Abstract
In this paper, we study a ratio-dependent predator-prey model with modied Holling-Tanner formalism, by using dynamical techniques and numerical continuation algorithms implemented in Matcont. We determine codim-1 and 2 bifurcation points and their corresponding normal form coecients. We also compute a curve of limit cycles of the system emanating from a Hopf point.
Highlights
BifurcationsWe shall study the stability and bifurcations of positive equilibria
U ), where all the parameters are positive
We investigate the complex dynamics of a ratio-dependent predator-prey model
Summary
We shall study the stability and bifurcations of positive equilibria. Where (λ1, λ2) is a parameter vector in a small neighbourhood of (0; 0) In this case, with the help of the transformation u = u1 + u∗, v = v1 + v∗, α1 = α∗1 + λ1 and α2 = α∗2 + λ2, system [2] can be written as: where So that: dx dt dx dt p0(λ) + a2(λ)x1 + b2(λ)x2 + p′11(λ)x21 +p′12(λ)x1x2 + p′22(λ)x22 + O( x 3) q0(λ) + c2(λ)x1 + d2(λ)x2 + q1′ 1(λ)x21 +q1′ 2(λ)x1x2 + q2′ 2(λ)x22 + O( x 3). = ρ2 = 0 exists a (due to the nondegeneracy C∞ function z1 defined in assumption), it follows from the a small neighbourhood of λ =. We have obtained the generic normal from of the Bogdanov-Takens bifurcation for the system(3.12)
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