# Asymptotic Stability of Steady States for Two-Component Logistic Map

Asymptotic Stability of Steady States for Two-Component Logistic Map

Consider two types of fish in a pond that do not prey on each other but compete for the available food. This system is governed by a generalization of the logistic equation, extended to two competing populations:

x

t

1

1

1

1

and

y

t

2

2

2

2

We restrict ourselves to the first quadrant of the plane, since and are species populations with positive values.

(x,y)

x

y

There are up to four steady states in the first quadrant:

S

1

x=0

y=0

S

2

x=0

y=

α

2

β

2

Δ(x,y)=0

2

S

3

x=

α

1

β

2

Δ(x,y)=0

1

y=0

S

4

x=

αβ-αγ

1

2

2

1

ββ-γγ

1

2

1

2

y=

αβ-αγ

2

1

1

2

ββ-γγ

1

2

1

2

Δ(x,y)=0

1

Δ(x,y)=0

2

S

1

S

2

S

3

S

4

αβ-αγ

1

2

2

1

ββ-γγ

1

2

1

2

αβ-αγ

2

1

1

2

ββ-γγ

1

2

1

2

For this Demonstration, we take the growth rate coefficients , , the self-inhibition parameters , , and the interaction parameters , .

α=0.55

1

α=0.65

2

β=0.55

1

β=0.5

2

γ=0.95

1

γ=0.9

2

The steady states are indicated in the plot as well as the nullclines: and (shown in blue and brown, respectively). is an unstable node, and are stable nodes, and is a saddle point. Coexistence is impossible, since the self-inhibition term is dominated by the interaction term or .

Δ(x,y)=0

1

Δ(x,y)=0

2

S

1

S

2

S

3

S

4

ββ=0.275<γγ=0.855

1

2

1

2

The separatrix curve is shown in black. This curve goes through and . You can select any initial condition (IC) in the first quadrant by dragging the locator. If the ICs are above the separatrix, the trajectories (shown in cyan) will eventually lead to steady-state . If the ICs are below the separatrix, the trajectories lead to steady state . Thus, the steady state for each pure species has a region of asymptotic stability or domain of attraction.

S

1

S

4

S

2

S

3