Stability and Dynamical Analysis of a Biological System

and Applied Analysis 3


Introduction
The upper layer of the ocean contains large volumes of drifting plankton, which can be divided into phytoplankton and zooplankton.Phytoplankton is the autotrophic component of the plankton community, which is consumed by zooplankton, most of which are too small to observe individually with the naked eye.Zooplankton, which are heterotrophic organisms in oceans, are also mostly invisible to the naked eye.Therefore, it is difficult and expensive to quantify plankton directly.Plankton not only play an important role in the marine system because they are at the bottom level of the food chain that supports commercial fisheries, but also play important roles in the cycling of many chemical elements, such as carbon, which may affect climate change [1].Furthermore, when plankton such as blue algae and dinoflagellates are present in large concentrations, the water appears to be discolored or murky, which is known as a red tide, and this can result in the death of marine and coastal species of fish, mammals, and other organisms [1].Thus, analyzing the dynamics of plankton using mathematical models is beneficial for understanding the features of plankton populations, which have enormous economic and ecological value.
However, the mechanism that leads to the occurrence of red tides is still an unsolved issue.Many models and theories have been proposed by mathematicians and ecologists to explain this phenomenon, but a general and correct explanation still remains a distant goal [2][3][4][5].
The popular mathematical model called a Gause-type predator-prey model is used to consider the phytoplanktonzooplankton interaction in the following form: where  = () and  = () are the population densities or biomass of phytoplankton and zooplankton at time , respectively.(, ) describes the intrinsic per capita growth rate of the phytoplankton, which may be a logistic growth function, exponential growth function, or other functions, where  is known as the environmental carrying capacity.() is the per unit-predator consumption rate of prey, which is commonly called the functional response.Some conventional forms of functional response include Holling types I, II, III, and IV and Ivlev type [6][7][8]. (0 <  < 1) is the conversion coefficient and  represents the per capita predator mortality, which is assumed to have a linear form, although other forms are possible [9,10].The global dynamics of model (1) with a logistic growth rate have been studied during the last three decades based on theoretical analysis and numerical simulations, and many results have been reported [11][12][13][14][15].In recent years, the Allee effect has been the focus of increasing interest and it is recognized to be an important phenomenon in many fields of ecology and conservation biology by more and more people [16][17][18][19][20][21].The Allee effect is named after W.C. Allee [22] and it describes a positive correlation between the density or number of population and individual fitness of population [16].Standard population models assume that the fitness of population increases as the population density or size declines [11-15, 23, 24], whereas Allee effect states that when a population is below a critical density or size, the population cannot sustain itself and this leads to extinction.Thus, the Allee effect increases the likelihood of extinction [25].Stephens et al. distinguished between a component Allee effect and a demographic Allee effect [17].However, conservation biologists are usually more interested in the demographic Allee effect because it ultimately governs the probability of the extinction or recovery of populations with low abundances [16].
Very recent ecological research has shown that two or more Allee effects can act on a single population simultaneously, which is known as the multiple (double) Allee effect [26,27].
There are many ways of describing the Allee effect [28], including the following differential equation: where  describes the growth rate and  is an auxiliary parameter where  > 0,  +  > 0. Indeed, it is considered that  represents the size of a fertile population and  is the nonfertile population, such as juvenile or oldest individuals [29].In this case,  (− <  < ) is called the Allee threshold because when the population density or size is below this threshold, the population is destined for extinction.When  > 0, (2) describes a strong Allee effect [8,30,31].In this case, the population growth rate decreases if the population size is below the threshold  and the population goes to extinction [29].In addition, (2) describes a weak Allee effect [29,[31][32][33] for  < 0.
It is obvious that (2) is equal to González-Olivares et al. [29] state that (3) describes a double Allee effect, that is, once in the factor () =  −  and the second time in the term () = /( + ) [34,35].In a marine environment, the plankton populations tend to move in horizontal and vertical directions due to the strong water current.This movement is usually modeled by a reaction-diffusion equation.In this study, we consider the following reaction-diffusion model with constant diffusion coefficient as well as a strong Allee effect in different spatial locations within a fixed smooth bounded domain Ω ∈   .We assume that the response function of the zooplankton follows the law of mass action [15] where  (0 <  < ) is the Allee threshold,  1 ,  2 are the diffusion coefficients of phytoplankton and zooplankton, respectively, Δ is the Laplacian operator, and  is the spatial habitat of two species, and we assume that the system is a close ecosystem and with a no-flux boundary condition.This paper is structured as follows.In Section 2, we analyze the basic dynamics of (4) including estimates of the solution and the local and global stability of equilibria.In Section 3, we provide the analysis of the Hopf bifurcation and the steady state bifurcation.A brief discussion and summary are given in Section 4.
From a biological viewpoint, this implies that if the initial population density is below the threshold , the phytoplankton become extinct so the zooplankton would become extinct.
The local stability of the steady state solutions can be analyzed as follows.
(4) (, V  ) is locally asymptotically stable for  <  <  and is unstable for  <  < , where Proof.The linearization of (4) at solution (, ) can be expressed as where According to Theorems 5.1.1 and 5.1.3from [37], we know that if all the eigenvalues of the operator  have negative real parts, then the solution (, ) is asymptotically stable; if there is at least one eigenvalue with a positive real part, then the solution (, ) is unstable; if some eigenvalues have zero real parts, then the stability cannot be determined using this method.
Let   ( = 0, 1, 2, . ..) be the eigenvalues of −Δ on Ω under a homogeneous Neumann boundary condition and Thus, it is known that  is an eigenvalue of  if and only if  is the eigenvalue of the matrix   = −   +  (,) for some  ≥ 0.

6
Abstract and Applied Analysis By taking the time derivative of , we have Due to the Neumann boundary condition, it can easily be derived that Further, If we choose  2 > 0 arbitrarily and  1 =  2 , then we can obtain Therefore, if  +  +  −  < 0, then / ≤ 0 and / = 0 if  = ,  = V  .This completes the proof.

Hopf Bifurcation Analysis.
In this section, we mainly analyze the properties of the Hopf bifurcation for system (4).
According to [38], we know that a Hopf bifurcation point  must satisfy the following conditions.

Lemma 8.
Let Ω be a bounded Lipschitz domain in   , and let  ∈ (Ω × ).If  ∈  1,2 (Ω) is a weak solution of the inequalities and if there is a constant  such that (, ) < 0 for  > , then  ≤  a.e. in Ω.
From Lemma 8, it can easily be derived that all nontrivial solutions of equation satisfy 0 ≤ () ≤ .

Discussion
Reaction-diffusion phytoplankton-zooplankton models with Allee effects have been studied extensively in recent years.In this study, we rigorously considered a Gause-type predatorprey model with a double Allee effect on prey, which was formulated as (4).It is known that the predator-prey model with the most usual form of Allee effect has a unique limit cycle, but the existence of two limit cycles was proved by González-Olivares et al. [29] with a double Allee effect.Thus, the double Allee effect produces different results with different mathematical expressions.
The paper [15] found that the system without Allee effect was always stable and without fluctuations, but in this paper the results of the stability of the equilibrium and the bifurcation analysis based on a rigorous theoretical analysis show that this system has complex spatiotemporal dynamics: for  0 () ≤ , the phytoplankton is destined to become extinct and leads to the extinction of zooplankton; after considering  the strong Allee effect in phytoplankton, extinction for both species is always a locally stable equilibrium.But for  > , which is the condition in which the interior equilibrium exists, the interior equilibrium is globally stable in some case and there always exist some other spatiotemporal patterns in other cases (Figure 3).Overall, our results indicate that the impact of the Allee effect increases the spatiotemporal complexity of the system.The mathematical form which expresses the double Allee effect has a strong impact of the dynamics of system.Thus, we think it is important for ecologists to be aware of the difference of the selection on the forms of Allee effect.
The limitations of our study are that we only consider a simplest phytoplankton-zooplankton interaction and the special formalization to describe the Allee effect.What is more is that, compared with the ODE dynamics, the results shown here are still coarse.Therefore, further research is still needed to elaborate a general theory on the influence of this ecological phenomenon.

Figure 2 :
Figure 2: Solution to ((, ), (, )) for (4).Phytoplankton is in (a) and zooplankton is in (b).The parameter values are the same as those given in Figure 1.