Dynamics Analysis of an Aquatic Ecological Model with Temperature Effect

Under the control framework of algae bloom in eutrophic lakes and reservoirs based on biological manipulation, the temperature variable is introduced into ecological modeling to show that it is a necessary condition for the rapid occurrence of algal blooms, and an aquatic ecological model with temperature effect is proposed to describe dynamic relationship between algae and biological manipulation predator. The mathematical theory work mainly investigates the existence and stability of some equilibrium points and some critical conditions for the occurrence of transcritical bifurcation and Hopf bifurcation. The numerical simulation mainly shows the dynamic evolution process of bifurcation dynamics, which can not only verify the validity and feasibility of these theoretical works but also analyze the influence of some key parameters on dynamic behavior evolution. Furthermore, It is worth emphasizing that temperature plays an important role in the coexistence of algae and biological manipulation predators. Moreover, the coexistence mode of algae and biological manipulation predators is discovered by means of dynamic bifurcation evolution. Finally, it is hoped that these research results can provide some reference for the study of aquatic ecosystems.


Introduction
As we all know, the temperature is an important environmental factor that determines the growth of the algal population, is one of the important factors that affect the growth of algal cells, the composition and content of biological ma-How to cite this paper: Yang cromolecules in cells, and is also a key ecological factor that affects the growth, reproduction and population succession of aquatic plants [1] [2]. Furthermore, suitable temperature is a necessary condition for algal bloom outbreaks and also an important environmental factor for the replacement of the dominant algal population [3] [4]. Thus, it is important to study the effect of temperature on algal growth and provide the theoretical basis for the prevention and treatment of water eutrophication.
Algae is a ubiquitous photosynthetic organism; the algal growth rate can increase with increasing temperature up to a certain limit; this is because that temperature strongly influences the cellular chemical composition and uptake of nutrients and also plays a significant role in algal growth [5]. The paper [6] pointed out that the ability to model algal productivity under transient conditions of temperature was critical for assessing the profitability and sustainability of full-scale algae cultivation outdoors. The paper [7] investigated the effect of temperature and irradiance on the growth and reproduction of the green macroalga and gave that a suitable temperature range over 21˚C -29˚C was more favorable for growth and reproduction. The paper [8] proposed that Microcystis aeruginosa had a wide range of adaptation to temperature, and the optimal growth temperature was 25˚C -30˚C. The paper [9] showed that the temperature condition that was most conducive to algal growth was 25˚C, and the optimal condition for algal toxins release was 28˚C. The paper [10] inquired into the effects of temperature on the germination of micro-propagules via laboratory experiments and indicated that sea temperature played a significant role in the germination of green algae. The paper [11] sought to elucidate the effects of temperature on algal growth rates, biomass accumulation, fatty acid production and composition and pointed out that temperature significantly impacted the overall productivity of algal biofuel systems by influencing species growth rates and fatty acid production.
The paper [12] studied the effects of different temperatures and illumination time on algal growth and obtained that the value of the algal growth rate constant was reduced to 0.812d −1 by lowering the water temperature to 16˚C. In conclusion, the algal population has strong adaptability to the environment, and temperature is one of the important ecological factors affecting algal growth.
In an aquatic ecosystem, the ecological effects between algae and biological manipulation predators are mutual, mainly including the harm of algal blooms to fish and the feeding and regulation of algae by biological manipulation predators [13]. The use of biological manipulation predators to control the excessive algal growth in eutrophic lakes was proposed and gradually attracted attention after the eutrophication of lakes became more and more common, and certain results in the control of cyanobacteria blooms have been achieved in some lakes at home and abroad [14] [15]. The paper [16] pointed out that silver carp completely eliminated cyanobacteria Microcystis by size and biovolume reduction. The paper [17] deemed that silver and bighead carps were just suitable for controlling cyanobacteria bloom not total algae biomass, and the application of  [18] believed that it was feasible to use silver carp and bighead carp to control Microcystis in eutrophic water. Furthermore, using ecological models to explore the dynamic relationship between algae and biological manipulation predators has also developed rapidly. The paper [19] showed that Microcystis aeruginosa aggregation could effectively control the dynamic feeding behavior of filter-feeding fish and provide shelter from predators. The paper [20] obtained that the filter-feeding fish population could be a crucial factor in controlling the proliferation of the algae population based on an algae-fish model. Therefore, fully understanding the ecological function relationship between algae and biological manipulation predator is a premise for better implementing algae control strategy with biological manipulation predators and ensuring algae control effect. In this paper, firstly, the temperature variable is introduced to build an aquatic ecological model which can describe the dynamic relationship between algae and biological manipulation predators. Secondly, mathematical theory works are implemented to obtain some critical threshold conditions by investigating the evolution process of some specific dynamic properties. Finally, the numerical simulation works are carried out to show the evolution process of dynamic properties and the influence mechanism of temperature variables. Generally speaking, the main purpose of this paper is to use an aquatic ecological model with temperature effect to explore the coexistence mode between algae and biological manipulation predators and reveal how temperature variable affect their dynamic relationships.

Aquatic Ecological Model
At present, the temperature has been regarded as a key factor for monitoring and predicting algal blooms. Furthermore, the reproduction rate of algae has an important relationship with temperature, and the appropriate temperature is a necessary condition for the rapid propagation and growth of the algae population. In order to better investigate the effect of temperature on the dynamic relationship between algae and biological manipulation predator, we will propose an aquatic ecological model with temperature effect, which can be described as follow: where ( ) N T and ( ) P T are density of algae population and biological manipulation predator (silver carp, bighead carp and anodonta woodiana elliptica) respectively. 1 r is maximum growth rate of algae population at reference temperature ref R (it is considered to be 25˚C). 1 k is the maximum environmental capacity for algae population. R represents actual temperature during the test run, which can be either a constant or a function of time t. min R is the lowest temperature value when the algal growth is 0, and it is generally considered to be 15˚C. 1 u is a respiratory rate, 1 m is a nonpredation-induced mortality of algae population, s is a sedimentation rate, 2 r is intrinsic growth rate of biological manipulation predator, 2 m is mortality rate of biological manipulation predator, e is energy conversion rate, a is capture rate of biological manipulation predator, b is a semi saturation coefficient, and α is assimilated food of catabolic loss during predation period.
For simplicity, we will replace the model (2.1) with the following variables: ( ) Then the model (2.1) can be rewritten as For model (2.2), we first discuss the existence and stability of all possible equilibrium points and explore the existence of a limit cycle with some key constraints. Then we give some critical conditions to demonstrate the occurrence of transcritical bifurcation and Hopf bifurcation. Finally, some numerical simulation results are implemented to verify the validity and feasibility of theoretical results. At the same time, through some numerical simulation results, the influence of temperature on dynamic relationship between algae and biological manipulation predators will be explored, and then ecological evolution significance represented by bifurcation dynamic evolution behavior is also given.

Equilibrium Points and Their Stability
In this section, some preliminary results shall be presented, including the existence and stability of all possible equilibrium points of the model (2.2).
We will consider the following equation to explore all possible equilibrium points of the model (2.2), It is easy to acquire that the model (2.2) always has trivial equilibrium point , and a biological manipulation predator extinction equilibrium point ( ) In view of the biological significance and the characteristics of the model (2.2), the interior equilibrium points are conditional.
In order to explore the existence and stability of all possible equilibrium points of the model (2.2), we will gradually give the following Theorems 1 -6.
According to the bi- This ends the proof.
Theorem 2. The model (2.2) has a boundary equilibrium point The eigenvalues of and 0 E is a saddle. This completes the proof.
Theorem 3. The model (2.2) always has a boundary equilibrium point ( ) Proof: The Jacobian matrix of the model (2.2) evaluated at 1 E is The eigenvalues of Proof: This theorem can be derived by flow analysis. When 2) has no internal equilibrium point, and has only two boundary equilibrium points 0 E and 1 E . It is easy to know that 0 E is always unstable and is a saddle, its unstable manifold is x-axis. Thus, we can divide the positive quadrant into the following three regions, Proof: The Jacobian matrix of the model (2.2) evaluated at E * is given by The determinant and the trace of matrix E J * are given by is globally asymptotically stable.
Proof: To prove the global asymptotic stability of the internal equilibrium point ( ) , E x y * * * , the following Lyapunov function was constructed: Based on the basic properties of the function, it can be concluded that ( ) , V x y is continuous for all 0 x > and 0 y > , and is computationally available: Then, we can obtain According to the above formula, it is obvious that the internal equilibrium point Proof: From the conclusion of the Theorem 5, we know that the internal equilibrium point E * may be an unstable focus. Now we will prove Theorem 7 by constructing an invariant region Ω , which consists of the following line 1 2 , L L , and , x y axis, It is easy to verify that < when M is a sufficiently large positive number,

Bifurcation Analysis
It is well known that the evolution process of bifurcation dynamics has important biological significance in the process of population dynamics. Therefore, we will explore some bifurcation dynamics behaviors of the model (2.2) and give some threshold conditions for specific bifurcation dynamics of the model (2.2).
H. Yang et al.

Transcritical Bifurcation
Here we will prove that the model ( Thus, based on Sotomayor's theorem we can deduce that the model (2.2) undergoes a transcritical bifurcation as the parameter p passes through a critical threshold TC p . This completes the proof.

Hopf Bifurcation
From the analysis of Theorem 5, we know that if  α β α β ∆ = − > If 0 l < , the limit cycle is stable; if 0 l > , the limit cycle is unstable. However, the expression for Lyapunov number l is rather cumbersome; we cannot directly judge the sign it, so we will give some numerical simulation results in Section 5.
Based on the mathematical theory, the existence and stability threshold conditions of all possible equilibrium points are given, and the critical conditions for inducing specific bifurcation dynamics of the model (2.2) are analyzed, which can provide a theoretical basis for subsequent numerical simulation work. Furthermore, it should also be emphasized that the key parameter p can seriously affect dynamic evolution characteristics of the model (2.2).

Numerical Simulations and Results
Now, we will investigate dynamic properties of the internal equilibrium point  . It is easy to find from Figure 1(b) that the density of algae x * increases with the increase of parameter p-value, and the growth is slow in the early stage and extremely fast in the later stage, which implies that when the value of parameter p exceeds a certain critical threshold, the density of algae x * will become larger and larger. Furthermore, it is obvious to know from Figure   1(a) that when the value of parameter p exceeds a certain critical threshold and is determined, the density of biological manipulation predator y * is a concave function of the density of algae x * , and y * can get a maximum value when the value of x * is ( ) , E x y * * * is stable, which means that algae and biological manipulation predators can form a constant steady-state coexistence mode. Furthermore, it is obvious to know from Figure 2(b) that model (2.2) has a limit cycle, which indicates that algae and biological manipulation In order to better understand how the value of parameter p affects the dynamic behavior evolution of the model (2.2), we give a bifurcation diagram of the model (2.2) in Figure 3. It can be seen clearly from Figure 3 and Figure 4 that if the value of parameter p is larger than 6.75 TC p = , the model (2.2) has only two boundary equilibrium points is unstable, which implies that biological manipulation predator will eventually approach extinction and algae will eventually approach the maximum biomass state, that is to say, biological manipulation predator and algae cannot form a final coexistence mode. If the value of parameter p is be-  , E x y * * * loses stability and a stable limit cycle appears. In other words, the model (2.2) undergoes a Hopf bifurcation; the numerical dynamic evolution process is shown in Figure 5.
Therefore, it is worth pointing out that the Hopf bifurcation can produce a periodic oscillation coexistence mode between biological manipulation predator and algae. Thus, the numerical simulation results not only prove the validity and feasibility of the theoretical derivation, but also directly show that the value of key parameter p seriously affects bifurcation dynamics evolution characteristics of the model (2.2).
It is obvious to find from   These results not only show that the model (2.1) has experienced a Hopf bifurcation as the value of parameter R increases, but also indicate that the increase in temperature is conducive to rapid growth of biological manipulation predators.
Based on the numerical simulation results, can clearly indicate that the results of theoretical derivation are effective and feasible. Furthermore, it should also be emphasized that temperature not only affects the dynamic evolution characteristics of the model (2.1), but also affects the biomass level of biological manipulation predator. Moreover, the model (2.2) has specific bifurcation dynamic behaviors (transcritical bifurcation and Hopf bifurcation) under the influence of the value of key parameter p; these two bifurcation dynamics behaviors lead to a constant steady-state coexistence mode and a periodic oscillation coexistence mode of algae and biological manipulation predator respectively.

Conclusions
Under the conceptual framework of biological control of cyanobacteria in eu-Journal of Applied Mathematics and Physics trophic lakes and reservoirs, based on the fact that temperature is an extremely important factor in determining ecology, which has an important relationship with algae proliferation rate, an aquatic ecological model with temperature effect is proposed to explore the coexistence modes of algae and biological manipulation predator and investigate how temperature affects their dynamic evolution.
Suppose temperature parameter R is a constant variable, which can approximately describe ecological culture system of algae and biological manipulation predator under laboratory conditions if temperature parameter R is a periodic function variable, which mainly represents the natural ecosystem of algae and biological manipulation predators in a naturally eutrophic lake.
Based on dynamic population theory, some threshold conditions are given to guarantee the existence and stability of all possible equilibrium points, and some critical conditions for the occurrence of transcritical bifurcation and Hopf bifurcation are also deduced. Furthermore, some key parameters affecting the dynamic evolution characteristics of the model (2.2) are found through theoretical derivation and numerical simulation. All in all, these results are the theoretical basis for subsequent numerical simulation work and abstractly display the influence of some parameters on the dynamic evolution of the model (2.2).
Through the numerical simulation test on dynamic behaviors of the model (2.1), the influence mechanism of temperature on the stable succession of aquatic ecosystem is discovered in Figure 6 and Figure 8, the coexistence mode of algae and biological manipulation predator can change from a constant steady-state mode to a periodic oscillation mode with the temperature increasing gradually, which also indirectly indicates that the appropriate temperature range is one of the key factors for algae and biological manipulation predator to form a stable coexistence mode, and the periodic oscillation coexistence mode is more favorable to control the growth rate of algae population by biological manipulation.
Based on the bifurcation dynamics evolution analysis of the model (2.2), it is worth pointing out that transcritical bifurcation can induce the appearance of the internal equilibrium point ( ) , E x y * * * , which represents the coexistence of algae and biological manipulation predator in a periodic oscillation mode, and completely changes the dynamic coexistence nature of algae and biological manipulation predator. Furthermore, when the value of control parameter p decreases and falls below a critical threshold, the coexistence mode of algae and biological manipulation predator has changed fundamentally again, periodic oscillation coexistence mode will replace the constant steady-state coexistence mode through a Hopf bifurcation. These results directly show that the value of key parameter p plays an important role in the bifurcation dynamic behavior evolution of the model (2.2). In general, some theoretical and numerical results obtained in this paper can provide a certain theoretical basis for the formation of a healthy and stable aquatic ecosystem and also provide certain numerical support for the feasibility of biological manipulation technology.
In the follow-up research works, firstly, we will introduce Arrhenius exponen-Journal of Applied Mathematics and Physics tial temperature function and partial normal distribution temperature function into ecological modeling and investigate the impact of different temperature function manifestations on the dynamic behavior of the model (2.1). Secondly, we will continue to deepen the environmental impact factors of such aquatic ecological models and then further explore the influence of various environmental factors on the dynamic relationship between algae and biological manipulation predators. Finally, we will further explore the dynamic pattern behavior of the model (2.1) with the help of these papers [25] [26] [27]. In a word, all these results are expected to be useful in studying the dynamic behavior of aquatic ecosystems.