Table 1 Nature of equilibria of Lotka-Volterra system (4) (refer to Supplementary information for the analytical solution).

From: Asymptotic stability of a modified Lotka-Volterra model with small immigrations

Model

Without immigrants

Small immigration

C(x) = c; D(y) = 0

C(x) = c/x; D(y) = 0

C(x) = 0; D(y) = d

C(x) = 0; D(y) = d/y

Type I (Linear) \(\{\begin{array}{c}\frac{{\rm{d}}x}{dt}=rx-axy+C(x)\\ \frac{{\rm{d}}y}{dt}=bxy-my+D(y)\end{array}\)

Unstable, limit cycle exists

Stable coexistence

Stable coexistence

Stable coexistence

Stable coexistence

Type II (Hyperbolic) \(\{\begin{array}{c}\frac{{\rm{d}}x}{dt}=rx-\frac{axy}{1+hx}+C(x)\\ \frac{{\rm{d}}y}{dt}=\frac{bxy}{1+hx}-my+D(y)\end{array}\)

Unstable

Locally asymptotically stable*

Locally asymptotically stable*

Locally asymptotically stable*

Locally asymptotically stable*

Type III (Sigmoid functional response) \(\{\begin{array}{c}\frac{{\rm{d}}x}{dt}=rx-\frac{a{x}^{2}y}{1+h{x}^{2}}+C(x)\\ \frac{{\rm{d}}y}{dt}=\frac{b{x}^{2}y}{1+h{x}^{2}}-my+D(y)\end{array}\)

Locally asymptotically stable (β > 0)

Locally asymptotically stable*

Locally asymptotically stable*

Locally asymptotically stable*

Locally asymptotically stable*

  1. \(\beta =(rb-hmr)\). *Note that the characteristic equation is given by \({{\rm{\lambda }}}^{2}-{\rm{Tr}}(J){\rm{\lambda }}+{\rm{Det}}(J)=0\). If \({\rm{Tr}}(J) < 0\) and \({\rm{Det}}(J) > 0\) where J is the Jacobian matrix, the steady state is locally asymptotically stable.
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