where r = ρ$_{m}$/ρ$_{d}$ . It is useful to rewrite these equations in the real space, where ¯ x refers to the conformal coordinates. Defining σ$_{m}$ = δρ$_{m}$ , σ$_{d}$ = δρ$_{d}$ , and assuming that the EOS of DE is constant w ′ = 0, we can change Eqs.( 5, 6) into, where ∇$_{x}$ = 1 $_{a}$∇$_{¯}$$_{x}$. The dot denotes the derivative with respect to the cosmic time. Ψ indicates the peculiar potential, which can be decomposed into Ψ = ψ$_{m}$ + ψ$_{d}$ , satisfying the Poisson equation [26], where σ$_{λ}$ represents the inhomogeneous fluctuation field and the subscript ” λ ” denotes DM or DE, respectively. We have included the correction from General Relativity. In a homogeneous and isotropic background < ψ$_{λ}$ > = 0, since < σ$_{λ}$ > = 0. For DE and DM, their peculiar potentials read [26] In this section we derive the Layer-Irvine equation [38] when there is an interaction between DE and DM. Layzer- Irvine equation describes how a collapsing system reaches a state of dynamical equilibrium in an expanding universe. 5 III. DERIVATION OF THE LAYZER-IRVINE EQUATION ψ$_{m}$ = − 4 πG ∫ dV ′ σ$_{m}$ | x − x ′ | , (10) ψ$_{d}$ = − 4 πG ∫ dV $_{′}$(1 + 3 w ) σ$_{d}$ | x − x ′ | . (11) ∇ 2 $_{λ}$ψ$_{λ}$ = 4 πG (1 + 3 w$_{λ}$ ) σ$_{λ}$, (9) ˙ σ$_{m}$ + 3 Hσ$_{m}$ + ∇$_{x}$ ( ρ$_{m}$V$_{m}$ ) = 3 H ( ξ$_{1}$σ$_{m}$ + ξ$_{2}$σ$_{d}$ ) , ∂ ∂t ( aV$_{m}$ ) = −∇$_{x}$ ( a Ψ) − 3 H ( ξ$_{1}$ + ξ$_{2}$/r )( aV$_{m}$ ); (7) ˙ σ$_{d}$ + 3 H (1 + C 2 $_{e}$) σ$_{d}$ + (1 + w ) ∇$_{x}$ ( ρ$_{d}$V$_{d}$ ) = − 3 H ( ξ$_{1}$σ$_{m}$ + ξ$_{2}$σ$_{d}$ ) , ∂ ∂t ( aV$_{d}$ ) = −∇$_{x}$ ( a Ψ) − C 2 e 1 + w ∇$_{x}$ · ( a Δ$_{d}$) + 3 H [ ( w − c 2 $_{a}$) + 1 + w − c 2 a 1 + w ( ξ$_{1}$r + ξ$_{2}$ ) ] ( aV$_{d}$ ); (8) Δ $^{′}$m + ∇$_{¯}$ x · V$_{m}$ = 3 H ξ$_{2}$ (Δ$_{d}$ − Δ$_{m}$) /r , V ′ m + H V$_{m}$ = −∇$_{¯}$ $_{x}$Ψ − 3 H ( ξ$_{1}$ + ξ$_{2}$/r ) V$_{m}$ ; (5) Δ $^{′}$d + (1 + w ) ∇$_{¯}$ x · V$_{d}$ = 3 H ( w − C 2 $_{e}$)Δ$_{d}$ + 3 H ξ$_{1}$r (Δ$_{d}$ − Δ$_{m}$) , V ′ d + H V$_{d}$ = −∇$_{¯}$ $_{x}$Ψ − C 2 e 1 + w ∇$_{¯}$ $_{x}$Δ$_{d}$ − w ′ 1 + w V$_{d}$ + 3 H { ( w − C 2 $_{a}$) + 1 + w − C 2 a 1 + w ( ξ$_{1}$r + ξ$_{2}$ ) } V d ; (6) a $^{2}$δQ0 I d ρ$_{d}$ ≈ − 3 H ( ξ$_{1}$ Δ$_{m}$ r + ξ$_{2}$ Δ$_{d}$) ,