ΦaPΦZ+ahP(y)ηZ(y)e0Z(y)ΦaPΦZ+ahP(y)+D22Z(y),x[0,L]×[0,L],P(x1,0,t)=P(x1,L,t),P(0,x2,t)=P(L,x2,t),Z(x1,0,t)=Z(x1,L,t),Z(0,x2,t)=Z(L,x2,t),x1,x2[0,L],P(x,0)0,Z(x,0)0,x[0,L]×[0,L],with Φf(x)=ΩΦ(xs)f(s)ds,ΩΦ(s)ds=1.

The P,Z represent phytoplankton and zooplankton respectively, and the explanations of the model are

Equilibrium

If we ignore the diffusion terms and the nonlocal predation, and only consider the left parts, the plankton model will degenerate into ordinary local differential equations dPdt=rP(1P)aPZZ+P,dZdt=eaPZZ+PηZ.And through defining F=Pf,f(P,Z)=r(1P)aZZ+P,G=Zg,g(P,Z)=eaPZ+Pη, we could get dPdt=Pf(P,Z),dZdt=Zg(P,Z).Obviously, the solution to the plankton model (4) is positive for any non-negative nontrivial initial conditions, which is consistent with the real world [39]. Furthermore, we can

Weakly nonlinear analysis

When the reaction equations are stable but the reaction-diffusion equations Turing unstable, the amplitude equations of (2) can be calculated. In the beginning, we expand the equations around the homogeneous steady state (P,Z) with truncated Taylor series approximation. But before deducing the extension of (2), we ought to firstly define T(P,Z)=eaPZ¯+P.Its third-order Taylor series approximation is δTTPδP+TZδZ¯+12TPPδP2+2TPZδPδZ¯+TZZδZ¯2+16TPPPδP3+3TPPZδP2δZ¯+3TPZZδPδZ¯2+TZZZδZ¯3.

And the

Numerical example

In this section, the numerical simulations are performed in a two-dimensional space (15 × 15) with spatial interval Δx=Δy=0.1, as well as temple interval Δt=0.01, and all simulations employ periodic boundary conditions. The diffusion operator is discretized by a five-point finite difference method, the left part Euler difference, and the integral fixed-point interpolation with RΩ<0.3. The nonlocal hunter function is the famous Gaussian function (Fig. 2), and if the variances is ξ2=0.02, the

Conclusion

We consider a bidimensional plankton community with nonlocal predation. The functional response is assigned the nonlocal Arditi-Ginzburg type II, which is reconstructed from the local predation function. If a Dirac Delta function is set as the nonlocal hunting function Φ, the integro-differential diffusion equations will degenerate into local ones. We choose the bidimensional Gaussian function, a radical function, as the nonlocal predation kernel function, and the variance plays a vital role on

CRediT authorship contribution statement

Zhi-bin Liu: Conceptualization, Methodology, Software, Investigation, Formal analysis, Writing – original draft. Shutang Liu: Software, Funding acquisition, Supervision. Wen Wang: Writing – review & editing, Investigation, Conceptualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China-Shandong Joint Fund (No. U1806203) and the National Natural Science Foundation of China (Nos. 61533011).

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