Dynamic Analysis of a Phytoplankton-Fish Model with the Impulsive Feedback Control Depending on the Fish Density and Its Changing Rate

This paper proposes a comprehensive ﬁshing strategy that takes into consideration the population density of ﬁsh and its current growth rate, which provides new ideas for ﬁshing strategies. Firstly, we establish a phytoplankton-ﬁsh model with the impulsive feedback control depending on the density and rate of change of the ﬁsh. Secondly, the complex phase and impulse sets of this model are divided into three cases, then the Poincar´e map for the model is deﬁned, and analyzed the properties of Poincar´e map. In addition, the suﬃcient and necessary conditions for the global asymptotic stability of the order-1 periodic solution and existence condition of order- k ( k ≥ 2) periodic solution are discussed. The action threshold depends on the density and rate of change of the ﬁsh, which is reasonable than earlier studies. The analysis method proposed in this paper also plays an important role in the analysis of impulse models with complex phase sets or impulse sets.


Introduction
Numerous plankton models have been developed over the years due to the importance of plankton, which includes both phytoplankton and zooplankton, to marine ecosystems, for example, the interaction model of phytoplankton and zooplankton with time lags proposed by [1] and the predation model of phytoplankton and fish with commercial value in [2], where the impact of different fishing levels on fisheries was discussed. Fish is not only a beneficial source of food, but also provides an important economic source. Recent studies [3][4][5][6][7][8][9] suggest continuous or over harvesting on a particular species may cause the extinction of that species of fish. This motivates researchers to conduct more in-depth research on fishing, such as how and when to implement capture strategies? Mathematical models are employed and play an important role in this kind of study, in which a basic assumption is that the capture is carried out as soon as the density of the population reaches a given threshold. Then the resulting models can be described by systems of impulsive differential equations [10][11][12][13][14][15][16], where the impulse set and phase set in the impulse model are two straight lines due to the assumption. Then theories of impulsive semi-dynamic systems, which is also known as a semi-continuous dynamic system can be applied to investigate the model dynamics, such as Zeng et al. generalised the Poincaré-Bendixon theorem of ordinary differential equations to explore the periodic solution of the model [17]. Jiang et al. obtained the order-1 periodic solutions using the Poincaré map [18,19], and Chen developed the idea of successor functions to study the mathematical models with pulse state feedback control [20]. Due to its importance in applications, in recent years, systems of impulsive differential equations have attracted more and more attention and been applied to different areas from population dynamics to chemical regulator systems [21][22][23][24][25][26][27][28]. Due to the challenges in analysing these models, most of existing models only considered the population size without considering the population growth rate when proposing a control strategy [29,30]. However it would be more reasonable if both growth rate and population size are taken into consideration when developing a capture strategy.
The main purpose of this research is twofold: (a) to improve a phytoplankton-fish model by proposing an impulse feedback control that incorporates not only the population density but also the population growth rate; (b) to propose a way for calculating the Poincaré map when the impulse set and phase set are curves instead of straight lines. More precisely, in this paper, we proposes a comprehensive control measure that takes into consideration the population density of fish and its current growth rate based on the fish density control model in [31]. The rest of paper is arranged as follows: In section 2, we propose a new phytoplankton-fish capture model with a new type of control strategy by taking into account the density of fish growth in addition to the current number of fish. In section 3, we construct the Poincaré maps for the model, and then some properties of Poincaré map are analyzed. Then this is followed by discussing the existence of the order-k(k ≥ 1) periodic solution of the model, and the uniqueness and global stability of order-1 periodic solution in section 4. We then carry out some numerical simulations in section 5 and finally conclude our study in section 6.

Model establishment
To formulate our model, we first introduce the following notations: Furthermore let au b+u represent the number of death fish due to the distribution of phytoplankton toxicants, and all parameters are positive and 0 < δ < 1. Then, Li, Liu and Cheng proposed the following phytoplankton-fish model with the impulsive feedback control [31].
Now it is known that the state pulse capture strategy in (1) has certain shortcomings as it only considers the population size and does not consider the population growth rate. To overcome these shortcomings, we propose a density and rate-dependent feedback control model, in which we will have curves as the pulse set and phase set, which is different from the existing works and brings extra difficulties in analysing the model. More precisely, the control threshold is described by where a 1 and b 1 are positive constants, satisfying a 1 + b 1 = 1, and they are the weight coefficients that control the rate of change in fish population and the rate of change in fish population density, respectively. Biologically, equation (2) implies that fish catching is allowed only when the fish population v satisfies a 1 v + b 1 dv dt ≥ H. Then we reach a model in the following form: which is known as a semi-continuous dynamic system [32,33]. Without the feedback control, the system (3) reduces to an ODE system A straightforward analysis or by [34] shows system (4) has two equilibrium points: a saddle point, O(0, 0) and a stable positive equilibrium determined by the intersection of the two nullclines Next we construct the Poincaré map to investigate the dynamics of (3).
3 Poincaré map of system (3) and its properties From biological point of view we only consider the properties of the model in region R + 2 = {(u, v), u ≥ 0, v ≥ 0}. To reveal the properties of the Poincaré map, without loss of generality, we assume the initial population size v + 0 of the fish population satisfies and the initial point .
Then, in order to define the impulse set and phase set of the model we introduce the following sets We further assume Σ N and Σ M intersect with L 2 at A + and A − respectively, then assume Σ N and Σ M intersect with x at B 1 (u 1 , 0) and B 2 (u 2 , 0), respectively. Then depending on the locations of Σ N and v * , namely the positive equilibrium point we can define the impulse set and phase set as follows.

Cases
In this case, E * is above the impulse set Σ M (see Fig. 1(a)). Thus, there exists a point T on the Σ N at which the trajectory of (3) is tangent to the Σ N . Denote the intersection between the trajectory and Σ M by Q 1 . Then we define the impulse set by } and the phase set by

Cases II
In this case, E * is below the impulse set Σ M (see Fig. 1(b)). We first assume that Γ M is the trajectory passing through M in the system. If there is a point W on the Σ M so that Γ W is just tangent to the Σ M , when v * < , there must be a point T on the Σ N so that Γ T is just tangent to the Σ N and Γ T tangent to the Σ M at the point Q 2 .Then, in this case, we define the impulse set by and the corresponding phase set by

Cases
In this case, E * is below the impulse set Σ M , as shown in Fig. 1(c). Thus, there is a trajectory denoted by Γ W − that is tangent to Σ M at W and intersects the Σ N at point P 1 and point P 2 , respectively. Thus we define the impulse set as and the phase set as Based on the above defined impulse and phase sets, we are now ready to construct the Poincaré map. To this end, let where 0 < u + k < +∞, and define the trajectory Then, there is It means that u + k+1 is determined by u + k . Since point A k+1 is on the impulse set, A k+1 jump to point From model (3), we obtain Let We define Then, according to the model (6), From (5) and (7), the Poincaré map expression of system (2) is , S   .
(9) Then we can show that it has the following properties summarised in Theorem 3.1.
it has a unique fixed point on (u T , +∞)(see Fig. 2(b)). When G m (u T ) < u T , it has a unique fixed point on (0, u T )(see Fig. 2(a)). When G m (u T ) = u T , u T is the fixed point.
Proof For any u +

is a linear center and
. Therefore, G m (S) is monotonically increasing on [u T , u 1 ).
When u + k 1 , u + k 2 ∈ (0, u T ), and satisfy u + will pass through L 2 and intersect Σ N at points , respectively. Here u + k i1 (i = 1, 2) ∈ [u T , u 1 ) and u + k 11 > u + k 21 . It can be seen that G m (u + k 1 ) > G m (u + k 2 ). Therefore G m (S) is monotonically decreasing on (0, u T ). (ii) Equation (5) suggests that w(u, v) is continuously differentiable. Thus G m (S) is continuously differentiable in the first quadrant.
(iii) When G m (u T ) = u T , then u T is the fixed point of function G m (S).
So there is at least oneũ ∈ (u T , S * ), satisfying G m (ũ) =ũ. When G m (u T ) < u T , let G m (u T ) = u 1 < u T . We know that G m (S) is monotonically decreasing on (0, u T ], so G m (u 1 ) > G m (u T ) = u 1 , and because G m (u T ) < u T , so there is at least onẽ u ∈ (u 1 , u T ) satisfies G m (ũ) =ũ.
From the above analysis, we know that G m (S) has at least one fixed point. Next we prove the uniqueness by contradiction. Assume the system has two fixed points,ũ 1 andũ 2 respectively, so that G m (ũ 1 ) =ũ 1 and G m (ũ 2 ) =ũ 2 . Letũ 1 <ũ 2 , we define Differentiate the last equation with respect to t. We obtain From system (3): , that is ) .
It is easy to know thatũ 1 µ( which is contradictory with So the fixed point is unique. > v * and Γ T intersects with Σ M (case II), G m (S) of system (3) has similar properties with case I.   > v * . Then, then trajectory Γ W − is tangent to Σ M at W and intersect Σ N at point P 1 (u P 1 , v P 1 ) and point P 2 (u P 2 , v P 2 ), respectively, where u P 1 > u P 2 . Furthermore, G m (S) has the following properties: (i) The domain of G m (S) is (0, u P 2 ] ∪ [u P 1 , u 1 ), and G m (S) is monotonically decreasing on (0, u P 2 ] and monotonically increasing on [u P 1 , u 1 ).
> v * and the trajectory Γ W − is tangent to Σ M at W and intersect the Σ N at point P 1 (u P 1 , v P 1 ) and point P 2 (u P 2 , v P 2 ), respectively, where u P 1 > u P 2 . Then take any point will pass through the L 2 intersect the Σ N at point A + and point A + (ii) Consider the relationship between G m (u P 2 ) and u P 2 : (a) When G m (u P 2 ) ≤ u P 2 (see Fig. 3(a)), we assume G m (u P 2 ) = u 1 ≤ u P 2 , we know that G m (S) is monotonically decreasing on (0, u P 2 ], so G m (u P 2 ) ≥ G m (u P 2 ) = u 1 , and because G m (u P 2 ) ≤ u P 2 , so there is a pointũ ∈ (u 1 , u P 2 ] and satisfies G m (ũ) =ũ. (b) When G m (u P 2 ) > u P 2 (see Fig. 3 ). Easy to get u k+1 < u + k < u k , that is u k+1 ̸ = u k , so there is noũ ∈ (u P 1 , u 1 ) and satisfies G m (ũ) =ũ. 4 The order-k(k ≥ 1) periodic solution of the semi-continuous dynamic system (3) and its stability Theorem 3.1 has proved that system (3) has a fixed point under certain conditions, that is, the system has an order-1 periodic solution. Below we will give more properties of the order-1 solution, and also the existence of the order-k solutions.
Proof. When G m (u Q 1 ) > u Q 1 , G m (S) has a fixed pointũ on (u Q 1 , u 1 ), that is, G m (ũ) =ũ from the (iii) of Theorem 3.1. For any point , which is G m (u + 0 ) = u + 1 , repeating the above process, further available, u + n = G n m (u + 0 ), n = 1, 2, · · · Then we have the following three cases: Following the same fashion, we obtain that Then we can get lim n→+∞ G m n (u 0 + ) =ũ.
The conclusion is proved.
Theorem 4.2. The semi-continuous dynamical system (3) has a stable order-1 periodic solution or an order-2 periodic solution when Proof. Take a point A + 0 (u + 0 , v + 0 ) on Σ N . Since E * is a center point, Γ A + 0 will intersect Σ M at point A 1 (u 1 , v 1 ), where u 1 = µ(u + 0 , v + 0 ). p 1 will reach point p + 1 (u + 1 , v + 1 ) by an impulse, so G m (u + 0 ) = u + 1 . Repeat the above process u + n = G n m (u + 0 ) = G m (G n−1 m (u + 0 )), (n = 1, 2, · · · ), when u Q 1 ≤ u + 0 < u 1 , G m (S) is increasing on [u Q 1 , u 1 ) and from the (i) of Theorem 3.1 we know there is no fixed point on [u Q 1 , u 1 ). Therefore, there is a positive integer i that satisfies this can be transformed into the above situation. So for any u + 0 ∈ (0, u 1 ), there is always be a i satisfies So we just consider the initial point Let G m (u + 0 ) ̸ = u + 0 and G 2 m (u + 0 ) ̸ = u + 0 . Consider the following four situations: According to the monotonicity of G m (S): Thus there is Proved by mathematical induction: Case II: Case III: . This situation is similar to Case I, it can get For Case I and Case III, there isũ ∈ (u Q 1 , G m (u Q 1 )), so that lim n→∞ u 2n That is to say, in these two cases, system (3) has a stable order-1 periodic solution.
For case II and case IV,ũ 1 ̸ =ũ 2 and lim n→∞ u 2n−1 + =ũ 1 , That is to say system (3) has a stable order-2 periodic solution in these two cases.
Then the necessary and sufficient condition for the order-1 periodic solution of system (3) to be globally stable is G 2 m (u + ) < u + for any u + ∈ (0, u Q 1 ].
Remark 4.1. Using the similar arguments, we can show that if G k−1 m (u Q 1 ) < u + m , where G m (u + m ) = u Q 1 , then the system (3) has a order-k(k ≥ 2) periodic solution.

Simulations and discussions.
In this section, we carry out some numerical simulations, illustrating our theoretical findings in previous sections, and then discuss these results from biological point of view. Figure 4(a) is the trajectory of the order-2 periodic solution of the model; Figure 4(b) and Figure 4(c) are the time series of phytoplankton density and fish density under the simulated order-2 periodic solution, respectively. The existence of the order-2 periodic solution of the system suggests that the rational fishing activities can make the system exist a benign ecological balance.  In figure 5, we numerically illustrate our mathematical findings in Theorem 3.2, corresponding to the cases in Fig. 3. In sub-figure a, we see the only order-1 periodic solution (see Fig. 5a), and in the case of Figure 3b, there is no order-1 periodic solution (see Fig. 5b). Fig. 6 illustrates the impulse and phase sets can change significantly with the weighting parameters a 1 and b 1 . That is to say, the comprehensive fishing measures that comprehensively consider the fish population density and the current fish growth rate are compared to the fishing measures that only consider the fish population density. The model will generate more complex pulse sets and phase sets. Below we discuss which of the two fishing measures is more biologically appropriate.  From Fig. 7) we see that no matter what the value of r is, the smaller the weight of a 1 , the greater the weight of b 1 , the longer the single period of the order-1 periodic solution, that is, the comprehensive control measures that take into account the fish population density and the current growth rate are longer than the cycle time of the system that only considers the fish population density. That is, the comprehensive fishing measures that comprehensively consider the fish population density and the current fish growth rate are more reasonable than the fishing measures that only consider the fish population density.
It can be seen from Figure 8 that the trajectories of different initial points eventually are stabilised. This illustrates the global asymptotic stability of the order-1 periodic solution. This provides theoretical support for the application of state-dependent feedback control in ecosystem balance.

Conclusion
We establish a phytoplankton-fish model with the impulsive feedback control depending on the density and rate of change of the fish. We have further studied the dynamics of the model, and numerical simulations show that the comprehensive fishing measures that consider the density of fish populations and the current growth rate of fish are more reasonable than the fishing measures that only consider the density of fish populations.
The results of this paper are an extension of literature [31], and we believe that the innovation of this model is to adopt dynamic thresholds determined by fish population density and current fish growth rate. This makes the pulse set and the phase set into two complex curves, which makes it difficult to study the dynamic properties of the model. With the help of Poincaré map, this paper gives a method to prove the existence and stability of periodic solutions of order-1 and the existence of periodic solutions of order-k(k ≥ 2). In a biological sense, the order-k(k ≥ 1) periodic solution of the state impulse feedback control system shows that under certain conditions, the phytoplankton population density and fish population density can maintain periodic oscillations, in other words, the population size can be controlled within a stable range. Numerical simulation of a 1 , b 1 , the influence of parameter changes on cycle changes. When a 1 = 1, b 1 = 0, the action threshold only depends on the fish density. This is the situation studied in the literature [31], which is a special case of this article.
The analysis method proposed in this paper also plays an important role in the analysis of pulse models with complex phase sets or impulse sets.