Role of Photosynthesis Processes in the Dynamics of the Plant Community

Abstract

The dynamics of the main photosynthetic structures are studied by mathematical modeling methods in this work. Chlorophyll portion variability in phytoplankton and formation of energy-intensive substances in the process of photosynthesis underlie the models. These cellular components are considered in terms of their participation in the growth of specific biomass. Computational experiments are conducted to simulate various degrees of environmental friendliness. The corresponding functions are built in accordance with seasonal fluctuations throughout the year in the Far East region of Russia. The stability of model solutions in long-term dynamics is also investigated. The models are tested for biological adequacy, and their effectiveness is compared. For the model selected as a result of the comparison, the optimal control problem was formulated and solved. This way reduces the space of the initial components of the model system. The main conclusion is that a step-by-step description of photosynthetic transformations gives a result close to the experimental description of phytoplankton production dynamics.

1. Introduction

Phytoplankton are a key link in the global carbon cycle. The mechanisms of primary production formation have been widely studied. Primary production is the result of the interaction of carbon dioxide and minerals when exposed to solar energy. Methods to calculate primary production using the assimilation number are known [1,2,3]. This scheme is simplified because it ignores the variability of photosynthetic substances in phytoplankton cells. Chlorophyll and other plant cell photosynthetic substances play an important role in primary production processes [4,5,6]. Hereinafter, we will refer to chlorophyll “a” as the main photosynthetic substance in cells. It actively absorbs solar energy. The absorbed photons are used to start the flow of electrons across the thylakoid membrane. The energy thus converted is used in the chemical reaction for the synthesis of organic compounds [7,8]. Chlorophyll grows together with biomass by metabolism. The ratio of chlorophyll to phytoplankton is usually called the chlorophyll quota [8]. It can be measured but is quite variable [9,10]. According to experts, the proportion can vary from 0.001 to 0.097 [11]. The ratio of chlorophyll to biomass varies depending on the state of cells and environmental conditions [12,13,14,15]. Such variations significantly affect the productivity of phytoplankton.

A substance of adenosine triphosphate (ATP) is a universal energy source for all organic biochemical processes. The functional indicators of primary production are traditionally assessed through the concentration of chlorophyll per unit of biomass [16]. An alternative method based on the amount of ATP is considered in several works [17,18,19,20]. Such works became relevant again after the methods for measuring ATP levels in cells were invented and improved [21]. In [19], a statistical modeling method for estimating the rate of biomass growth is presented. The method is built on the basis of measurements of the proportion of chlorophyll per unit of biomass (chlorophyll quota) and estimates of the proportion of ATP.

We developed mathematical models that are based on the chlorophyll quota and on the concentration of energy-intensive substances. Similar to many researchers [22,23,24,25,26], we based our constructions on the Droop model [27,28]. The functioning of phytoplankton in the aquatic environment is traditionally described by the Monod dependence [29,30]. At the same time, microorganism growth continues for some time after the nutrient resources are depleted. This phenomenon is important for the photosynthesis process. A detailed analysis of these mechanisms is discussed in [23]. Monod’s kinetics cannot imitate this phenomenon. Droop’s concept allows separating the nutrient uptake rate from the growth rate. Many authors use it for a detailed description of the nutritionally limited reproduction of phytoplankton [31,32,33]. Our models describe an important impact of energy-intensive substance concentration and chlorophyll quote on biomass dynamics.

2. Droop Model

The Droop model for describing the growth of phytoplankton in a chemostat [34] is chosen as the base model. This model is a system of differential equations. For an open system, it has the following form:

$$\dot{p}=\left(\mu \left(q\right)-D\right)p,$$

(1)

$$\dot{s}=D\left(s_{in}-s\right)-\nu \left(s, q\right)p,$$

(2)

$$\dot{q}=\nu \left(s, q\right)-\mu \left(q\right)q.$$

(3)

The model has the following denotations:

• s is the concentration of the nutrient substance;
• p is the phytoplankton biomass in volume unity;
• ν is the pace of nutrient assimilation;
• q is the cell quota;
• μ is the rate of biomass growth;
• D is the rate of water flow;
• s_in is the concentration of the nutrient substance in the inner flow.

Cell quota is the concentration of a mineral substance in a phytoplankton cell.

The pace of nutrient assimilation is described as function

$$\nu \left(s,q\right)={\nu }_0\frac{s}{s+s_0}\left(1-\frac{q}{Q_1}\right)$$

(4)

The rate of biomass growth is described as function

$$\mu \left(q\right)={\mu }_0\left(1-\frac{Q_0}{q}\right)$$

(5)

Function μ(q) describes the velocity of phytoplankton growth, and ${\mu }_0$ is the maximum of this velocity. Parameters $Q_0$ and $Q_1$ are the minimum and maximum of cell quota q, respectively. At $q<Q_0$, biomass growth stops. The form of function μ(q) is the traditional form of the Droop model [35]. We propose that the next conditions are true: all constant parameters have positive values, and $Q_0<Q_1$.

We list the properties of the Droop model that we used in our study.

The equilibria are found in the system of the following equations:

$$\left(\mu \left(q\right)-D\right)p=0, D\left(s_{in}-s\right)-\nu \left(s, q\right)p=0, \nu \left(s, q\right)-\mu \left(q\right)q=0.$$

Single trivial equilibrium $\left(p_e=0, s_e=s_{in}, q_e\right)$ always exists. Single positive equilibrium $(p_{*}>0, s_{*}, q_{*})$ exists under the following conditions:

$$D\le {\mu }_0\left(1-\frac{Q_0}{Q_1}\right), D{q}_{*}\le \nu \left(s_{in},q_{*}\right) \mathrm{with} q_{*}={\mu }^{-1}\left(D\right).$$

The Droop model has an attractor. A common content of the nutrients is denoted as $x\left(t\right)=p\left(t\right)·q\left(t\right)+s\left(t\right)$. Model (1)–(3) yields property $\underset{t\to \infty }{\lim }x\left(t\right)=s_{in}$. The proof follows from equation $\dot{x}=D{s}_{in}-Dx$ as a corollary of the system (1)–(3). We denote this attractor as set

$$A=\left\{\left(p,s, q\right)\in R_{+}^3: p·q+s=s_{in}, q\in \left[Q_0,Q_1\right] \right\}$$

(6)

Two possible equilibria are contained in set A.

3. Droop Model Modification with Chlorophyll Quota (Model C)

Our models are extensions of the Droop model. Bernard proposed a modification of the Droop model [23]. The effect of lighting on phytoplankton growth is described using the following restricted function: $\mu \left(q\right)={\mu }_0\left(1-\frac{Q_0}{q}\right)\cdot \chi \left(\cdot \right)$.

Function $\chi \left(\cdot \right)$ can take different forms. In [8], a transition is made from the rate of biomass growth to the rate of carbon uptake per unit of chlorophyll. Thus, the dynamic process is transferred from the mass balance to the category of photosynthetic transformations. At this stage, the authors introduce the concept of the chlorophyll quota as the ratio of the total concentration of chlorophyll in the system to the number of cells containing chlorophyll. Many researchers have indicated that the chlorophyll quota is somewhat dependent on the cell quota [36]. Our modification of the Droop model uses this proposition.

The modification of Model C has three Equations (1)–(3) from the Droop model, to which the chlorophyll quota dynamics relation is added as follows:

$$\dot{p}=\left(\mu \left(q,c,I,T\right)-D\right)p,$$

(7)

$$\dot{s}=D\left(s_{in}-s\right)-\nu \left(s, q\right)p,$$

(8)

$$\dot{q}=\nu \left(s, q\right)-\mu \left(q,c,I,T\right)q.$$

(9)

$$\dot{c}=c_m\gamma \left(q,c\right)-\rho c.$$

(10)

Chlorophyll quota $c\in [0,c_m]$. The normalized proportion of chlorophyll tends to approach the cell quota. Chlorophyll consumption by the photosynthesis process [37] is also considered.

The system of Equations (7)–(10) is Model C.

Function $\nu \left(s, q\right)$ has the form of (4). Growth rate $\mu$ of phytoplankton biomass has the following form:

$$\mu \left(q,c,I,T\right)={\mu }_0\left(1-\frac{Q_0}{q}\right)·\left(\alpha \phi \left(I\right)·\frac{c}{c_m}+{\alpha }_0\right)·\psi \left(T\right)$$

(11)

Parameters $\alpha$ and ${\alpha }_0$ are associated with the rate of biomass growth during the light and dark stages of photosynthesis, respectively.

In addition to the mineral nutrition constantly entering the system, photosynthetically active radiation (PAR) and water temperature are among the influencing factors. The growth rate depends on PAR as function $\phi \left(I\right)=\frac{I}{I+I_0}$ and on water temperature as function $\psi \left(T\right)=\exp \left(-{\left(\frac{T-T_{opt}}{T_1-T_0}\right)}^2\right)$.

The influence of cell quota on chlorophyll quota is regulated by the following function:

$$\gamma \left(q,c\right)={\gamma }_0\left(\frac{q}{Q_1}-\frac{c}{c_m}\right)$$

(12)

3.1. Existence and Stability of Equilibrium at Model C for Constant Environmental Parameters

We propose that parameters $I, T$ of the environment are constants in this subsection. We lose these parameters from the function in this subsection.

We have to define the possible equilibrium of the next system:

$$\left(\mu \left(q,c\right)-D\right)p=0,$$

(13)

$$D\left(s_{in}-s\right)-v\left(s, q\right)p=0,$$

(14)

$$v\left(s, q\right)-\mu \left(q,c\right)q=0,$$

(15)

$$c_m\gamma \left(q,c\right)-\rho c=0.$$

(16)

Theorem 1.

Single trivial equilibrium$\left(p_e=0,s_e=s_{in}, q_e, c_e\right)$always exists. Single positive equilibrium$(p_{*}>0, s_{*}, q_{*}, c_{*})$exists under the following inequalities:

$$\mu \left(Q_1,c\left(Q_1\right)\right)-D\ge 0,$$

$$\nu \left(s_{in}, q_{*}\right)-\mu \left(q_{*},c_{*}\right)q_{*}\ge 0.$$

Values $q_{*}, c_{*}$are defined unambiguously in the following system of equations:

$$\mu \left(q,c\right)-D=0,$$

$$c_m\gamma \left(q,c\right)-\rho c=0.$$

Proof. 

(1) Trivial equilibrium. Linear function $c\left(q\right)$ is defined in Equation (16) of system (13)–(16). Unique value $q_e$ is defined from Equation (15) of system (13)–(16) for $s_e=s_{in}$. The left part of this equation has properties of strictly decreasing with regard to q and different signs in interval $q\in \left[Q_0,Q_1\right]$.

(2) Positive equilibrium $(p_{*}>0, s_{*}, q_{*}, c_{*})$ is defined in equation system (13)–(16) for $p>0$ . Linear function $c\left(q\right)$ is defined in Equation (16) of system (13)–(16). It is an increasing function. Unique $q_{*}$ is defined in Equation (13) if the left part of this equation has a nonnegative value for $q=Q_1$

$$\mu \left(Q_1,c\left(Q_1\right)\right)-D\ge 0.$$

The components $q_{*}, c_{*}=c\left(q_{*}\right)$ must perform the following inequation:

$$\nu \left(s_{in}, q_{*}\right)-\mu \left(q_{*},c_{*}\right)q_{*}\ge 0.$$

We then have unique $s_{*}\in \left[0, s_{in}\right]$ from Equation (15). Component $p_{*}$ is calculated in Equation (14). To this end, we have equilibrium $\left(p_{*}, s_{*}, q_{*}, c_{*}\right)$. □

Two possible equilibria belong to set A (6). One trivial equilibrium $p=0$ and no more than one positive equilibrium $p>0$ exist for positive constant parameters in Model C. The stability of these solutions can be studied in the range of changes in key parameters using a linearized system by the Lyapunov method [38,39]. The illustration of these dynamical properties is shown in Figure 1 in projection on plane $\left(\alpha ,Q_0\right)$.

The area of instability of trivial equilibrium and asymptotical stability positive equilibrium has a blue color. The “green” area of parameter variation corresponds to the existence of both equilibria. The trivial equilibrium is stable, and the positive equilibrium is unstable. A stable trivial equilibrium exists in the “red” set.

Depending on the area in which the model parameters fall, the phase portrait of the main components $\left(p, s, q\right)$ of the model system changes. The phase portraits of system (7)–(10) are shown in Figure 2 for the points indicated by numbers 1–3 in Figure 1.

The solution of Model C is calculated from different points of the area of the initial approximation. The system parameters are set based on hitting each of the designated stability areas. Parameter $\alpha$ is constant for all solutions. Parameter $Q_0$ of the minimal cell quota is variable. In all cases, the solution asymptotically tends to the attractor (Section 2). For the “blue” region, solutions from different initial states tend to a positive equilibrium. The “attraction” of the equilibrium point has a global impact on the dynamics of the solution. In the “green” region, solutions from different initial states tend to different points belonging to the attractor and then “wander” around the attractor. In the “red” region, the “attraction” of the trivial equilibrium is global in nature. Thus, the minimal cell quota has a big role in the quality characteristics of solutions.

4. Model with Separation of Light and Dark Stages of Photosynthesis (Model E)

The light stage of photosynthesis includes an accumulation of energy-intensive compounds. These substances are then involved in the synthesis of organic matter at the dark stage of photosynthesis. Model E has the three equations of the Droop model (1)–(3) and the following equations:

$$\dot{p}=\left(\mu \left(q,r,T\right)-D\right)p,$$

(17)

$$\dot{s}=D\left(s_{in}-s\right)-\nu \left(s, q\right)p,$$

(18)

$$\dot{q}=\nu \left(s, q\right)-\mu \left(q\right)q.$$

(19)

$$\dot{c}=c_m\gamma \left(q,c\right)-\delta \left(q,I\right)·c,$$

(20)

$$\dot{r}=\epsilon \delta \left(q,I\right)·c-\mu \left(q,r,T\right)r.$$

(21)

Parameter $r\ge r_0$ has a sense of growth pool. It reflects the amount of the energy-intensive substances (ATP) in biomass. Parameter $r_0$ denotes a value of the energy pool below which the fusion reaction does not occur. We have the coefficient of growth biomass in the following form:

$$\delta \left(q,I\right)={\delta }_0·\left(1-\frac{Q_0}{q}\right)·\phi \left(I\right),$$

(22)

$$\nu \left(s,q\right)={\nu }_0\frac{s}{s+s_0}\left(1-\frac{q}{Q_1}\right),$$

(23)

$$\gamma \left(q,c\right)={\gamma }_0\left(\frac{q}{Q_1}-\frac{c}{c_m}\right),$$

(24)

$$\mu \left(q,r,T\right)={\mu }_0·\left(1-\frac{Q_0}{q}\right)·\beta \left(1-\frac{r_0}{r}\right)·\psi \left(T\right).$$

(25)

Function $\beta \left(1-\frac{r_0}{r}\right)$ describes an influence of $r$ on biomass growth similar to the influence of the cell quota.

The system of Equations (17)–(21) is Model E.

Existence and Stability of Equilibrium in Model E for Constant Environmental

We propose that parameters $I, T$ of the environment are constants in this subsection, as in Section 3.1.

To define the possible equilibrium, we have the following system:

$$\left(\mu \left(q,r,T\right)-D\right)p=0,$$

(26)

$$D\left(s_{in}-s\right)-\nu \left(s, q\right)p=0,$$

(27)

$$\nu \left(s, q\right)-\mu \left(q,r,T\right)q=0,$$

(28)

$$c_m\gamma \left(q,c\right)-\delta \left(q,I\right)c=0,$$

(29)

$$\epsilon \delta \left(q,I\right)·c-\mu \left(q,r,T\right)r=0.$$

(30)

Theorem 2.

Model E has no more than one trivial equilibrium$\left(p_e=0,s_e=s_{in}, q_e, c_e,r_e\right)$. This model has no more than one positive equilibrium$\left(p_{*}>0, s_{*}, q_{*}, c_{*},r_{*}\right)$.

Proof. 

(1) Parameters $\left(q_e, c_e,r_e\right)$ of the trivial equilibrium are defined from Equations (28)–(30) as follows:

$$\nu \left(s_{in}, q\right)-\mu \left(q,r,T\right)q=0,$$

(31)

$$c_m\gamma \left(q,c\right)-\frac{1}{\epsilon q}\nu \left(s_{in}, q\right)r=0,$$

(32)

$$\epsilon \delta \left(q,I\right)·c-\mu \left(q,r,T\right)r=0.$$

(33)

Equation (32) is obtained by expressing $\nu \left(s_{in},q\right)$ from Equation (31) and subsequent expressing δ(q,I) in terms of $\nu \left(s_{in},q\right)$ with (33). The left side of Equation (31) is strictly decreasing in q and r. The strict decrease of function r(q) follows from (31). Function c is expressed in terms of q and r from Equation (33). The resulting expression leads to the fact that c depends directly only on r(q). We obtain a strict decrease in functions c in q. In total, we obtain the following expression: $c_m\gamma \left(q,c\left(q\right)\right)-\frac{1}{\epsilon q}\nu \left(s_{in}, q\right)r\left(q\right)=0$. The function on the left side is strictly increasing. This equation has no more than one solution $q_e$. No more than one $c_e,r_e$ is defined from this fact.

(2) Positive equilibrium $(p_{*}>0, s_{*}, q_{*}, c_{*},r_{*})$ is defined in the following equation system

$$\mu \left(q,r,T\right)-D=0,$$

(34)

$$\left(s_{in}-s\right)-qp=0,$$

(35)

$$v\left(s, q\right)-Dq=0,$$

(36)

$$c_1\gamma \left(q,c\right)-\frac{1}{\epsilon }Dr=0,$$

(37)

$$\epsilon \delta \left(q,I\right)·c-Dr=0.$$

(38)

This system is received from system (26)–(30). The scheme of equilibrium definition is next. Strictly increasing function $q=q\left(s\right)$ is denoted in Equation (36) of this system. Furthermore, strictly decreasing functions $r=r\left(s\right)$, $p=p\left(s\right)$ and $c=c\left(s\right)$ are denoted in Equations (34), (35) and (38), respectively. These functions are introduced into Equation (37). The function in the left part is a strictly increasing function of variable s. Therefore, Equation (37) has no more than one solution. □

Two possible equilibria belong to set A (6). One of them is trivial equilibrium $p=0$, and the other one is positive equilibrium $p>0$.

In addition to the minimum cell quota, numerical experiments have shown a significant effect on the stability of the solutions of parameter β. The domains of existence and stability of both solutions are in the plane of the ratios of these parameters (Figure 3).

Solutions are defined as a trivial or positive equilibrium. The farther the matching parameters point is from the region of instability, the more obvious the tendency toward attraction by some equilibrium is. The phase portraits of the main components of Model E illustrate different behaviors of the solution trajectories depending on the area from which the model parameters are selected (Figure 4).

The dynamics of the solutions in Models C and E have similar properties. Droop submodel makes some general basis in these models. The zone of instability is smaller for Model E. It represents a very narrow boundary region in which the Jacobian eigenvalues vanish. The complex dynamics can be in this “green” set on plane $\left(\beta ,Q_0\right)$ parameters. The attractor largely affects the dynamics of solutions. These properties connect with the existence and stability of the equilibrium solutions. In general, the solutions’ dynamics are subject to attraction by the attractor of the Droop submodel.

The behavior of the solutions of both models is illustrated with consistent coefficients near the attractor.

Figure 5 shows the dynamics of biomass along the attractor under constant external conditions calculated using Model C (left) and Model E (right). According to Model C, the quasi-stationary state of biomass is reached nearly four times longer. The dynamics of other indicators on the attractor are shown in Figure 6.

Figure 6 shows that Model E makes it possible to reach the equilibrium state of the system in a shorter period. Obviously, the reason is in the structure of the equations of Model E. The chlorophyll quota directly affects the growth of biomass. The configuration Model E allows cellular structures to quickly “adjust” to the biomass changes that occurred in the previous time step. Sequential (one after another) changes in the main components of the system allow approaching equilibrium faster.

5. Model E: The Optimal Solution on The Attractor

Studying the behavior of solutions on attractor A (6) gives a more complete picture of the properties of Model E. The trajectories under the additional condition of optimality are considered according to the criterion of the largest increase in phytoplankton biomass p for a fixed time.

We transform the Droop model for solutions on attractor A.

Lemma 1.

The Droop model on attractor A is transformed to the following form:

$$\dot{p}=\left(\mu \left(q,r,T\right)-D\right)p,$$

(39)

$$\dot{q}=\tilde{\nu }\left(p, q\right)-\mu \left(q,r,T\right)q,$$

(40)

where

$$\tilde{\nu }\left(p, q\right)=\nu \left(s_{in}-pq, q\right).$$

(41)

If the initial conditions for solving the system of Equations (39) and (40) are positive and admissible, then the solution satisfies the condition $p\left(t\right)q\left(t\right)\in \left(0, s_{in}\right]$ for any t.

Proof. 

According to the definition of attractor A (6), formula $pq+s=s_{in}$ is valid. From here, we substitute s into $\nu \left(s, q\right)$ in Equation (3) of the Droop model (1)–(3). Then, Equation (2) can be excluded from this system. Equations (39) and (40) are obtained.

Owing to the validity of Formula (41), the solution of system (39) and (40) at any time t satisfies the following condition: ${\partial }_t\left(pq\right)=\left(\nu \left(p, q\right)-Dq\right)p$, and $\tilde{\nu }\left(p, q\right)=0$ for $s_{in}-pq=0$. This condition implies the second part of the Lemma assertion. □

5.1. Formulation of the Optimal Control Problem

The dynamics of phytoplankton biomass are assumed to be modeled on the interval $\left[0, t_e\right]$. We will proceed from the condition of phytoplankton biomass maximization at final time $p\left(t_e\right)\to max$. This condition is equivalent to condition ${{\displaystyle \int }}_0^{t_e}\dot{p}\left(t\right)dt={{\displaystyle \int }}_0^{t_e}\left(\mu \left(q,r,T\right)-D\right)p_{ }dt\to max$. Equations (17)–(21) are considered for c and r in the form of the comparative smallness of $\dot{r}$ in Equation (21). Then, we obtain the following optimization criterion:

$${{\displaystyle \int }}_0^{t_e}\{\left(\mu \left(q,r,T\right)-D\right)p-\omega {\left(\epsilon \delta \left(q,I\right)·c-\mu \left(q,r,T\right)r\right)}^2\}dt\underset{c\in \left[0,c_m\right], r\ge r_0}{\to }max,$$

(42)

where ω is a relatively small positive number.

The relevant restrictions are given in Equations (39) and (40) under the given initial conditions for p and q.

Problem (39), (40) and (42) is a Lagrange problem [40,41] and is solved on the basis of the Pontryagin maximum principle [42] according to the Lagrange approach.

5.2. Solution of the Optimal Control Problem

The Lagrange function is composed as follows:

$$L\left(p,q,c,r,\lambda \right)=({\lambda }_1-1)\left(\mu \left(q,r,T\right)-D\right)p+\omega {\left(\epsilon \delta \left(q,I\right)·c-\mu \left(q,r,T\right)r\right)}^2+{\lambda }_2\left[\tilde{\nu }\left(p, q\right)-\mu \left(q,r,T\right)q\right]$$

Then, the Euler–Lagrange equations are as follows:

$${\dot{\lambda }}_1=\left({\lambda }_1-1\right)\left(\mu \left(q,r,T\right)-D\right)+{\lambda }_2{\partial }_p\tilde{\nu }\left(p, q\right)$$

$${\dot{\lambda }}_2=\left({\lambda }_1-1\right){\partial }_q\mu \left(q,r,T\right)p+2\omega \left(\epsilon \delta \left(q,I\right)c-\mu \left(q,r,T\right)r\right)\left(\epsilon {\partial }_q\delta \left(q,I\right)c-{\partial }_q\mu \left(q,r,T\right)r\right)+{\lambda }_2\left[{\partial }_q\tilde{\nu }\left(p, q\right)-{\partial }_q\left(\mu \left(q,r,T\right)q\right)\right]$$

under conditions

$${\lambda }_1\left(t_e\right)=0, {\lambda }_2\left(t_e\right)=0$$

Functions c and r are determined from the condition of minimizing L with respect to these variables.

5.3. Optimal Control Problem Modification

The condition of minimizing c is calculated from the equation $\epsilon \delta \left(q,I\right)·c-\mu \left(q,r,T\right)r=0$ for $q>Q_0$, and $\frac{\mu \left(q,r,T\right)r}{\epsilon \delta \left(q,I\right)}\le c_m$. Value $r_m$ is the value of r at which the last inequality becomes an equality.

The minimum condition for L with respect to r turns into a condition for minimizing the expression with respect to r as follows:

$$L\left(p,q,c,r,\lambda \right)=({\lambda }_1-1)\left(\mu \left(q,r,T\right)-D\right)p+{\lambda }_2\left[\tilde{\nu }\left(p, q\right)-\mu \left(q,r,T\right)q\right]$$

It boils down to minimizing over r as follows:

$$\left[({\lambda }_1-1\right)p-{\lambda }_{2}q]\mu \left(q,r,T\right)$$

We denote $\xi =\left[({\lambda }_1-1\right)p-{\lambda }_{2}q]$. Then, the following condition is satisfied:

$$r=\left\{\begin{array}{c}{r}_m, \xi <0 \\ r_0, \xi \ge 0\end{array}\right.$$

Next, the equations for ${\lambda }_{1,2}$ are simplified as follows:

$${\dot{\lambda }}_1=\left({\lambda }_1-1\right)\left(\mu \left(q,r,T\right)-D\right)+{\lambda }_2{\partial }_p\tilde{\nu }\left(p, q\right)$$

$${\dot{\lambda }}_2=\left({\lambda }_1-1\right){\partial }_q\mu \left(q,r,T\right)p+{\lambda }_2\left[{\partial }_q\tilde{\nu }\left(p, q\right)-{\partial }_q\left(\mu \left(q,r,T\right)q\right)\right]$$

Finally, problem (39), (40) and (42) acquires the following form.

Theorem 3.

To solve problem (39), (40) and (42), the following boundary value problem needs to be solved:

$$\dot{p}=\left(\mu \left(q,r,T\right)-D\right)p,$$

(43)

$$\dot{q}=\tilde{\nu }\left(p, q\right)-\mu \left(q,r,T\right)q,$$

(44)

$${\dot{\lambda }}_1=\left({\lambda }_1-1\right)\left(\mu \left(q,r,T\right)-D\right)+{\lambda }_2{\partial }_p\tilde{\nu }\left(p, q\right),$$

(45)

$${\dot{\lambda }}_2=\left({\lambda }_1-1\right){\partial }_q\mu \left(q,r,T\right)p+{\lambda }_2\left[{\partial }_q\tilde{\nu }\left(p, q\right)-{\partial }_q\left(\mu \left(q,r,T\right)q\right)\right]$$

(46)

for a given function$r\left(t\right)=\left\{\begin{array}{c}{r}_m, \xi \left(t\right)<0 \\ r_0, \xi \left(t\right)\ge 0\end{array}\right.$with$\xi \left(t\right)=\left[({\lambda }_1\left(t\right)-1\right)p\left(t\right)-{\lambda }_2\left(t\right)q\left(t\right)]$.

The boundary conditions are as follows:

$$p\left(0\right)=p_0, q\left(0\right)=q_0, {\lambda }_1\left(t_e\right)=0, {\lambda }_2\left(t_e\right)=0.$$

The functions included in the equations correspond to Model E (Formulas (22)–(25)) with a change in (41).

Proof. 

The proof is given before the statement of the theorem. □

Test calculations for problem (43)–(46) are conducted under the following conditions: the left boundary conditions are chosen to ensure that the initial point lays on the attractor. The right boundary conditions for ${\lambda }_{1,2}$ are set equal to zero. The values of other parameters are specified in Section 6. The calculations are made under constant external conditions. Values φ and ψ are constant and correspond to the average daily values of light irradiance and temperature. Our examples show that the solution to the boundary value problem in this case can be found using standard numerical procedures. Some initial values may require some regularization.

Additional details on the selected values of environmental indicators and their changes during the daily and annual cycle are described in Section 6, which is devoted to numerical experiments.

6. Calculation Experiments

The comparative analysis of solutions’ properties of Models C and E is the basic content of computer experiments. Research data are defined in

Table 1. Data for research.

Name	Symbol	Unit of measurement	Value
Maximal of growth velocity	μ0	1/hour	0.09
Semi-saturation constant:
by nutrients	s0	g/m3	0.014
by illumination	I0	E */(m2 h)	1.25
Optimum temperature for phytoplankton growth	Topt	°C	10.0
Minimum temperature for phytoplankton growth	T0	°C	0.0
Maximum temperature for phytoplankton growth	T1	°C	20.0
Maximum nutrient absorption rate	ν0	1/hour	1.95·10−5
Flow rate	D	1/hour	0.00145
Concentration of nutrients in the input stream	sin	g/m3	0.022
Minimum cell quota	Q0	–	0.0015
Maximum cell quota	Q1	–	0.0075
Proportion of biomass associated with the light stage of photosynthesis	α	–	0.037
Proportion of biomass associated with the dark stage of photosynthesis	α0	–	0.025
Maximum proportion of chlorophyll in phytoplankton	cm	–	0.04
Proportion of chlorophyll consumed in the light stage of photosynthesis	ρ	–	1·10−3
Maximum rate of chlorophyll quota change	γ0	1/hour	1·10−3
Proportion of phytoplankton biomass generated from stored energy-intensive substances	β	–	0.035
Maximum of proportion of chlorophyll consumed in the light stage of photosynthesis	δ0	–	0.04
Proportion of chlorophyll involved in energy-intensive substances	ε	–	1
Minimum proportion of energy-intensive substances required to initiate enzymatic reaction	r0	–	0.002

* E denotes one Einstein = one mole photons.

from references [43,44,45].

In the calculations, the qualitative properties of the dynamics of the main model variables are analyzed depending on significant parameters. These parameters are α, β and Q₀. All parameters are involved in the function of the specific biomass growth rate. Parameter $\alpha$ is “responsible” for the influence of illumination in the first model, and parameter $\beta$ characterizes the influence of ATP in the second model. Preliminary numerical experiments show the greatest significance of parameters α and β (apart from Q₀) for the dynamics of the described processes.

6.1. Dynamics in a Variable Environment: Influence of Light and Darkness

The behavior of models on an attractor under constant external conditions is considered (Section 4). Notably, these models are sensitive to changes in external conditions. We simulate the conditions similar to a laboratory experiment [20]. At some initial concentrations, the dynamics of the system are calculated sequentially over a time interval corresponding to a day and 100 days. The illumination function is modeled as follows: the light level was constant (60 Ein·m⁻³day⁻¹) for 12 h; this period was followed by a period of complete darkness of 12 h. In this case, the temperature of the medium is assumed to be constant (7 °C).

The numerical experiment demonstrates the behavior of the model components corresponding to the description of the results of a laboratory experiment under similar conditions. The daily cycle is divided into equal periods of presence/absence of light. During the light period (Figure 7, left), a decrease in the proportion of chlorophyll is observed. This observation completely agrees with the observations described in [20]. In the biological sense, such a process is associated with an increase in biomass and the dispersion of the proportion of chlorophyll in it. The model behavior of the chlorophyll quota is due to the fact that it tends to correspond to the intracellular quota of nutrients. The cellular pool of nutrients is consumed during the progressive growth of biomass. The value of the chlorophyll quota decreases following the cell quota to its minimum value, below which the biomass growth stops. Next, the consumption of chlorophyll for the formation of ATP stops. In a 100-day period, a weak trend toward a decrease in the chlorophyll quota in combination with daily fluctuations is observed after a certain surge (Figure 7, right).

The proportion of ATP changes in antiphase with respect to the chlorophyll quota. This is also in line with the findings presented in [20]. Biologically, during the light period, the energy to start the enzymatic reaction comes from the capture of photons by the reaction centers of the chlorophyll molecules. Therefore, energy-intensive substances acquire the ability to accumulate. During the dark period, energy for chemical transformations can only be obtained by releasing it as a result of the breakdown of ATP. Accordingly, its pool is reduced (Figure 8, left).

In the long-term light/dark alternation, the ATP pool tends to decrease (Figure 8, right). With a long-term progressive increase in biomass (Figure 9, left), the trend toward a decrease in both indicators appears to be adequate (Figure 7 and Figure 8, right).

Other numerical experiments have shown that phytoplankton acquire a tendency for extinction with a significant deterioration in external conditions, such as a change in temperature relative to the optimum, a reduction in the period of illumination and a decrease in the level of illumination. The dynamics of plankton with a threefold reduction in the abovementioned indicators are shown in Figure 9 (right).

At the same time, the chlorophyll quota and the proportion of ATP reveal an interesting trend (Figure 10). First, the indicators go in the antiphase. Under relatively high initial concentrations of phytoplankton and nutrients, an increase in biomass is observed at the first stage. The chlorophyll quota increases with some lag. This lag is necessary for the formation of chlorophyll in the newly formed biomass. The rapid consumption of ATP is due to the low level of light energy coming from outside. Then, the trend changes. Both indicators show weak growth. However, a slight accumulation of ATP and chlorophyll does not cope with the lack of energy and unfavorable temperature. Phytoplankton biomass is decreasing (Figure 9, right).

6.2. Annual Cycle

6.2.1. Solution of the Initial Problem for the Annual Period

Under changing environmental conditions, the model systems of equations cease to be autonomous. The annual dynamics calculated for both models undergo two characteristic bursts, namely, during spring and autumn.

The annual cycle in Figure 11 shows the dynamics of phytoplankton biomass, which are typical for the Far Eastern seas of Russia. The simulation of the annual cycle is provided by seasonal changes in ambient temperature and illumination. The daily change in illumination is considered.

The increase in biomass during spring is higher than that during autumn [46,47]. In the Far East region, the temperature regime has some peculiarities. In the Far Eastern seas, cold-loving species (mainly diatoms) [48] predominate, and a certain proportion is occupied by more thermophilic (dinophyte) species [49]. The most favorable temperature for the entire plant complex forms in late summer–early autumn up to its middle. The increase in biomass during spring is associated with the consumption of nutrients stored during winter. The numerical experiment simulates the natural conditions of the region. The external illumination function is a parabolic curve with a maximum in mid-July. Temperature is a similar smooth function with a maximum in the second half of August and relatively constant values throughout the year except for summer. In accordance with the winter accumulation of mineral nutrition, their initial concentrations are higher than the average observed concentrations.

The temperature and light functions are based on real data provided by the Center for Multifunctional Satellite Environmental Monitoring at the Institute of Automation and Control Processes, the Far Eastern Branch of the Russian Academy of Sciences (IACP FEB RAS), in Vladivostok [50]. Initial approximations for the numerical solution of model systems are formed on the basis of remote sensing data. The species composition of phytoplankton in the Far Eastern seas was formed under the influence of regional natural conditions and is tolerant to low temperatures. Biomass growth is quite intensive throughout the year. This phenomenon is the reason for the intensive consumption of nutrients. Both models support this deduction. The concentration of nutrients throughout the entire vegetative cycle approaches its value in the input flow (Figure 11, Part 1).

The value of the cell quota fluctuates between the minimum and maximum values, and these fluctuations occur in antiphase to the volume of biomass, especially during periods of its rise. With intensive growth of phytoplankton, the intracellular pool of nutrients is dispersed over an increasing number of cells. Therefore, the value of the cell quota decreases. Daily fluctuations in the chlorophyll quota (Figure 11, Part 2) become more pronounced. As already mentioned [9], this observation is consistent with experimental observations.

The dynamics of the share of ATP in phytoplankton repeat the annual dynamics of the chlorophyll quota with some time lag. Fluctuations around daily averages are much less noticeable than the variability of the chlorophyll quota. This result is also confirmed by the conclusions contained in [20].

6.2.2. Solution of the Problem of Optimal Control in the Annual Period

Annual fluctuations in biomass can also be considered a solution to the optimal control problem (Section 5). By modeling the external conditions in the manner described above, the biomass dynamics presented in Figure 12 can be obtained.

The obtained solution of the boundary value problem (43)–(46) quite closely repeats the solution of the initial Cauchy problem for Models C and E with characteristic bursts in biomass during spring and autumn. Moreover, the maximum values of seasonal concentrations are either comparable in absolute values (spring outbreak) or equal (autumn outbreak).

Thus, under the condition that the initial-boundary conditions belong to the attractor of system (26)–(30), the numerical solution of the optimal control problem (43)–(46) can be used to predict phytoplankton dynamics. In this case, the dimension of the space of the system components decreases. At the same time, the numerical solution of boundary value problems is associated with problems of convergence of computational procedures. This restriction requires additional regularization algorithms, which cannot always be effectively applied.

6.3. Climate Fluctuations

The illumination and temperature functions correspond to the climatic features of the Far Eastern seas. In this case, both models demonstrate adequate behavior of the phytoplankton biomass. The solution of Model E is characterized by smoother dynamics than that of Model C. This result is due to the structure of the model equations in Model E. Coefficient ρ at the chlorophyll quota ceases to be constant and becomes function $\delta \left(q,I\right)$ depending on external parameters. Thus, a large proportion of the dispersion due to high gradients of the illumination function is accounted for by chlorophyll. It is partially repaid due to the balance correlation with the share of ATP. This transition from a constant coefficient to function $\delta \left(q,I\right)$ provides more smoothness for the biomass growth function. Thus, Model E appears to be preferable for understanding the trend of biomass changes. When modeling anomalous scenarios, the emerging complex dynamics of phytoplankton are explained by changes in external parameters and not by the properties of the model itself. On the basis of Model E, several calculations are undertaken to simulate climatic fluctuations during a given season. Figure 13 shows the results of changes in phytoplankton dynamics, which correspond to a 2 °C cooling in winter (left) and a 2 °C warming in summer. The general conclusion is that the deviation of average temperatures from the optimum within the tolerance interval negatively affects the growth rate. However, in general, such temperature fluctuations insignificantly affect the average values of biomass during the year. Thus, Model E adequately reflects the regulatory role of the temperature factor in the phytoplankton growth process [51]. This fact is a qualitative conclusion of most experimental observations [52,53,54].

The higher sensitivity of Model E is related to parameter β. Even with a slight increase in its value (above 10⁻³), the specific biomass increases markedly (Figure 14). At the same time, the temperature still insignificantly affects the annual dynamics.

6.4. Long-Term Dynamics

Figure 15 shows changes in biomass dynamics over many years. Scenarios corresponding to different values of β in the presence/absence of climate change are considered.

The results of model calculations are as follows: if the value of β is small, and the replenishment of the nutrient resource is limited, then a gradual extinction of the phytoplankton biomass occurs. Numerical experiments have shown that the negative gradient of the growth function is higher when the value of β is smaller. With a slight increase in the proportion of ATP that stimulates the synthesis reaction, the downtrend gradually breaks and turns into an exponential one. With a larger increase in β, the dynamics become stable, including a short-term surge, and then a transition takes place to a quasi-stationary regime. In this case, the step in β affects the maximum biomass value at the burst stage and the average quasi-stationary value but does not affect the dynamic contour itself (Figure 15). Among the endogenous parameters of the model, the value of the minimum cell quota Q₀ has a comparable impact. The influence of this parameter does not need to be considered separately given that it has a clear biological meaning. The growth is slower when the minimum cellular quota of nutrients required to start it is larger, and vice versa. The cumulative effect of parameters β and Q₀ on the equilibrium values of the specific biomass is shown in Figure 16.

7. Conclusions

The presented models are based on Droop’s concept of the relation [28] between the growth rate and intracellular nutrient quota. This model allows describing the growth of biomass after the cessation of the supply of mineral nutrition to the system.

In addition to the Droop model, both proposed models are based on the concept of chlorophyll quota [55,56]. Experimental studies [37] suggest a relationship between the proportion of chlorophyll in phytoplankton and its cell quota. Specifically, a linear relationship exists between indicators with a certain proportionality coefficient. The presented models allow avoiding simplification and describing the relationship between chlorophyll and cell quota with the mechanics of primary production processes.

Model C has complex dynamics when stable equilibria do not exist. These dynamics are submitted to the attractor of the Droop model (6).

Model E describes the stage of accumulation of the internal energy reserve necessary to start the enzymatic reaction. This model system includes the percentage of ATP in biomass. The form of Droop dependence is borrowed to describe the growth of biomass in the dark stage of photosynthesis (Model E), when illumination as a necessary resource dries up. Then, the only source of energy needed to start the enzymatic reaction is the energy-intensive substances stored in the light stage of photosynthesis [57,58]. Modern research methods allow measuring this quantity directly [17,18]. Organic substances are formed as a result of the enzymatic reaction. These substances are the building blocks of living cells [59,60].

The behavior of the chlorophyll quota and the proportion of ATP in the biomass, which is calculated according to Model E, agrees with the results of laboratory experiments presented in [20]. In the absence of light, phytoplankton cells use ATP as an energy source. Chlorophyll begins to be consumed only when a sufficient amount of energy accumulates. The reaction of organic matter synthesis is slower. Therefore, chlorophyll is also consumed slowly. Consequently, its highest concentration in phytoplankton cells is recorded in absolute darkness [61] (Figure 7, left). On the contrary, the ATP concentration drops to a minimum (Figure 8, left), considering that the consumption of energy-intensive substances increases in the dark due to the absence of other energy sources [62].

Modeling of the annual cycle shows adequate dynamics in both models (Figure 13). Seasonal habitat changes lead to bursts in biomass productivity during spring and autumn. This result agrees with known field observations [46,47]. Given the availability of data from remote observations and the results of laboratory experiments, the presented models can be used to assess the biological productivity of aquatic ecosystems.

Model E can be transformed by solving the optimal control problem. The application of optimal management methods is relevant if the plant community is considered as a certain population striving for the maximum possible increase in the total biomass. The described and investigated approach (Section 5) allows reducing the space of the original components of the model. This way can greatly simplify the collection of experimental data in the practical use of models.

The applicability conditions of the models are shown in

Table 2. Terms of use for models.

	Model C	Model E	Optimal Control Problem (Model E)
Formulation of the problem	initial value problem	initial value problem	boundary value problem
Initial-boundary conditions:
biomass concentration	necessary	necessary	necessary
nutrient concentration	necessary	necessary	-
cell quota	necessary	necessary	necessary
proportion of ATP	-	necessary	-

.

If it is not possible to measure the proportion of ATP in biomass, Model C can be used. The cell quota is a species-specific characteristic, so it is easier to measure.