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Generative adversarial network based data augmentation to improve cervical cell classification model

  • These two authors contributed equally.
  • The survival rate of cervical cancer can be improved by the early screening. However, the screening is a heavy task for pathologists. Thus, automatic cervical cell classification model is proposed to assist pathologists in screening. In cervical cell classification, the number of abnormal cells is small, meanwhile, the ratio between the number of abnormal cells and the number of normal cells is small too. In order to deal with the small sample and class imbalance problem, a generative adversarial network (GAN) trained by images of abnormal cells is proposed to obtain the generated images of abnormal cells. Using both generated images and real images, a convolutional neural network (CNN) is trained. We design four experiments, including 1) training the CNN by under-sampled images of normal cells and the real images of abnormal cells, 2) pre-training the CNN by other dataset and fine-tuning it by real images of cells, 3) training the CNN by generated images of abnormal cells and the real images, 4) pre-training the CNN by generated images of abnormal cells and fine-tuning it by real images of cells. Comparing these experimental results, we find that 1) GAN generated images of abnormal cells can effectively solve the problem of small sample and class imbalance in cervical cell classification; 2) CNN model pre-trained by generated images and fine-tuned by real images achieves the best performance whose AUC value is 0.984.

    Citation: Suxiang Yu, Shuai Zhang, Bin Wang, Hua Dun, Long Xu, Xin Huang, Ermin Shi, Xinxing Feng. Generative adversarial network based data augmentation to improve cervical cell classification model[J]. Mathematical Biosciences and Engineering, 2021, 18(2): 1740-1752. doi: 10.3934/mbe.2021090

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  • The survival rate of cervical cancer can be improved by the early screening. However, the screening is a heavy task for pathologists. Thus, automatic cervical cell classification model is proposed to assist pathologists in screening. In cervical cell classification, the number of abnormal cells is small, meanwhile, the ratio between the number of abnormal cells and the number of normal cells is small too. In order to deal with the small sample and class imbalance problem, a generative adversarial network (GAN) trained by images of abnormal cells is proposed to obtain the generated images of abnormal cells. Using both generated images and real images, a convolutional neural network (CNN) is trained. We design four experiments, including 1) training the CNN by under-sampled images of normal cells and the real images of abnormal cells, 2) pre-training the CNN by other dataset and fine-tuning it by real images of cells, 3) training the CNN by generated images of abnormal cells and the real images, 4) pre-training the CNN by generated images of abnormal cells and fine-tuning it by real images of cells. Comparing these experimental results, we find that 1) GAN generated images of abnormal cells can effectively solve the problem of small sample and class imbalance in cervical cell classification; 2) CNN model pre-trained by generated images and fine-tuned by real images achieves the best performance whose AUC value is 0.984.



    Phytoplankton is the primary producer of water, it not only provides natural live bait for fish directly or indirectly, but also is the main producer of dissolved oxygen in water body, which is the basis of water productivity. Its quantity determines the density of herbivorous zooplankton, and also determines the output of herbivorous fish. Therefore, the amount of harvested adult fish yields basically depends on the phytoplankton density. Fish is not only a beneficial food source for human beings, but also an important economic source for fishermen. So, it is of great significance to study phytoplankton-fish models in Marine ecosystems. For example, the commercially valuable phytoplankton-fish predation model proposed in [1] discusses the impact of different factors on the level of fishery. Recent studies [2,3,4,5] suggest that persistent or overfishing of specific fish species may lead to the extinction of this species. This prompted the researchers to conduct a more in-depth study of the dynamic properties of the fish and their harvesting.

    The basic mathematical tool for designing impulse control system is impulse differential equation theory. Impulse control is a normal form of control based on impulse differential equations [6]. There are many transient changes in nature, such as pest control, glucose regulation, vaccination, etc. [7,8,9]. When these situations occur, they are not disordered, they develop and change with regularity. To describe them, scholars put forward the time impulse differential equation [10,11]. Not all the changes are regular like green tide bursts, there are also some changes that are irregular and not limited by time. Scholars from mathematics put forward the state impulse differential equation model to describe them [12,13]. In recent years, pulsed semi-continuous dynamical systems are widely used in population dynamics [14,15]. For example, Zeng et al. [16] generalized the Poincarˊe-Bendixon theorem for ordinary differential equations, and explored the properties of order-1 periodic solutions of predator-prey models with state feedback control. Chen et al. [17] first proposed the method of applying successor functions to study the dynamical simplicity of semi-continuous dynamical systems. Jiang et al. [18,19] applied Poincarˊe map to obtain the existence and stability conditions of the order-1 periodic solution of the impulsive semi-dynamic system. Hou et al. [20] investigated a class of predator-prey models with shelter and nonlinear impulse feedback control using Poincarˊe map. Due to the importance of pulsed semi-continuous dynamic systems in population, impulsive differential equations have attracted more and more attention in recent years, and have been applied in various fields from population dynamics to chemical regulation systems [21,22,23,24,25,26,27,28].

    In previous studies, most existing models only consider population size, rather than population growth rate, when proposing control strategies [29,30]. Usually, when a population reaches a threshold, the number of organisms released during the pulse is independent of population density. A biological point of view, a fixed threshold cannot be combined with the actual situation of the biological population to determine the control strategy. In reality, there are two situations of population growth: 1) The population density is small, but the rate of change is high, which often occurs in the early stage of population growth; 2) The population density is large, but the rate of change is small. Therefore, the threshold of the model needs to consider both the population density and the rate of change, which is more realistic. In the process of biological population control, it is necessary to comprehensively consider different factors, so that the adopted control method is more in line with the development law of biological populations [31]. Based on the above analysis, this paper proposes and studies a comprehensive control model that comprehensively considers fish population density and its current growth rate, and deeply studies the complex dynamic properties of the model, showing the impact of dynamic thresholds on system dynamics.

    The rest of this article is organized as follows: In Section 2, we present a new phytoplankton-fish capture model with a new control strategy that takes into account the density of fish growth as well as the current fish population. In Section 3, we construct the Poincarˊe map for the model, and then analyze some properties of the Poincarˊe map. Then, in Section 4, we discuss the existence of periodic solutions of order-k(k1) of the model, as well as the uniqueness and global stability of periodic solutions of order-1. Then, some numerical simulations are performed in Section 5. Finally, our findings are discussed in Section 6.

    To formulate our model, we first introduce the following notations:

    Table 1.  Biological significance of parameters.
    parameters biological significance
    u(t)R+: the population of phytoplankton at time tR+
    v(t)R+: the population of fish at time tR+
    r: the intrinsic growth rate of phytoplankton
    α: the absorption rate of phytoplankton by fish
    β: the conversion rate of biomass
    d: the mortality rate of fish
    a: the rate at which phytoplankton release toxins
    b: the half-saturation constant
    H: the threshold at which fish are allowed to be caught
    δ: the maximum capture rate,0<δ<1
    γ: the half-saturation constant,γ>0
    τ: the reduced number of phytoplankton,τ>0
    θ: the morphology parameters,θ>0

     | Show Table
    DownLoad: CSV

    Furthermore let aub+u represent the number of fish, due to the distribution of phytoplankton toxicants, and all parameters are positive and 0<δ<1. Then, Li et al. proposed the following phytoplankton-fish model with the impulsive feedback control [32]

    {dudt=(rαv(t))u(t),dvdt=(βu(t)dau(t)b+u(t))v(t),}v<H,u(t+)=u(t)(1δu(t)u(t)+γ)v(t+)=v(t)τ1+θv(t),}v=H, (2.1)

    Now, it is known that the state pulse capture strategy in (2.1) has certain shortcomings, as it only considers the population size and does not consider the population growth rate. In addition, the control of the pulse model in literature [33,34] is also single. We propose a feedback control model that considers both population density and population rate, which brings difficulties to the analysis of the model. In literature [32,33,34], the pulse set and phase set are straight lines on the plane, while in this paper, the phase set is a curve and the determination of phase set is complicated, which also adds a lot of difficulties to the numerical simulation. More precisely, the control threshold is described by

    a1v+b1dvdt=H, (2.2)

    where a1 and b1 are positive constants, satisfying a1+b1=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. In real life, we should fish reasonably when a1v+b1dvdtH is met. We only assumes that fishing is carried out when a1v+b1dvdt=H is reached. Then, we reach a model in the following form:

    {dudt=(rαv(t))u(t),dvdt=(βu(t)dau(t)b+u(t))v(t),}a1v+b1dvdt<H,u(t+)=u(t)(1δu(t)u(t)+γ),v(t+)=v(t)τ1+θv(t),}a1v+b1dvdt=H, (2.3)

    which is known as a semi-continuous dynamic system [33,35]. Without the feedback control, the system (2.3) reduces to an ODE system

    {dudt=(rαv(t))u(t),dvdt=(βu(t)dau(t)b+u(t))v(t). (2.4)

    According to reference [24], we know that system (2.4) has two equilibrium points in the range of real numbers: a saddle point, O(0,0) and a stable positive equilibrium E(u,v), if and only if, (d+aβb)24βbd and d+a>βb. Where,

    v=rα,   u=d+aβb+(d+aβb)2+4βbd2β

    determined by the intersection of the two nullclines

    L1:ν=rα;    L2:u=d+aβb+(d+aβb)2+4βbd2β.

    Next, we construct the Poincarˊe map to investigate the dynamics of system (2.3).

    [Note]:When dudt<0 and dvdt<0, the growth rate of its phytoplankton and fish populations is negative. Mathematically speaking, their numbers gradually decrease, and with the increase of time, they will become extinct without control.

    By using the Eq (2.2) and the second equation in (2.3), we obtain

    v=Ha1+b1βub1dab1ub+u. (3.1)

    From a biological point of view, we only consider the properties of the model in region R+2={(u,v),u0,v0}. To reveal the properties of the Poincarˊe map, without loss of generality, we assume the initial population size v+0 of the fish population satisfies

    a1v+0+b1(dv+0/dt)<H,

    and the initial point (u+0,v+0) satisfies

    M(v)=Ha1+b1βσ(u)b1dab1σ(u)b+σ(u),

    where M(v)=(θv1)+(θv1)2+4θ(τ+v)2θ. Then, in order to define the impulse set and phase set of the model, we introduce the following sets:

    ΣN={(u,v),u0,v0,M(v)=Ha1+b1βσ(u)b1dab1σ(u)b+σ(u)},ΣM={(u,v),u0,v0,v=Ha1+b1βub1dab1ub+u},

    where

    σ(u)=γ+u+4γδu+γ2+2γu+u22(1δ).

    We further assume ΣN and ΣM intersect with L2 at A+ and A, respectively, then assume ΣN and ΣM intersect with x at B1(u1,0) and B2(u2,0), respectively. Then, depending on the locations of ΣN and v, namely, the positive equilibrium point, we can define the impulse set and phase set of the Poincarˊe map.

    In this case, E is above the impulse set ΣM (see Figure 1(a)). Thus, there exists a point T on the ΣN, at which the trajectory of (2.3) is tangent to the ΣN. Denote the intersection between the trajectory and ΣM by Q1. Then, we define the impulse set by

    M1={(u,v)|v=Ha1+b1βub1dab1ub+u,uuQ1},
    Figure 1.  The phase set and impulsive set for Cases Ⅰ, Ⅱ and Ⅲ, respectively.

    and the phase set by

    N1={(u,v)|v=Ha1+b1βσ(u)b1dab1σ(u)b+σ(u),uuQ1(1δuQ1uQ1+γ)}.

    In this case, E is below the impulse set ΣM (see Figure 1(b)). We first assume that ΓM is the trajectory passing through M in the system. If there is a point W on the ΣM, so that ΓW is just tangent to the ΣM, when v<Ha1+b1βub1dab1ub+u and vW>Ha1+b1βσ(u)b1dab1σ(u)b+σ(u), there must be a point T on the ΣN so that ΓT is just tangent to the ΣN and ΓT tangent to the ΣM at the point Q2. Then, in this case, we define the impulse set by

    M2={(u,v)|v=Ha1+b1βub1dab1ub+u,uuQ2},

    and the corresponding phase set by

    N2={(u,v)|v=Ha1+b1βσ(u)b1dab1σ(u)b+σ(u),uuQ2(1δuQ2uQ2+γ)}.

    In this case, E is below the impulse set ΣM, as shown in Figure 1(c). Thus, there is a trajectory denoted by ΓW that is tangent to ΣM at W, and intersects the ΣN at point P1 and point P2, respectively. Thus, we define the impulse set as

    M3={(u,v)|v=Ha1+b1βub1dab1ub+u,uuW},

    and the phase set as

    N3={(u,v)|v=Ha1+b1βσ(u)b1dab1σ(u)b+σ(u),u(0,uP2][uP1,+)}.

    Based on the above defined impulse and phase sets, we are now ready to construct the Poincarˊe map. To this end, let

    A+k=(u+k,v+k)=(u+k,Ha1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k))ΣN,

    where 0<u+k<+, and define the trajectory

    π(t,t0,A+k)Δ=(u(t,t0,A+k),v(t,t0,A+k))

    passing through A+k that will reach the ΣM at point Ak+1(uk+1,vk+1) at time t1, where

    vk+1=v(t1,t0,(u+k,v+k))=Ha1+b1βuk+1b1dab1uk+1b+uk+1.

    Then, there is

    u+k+1=u(t1,t0,(u+k,Ha1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k)))Δ=u(u+k,Ha1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k))Δ=P(u+k).

    It means that u+k+1 is determined by u+k. Since point Ak+1 is on the impulse set, Ak+1 jump to point

    A+k+1(u+k+1,Ha1+b1βσ(u+k+1)b1dab1σ(u+k+1)b+σ(u+k+1)),

    where

    u+k+1=(1δuk+1uk+1+γ)uk+1=(1δP(u+k)P(u+k)+γ)P(u+k)Δ=Gm(u+k). (3.2)

    From model (2.3), we obtain

    {dudv=(rαv)u(βudaub+u)vw(u,v),u(Ha1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k))=u+0. (3.3)

    Let v+0=Ha1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k), u+0=S, then we obtain that (u+0,v+0) is in ΣN.

    We define

    u(v)=u(S,Ha1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k))Δ=u(S,v).

    Then, according to the model (3.2),

    u(S,v)=S+vHa1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k)ω(u(S,s),s)ds. (3.4)

    From Eqs (3.2) and (3.3), the Poincarˊe map expression of system (2.2) is

    Gm(S)=(1δu(S,Ha1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k)        )u(S,Ha1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k)      )+γ        )u(S,Ha1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k)          ). (3.5)

    Then, we can show that it has the following properties in Theorem 3.1.

    Theorem 3.1. If Ha1+b1βub1dab1ub+u<v, then the trajectory ΓT intersects with ΣM at point Q1(uQ1,vQ1), and we have:

    (i) The domain of Gm(S) is (0,u1), Gm(S) is monotonically decreasing on (0,uT], and monotonically increasing on (uT,u1).

    (ii) Gm(S) is continuously differentiable on (0,u1).

    (iii) When Gm(uT)>uT, it has a unique fixed point on (uT,+) (see Figure 2(b)). When Gm(uT)<uT, it has a unique fixed point on (0,uT) (see Figure 2(a)). When Gm(uT)=uT, uT is the fixed point.

    Figure 2.  The Poincarˊe map Gm(S) related to the impulsive point series S. The parameter values are as follows: r=1.444, α=0.1, β=0.1, d=0.5, a=0.1, b=1, δ=0.4, γ=1. (c)a1=0.8, b1=0.2, H=16, θ=0.6, τ=16. (b)H=15, θ=1, τ=18.

    Proof. (ⅰ) Note that E(u,v) is a linear center and Ha1+b1βub1dab1ub+u<v. Taking any point A+k(u+k,v+k) on ΣN, the trajectory ΓA+k will reach the M1. So the domain of Gm(S) is (0,u1).

    For any u+k1,u+k2[uT,u1), and where u+k1<u+k2, it is easy to get

    μ(u+k1,v+k1)<μ(u+k2,v+k1),

    which implies that there is Gm(u+k1)<Gm(u+k2). Therefore, Gm(S) is monotonically increasing on [uT,u1).

    When u+k1, u+k2(0,uT), u+k1<u+k2. The trajectory ΓA+k1 from point A+k1(u+k1,v+k1) and the trajectory ΓA+k2 from point A+k2(u+k2,v+k2) will pass through L2 and intersect ΣN at points A+k11(u+k11,v+k11) and A+k21(u+k21,v+k21), respectively. Here u+ki1(i=1,2)[uT,u1) and u+k11>u+k21. It can be seen that Gm(u+k1)>Gm(u+k2). Therefore, Gm(S) is monotonically decreasing on (0,uT).

    (ⅱ) Equation (3.2) suggests that w(u,v) is continuously differentiable. Thus, Gm(S) is continuously differentiable in the first quadrant.

    (ⅲ) When Gm(uT)=uT, then uT is the fixed point of function Gm(S).

    If Gm(uT)>uT, Gm(uT)uT>0. Since 0<δ<1, there must be a value S, such that

    Gm(S)=S(1δSS+γ)=S(1δδγS+γ)<S(1δ)<S.

    So there is at least one ˜u(uT,S), satisfying Gm(˜u)=˜u.

    When Gm(uT)<uT, let Gm(uT)=u1<uT. We know that Gm(S) is monotonically decreasing on (0,uT], so Gm(u1)>Gm(uT)=u1, and because Gm(uT)<uT, so there is at least one ˜u(u1,uT), satisfies Gm(˜u)=˜u.

    From the above analysis, we know that Gm(S) has at least one fixed point. Next, we prove the uniqueness by contradiction. Assume the system has two fixed points, ˜u1 and ˜u2 respectively, so that Gm(˜u1)=˜u1 and Gm(˜u2)=˜u2. Let ˜u1<˜u2, we define

    d˜u1˜u2(u)=u(v,˜u2)u(v,˜u1).

    Differentiate the last equation with respect to t obtain

    d˜u1˜u2(u)=u(v,˜u2)u(v,˜u1)=rαvv[˜u2β˜u2da˜u2b+˜u2˜u1β˜u1da˜u1b+˜u1].

    Let

    g(u)=uβudaub+u,

    then

    g(u)=da(ub+u)2(βudaub+u)2<0,

    so

    g(˜u2)<g(˜u1),

    that is

    d˜u1˜u2(u)<0,

    i.e.

    d˜u1˜u2(a1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k))>d˜u1˜u2(a1+b1βu+kb1dab1u+kb+u+k).

    From system (2.3):

    ˜u1=μ(˜u1)(1δμ(˜u1)μ(˜u1)+γ)=μ(˜u1)(1δ+δγμ(˜u1)+γ)=[μ(˜u2)d˜u1˜u2(a1+b1βu+kb1dab1u+kb+u+k)](1δ+δγμ(˜u1)+γ)=μ(˜u2)(1δ+δγμ(˜u1)+γ)d˜u1˜u2(a1+b1βu+kb1dab1u+kb+u+k)(1δ+δγμ(˜u1)+γ)>μ(˜u2)(1δ+δγμ(˜u2)+γ)˜u1d˜u1˜u2(a1+b1βu+kb1dab1u+kb+u+k)μ(˜u1)=˜u2˜u1d˜u1˜u2(a1+b1βu+kb1dab1u+kb+u+k)μ(˜u1),

    that is

    ˜u1d˜u1˜u2(a1+b1βu+kb1dab1u+kb+u+k)μ(˜u1)>˜u2˜u1=d˜u1˜u2(a1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k)).

    It is easy to know that ˜u1μ(˜u1)<1 if a1+b1βub1dab1ub+u<v, so

    d˜u1˜u2(a1+b1βu+kb1dab1u+kb+u+k)>˜u1d˜u1˜u2(a1+b1βu+kb1dab1u+kb+u+k)μ(˜u1)>d˜u1˜u2(a1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k)),

    which is contradictory with

    d˜u1˜u2(a1+b1βσ(u+k)b1dab1σ(u+k)b+σ(u+k))>d˜u1˜u2(a1+b1βu+kb1dab1u+kb+u+k).

    The fixed point is unique.

    When Ha1+b1βub1dab1ub+u>v and ΓT intersects with ΣM (case Ⅱ), Gm(S) of system (2.3) has similar properties with case Ⅰ.

    Theorem 3.2. Suppose Ha1+b1βub1dab1ub+u>v. Then, then trajectory ΓW is tangent to ΣM at W and intersect ΣN at point P1(uP1,vP1) and point P2(uP2,vP2), respectively, where uP1>uP2. Furthermore, Gm(S) has the following properties:

    (i) The domain of Gm(S) is (0,uP2][uP1,u1), and Gm(S) is monotonically decreasing on (0,uP2] and monotonically increasing on [uP1,u1).

    (ii) When Gm(uP2)uP2, there is a unique fixed point on (0,uP2] (see Figure 3(a)). When Gm(uP2)>uP2, there is no fixed point (see Figure 3(b)).

    Figure 3.  The Poincarˊe map Gm(S) related to the impulsive point series S. The parameter values are as follows: r=1.444, α=0.1, β=0.15, d=0.5, a=0.5, b=1, γ=1. (a) a1=0.6, b1=0.4, H=13, δ=0.7, θ=1, τ=18. (b) a1=0.8, b1=0.2, H=16, δ=0.4, θ=0.6, τ=65.

    Proof. (ⅰ) Because E(u,v) is the center point, if Ha1+b1βub1dab1ub+u>v and the trajectory ΓW is tangent to ΣM at W and intersect with the ΣN at point P1(uP1,vP1) and point P2(uP2,vP2), respectively, uP1>uP2. Then, take any point A+k(u+k,v+k) in ΣN, if u+k(0,uP2][uP1,+), the trajectory ΓA+k will reach the M3 at point Ak+1(uk,vk), if u+k(uP2,uP1) the ΓA+k has no intersection with the M3. So the domain of Gm(S) is (0,uP2][uP1,u1).

    For any u+k1,u+k2[uP1,u1) and u+k1<u+k2 easy to get

    μ(u+k1,v+k1)<μ(u+k2,v+k2),

    so Gm(u+k1)<Gm(u+k2). Therefore, Gm(S) is monotonically increasing on [uP1,u1).

    When u+k1, u+k2(0,uP2], u+k1<u+k2. The trajectory from A+k1(u+k1,v+k1) and A+k2(u+k2,v+k2) will pass through the L2 intersect with the ΣN at point A+k11(u+k11,v+k11) and point A+k21(u+k21,v+k21), respectively. Then, u+ki1(i=1,2)[u32,u1) and u+k11>u+k21. So Gm(u+k1)>Gm(u+k2), therefore, Gm(S) is monotonically decreasing on (0,uP2].

    (ⅱ) Consider the relationship between Gm(uP2) and uP2:

    (a) When Gm(uP2)uP2 (see Figure 3(a)), we assume Gm(uP2)=u1uP2. We know that Gm(S) is monotonically decreasing on (0,uP2], so Gm(uP2)Gm(uP2)=u1, and because Gm(uP2)uP2, so there is a point ˜u(u1,uP2] satisfies Gm(˜u)=˜u.

    (b) When Gm(uP2)>uP2 (see Figure 3(b)), there is no ˜u(0,uP2] to satisfies Gm(˜u)=˜u.

    For any uk(uP1,u1), trajectory of point Ak(uk,vk) on ΣN is tangent to the ΣM at point A+k(u+k,v+k). The point A+k(u+k,v+k) will be pulsed to Ak+1(uk+1,vk+1). To get uk+1<u+k<uk, that is uk+1uk, so there is no ˜u(uP1,u1) satisfies Gm(˜u)=˜u.

    Theorem 3.1 has proved that system (2.3) 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.

    Theorem 4.1. The order-1 periodic solution of system (2.3) is globally asymptotically stable if Ha1+b1βub1dab1ub+u<v and Gm(uQ1)>uQ1.

    Proof. When Gm(uQ1)>uQ1, Gm(S) has a fixed point ˜u on (uQ1,u1), that is, Gm(˜u)=˜u from the (ⅲ) of Theorem 3.1. For any point A+0(u+0,v+0) in ΣN, where u+0>uQ1, ΓA+0 will intersect ΣM at the point p+1(u+1,v+1), which is Gm(u+0)=u+1, repeating the above process,

    Gm(Gm(u+0))=G2m(u+0),

    that is,

    Gm(u+1)=u+2,

    further available,

    u+n=Gnm(u+0),n=1,2,

    Then, we have the following three cases:

    Case 1: uQ1<u+0˜u. Since Gm(uQ1)>uQ1 and Gm(S) is monotonously increasing on (uQ1,+). Let Gm(u+i)=u+i+1 yields

    u+0<Gm(u+0)=u+1Gm(˜u)=˜u,u+0<Gm(u+0)<Gm(u+1)=Gm2(u+0)Gm(˜u)=˜u,

    Following the same fashion, we obtain that

    u+0<Gm(u+0)<...<Gnm(u+0)<...<˜u.

    Then we can get

    limn+Gmn(u0+)=˜u.

    Case 2: ˜u<u+0<u1. In this case, we have

    ˜u=Gm(˜u)<Gm(u+0),

    and

    ˜u=Gm(˜u)<Gm2(u+0)<G(u+0).

    By mathematical induction,

    ˜u=Gm(˜u)<...<Gmn(u+0)<Gmn1(u+0)<

    Thus,

    limn+Gmn(u0+)=˜u.

    Case 3: 0<u+0<uQ1. Since Gm(uQ1)>uQ1 and Gm(S) is decreasing on (0,uQ1), we obtain that Gm(u+0)>Gm(uQ1)>uQ1 for any u+0(0,uQ1). So, we can conclude that Gm(u+0)>uQ1. When uQ1<Gm(u+0)<˜u, this is the situation in Case 1; when Gm(u+0)>˜u, this is Case 2 above.

    Thus, we always have

    limn+Gmn(u0+)=˜u.

    The conclusion is proved.

    Theorem 4.2. The semi-continuous dynamical system (2.3) has a stable order-1 periodic solution or an order-2 periodic solution when Ha1+b1βub1dab1ub+u<v, Gm(uQ1)<uQ1 and G2m(uQ1)<uQ1.

    Proof. Take a point A+0(u+0,v+0) on ΣN. Since E is a center point, ΓA+0 will intersect ΣM at point A1(u1,v1), where u1=μ(u+0,v+0). p1 will reach point p+1(u+1,v+1) by an impulse, so Gm(u+0)=u+1. Repeat the above process

    u+n=Gnm(u+0)=Gm(Gn1m(u+0)),  (n=1,2,).

    When uQ1u+0<u1, Gm(S) is increasing on [uQ1,u1) and from the (i) of Theorem 3.1, we know there is no fixed point on [uQ1,u1). Therefore, there is a positive integer i that satisfies

    u+i1=Gm(u+i2)<uQ1

    and

    u+i=Gm(u+i1)<Gm(uQ1)<uQ1.

    When 0<u+0<uQ1, the ΓA+0 will pass through the L2 and intersects the ΣN at p0(u0,Hτ1+θH), where 0<u0<uQ1, this can be transformed into the above situation.

    So, for any u+0(0,u1), there always be i satisfies

    Gm(uQ1)<Gmi(u+0)<uQ1(i1).

    So, we just consider the initial point p+0(u+0,v+0), where uQ1u+0<Gm(uQ1). Since Gm(S) is monotonically decreasing on [Gm(uQ1),uQ1], we have

    Gm[Gm(uQ1),uQ1][Gm(uQ1),uQ1].

    Let Gm(u+0)u+0 and G2m(u+0)u+0. Consider the following four situations:

    Case Ⅰ: uQ1u+0>Gm2(u+0)>Gm(u+0)Gm(uQ1). According to the monotonicity of Gm(S):

    u+2=Gm(u+1)>Gm(u+2)=u+3>Gm(u+0)=u+1,

    furthermore,

    u+4=Gm(u+3)<Gm(u+0)=u+0.

    Thus, there is

    uQ1u+0>u+2>u+4>u+3>u+1Gm(uQ1).

    Proved by mathematical induction:

    uQ1u+0>u+2>...>u+2n>u+2n+2>...>u+2n+1>u+2n1>...>u+1Gm(uQ1).

    Case Ⅱ: uQ1u+1>u+0>u+2Gm(uQ1). Gm(S) is monotonically decreasing on [Gm(uQ1),uQ1]. So,

    u+3=Gm(u+2)>Gm(u+0)=u+1>Gm(u+1)=u+2,

    and

    u+4=Gm(u+3)<Gm(u+1)=u+2<Gm(u+2)=u+3,

    then

    uQ1u+3>u+1>u+0>u+2>u+4G(uQ1),

    so

    uQ1...>u+2n+1>u+2n1>...>u+3>u+1>u+0>u+2>u+4>...>u+2n>...G(uQ1).

    Case Ⅲ: uQ1u+1>u+2>u+0Gm(uQ1).

    To get

    uQ1u+1>u+3>...>u+2n1>...>u+2n+1>u+2n>...>u+2>u+0Gm(uQ1).

    Case Ⅳ: uQ1u+2>u+0>u+1Gm(uQ1).

    This situation is similar to Case Ⅰ, can get

    Gm(uQ1)u+2n+1<u+2n1<...<u+1<u+0<u+2<...<u+2n<u+2n+1<...uQ1.

    For Case Ⅰ and Case Ⅲ, there is ˜u(uQ1,Gm(uQ1)), so that

    limnu2n+=limnu2n1+=˜u,

    That is to say, in these two cases, system (2.3) has a stable order-1 periodic solution.

    For case Ⅱ and case Ⅳ, ~u1~u2 and

    limnu2n1+=˜u1,limnu2n+=˜u2,

    That is to say, system (2.3) has a stable order-2 periodic solution in these two cases.

    Theorem 4.3. Let H1<uQ1 and Gm(uQ1)<uQ1. Then, the necessary and sufficient condition for the order-1 periodic solution of system (2.3) to be globally stable is G2m(u+)<u+ for any u+(0,uQ1].

    Proof. Sufficiency:

    When Gm(uQ1)<uQ1, there exists ˜u(Gm(uQ1),uQ1) which satisfies Gm(˜u)=˜u.

    For any u+(˜u,uQ1), we make u+1=Gm(u+) and u+2=Gm(u+1)=G2m(u+), since G2m(u+)<u+<uQ1 to get ˜u>u+1>Gm(uQ1), uQ1>u+2>u+4>˜u>u+3>u+1>Gm(uQ1), so, uQ1>u+2n>˜u>u+2n+1>Gm(uQ1).

    So

    limnu2n+=limnu2n+1+=˜u.

    Necessity: If G2m(u+)<u+ is not true for any u+[˜u,uQ1], there exists a maximum u0[˜u,uQ1] satisfies Gm2(u0)u0. There u1 and G2m(u1)<u1 for any ˜uε<u1<˜u+ε, where ε>0. From the continuity of G2m(u) on the [u0,u1] there is at least one u[u0,u1] and Gm2(u)=u. This is contradictory.

    Theorem 4.4. If Gm(uQ1)<uQ1 and there is u+m, such that u+m=min{u+:Gm(u+)=uQ1}, the system (2.3) has a order-3 periodic solution if G2m(uQ1)<u+m.

    Proof. If Gm(uQ1)<uQ1, there is a order-1 periodic solution in (Gm(uQ1),uQ1), i.e., Gm(˜u)=˜u, here, ˜u(Gm(uQ1),uQ1). Since Gm(˜u)=˜u, so there is u+m such that u+m(0,˜u) and Gm(u+m)=uQ1. Further, G3m(u+m)=G2m(uQ1)<u+m, on the other hand, limx0G3m(x)>x, so there is at least one value of u, such that Gm3(u)=u.

    Remark 4.1. Using the similar arguments, we can show that if Gk1m(uQ1)<u+m, where Gm(u+m)=uQ1, then the system (2.3) has an order-k (k2) periodic solution.

    [Note]: Global asymptotic stability means: A solution u(t) is called asymptotically stable if it is stable and there exists a number ε0>0, such that, for any other solution v(t) with u(t0)v(t0)<ε0, the following holds:limtu(t)v(t)=0 [35].

    In this section, we carry out some numerical simulations, illustrating our theoretical findings in previous sections and then discuss these results from a biological point of view.

    Figure 4(a) is the trajectory of the order-2 periodic solution of the model. Figure 4(b), (c) is 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 indicates that after two fishing in a period of time, the population returns to the initial point, so the population shows this stable periodic change.

    Figure 4.  The order-2 periodic solution of Cases Ⅲ.

    In Figure 5(a), we numerically illustrate our mathematical findings in Theorem 3.2, corresponding to the cases in Figure 3. In sub-figure a, we see the only order-1 periodic solution (see Figure 5(a)), and in the case of Figure 3(b), there is no order-1 periodic solution (see Figure 5(b)).

    Figure 5.  Periodic solutions in the case of Figures 3 (a) and (b), respectively.

    Figure 6 shows that when the weight parameters a1 and b1 change, the pulse and phase sets change significantly. 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.

    Figure 6.  The impulse and phase sets of the system (2.3) with different weighted parameters a1 and b1.

    From Figure 7, we see that, no matter what the value of r is, the smaller the weight of a1, the greater the weight of b1, 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.

    Figure 7.  The time t of a single period of the order-1 periodic solution with the growth rate r of phytoplankton.

    It can be seen from Figure 8 that the trajectories of different initial points, eventually. 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.

    Figure 8.  The path curve of system (2.3) starting from the points a and b.

    In this paper, we propose an integrated control model that takes into account fish population densities and their current growth rates. The control form adopted is more in line with the development law of biological populations, so that fish can be harvested in time, and more economic benefits can be obtained. We delve into the complex dynamics of the model and illustrate the effect of dynamic thresholds on system dynamics. In this paper, the existence and stability of periodic solutions of order-1, and the existence of periodic solutions of order-k (k1) are proved. The correctness of the theoretical results was verified by numerical simulation, indicating that the phytoplankton and fish population densities can maintain periodic oscillations through effective control strategies, that is, the population size can be controlled within a stable range.

    The results of this paper are an extension of literature [32], and we believe that the innovation of this model is as follows:

    1) This paper employs dynamic thresholds determined by fish population density and current fish growth rate. Fixed thresholds are a special case of dynamic thresholds, which are extensions of fixed thresholds. For example, when a1=1, b1=0 in model (2.3), it is the case of a fixed threshold.

    2) The dynamic properties of the model can be better investigated using the method of Poincarˊe map. The application of control methods and research methods make the model (2.3) have more complex dynamics than the literature [32] as Theorems 3.1 and 3.2.

    Due to the synthesis of various factors affecting the development of the population, the control method adopted in this paper is more realistic, and the obtained research results have practical guiding significance. The analysis method proposed in this paper also has reference value in analyzing impulse models with complex phase sets or impulse sets.

    The paper was supported by the National Natural Science Foundation of China (No. 11872335).

    Declare that there no of interest regarding the publication of this paper.



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