Analyze a two-species competition model

Our goal is to understand the stability of equilibrium points.

The model is described by the following system of differential

equations: dx

dt = x(λ − ax − by )

dy

dt = y (μ − cx − dy )

where:

▶ λ, μ: Growth rates of species x and y

▶ a, d: Intraspecific competition coefficients

▶ b, c: Interspecific competition coefficients

Case 2: Initial condition dependent survival

Conditions

The conditions for this case are:

▶ cλ − aμ > 0

▶ bμ − dλ > 0

Interpretation

▶ cλ − aμ > 0: The effect of parameter c on λ exceeds the

effect of parameter a on μ.

▶ bμ − dλ > 0: The effect of parameter b on μ exceeds the

effect of parameter d on λ.

Equilibria:

▶ There is an equilibrium (x∞, y∞) in the first quadrant.

▶ (x∞, y∞) is a saddle point.

▶ (K , 0) and (0, M) are asymptotically stable nodes.

▶ Stability Analysis:

▶ The determinant of the community matrix is negative for

(x∞, y∞).

▶ The community matrices for (K , 0) and (0, M) have positive

determinants and negative traces.

We can further verify this from the behaviour of trajectories.

There are three more cases that you can explore.