Analysis of a Delayed Predator–Prey System with Harvesting

A Appendix

The local parameterization method given in Ref. [30] is briefly introduced as follows.

Consider the differential-algebra system

where f:Rn→Rn, g:Rn→Rm (n>m), f and g are continuously differentiable functions, f=(f1,f2,⋯,fn)T, g=(g1,g2,⋯,gm)T. Let X‾∗ be an equilibrium point of (30).

If the differential-algebra system (30) satisfies the following three conditions: ( H1)  rank DX‾g(X‾∗)=m , ( H2)  DX‾g(X‾∗)=(0,P)m×n , where Pm×m is a nonsingular matrix,

then we can use the local parameterization for (30):

where U0=In−m0n×(n−m), V0=0Imn×m, Y(t)=(y1(t),y2(t),⋯,yn−m(t))T∈R(n−m), h(Y) is a smooth mapping from R(n−m) into Rm.

Substituting X‾(t)=ψ(Y(t)) into the first equation of (30),

Differentiating both sides of eq. (31) with respect to Y,

Then left multiply matrix U0T to the both sides of (34), we have

Differentiating both sides of eq. (32) with respect to Y, we have

It follows from eqs (35) and (36) that

Substituting eq. (37) to eq. (33),

By eqs (33), (36) and (38), we can obtain

Hence, the differential-algebra system (30) is equivalent to the parameterized system

Since X‾∗ is the equilibrium point of (30), the following equation holds near X‾∗:

which shows that the equilibrium X‾∗ of the differential-algebra system (30) corresponds to the equilibrium Y=0 of the local parameterized system (40). According to Taylor series expansion approach, we can calculate the Taylor expansions of U0Tf(ψ(Y(t))) at the point X‾∗, and then, in view of eq. (41), the parameterized system (40) can be rewritten as

where U0TDX‾f(X‾∗)=(DX‾f1(X‾∗),DX‾f2(X‾∗),⋯,DX‾fn−m(X‾∗))T.

B Appendix

To verify the transversality conditions for Hopf bifurcations in model (5), we need the following preparations.

Differentiating the both sides of eq. (14) regarding τ, noticing that λ is a function of τ, we derive

which leads to

Letting Θ=ky∗−Mx∗y∗ and substituting λ=iω into the above equation,

By combining with eqs (16) and (17), we can calculate that

From eq. (43),

C Appendix

The coefficients g20, g11, g02 and g21 in eq. (27) are calculated as follows.

In view of (26), we derive

where

From eq. (25) and the center manifold reduction for Hopf bifurcation in Ref. [24], we have

Substituting eq. (46) to eq. (45), we get

i.e.,

Comparing the coefficients of eqs (27) and (47), we have

Now we have derived the concrete expressions of g20, g11 and g02. However, from eq. (48), we can see that W20(θ) and W11(θ) appear in g21. In order to obtain the concrete expression of g21, we still need to calculate W20(θ) and W11(θ).

According to the algorithms of Hassard et al.[24], we have

where

with

Furthermore, we can calculate the concrete expressions of matrices G1 and G2, which take the form of