Table 2.

LAS and biological existence conditions for the equilibria of system (16).

Equilibrium points	Biological existence conditions	LAS conditions
E 0(0,0)	Always exists	Unstable point
E 1(Λ, 0)	Always exists	(δ/(μ − νδ)) > Λ
E 2(P∗, T∗)	(δ/(μ − νδ)) > Λ	For i=1,2,…, m(α1+α2), either the eigenvalues λi obtained from the characteristic equation λm(α1+α2) − λmα2βP(A1(A2+1) − 1)+((βPδ2A1)/(μΛ(1 − A1)))=0 satisfy |arg(λ)| > (1/m)(π/2) or the angles θi obtained from the equation (((sinmα2θ)(α2/α1)sinmα1θ)/((sinm(α1+α2)θ)(1+(α2/α1))))=(βPδ2A1/μΛ(1 − A1))/(βP(A1(A2+1) − 1))(1+(α2/α1)) satisfy |θi| > (π/2m)

where (P^∗, T^∗) and (A₁, A₂) are defined in (19) and (24), respectively.