# Stability and Bifurcation for a Single-Species Model with Delay Weak Kernel and Constant Rate Harvesting

Authors: Xiangrui Li; Shuibo Huang
Journal: Complexity
 (2019)

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## Abstract

In this paper, we consider the effect of constant rate harvesting on the dynamics of a single-species model with a delay weak kernel. By a simple transformation, the single-species model is transformed into a two-dimensional system. The existence and the stability of possible equilibria under different conditions are carried out by analysing the two-dimensional system. We show that there exists a critical harvesting value such that the population goes extinct in finite time if the constant rate harvestingu is greater than the critical value, and there exists a degenerate critical point or a saddle-node bifurcation when the constant rate harvesting u equals the critical value. When the constant rate harvesting u is less than the critical value, sufficient conditions about the existence of the Hopf bifurcation are derived by topological normal form for the Hopf bifurcation and calculating the first Lyapunov coefficient. The key results obtained in the present paper are illustrated using numerical simulations. These results indicate that it is important to select the appropriate constant rate harvesting u.


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## Body

## 1. Introduction

Ecological population dynamics is an important research field of mathematical biology. The single-species model is the cornerstone of research for mathematical biology. As early as 1798, Malthus [1] put forward the famous Malthus population model, that is,(1)dNtdt=rNt,where Nt represents the unit density of population at time t, and r>0 is the intrinsic rate of growth for population. Considering the carrying capacity of the environment, Verhulst [2] improved the Malthus population model to logistic equation in 1838:(2)dNtdt=rNt1−NtK,where K is the maximum carrying capacity of the environment. Logistic equation successfully predicted the total population of the world and was widely used to study the dynamical behavior of single species. Especially because of its simple form, clear biological meaning of model parameters, and clear dynamic behavior, it has very important applications in many fields such as ecology [3], biological resource management [4], life science [5], cell and molecular biology [6], biostatistics [7, 8], stock market [9], and medicine [10–12]. Other continuous single-species models include Gompertz model, Food-limit model, Allee model, Rosenzweig model, and Beverton–Holt model (see [13] and references therein).Time-delay phenomenon is an important factor affecting the stability of systems. Many scholars in the field of mathematical biology solved biological problems by studying this phenomenon, for example, discrete delay [14], delay dependent parameters [15], distributed delays [16], variable delays [17], and stochastic model with delays [18–22]. The logistic growth model with discrete delay τ (Hutchinson [23]) is governed by(3)dNtdt=rNt1−Nt−τK.In 1934, in order to solve the problem that Verhulst’s logistic equation could not be applied due to population disappearance, Volterra [24] established a more accurate model than the former model:(4)dNtdt=Nt1−1K∫−∞tGt−sNsds,where Gt, called the delay kernel, is a weighting factor which says how much emphasis should be given to the size of the population at earlier times to determine the present effect on resource availability. The delay kernel is usually normalized so that ∫0∞Gudu=1. Two special cases,(5)Gu=αe−αu,Gu=α2ue−αu,are called weak delay kernel and strong delay kernel, respectively. Let Gu=αe−αu, Volterra’s model [25] follows the form(6)dNtdt=rNt1−1K∫−∞tαe−αt−sNsds.In 1977, Cushing [25] proposed the following single-species model with weak delay kernel:(7)dxtdt=rxt1−axt−ω∫−∞tαe−αt−sxsds,and detailed qualitative results of this model were obtained.The study concerning population dynamics with harvesting is a subject of mathematical bioeconomics. The study on mathematical models of bioeconomics for resource management can be found in [4, 26–34]. Renewable resources, which are vital to human survival, are increasingly being overexploited. In the light of the reduction of resource stocks and the deterioration of the environment, the exploitation and management of renewable resources has become a major issue concerned nowadays. In order to protect the natural environment which human beings depend on, the development and utilization of renewable resources must be moderate. The maximum sustained yield (MSY) with the minimum effort is what we want. The MSY is the largest harvest rate that can be sustained indefinitely. If the harvest rate of a population is more than its MSY, the population will go extinct. So the determination of the MSY is very important. Generally, harvesting strategies include constant harvesting, linear harvesting, time-dependent harvesting, impulsive harvesting, and seasonal harvesting. Which harvesting strategy can be used to achieve the best economic benefits under the premise of ensuring the sustainable development of species and ecological environment? It is a question we need to answer. Harvesting models are introduced by Kot [3] and Murray [35]. A model with a constant rate harvesting Y0 was studied by Brauer and Sanchez [36]. The model is(8)dNtdt=rNt1−NtK−Y0.Ludwig et al. [37] considered the following equation with harvesting function pN on the budworm population dynamics:(9)dNtdt=rBNt1−NtKB−pNt.Population models with different harvesting strategies have been studied by scholars, for example, impulsive harvesting [38–44], constant rate harvesting [45–48], optimal harvesting strategy [49], linear harvesting [50, 51], and nonlinear harvesting [52, 53].Volterra’s model [24] and Cushing’s model [25] can sufficiently describe the deterministic dynamical behaviors. However, it is inevitable for us to encounter some questions for resource management. Hence, the models do not effectively predict population behaviors in reality. Due to the size of the population at earlier times to determine the present effect on resource availability, motivated by the above work, introducing a constant rate harvesting u into the Volterra’s model [24], we consider a single-species model with a delay weak kernel and a constant rate harvesting:(10)dxtdt=rxt1−1K∫−∞tαe−αt−sxsds−u,where xt represents the unit density of the species at time t, r>0 represents the intrinsic growth rate of the species (reflecting the characteristics of the species itself), K>0 is carrying capacity, αe−αt−s is a common weak kernel function, and u stands for constant rate harvesting, here α and u are positive constants.Letyt=∫−∞tαe−αt−sxsds, we have the following system:(11)dxtdt=rxt1−1Kyt−u,dytdt=αxt−αyt.The purpose of this paper is to study the effect of constant rate harvestingu in system (11). The present paper is built up as follows. In Section 2, we give a simple analysis of system (11) and some lemmas and a definition are given. In Section 3, the existence of degenerate critical point and saddle-node bifurcation of system (11) is analysed. The stability of the equilibria and the existence of Hopf bifurcation of system (11) are discussed in Section 4. In order to illustrate the theoretical analysis, Section 5 gives some examples and their simulations. Finally, a brief conclusion is given in Section 6.


## 2. A Simple Analysis of System and Preliminaries

In this section, a simple analysis of system (11) is performed, and some lemmas and a definition are given.Let the right-hand side of system (11) equal to zero, that is,(12)dxtdt=rxt1−1Kyt−u≔Pxt,yt=0,dytdt=αxt−αyt≔Qxt,yt=0.By the second formula of system (12) αxt−αyt=0, i.e., xt=yt, and substituting it into the first formula, we obtain the quadratic equation rxt1−1/Kxt−u=0. In this system, denote Δ as Δ=r2−4ur/K, which is a discriminant of quadratic equation rxt1−1/Kxt−u=0. Obviously, the equation rxt1−1/Kxt−u=0 has two different real roots if Δ>0, i.e., u<1/4Kr; the equation rxt1−1/Kxt−u=0 has two identical real roots if Δ=0, i.e., u=1/4Kr; the equation rxt1−1/Kxt−u=0 has no real roots for Δ<0u>1/4Kr. From the above analysis, we can find dxt/dt<0 as u>1/4Kr, and there is no equilibrium in system (11) in this case.Furthermore, we get the Jacobian matrix of system (11):(13)A=∂P∂x∂P∂y∂Q∂x∂Q∂y=r1−1Kyt−rKxtα−α.Naturally, the determinantD of the Jacobian matrix A is written by the following calculation:(14)D=detA=r1−1Kyt−rKxtα−α.The traceT of the matrix A also can be obtained as follows:(15)T=trA=∂P∂x+∂Q∂y=r1−1Kyt−α.Because system (11) has no equilibrium and the population xt is reduced to extinction in finite time if u>1/4Kr, we will not discuss this case. Therefore, we analyse the stability of the equilibria and the existence of bifurcations of system (11) under two different cases: Δ=0 and Δ>0.Before the main results are given, some lemmas and a definition are introduced.Lemma 1 (see [54]).
Given the rank of the matrixA is equal to 1 (i.e., rankA=1) and the trace of the matrix A is equal to 0 (that is T=trA=0) in system as follows:(16)dxdt=ft,x.System (16) can be transformed into the following system by a one-to-one transformation in a neighborhood of the origin O:(17)dxdt=y,dydt=Q2x,y,where Q2x,y is analytic, with terms of order no less than 2. Suppose(18)Q2x,y=alxl1+hx+bnxny1+gx+y2fx,y,where hx, gx, and fx,y are analytic, h0=g0=0, l≥2, al≠0, n is a natural number, and O is the lth critical point.Ifl=2m+1 is an odd number, let Δl=bn2+4m+1al, then the following holds: (1)
Ifal>0, then the critical point O is a saddle, its index IO=−1(2)
If (i)al<0, bn≠0, n is an even number, n<m, or (ii) al<0, bn≠0, n is an even number, n=m, Δl≥0, then the critical point O is a node, its index IO=+1(3)
If (i)al<0, bn≠0, n is an odd number, n<m, or (ii) al<0, bn≠0, n is an odd number, n=m, Δl≥0, then O is a critical point with an elliptic domain, its index IO=+1(4)
If (i)al<0, bn=0, or al<0, bn≠0, and n>m, or (ii) al<0, bn≠0, n=m, Δl<0, then the critical point O is a center or focal, its index IO=+1Ifl=2m is an even number, then the following holds:(1)
Ifbn=0 or bn≠0 and n≥m, then O is a degenerate critical point, its index IO=0(2)
Ifbn≠0, 1≤n<m, then the critical point O is a saddle-node, its index IO=0Lemma 2 (see [54]).
Assume that the rank of matrixA is equal to 1, i.e., rankA=1 and the trace of matrix A satisfies trA=T≠0 in system (16), system (16) can be transformed into(19)dxdt=P2x,y=Px,y,dydt=y+Q2x,y=Qx,y,where P2 and Q2 are analytic functions of order no less than 2. So Q=0 ensures the existence of Gx (or the existence of Gy). If Gx=alxl+⋯, l≥2 al≠0, then the following properties are satisfied:(1)
Ifl is an odd number and al>0, then O is a node, the index IO=+1(2)
Ifl is an odd number and al<0, then O is a saddle point, its index IO=−1(3)
Ifl is an even number, then O is a saddle-node, the index IO=0Lemma 3 (see [55]).
Assuming thatγ=0, we stand a function φx,0 for(20)φx,0=12Bx,x+16Cx,x,x+Ox4.In this formula,Bδ,η and Cδ,η,ζ are symmetric multilinear vector functions. We have(21)Biδ,η=∑j,k=12∂2φiξ,0∂ξj∂ξkξ=0δjηk,i=1,2,(22)Ciδ,η,ζ=∑j,k,l=12∂3φiξ,0∂ξj∂ξk∂ξlξ=0δjηkζl,i=1,2.Moreover, the Taylor coefficientsgkl of the equation(23)Bzq+z¯q¯,zq+z¯q¯=z2Bq,q+2zz¯Bq,q¯+z¯2Bq,q¯,can be denoted by the following equations:(24)g20=p,Bq,q,g11=p,Bq,q¯,g02=p,Bq¯,q¯,g21=p,Cq,q,q¯,where the vector q¯ is the conjugate vector of q and the vector z¯ is the conjugate vector of z.Definition 1 (see [55]).
The first Lyapunov coefficient denotes asl1β, and when β=0, the following statement is completely true:(25)l10=12ω02Reig20g11+ωg21,where the value of l10 depends on the normalization of the eigenvectors p and q. The most important thing in bifurcation analysis is that the sign of l10 does not change under p,q=1.Lemma 4 (see [55]).
Suppose a two-dimensional system,(26)dxdt=fx,γ,with smooth f, has for all sufficiently small γ the equilibrium x=0 with eigenvalues,(27)λ1,2γ=μγ±iωγ,where μ0=0,ω0=ω0>0. Let the following conditions be satisfied:(1)
l10≠0, where l1 is the first Lyapunov coefficient(2)
μ′0≠0
Then, there are invertible coordinate and parameter changes and a time reparameterization transforming into(28)ddτy1y2=β−11βy1y2±y12+y22y1y2+Oy4.Lemma 5 (see [55]).
Any generic two-dimensional, one-parameter system (topological normal form for the Hopf bifurcation):(29)dxdt=fx,γ,having at γ=0 the equilibrium x=0 with eigenvalues,(30)λ1,20=±iω0,ω0>0,is locally topologically equivalent near the origin to one of the following normal forms:(31)dy1dtdy2dt=β−11βy1y2±y12+y22y1y2.


## 3. Degenerate Critical Point and Saddle-Node Bifurcation

In this section, we investigate the properties of the equilibrium underΔ=0, i.e., u=1/4Kr. System (11) has two different cases for the discriminant Δ=0. One case is linear equations of system (11) with one zero eigenvalue and one nonzero eigenvalue, and the other case is linear equations of system (11) with two zero eigenvalues.Ifu=1/4Kr, the solution of the equation (12) is(32)x1∗=y1∗=K2.So system (11) has a unique equilibrium E1x1∗,y1∗=K/2,K/2. Substituting x1∗=y1∗=K/2 into (13) and (14), A1∗ and D1∗ are as follows:(33)A1∗=r2−r2α−α,D1∗=r2−r2α−α=0.Consequently,(34)detλI−A1∗=λI−A1∗=λ−r2r2−αλ+α=λ2+α−r2λ,where I denotes the identity matrix. Calculating λI−A1∗=0, one can obtain the eigenvalues of matrix A1∗: λ1=0 and λ2=r/2−α.DenoteX1 and Y1 as X1=xt−K/2 and Y1=yt−K/2, system (11) can be transformed into(35)dX1dt=r2X1−Y1−rKX1Y1,dY1dt=αX1−Y1.Theorem 1.
Ifu=1/4Kr and α=r/2, then the unique equilibrium E1K/2,K/2 of system (11) is a degenerate critical point and system (11) will not admit any closed orbits. If u=1/4Kr and α≠r/2, then the unique equilibrium E1K/2,K/2 of system (11) is a saddle-node and system (11) will not also admit any closed orbits.Proof.
It is easy to obtain thatλ1=λ2=0 if u=1/4Kr and α=r/2; in the other word, the linear equations of system (11) has two zero eigenvalues.
In this case, substitutingα=r/2 into system (35), we get(36)dX1dt=r2X1−Y1−rKX1Y1,dY1dt=r2X1−Y1.
Let(37)sv=02r1−1X1Y1,that is, s=2/rY1 and v=X1−Y1.
Take the derivatives ofs and v, respectively, and the following system can be obtained:(38)dsdt=v,dvdt=−r34Ks2−r22Ksv.
SinceO is an isolated equilibrium of system (38), it is assumed(39)dvdt=−r34Ks2−r22Ksv=alsl1+hs+bnsnv1+gs+v2fs,v,where hs, gs, and fs,v are analytic, h0=g0=0. By Lemma 1, Q2s,v=alsl+bnsnv. On account of l=2=2m is an even number, b1=−r2/2K≠0 and 1=n=m, we can obtain that the equilibrium E1K/2,K/2 is a degenerate critical point by Lemma 1. It is a degenerate critical point whose index is not one, so it has no closed orbits if system (11) has a unique equilibrium.
Ifu=1/4Kr and α≠r/2 in system (11), then the linear equations of system (11) has one zero eigenvalue and one nonzero eigenvalue.
Let(40)wv=4αr2α−r−22α−r1−1X1Y1,and take the derivatives of w and v, system (35) is replaced by(41)dwdt=−αr2K2α−rw2+2αr2α+rK2α−r2wv−8α2rK2α−r3v2,dvdt=−r34Kw2+r2−αv+r22α+r2K2α−rwv−2αr2K2α−r2v2.
We know thatdv/dt=0 can ensure the existence of Gw by Lemma 2. Let dv/dt=0, we have(42)v=2r−2αr34Kw2−r22α+r2K2α−rwv−2αr2K2α−r2v2.
Substituting equation (42) to(43)dwdt=−αr2K2α−rw2+2αr2α+rK2α−r2wv−8α2rK2α−r3v2,we have(44)dwdt=−αr2K2α−rw2+O3.
Becausel=2 is an even number, we can obtain that the equilibrium E1K/2,K/2 is a saddle-node by Lemma 2. Because the unique equilibrium point E1K/2,K/2 is a saddle-node whose index is not one, system (11) does not admit any closed orbits.


## 4. Hopf Bifurcation

In this section, the caseΔ>0, that is u<1/4Kr, is discussed. If u<1/4Kr, system (11) has two equilibria:(45)E2x2∗,y2∗=K2−K21−4uKr,K2−K21−4uKr,E3x3∗,y3∗=K2+K21−4uKr,K2+K21−4uKr.We analyse and discuss qualitative characteristics of these two equilibria, respectively.Theorem 2.
The equilibriumE2x2∗,y2∗ is a saddle point if u<1/4Kr in system (11).Proof.
The determinantD2∗ of the linear matrix corresponding to E2x2∗,y2∗ of system (11) is as follows:(46)D2∗=r21+1−4uKr−r21−1−4uKrα−α=−αr1−4uKr,where r>0 and α is a positive constant, so D2∗<0. Obviously, E2x2∗,y2∗ is a saddle point.Theorem 3.
Ifu<1/4Kr in system (11), then the following holds:(1)
If2α>r1−1−4u/Kr, then E3x3∗,y3∗ is asymptotically stable(2)
If2α<r1−1−4u/Kr, then E3x3∗,y3∗ is unstable(3)
If2α=r1−1−4u/Kr, a unique and stable limit cycle bifurcation emerges via the Hopf bifurcation from the equilibrium E3x3∗,y3∗ for small enough:(47)T=r21−1−4uKr−α>0.Proof.
The linear matrixA3∗ corresponding to the equilibrium E3x3∗,y3∗ of system (11) is governed by(48)A3∗=r21−1−4uKr−r21+1−4uKrα−α,D3∗=αr1−4u/Kr is the determinant of Jacobian matrix A3∗, where r>0 and α is a positive constant, so D3∗>0.
It is easy to derive the traceT of the matrix A3∗ from the following formula:(49)T=r21−1−4uKr−α.
Obviously, ifT<0, the real part Reλ of the eigenvalues of A3∗ satisfies Reλi<0i=1,2, so E3x3∗,y3∗ is asymptotically stable; if T>0, the real part Reλ of the eigenvalues of A3∗ satisfies Reλi>0i=1,2, so E3x3∗,y3∗ is unstable. This completes the proofs of 1 and 2.
Denote(50)μα=T2=r41−1−4uKr−α2.
Let(51)α0=r21−1−4uKr,we have μα0=0. The condition (2) of Lemma 4 is easy to verify:(52)μ′α0=−12<0.
Moreover,(53)ω2α0=r221−1−4uKr1−4uKr>0.
To compute the first Lyapunov coefficient, denoteX2 and Y2 as(54)X2=xt−K21+1−4uKr,Y2=yt−K21+1−4uKr.
System (11) is converted into(55)dX2dt=r21−1−4uKrX2−r21+1−4uKrY2−rKX2Y2,dY2dt=r21−1−4uKrX2−r21−1−4uKrY2.
Perform the transformation(56)M:ξ1=X2−Y2,ξ2=Y2.
The transformationM transforms the system into(57)dξ1dt=−arξ2−rKξ1ξ2−rKξ22≡F1ξ1,ξ2,dξ2dt=r21−aξ1≡F2ξ1,ξ2,where a=1−4u/Kr,0<a<1. The Jacobian matrix A4∗ of system (57) at O0,0 is(58)A4∗=0−arr21−a0.
The determinantD4∗ of the Jacobian matrix A4∗ evaluates(59)D4∗=0−arr21−a0.
By(60)detλI−D4∗=λar−r21−aλ=λ2+r221−1−4uKr1−4uKr=0,we can calculate eigenvalues(61)λ1,2=±a1−a2ri.
Thus, it is easy to find complex vectors(62)q∼a1−a2−1−a2a1−a2i,p∼1−a2a1−a2−a1−a2i.
In order to accomplish the normalizationp,q=1 from the expressions (62), we can take, for example,(63)q=a1−a2−1−a2a1−a2i,p=1a1−a2/2a1−a/21−a2a1−a2−a1−a2i.
To computeBδ,η and Cδ,η,ζ in Lemma 3, let δ=δ1,δ2T, η=η1,η2T, and ζ=ζ1,ζ2T. On the basis of the (21) and (22), we find(64)Bδ,η=−2rKδ2η2−rKδ1η2+δ2η10,(65)Cδ,η,ζ=00.
Substituting the value of the vectorq and the conjugate vector q¯ of q into equations (64) and (65), we obtain(66)Bq,q=ar1−a34K+ar1−a22Ka1−a2i0,Bq,q¯=−ar1−a34K0,Cq,q,q¯=00.
Basing on Lemma3, we can calculate(67)g20=p,Bq,q=r1−a2+21−aa1−a/2ri4K,(68)g11=p,Bq,q¯=−r1−a24K,(69)g21=p,Cq,q,q¯=0.
Finally, the first Lyapunov coefficient is calculated; as the following equation shows, substitute the formulas (67)–(69) into formula (25):(70)l1α0=12ω2Reig20g11+ωg21=1−a28aK2a1−a2>0.
It is clear that the condition (1) of Lemma4 holds as l1α0>0. Therefore, by Lemmas 4 and 5, there exists a unique and stable limit cycle bifurcation via the Hopf bifurcation from the equilibrium E3x3∗,y3∗ for small enough T=r/21−1−4u/Kr−α>0,
For simplicity, we useα as bifurcation parameter in this section. If u is chosen as bifurcation parameter, the conditions obtained can be converted. So we have the following remark:Remark 1.
Ifu<1/4Kr in system (11), then E2x2∗,y2∗ is a saddle and u=αK1−α/r is a bifurcation point. The following holds:(1)
E3x3∗,y3∗ is stable if u<minαK1−α/r,1/4Kr(2)
E3x3∗,y3∗ is unstable if 1/4Kr>u>αK1−α/r, a unique and stable limit cycle bifurcation emerges near E3x3∗,y3∗ for small enough Tu0=u0−αK1−α/r>0.


## 5. Some Examples and Their Simulations

The main results obtained are listed in Table1 by the previous analysis.Table 1
The results under different conditions.
Conditions
Subconditions
Equilibria
Stability
u
=
K
r
/
4
α
=
r
/
2
E
1
K
/
2
,
K
/
2
E
1
K
/
2
,
K
/
2 is a degenerate critical point, system (11) has no limit circle
α
≠
r
/
2
E
1
K
/
2
,
K
/
2
E
1
K
/
2
,
K
/
2 is a saddle-node, system (11) has no limit circle
u
<
K
r
/
4
2
α
>
r
1
−
1
−
4
u
/
K
r
E
2
x
2
∗
,
y
2
∗, E3x3∗,y3∗
E
2
x
2
∗
,
y
2
∗ is a saddle, E3x3∗,y3∗ is asymptotically stable
2
α
<
r
1
−
1
−
4
u
/
K
r
E
2
x
2
∗
,
y
2
∗, E3x3∗,y3∗
E
2
x
2
∗
,
y
2
∗ is a saddle, E3x3∗,y3∗ is unstable
2
α
=
r
1
−
1
−
4
u
/
K
r
E
2
x
2
∗
,
y
2
∗, E3x3∗,y3∗
E
2
x
2
∗
,
y
2
∗ is a saddle, a Hopf bifurcation occurs near E3x3∗,y3∗ for small enough T=r/21−1−4u/Kr−α>0
u
>
K
r
/
4
System (11) has no equilibrium point and the population xt is reduced to extinction in finite timeFor further verification of the results obtained, we offer an example as follows: letr=2 and K=8, system (11) becomes(71)dxtdt=2xt1−18yt−u,dytdt=αxt−αyt.System (71) has a unique equilibrium point x1∗=y1∗=K/2=4 if u=4. Let α=1=r/2, E1x1∗,y1∗ is a degenerate critical point as Theorem 1 shows. By Section 3, system (11) becomes system (38) by coordinate translation and nonsingular linear transformation. By the numerical simulation of the transformation system (38), we confirm this case, the vector fields can be found by Figure 1(a). Furthermore, let α=0.6≠r/2 and E1x1∗,y1∗ is saddle-node as Theorem 1 shows. Similarly, by the numerical simulation of the transformation system, Figure 1(b) verifies this result.Figure 1
The vector fields for the transformation system of model (71) when u=Kr/4; here u=4,K=8, and r=2. (a) The vector fields for the transformation system of model (71) for α=1=r/2; (b) the vector fields for the transformation system of model (71) for α=0.6≠r/2.
(a)
(b)Ifu<1/4Kr, there are two equilibria in system (11):(72)E2x2∗,y2∗=K2−K21−4uKr,K2−K21−4uKr,E3x3∗,y3∗=K2+K21−4uKr,K2+K21−4uKr.Letu=2, K=8, and r=2, we obtain x2∗=y2∗≈1.1716 and x3∗=y3∗≈6.828 by performing some simple calculations. We know that E2x2∗,y2∗ is saddle point by Theorem 2, and the case is shown in Figure 2. Furthermore, by Theorem 3, Hopf bifurcation occurs near E3x3∗,y3∗ in system (71) with α=0.292 holding small enough T=r/21−1−4u/Kr−α>0; the blue part in Figure 3(a) is the trajectory of system (71) with initial value (7, 6.5), and it can be seen that system (71) has periodic solution by Figure 3(a); E3x3∗,y3∗ is unstable for α=0.25 satisfying T=r/21−1−4u/Kr−α>0, and the trajectory of system (71) with initial value (7, 6.5) goes out from the inside in anticlockwise direction as Figure 3(b) shows; E3x3∗,y3∗ is stable for α=0.4 satisfying T=r/21−1−4u/Kr−α<0, we can find that the trajectory of system (71) with initial value (7, 6.5) rotates from the outside to the inside in a counterclockwise direction by Figure 3(c).Figure 2
The vector fields of model (71) when u<Kr/4, here u=2,K=8, and r=2.Figure 3
The vector fields of model (71) when u<Kr/4, here u=2,K=8, and r=2; the blue part is the trajectory with initial value 7,6.5. (a) The vector fields of model (71) for α=0.292 which holds T=r/21−1−4u/Kr−α>0 near T=0; (b) the vector fields of model (71) for α=0.25 which holds T>0; (c) the vector fields of model (71) for α=0.4 which holds T<0.
(a)
(b)
(c)The time series and portrait phase of system (71) in Figure 4 show that there occurs Hopf bifurcation periodic solution near x3∗,y3∗ for small enough T=r/21−1−4u/Kr−α>0. Figure 5 reveals the instability of x3∗,y3∗ for T=r/21−1−4u/Kr−α>0. Figure 6 illustrates x3∗,y3∗ is stable if T=r/21−1−4u/Kr−α>0.Figure 4
The time series and portrait phase of model (71) when u<Kr/4; here u=2,K=8,r=2, and α=0.2928. (a) The time series of xt; (b) the time series of yt; (c) the portrait phase.
(a)
(b)
(c)Figure 5
The time series and portrait phase of model (71) when u<Kr/4; here u=2,K=8,r=2, and α=0.29. (a) The time series of xt; (b) the time series of yt; (c) the portrait phase.
(a)
(b)
(c)Figure 6
The time series and portrait phase of model (71) when u<Kr/4; here u=2,K=8,r=2, and α=0.6. (a) The time series of xt; (b) the time series of yt; (c) the portrait phase.
(a)
(b)
(c)By Remark1, if u<1/4Kr in system (11), then u=αK1−α/r is a bifurcation point. Let α=0.3, K=8, and r=2, then we can obtain u=2.04 is bifurcation point by simple calculation. The time series and portrait phase of system (71) in Figure 7 shows that there occurs Hopf bifurcation periodic solution near x3∗,y3∗ for small enough Tu0=u0−αK1−α/r>0. The instability of x3∗,y3∗ for Tu0=u0−αK1−α/r>0. is shown in Figure 8. Figure 9 illustrates x3∗,y3∗ is stable if Tu0=u0−αK1−α/r>0.Figure 7
The time series and portrait phase of model (71) when u<Kr/4; here α=0.3,K=8,r=2, and (u) is close enough to 2.04. (a) The time series of xt; (b) the time series of yt; (c) the portrait phase.
(a)
(b)
(c)Figure 8
The time series and portrait phase of model (71) when u<Kr/4, here α=0.3,K=8,r=2 and u=2.1. (a) The time series of xt; (b) the time series of yt; (c) the portrait phase.
(a)
(b)
(c)Figure 9
The time series and portrait phase of model (71) when u<Kr/4; here α=0.3,K=8,r=2, and u=1. (a) The time series of xt; (b) the time series of yt; (c) the portrait phase.
(a)
(b)
(c)


## 6. Conclusions

A single-species model with a delay weak kernel and a constant harvesting rate is established in this paper. Sufficient conditions of the existence and stability of equilibria in two different cases have been analysed. We know that system (11) is unstable when constant rate harvesting u>Kr/4, that is, harvesting rates on population exceed the MSY, which will reduce the population to extinction in finite time, since dx/dt<0. If the constant rate harvesting u is equal to the critical value, system (11) has a unique equilibrium point which is a degenerate critical point or a saddle-node. When the constant rate harvesting u is less than the critical value, by topological normal form for the Hopf bifurcation and calculating the first Lyapunov coefficient, sufficient conditions about the existence of the Hopf bifurcation are obtained.We can see that system (11) has a stable equilibrium when u<minαK1−α/r,1/4Kr by Remark 1. That is, sufficiently low harvesting rates can be sustained in perpetuity. When 1/4Kr>u>αK1−α/r, system (11) emerges a unique and stable limit cycle bifurcation near E3x3∗,y3∗ for small enough Tu0=u0−αK1−α/r>0, that is, system (11) has a stable periodic solution and population will survive. Therefore, in order to ensure the permanence of species, the control of sustained harvest is very important.Our work on system (11) reveals that the single-species system with a constant rate harvesting is interesting and rich in dynamics. It shows that the intensity of the harvest has effect on the dynamics of system, including the existence and stability of equilibria and Hopf bifurcation. Excessive harvest will lead to extinction of species, and it has a serious impact on biodiversity. Biologically, there is still a lot of work to be done in this area. For example, the study about two-parameter bifurcations of equilibria is a very meaningful work, and it would also be interesting to study how linear rate harvesting and nonlinear rate harvesting in population impact on the dynamics of system and what is the difference between constant rate harvesting and other ways of harvesting. We will consider these issues in our future work.



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*Source: 1810385-2019-12-10.xml*